'Procesamiento de señales de georadar:...
Transcript of 'Procesamiento de señales de georadar:...
Di r ecci ó n:Di r ecci ó n: Biblioteca Central Dr. Luis F. Leloir, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Intendente Güiraldes 2160 - C1428EGA - Tel. (++54 +11) 4789-9293
Co nta cto :Co nta cto : [email protected]
Tesis Doctoral
Procesamiento de señales deProcesamiento de señales degeoradar: implementación delgeoradar: implementación del
método de arreglos sintéticos demétodo de arreglos sintéticos deantenas emisorasantenas emisoras
Cedrina, Lorena Valeria
2010
Este documento forma parte de la colección de tesis doctorales y de maestría de la BibliotecaCentral Dr. Luis Federico Leloir, disponible en digital.bl.fcen.uba.ar. Su utilización debe seracompañada por la cita bibliográfica con reconocimiento de la fuente.
This document is part of the doctoral theses collection of the Central Library Dr. Luis FedericoLeloir, available in digital.bl.fcen.uba.ar. It should be used accompanied by the correspondingcitation acknowledging the source.
Cita tipo APA:
Cedrina, Lorena Valeria. (2010). Procesamiento de señales de georadar: implementación delmétodo de arreglos sintéticos de antenas emisoras. Facultad de Ciencias Exactas y Naturales.Universidad de Buenos Aires.
Cita tipo Chicago:
Cedrina, Lorena Valeria. "Procesamiento de señales de georadar: implementación del método dearreglos sintéticos de antenas emisoras". Facultad de Ciencias Exactas y Naturales.Universidad de Buenos Aires. 2010.
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♣rt♠♥t♦ ís
Pr♦s♠♥t♦ ñs ♦rr
♠♣♠♥tó♥ ét♦♦ rr♦s ♥tét♦s
♥t♥s ♠s♦rs
ss ♣rs♥t ♣r ♦♣tr tít♦ ♦t♦r ❯♥rs
♥♦s rs ♥ ár ♥s íss
♦r♥ ❱r r♥
rt♦r ss r ♥ s
rt♦r sst♥t r ést♦r ♦♥♦♠♦
♦♥sr♦ st♦s r rt rrr♦
r r♦ r♣♦ ♦ís ♣ ② ♠♥t
♥♦s rs ♣t♠r
Pr♦s♠♥t♦ ñs ♦rr♠♣♠♥tó♥ ét♦♦ rr♦s
♥tét♦s ♥t♥s ♠s♦rs
♣r♥♣ ♦t♦ st ss s ♥tr♦r ♠ét♦♦ rr♦s s♥té
t♦s ♠s♦rs ♦rr ② srr♦r s té♥s ② rr♠♥ts áss
♣r r t s ♠♣♠♥tó♥ ♦s ♠ét♦♦s ss ♦rtr s♠
♣ ♣r♠t♥ rs♦r ♠② rss st♦♥s ①♣r♠♥ts ♥ ♣rtr
♦s st♦s r③♦s ♥ ③♦♥ P♦ ♥♦ ♣r♠tr♦♥ rtr③r
♠♥t strtrs rq♦ós ♦♥ ♦ ♦♥trst ♦♥ ♠♦ r♥
♥t ♥ ♥s st♦♥s s sñs s♦♥ ♠② és ② s ♥sr♦ t③r
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s ♠t♦♦♦ís ss ♦rtr ♠út♣ ts ♦♠♦ ♣♥t♦ ♠♦
♦♠ú♥ t♠é♥ ♣rs♥t♥ ♠t♦♥s ② q ♥ ♥♦s s♦s s ♠♦rs
q s ♦r♥ ♥♦ s♦♥ s♥ts
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♦rr ♥ q s ♦♥sr♥ rr♦s ♥t♥s ♣r ♦t♥r ③♦♥s
♠♥s ♠ás strs ♦♥♥tr♥♦ ♥rí s♦r ♦s ♥♦s ♥trés
② s♠♥②é♥♦ ♥ s ③♦♥s ♣rérs ♥r♦rs sñs s♥rs
♥t st ♠t♦♦♦í ♦rtr ♠út♣ s ♦r ♠♦rr ró♥
♥t♥s sñ ♣r♠r rs♣t♦ ♦trs sñs s♥rs sí
♦♠♦ s ♦♥t♥ tr ♦s rst♦s ♦t♥♦s ♥ ♣rs♥t tr♦
♠♦strr♦♥ ♥♦s s♥t♦s ♥ ♥t♦ ♣ó♥ st ♥♦
♠ét♦♦ ♥ ♦rr ♣r str rt♦rs q ♥r♥ s♦♠♥t sñs
♠② t♥s ② ♣r ♠♦rr ♥ ♦r♠ st sñs ♦r♥s ♥ st♥t♦s
♦t♦s ♦ q ♦♥stt② ♥ ♣s♦ ♥♠♥t ♣r ♦♥t♥r ♦♥ srr♦♦
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Prs s ♦rr Pr♦s♠♥t♦ rrr♠s rr♦s s♥té
t♦s ♦rtr ♠út♣ ét♦♦ ♣r rtr③ó♥ strtrs
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tt♦♥ ❯s s♥ ♦st ♠t♦s ♦ s♦♥ ♠♥② r♥t ①♣r♠♥t
stt♦♥s ♥ ♣rtr sts ♥ t r ♦ P♦ ♥♦ ♦ q
t② rtr③♥ r♦♦ strtrs t ♦ ♦♥trst rs♣t t♦ t
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♦tr ♠t♦♦♦s t ♠t♣ ♦st t♦ s t♦ ♠♣r♦ ts
st② t t s ♠t♦st ♠t♦♦♦s s s ♦♠♠♦♥ ♠♣♦♥t
s♦ ♠tt♦♥s s♥ ♥ s♦♠ ss t ♠♣r♦♠♥ts r ♥♦t s♥t
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t♦ ♠♣♦②s ♠ttr rr②s t♦ ♥rr♦ t ♠♥t♦♥ r ts ♦♥♥tr
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♥rs t s♥ t♦ ♥♦s rt♦ ♥ t tr ♦r♥ ♦ t ♥ts ts
s♥♥t② ♠♣r♦♥ t ♥t♦♥ ♦ r② s♥s ♥ t rs♦t♦♥
♦ s♥s ♦r♥t t r♥t ♦ts
②♦rs r♦♥ P♥trt♥ r rr♠ ♣r♦ss♥ ②♥tt
rr②s t♣ ♦st t♦ ♦r rtr③♥ strtrs
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t♦s
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ét♦♦ ♦rr
q♣♦s ♦rr
Pr♦♣ó♥ sñs ♦rr
t♦♦♦í ♦rtr s♠♣
ñs ♣r♦s ♣♦r st♥t♦s t♣♦s rt♦rs
♠♣♦ s♦♥♦ ♦rtr s♠♣
Pr♦s♠♥t♦ ás♦ ♣r t♦s ♦rtr s♠♣
t♦♦♦í ♠ró♥ sñs
t♦♦♦ís ♦rtr ♠út♣ ♦♠♠♦♥ ♣♦♥t
♣ó♥ ♦s ♠ét♦♦s ss ♥ ♦rr
t♦ rq♦ó♦ P♦ ♥♦
♦③ó♥ ♥ strtr ♣rs
tó♥ ss ♥trrs
ét♦♦ ♦rtr s♠♣
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ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦rr
rr♦s ♠s♦rs
ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦rr
rtr③ó♥ ♠♣♦ rr♦
s♣st ♠ét♦♦ ♥ s♦s ás♦s
Pr♦s♠♥t♦ ♣r ♠♦rr sñs ♦♠♣s
♣ó♥ ♠ét♦♦ s♦s ♦♥ ♥♦s ♠út♣s
t♦s s♠♦s
t♦s ①♣r♠♥ts
rtr③ó♥ ♣rs t♣
♣ó♥ sñs ♣r♦s ♣♦r ss
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♥ó♥ ♥♦rs
♦♠♣ró♥ ♥tr ♦s ♠ét♦♦s ② P
t♦s s♠♦s
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t♦r ①t♥s♦
t♦s ①♣r♠♥ts
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s♠♥ rst♦s ② ♦♥s♦♥s
trs í♥s tr♦
♠ó♥ ♥♠ér rrr♠s
rr♦s ♥t♥s ♥ ♥ ♠♦ ♥♦r♠
Pr♦r♠ rr②
Pr♦r♠ ♥t♦r♥♦
r♠♥t♦s
♦rí
♣ít♦
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♠ét♦♦ ♦rr s ♥ té♥ ♣r♦s♣ó♥ ♥♦ ♥s t r
s♦ó♥ q ♣r♠t ♥str s ♣s s♣r♦rs s♦ ♥ t
♦♥stt② ♥♦ ♦s ♠ét♦♦s ♠ás t③♦s ♥ ♥st♦♥s ♦íss ♦♥
♣♦♥s ♥ rss árs ♦♠♦ ♣♦r ♠♣♦ ♦♦í ❬❪ ❬❪ ❬❪ rq♦♦
í ❬❪ ❬❪ ❬❪ ♥♥rí ❬❪ ❬❪ ❬❪ ❬❪ ② ♠♦♥t♦r♦ strrá♥ ❬❪
❬❪ ❬❪ ♥ ♠ét♦♦ ♦rr r♦♥ ♣♥trt♥ rr P s ♠t♥
♣s♦s tr♦♠♥ét♦s s s♣r s♦ ♦s s s ♣r♦♣♥
trés ♠s♠♦ ♦♥ s r♥ ② rrt♥ ♥ s sss ♥trss ♥
s♣r s♦ s rstr ♠♣t sñ r ♦ r♦
t♦♦ st ♣r♦s♦ ♦♠♦ ♥ó♥ t♠♣♦ ♦ s ♥③♥ ♦s t♠♣♦s
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♥ ♣rtr s s ♦♥♦ ♦ ♣r♦♣ó♥ ♥ ♦s ♠♦s s ♣♦s
st♠r s ♣r♦♥s s ♥trss ♣rtr ♦s t♠♣♦s
trá♥st♦
♦s q♣♦s rr ♣r ♣r♦s♣ó♥ ♦ís s♠♥t t③♥ r
♥s ♥tr ③ ② ③ ♥ r♥s ♠②♦rs ♦♥ts
♦♥ ♠♥♦rs ♣r♦♥ ♠♦r rs♦ó♥ ♥r♠♥t♦ s♦ró♥ ②
s♣rsó♥ ♥ s sñs tr♥s♠ts r♥ ♣r♦♥ ♣♥tró♥
♥t♦ ♠②♦r s r♥ ♥t♥ ♠♥♦s ♣r♦♥ s ③♦♥ ♣♦s
sr st r♥♦ ♣r♦♥s q s ♣♥ rtr③r ♣♥
♦♠♣♦só♥ ♠tr③ ♦♥ s ♥ ♦s ♦t♦s ♦ strt♦s
♥trés ♣♦r ♠♣♦ ♦♥t♥♦ rs ♠ s♦ t
② s rtrísts s ♥t♥s rr ♥ ♣♦♥s ♦íss s
t ♥③r ♣♥tr♦♥s s ♥s ♥s ♥tí♠tr♦s ♣r s
r♥s ♠ás ts st ♥s ♥s ♠tr♦s ♣r s r♥s
♠ás s
♥ ♠②♦rí s ♣♦♥s ♦rr st♥ ♥tr s ♥t♥s
♠s♦r ② r♣t♦r ♦st s ♠♥t♥ ♦♥st♥t r♥t s♦♥♦ ♠t♦♦
♦í ♦rtr s♠♣ ♦ s♥ ♦st ② ♣r ♥t♥s s s♣③
♦ r♦ í♥s ♣rs q s str②♥ ♥ trr♥♦ ♣r rr ♦♠♣
t♠♥t ár st♦ ♦♥ st ♠t♦♦♦í s ♣♥ ♥③r r♥s
♣♦r♦♥s s♦ ♥ t♠♣♦s rt♠♥t ♦rt♦s ② ♦♥ ♥
♥tó♥ s strtrs ♥trrs ❬❪ ❬❪ ♥ ♣rtr qs
ó♥ ♥ rs ♦♥ t ♥s í♥s ♣rs ♣ r ♠á♥s ♦s
♥♦s r♥ ❬❪ ❬❪
♥ ♠r♦ s ♥t♥s ♦rr t♥♥ rt ♠t ♦♥ ♦
♥♦s ♠♥ó♥ st♥t ♠♣♦s ❬❪ ❬❪ ♦♠♦ ♦♥s♥ ♥ ró♥
♠♣♦rt♥t ♥rí tr♥s♠t s ♣r r ♠♥♦ ♠s♦r r
t♦r r♣t♦r r♥♦ s ♣♦ss tó♥ s♣♠♥t ♥
s♦s ♦♥ ró♥ sñ r♦ ♦♠♦ ♦rr ♣♦r ♠♣♦ ♥ ♦♥♦♥s
t s♦ró♥ ♦ ♦♥trst ♦ ♥♦s ♣r♦♥♦s st ♣ér ♥rí
r t ♠t♦♦♦í ♦rr ♥♦ s t③♥ ♥t♥s
♠s♦rs ② r♣t♦rs ú♥s ② st♥ ♥tr s
tó♥ ♥tr♣rtó♥ sñs ♦t♦s ♥trr♦s s ♣♥ ♠
♦rr t③♥♦ ♠ét♦♦s ♦rtr ♠út♣ ♦♠♦ ♣♦r ♠♣♦ ♦s ♠ét♦♦s
♣♥t♦ ♠♦ ♦♠ú♥ ♦♠♠♦♥ ♠♣♦♥t P ❬❪ ❬❪ ❲ ♥
rt♦♥ ♥ rrt♦♥ ❬❪ ❱ ♠♣t rt♦♥ t ♦st ❬❪ ❬❪
♦ ❱ ♠♣t rt♦♥ t ♥ ❬❪
♥ s ♠t♦♦♦ís qsó♥ ♦rtr ♠út♣ s ♥t♥s ♠
s♦r ② r♣t♦r s ♦♦♥ s♦r s♣r ♦♥ ♥ s♣♦só♥ ♣rtr
♣r ♥♦ ♦s st♥t♦s ♠ét♦♦s ② s rí st♥ ♥tr s
st ♠♥r s ①♠♥ r♣ts s ♠s♠ ♣♦ró♥ s♦ ♣r♦
♠♥t♦ s r♣t ♥ ♣♦só♥ s♦ ♥str ♦ s ♣r♦s♥ ♦s
t♦s qr♦s ♣r♦r♥♦ ♦t♥r sñs ♠ás ♥t♥ss ② rs q ♥♦
s t③♥ ♠t♦♦♦ís ♦rtr s♠♣ ♥ ♥r qsó♥ ♦♥
♦rtr ♠út♣ s ♣ ♥ ♥ú♠r♦ ♣qñ♦ ♣♥t♦s ♣r ♦t♥r
♦ ♣r♦♣ó♥ ② ♣r♦♥ s ♣r♥♣s ♥trss ♦
r♦ ♥s ♣♦s í♥s ♦ ♠② rr ③ ♦ r♦ í♥s ♣rs ♣r
♦t♥r ♠♣♦ ♦s ② ♣r ♦rr ♥ ♠♣ ♦♠♣t♦ ss♦
♦s ♠ét♦♦s ♦rtr ♠út♣ s♥ r♦r③r s sñs ♣r♠rs ♥
ró♥ r♦ ② s sñs s♥rs ♠♦r♥♦ ♣♥tró♥ ② ♦
r♥ tr s sñs ♦t♥s t♦r ♠t♥t ♠ás ♠♣♦rt♥t
st♦s ♠ét♦♦s s q rqr♥ t♠♣♦s r♦s qsó♥ ② ♣r♦s♠♥t♦
t♦s
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♥rí s♣♦♥ s♦r ♦s ♥♦s ♥trés ♠♥t ♥r♠♥t♦
rt ♦s ♠♣♦s tr♥s♠t♦s ♣♦r q♣♦ st ♥r♠♥t♦ s
♣ ♦rr t③♥♦ ♥ r♣♦ ♥t♥s ♠s♦rs r♥s ♥tr sí s q
♦r♠♥ ♥ rr♦ ♥ st s♦ s ♣♥ s♦♥r s r♦♥s s
st♥ ② ♠♣t ♥tr s ♥t♥s ♠♥r ♥♦str ♠♥t
♦s ♠♣♦s tr♥s♠t♦s ② rr♦s ♥♦ ♦s ♥♦s P♦r ♦tr♦
♦ s ♣♦s ♦t♥r rst♦s s♠rs ♦♥ ♥ ú♥♦ ♠s♦r ♦♦á♥♦♦
ss♠♥t ♥ s ♣♦s♦♥s q rí♥ ♦♣r s ♦♠♣♦♥♥ts rs
rr♦ ② ♦ s♣r♣♦♥♥♦ ♦s ♠♣♦s ♦s rstr♦s ♥s ♣r
s♥tt③r rs♣st t♦t rr♦
♥ ár ♦rr s ♥ r③♦ st♥t♦s t♣♦s st♦s q
♥♦r♥ ♦♥♥t♦s ♠s♦rs ♦ r♣t♦rs ♥ ♥ r♥ ♣rt ♦s s♦s
s t③♥ ♥s ♥t♥s q s♣r♥ ♦ r♥ ♥ ♦r♠ ♦♥st
② t♦♠át ♦♥ ♦t♦ ♠♥♠③r ♦s t♠♣♦s ♣r♦s♣ó♥ t♥t♦
♣r s♦♥♦ árs ①t♥ss ♦♠♦ ♣r ♦t♥r srs t♦s ♦♥ t
♥s s♣ ❬❪ ❬❪ P♦r ♠♣♦ ♥ t♦ t ❬❪ s s ♥ ♦♥♥t♦
♥t♥s ♦r♠♦ ♣♦r ♥♦ ♠s♦rs ② ♥♦ r♣t♦rs q sr s♣rs
s♥♠♥t ♣r♠t qrr t♦s ♥ ♦♥ró♥ P
P♦r s ♦ sí♥tss ♠♣♦s s ♥ ♠t♦♦♦í st♥t t③
♥ ár ♦ís sís♠ ❬❪ ❬❪ ② ♥ rrs ér♦s ♦s ♥ trr
♦♥s ② stéts ❬❪ ❬❪ P♦r ♦♥trr♦ ♥ ár ♦rr ①st ♥
♥t ♣♦♥s rt♠♥t ss rs♣t♦ ♦♥ ♥ ♣r
♣r♦♣♦ró♥ rst♦s ♣r♠♥rs ♥ s♣ ♣ ♠♥♦♥rs
tr♦ s♥ t ❬❪ q♥s ♥ srr♦♦ ♥ q♣♦ q
♥r ♦♥s ♣r♦①♠♠♥t ♣♥s ♣r♦r♥♦ ♣r♦r ♠t♦♦♦ís
♣r♦s♠♥t♦ t♦s ♣rtr s♥tét ♠r♠♥t s t ❬❪
sr♥ ♣r♦r t♦ ♥ ♦♥♦♠♥t♦s ♥③♦ ♥ ár
rrs ér♦s ♣r s ♣tó♥ sst♠s ♦rr ss♣♥♦s ② ♥
♣rtr s♦ rr♦s ♥s ♠s♦rs ♦ r♣t♦rs ♥ ❬❪
♥ ♣rs♥t♦ ♥ ♠t♦♦♦í ♣r ♣♦♥r ♥ ♥ú♠r♦ ♦♥s ♦♠♣♦♥♥
ts ♥ s ♥ s♦ ♥ ♠♦♦ ♣s ♦r③♦♥ts s♣♦♥♥♦ rts
rtrísts ♥ s sñs ② t ❬❪ ♦♥sr♥ ♥ ♣r♦♠♥t♦
♣rtr s♥tét ♦♥ ♦t♦ ♦t♥r ♠á♥s ♠♥s ♥t♣rs♦
♥s s♣rs ♥♠♥t t③ ② Prr♦ ❬❪ r③♥ ♥ ♥áss s♦r
♥♥ ♦s ♣rá♠tr♦s ♥ rr♦ ♥t♥s ♥ú♠r♦ ♥ts
s♣ró♥ ② s♣③♠♥t♦ t♠♣♦r rt♦ ♥tr ♥ts ♥ ♦s ♣tr♦♥s
ró♥ ♦s t♦rs ♠str♥ ♥s ♠♦rs ♦t♥s ♥ sñs
rt♦rs ♣♥♦s ② ♣♦ ♥♥ó♥ rs♣t♦ s sñs ♦rtr
s♠♣
♥q ♦s st♦s t♦s ♠str♥ ♥♦s ♥s ♠t♦♦ó♦s r
♦♥♦s ♦♥ sñs P qrs ♦♥ ♠út♣s ♥t♥s ♦s ♠s♠♦s ♥♦
♥ ♥t ♥ ♠t♦♦♦í ♥r ② ♥♠♥t q ♣r♠t ♥ ♣
ó♥ sst♠át ♦s rr♦s s♥tét♦s ♥ ár P♦r ♠♣♦ tr♦
③ ② Prr♦ ❬❪ s ♠t ♣tr♦♥s ró♥ ♥ í♦ ② ♥ ró♥
♠♣♦ ♥♦ ♥ st ♥áss ♣r♠t ♦t♥r ♥s rtrísts
♥rs ♣tró♥ ró♥ st♦s rst♦s s♦♥ t ♠t ②
q ♦s st♦s ♦ís♦s t♠♥t s r③♥ ♥ ♠♦s ♦♠♣♦s ♦♥
♥♦s str♦s ♥ t♦♦ r♥♦ st♥s s ♠♣♦ r♥♦ st
♠♣♦ ♥♦ s♠s♠♦ ♥♦ s trt♥ s♦s ♦♥ ♥♦s ♠ás ♥rs ♦♥
t♠ñ♦ ♦r♠ ② ♦r♥tó♥ rtrr ♦s q ♣r♥ ♠② r♥t♠♥t
♥ ♦s st♦s ♦ís♦s
t♦s
♦t♦ st ss s srr♦r ♠♣♠♥tr ♠ét♦♦ rr♦s
s♥tét♦s ♠s♦rs ♦rr ② srr♦r s té♥s ② ♦s ♣r♦r♠s
♦♠♣t♦♥s ♥sr♦s ♣r ♣ó♥ ♠ét♦♦ ② ♣r♦s♠♥t♦
♦s t♦s
Pr st♦ s ♥ sr ♦s ♣s♦s q s sr♥ ♦♥t♥ó♥
r rtr③ó♥ strtrs ♦♥ ró♥ sñ r♦
s♥♦ ♦s ♠ét♦♦s ♠ás ss ♦st ♦ ② ♠ró♥ t♦s sí
♦♠♦ ♠♦♦ ♣r ♥tr ♥♦s ② ♠ét♦♦s s③ó♥
♥③r ♦s ♠♣♦s ♥r♦s ♣♦r rr♦s ♣♦♦s ♦s s♦r
s♣r s♦ ♣♦r ♠♦ s♠♦♥s ♥♠érs ♦♠♦ ♥ó♥
♦s ♣rá♠tr♦s rr♦ ♠ás r♥ts ② st♥ ♥tr
rr♦ ② ♥♦ ♥ ♦t♥r ♣ts q ♣r♠t♥ sñr rr
♦s ② ♠t♦♦♦ís ♣r♦♣♦s ♣r ♥r♠♥tr rt ♦s
♠♣♦s tr♥s♠t♦s ♠♥r q ♦s ♥♦s s♥ ♠♥♦s
♠♥t
str rs♣st ♠ét♦♦ rr♦s s♥tét♦s ♣♦r ♠♦
♠♦♦s ♥♠ér♦s
srr♦r ó♦s ♦♠♣t♦♥s q t♥ ♣ó♥ ♠ét♦
♦ rr♦s s♥tét♦s srr♦r ♠ás ♥t♦r♥♦s rá♦s q s♠
♣q♥ t③ó♥ ♦s ó♦s
r s ♠♦rs ♦t♥s ♦♥ ♠ét♦♦ t♥t♦ tt ♦♠♦
♥ttt♠♥t ♦♠♣rr ♦s rst♦s ♦t♥♦s ♦♥ ♦s ♦tr♦s
♠ét♦♦s ♦rtr s♠♣ ② ♠út♣
♣r ♠ét♦♦ s♦s ①♣r♠♥ts ♥ ♦s q ♦s s♦♥♦s r③
♦s ♦♥ ♥ ú♥♦ ♠s♦r rr♦♥ rst♦s ♣♦♦ stst♦r♦s
♦♥t♥♦ ss
♦♥t♥ó♥ ♣rs♥t ♥tr♦ó♥ ♣ít♦ ♥ ♣ít♦
st ss s ♣rs♥t♥ ♦s ♥♠♥t♦s tór♦s ② ♦s ♣r♥♣♦s ♥♦
♥♠♥t♦ ♠ét♦♦ ♦rr sr♥ s ♠t♦♦♦ís ♦rtr
s♠♣ ② ♠út♣ ② ♠t♦♦♦í ♠ró♥ t♦s
♥ ♣ít♦ s ♣rs♥t ♥ sr tr♦s r③♦s ♥ st♦
rq♦ó♦ P♦ ♥♦ ♦♥ s ♣♥ s té♥s ♠ró♥ P
② ♠♦♦ ♣r ♠♦rr ♥tr♣rtó♥ ♥ s♦s ①♣r♠♥ts ♦♥
ró♥ sñ r♦
♥ ♣ít♦ s ♣rs♥t ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs
♦rr ♣rt ♦s ♥♠♥t♦s t♦rí rr♦s ♥t♥s
t♣♦ ♣♦♦ ♥ í♦ ② ♦ s st ♠♣♦ rr♦ ♥ ♥ s♠s♣♦
♠str ♣ó♥ ♠ét♦♦ s♦s s♠♣s ♦♥ t♦s s♠♦s ② s
♣rs♥t ♥ ♠t♦♦♦í ♣r ♠♦rr ♦♥t♥ tr s sñs
♥ ♣ít♦ s ♠str♥ ♦s rst♦s ♣ó♥ ♠ét♦♦
st♥t♦s s♦s ♦♠♣♦s ♦♥ sñs ♠út♣s t♥t♦ s♥tét♦s ♦♠♦ ①♣
r♠♥ts
♥ ♣ít♦ s ú♥ ♥ttt♠♥t s ♠♦rs ♦t♥s ♥
♣ó♥ ♠ét♦♦ ♣♦r ♠♦ rs♦s ♥♦rs
♠♥ ♦♠♣r♥ ♦s rst♦s ♦t♥♦s ♦♥ ♠ét♦♦ rr♦s s♥té
t♦s ♠s♦rs ♦rr ② ♦♥ ♦s ♠ét♦♦s ② P ♦♠♣ró♥
s r③ t♥t♦ ♦♥ t♦s s♠♦s ② ♦♠♦ ①♣r♠♥ts
♥ ♣ít♦ s rs♠♥ ♦s rst♦s ♦t♥♦s ♥ ss s ♣r
s♥t♥ s ♦♥s♦♥s ♥rs ② s sr♥ s í♥s tr♦ tr♦
♥ ♦s ♣é♥s ② s ♥ ♦s ts s s♠♦♥s ♥♠érs
r③s ② s ♣rs♥t♥ ♦♥t♥♦s tór♦s q ♦♠♣♠♥t♥ ♦♥t♥♦
ss
♥♠♥t ♥ ♦s ♣é♥s ② s sr♥ ♦s ♣r♦r♠s ♦♠♣
t♦♥s srr♦♦s ♥ ♠r♦ st ss ♥ st♦s ♣r♦r♠s s♦♥
♣rt ♠♣♦rt♥t s ♦♥t♥♦ ♦s ♠s♠♦s ♥ s♦ ♦♦♦s ♥ ♣é♥s
♣r ♦t♥r ♥ ♠②♦r ♦♥t♥ ♥ srr♦♦ st ss
♣ít♦
ét♦♦ ♦rr
♥ st ♣ít♦ s sr♥ ♦s ♣r♥♣♦s ♥♠♥ts ♠ét♦♦
♦rr ♦s ♦♠♣♦♥♥ts ás♦s q♣♦ ② s ♣rs♥t♥ s ♠t♦♦♦ís
qsó♥ ② ♣r♦s♠♥t♦ ♠ás t③s
q♣♦s ♦rr
♦s ♦♠♣♦♥♥ts ♣r♥♣s ♥ q♣♦ ♦rr s♦♥ sst♠ ♠
s♦r sst♠ r♣t♦r ② ♥ ♦♥tr♦ ② rstr♦ r ♥
♦♥tr♦ ♥t ♦♥ ♥ ♥r♦r sñs ♥ sst♠ rstr♦ ② ♣r♠t
s♦♥r ♦s ♣rá♠tr♦s ♠só♥ ② r♣ó♥ s sñs ♥r♦r
♥í ♥ sñ étr ♥t♥ ♠s♦r ② ♦r♥ q ést ♣r♦③ ♣s♦s
tr♦♠♥ét♦s q s tr♥s♠t♥ s♦ ♦s ♣s♦s s ♣r♦♣♥
r sq♠ ♦s ♦♠♣♦♥♥ts ♥ q♣♦ ♦
rr
♥trtú♥ ♦♥ s st♥ts s♦♥t♥s ♥ ss♦ ♥t♥ r♣
t♦r ♣t sts sñs s tr♥s♦r♠ ♥ sñs étrs ② s rstr ♥
♥ ♦♥tr♦ ♥ ♥ó♥ t♠♣♦ ♥ ♦s♦♥s ♥ ♦♥tr♦ t
♥ ♥♦r♣♦r ♥ ♣♥t ♣r s③ó♥ ♦s t♦s ② ♥t ♦♥
♣ r③r ♥ ♣r♦s♠♥t♦ ♣r♠♥r ♦s t♦s ♥ s♠tá♥♦
♦♥ s qsó♥ ♥ ♥♦s s♦s st♦ ♣r♠t ♦t♥r ♥ ♥tr♣rtó♥
♣r♠♥r ♦s t♦s ♣r♦ ♥ ♥r ♣r♦s♠♥t♦ ♥tr♣rtó♥ s
r③♥ ♥ ♥ t♣ ♣♦str♦r ♦ r tr♥sr♦ ♦s ♠s♠♦s ♥
♦♠♣t♦r
r q♣♦ rr ♥ ♦♥ró♥ ♦rtr s♠
♣
♦s q♣♦s ♦rr ♣♥ t♥r ♠♦♥t ♠♦♥♦stát♦ ♥ q s
♥t♥s ♠s♦r ② r♣t♦r s ♥ ♥ ♠s♠ ♥ ♦ ♠♦♥t stá
t♦ ♥ q s ♥ ♥ ♥s s♣rs st út♠♦ ♣r♠t ♣r
♣r♦♠♥t♦s qsó♥ t♥t♦ ♦rtr s♠♣ ♦♠♦ ♦rtr
♠út♣ ♥ r s ♠str ♥ q♣♦ ♠♦♥t stát♦ ♦♥
r♦ ♣r qrr t♦s ♦♥ ♥ s♣ró♥ ♦♥st♥t ♥tr s ♥t♥s
♠s♦r ② r♣t♦r
Pr♦♣ó♥ sñs ♦rr
♥ ♥r ♠ét♦♦ ♦rr s ♣ ♥ ♠♦s ♥trs ♦s ás
rst♥ s♥t♠♥t ♦♠♣♦s ♦♠♦ ♣r q ♣r♦♠ tr♦♠♥ét
♦ ♦♠♣t♦ ♣ sr rst♦ ♥ ♦r♠ ♥ít ♥ ♠r♦ ♦♥sr♥♦
♠♦♦s ♥ít♦s s♠♣s s ♣♦s ♦t♥r ♥ ♥ ♣r♦①♠ó♥
♥s s rtrísts ♣r♥♣s s sñs q s ♣r♦♣♥ ♥
ss♦ ♥trtú♥ ♦♥ s ♥trss ♥trrs
♣r♦♣ó♥ ♦♥s tr♦♠♥éts ♥ ♥ ♠♦ ♠tr s ♣
srr ♣♦r ♠♦ s ♦♥s ① ❬❪
~∇× ~E = −∂ ~B
∂t
~∇× ~H = ~J +∂ ~D
∂t
~∇ · ~D = q
~∇ · ~B = 0
♦♥ ~E s ♥t♥s ♠♣♦ étr♦ ~B s ♥s ♦ ♠
♥ét♦ ~H ♥ ♥t♥s ♠♣♦ ♠♥ét♦ ~D s t♦r s♣③♠♥t♦
étr♦ ~J s t♦r ♥s ♦rr♥t étr q s ♥s
r étr ② t s t♠♣♦
s r♦♥s ♦♥sttts ② ② ♠ sr♥ rs♣st
♦s ♠♦s ♠trs ♥ ♣rs♥ ♠♣♦s tr♦♠♥ét♦s ♥ ♠♦s
♥s sótr♦♣♦s s ♣♥ ①♣rsr ♦♠♦ ❬❪
~J = σ ~E
~D = ǫ ~E
~B = µ ~H
♦♥ ǫ σ ② µ s♦♥ srs ② r♣rs♥t♥ ♣r♠t ♦♥t
② ♣r♠ ♠♥ét rs♣t♠♥t
s ♦♥sr ♠♦♦ s♠♣ ♥ ♦♥ ♣♥ ♠♦♥♦r♦♠át
r♥ ♥r ω s ♣♦s ♦t♥r ♥ st♠ó♥ ♦
♣r♦♣ó♥ ♦♥ ♥ s♦ ♠♦s ♦♥ s ♣érs σωǫ
≪ 1
♦♥ s ♣r♦♣ ♦♥ ♦ s ❬❪
v≈ 1√µǫ
♦♥ ǫ = ǫ0ǫr ② µ = µ0µr ♦♥ ǫ0 ② µ0 ♣r♠t ② ♣r♠
♠♥ét í♦ rs♣t♠♥t ② µr ② ǫr ♦s ♦rs rt♦s ♦rrs♣♦♥
♥ts Pr ♥ ♠♦ ♥♦ ♠♥ét♦ s ♣ ①♣rsr ♦♠♦
v =c√ǫr
♦♥ c = 1/√µ0ǫ0 s ♦ ③ ♥ í♦ ♦♥t ♦♥
♥ ♠♦ s ♣ ①♣rsr ♥ tér♠♥♦s ♦ ♣r♦♣ó♥
λ =v
f
♦♥ f s r♥ f = ω/2π s ♦♥s ② ♥♥
ró♥ ♥tr ♦s ♣r♥♣s ♣rá♠tr♦s q sr♥ ♣r♦♣ó♥
sñ ② s ♣r♦♣s tr♦♠♥éts ♠♦ ♥ t s
♠str♥ ♦rs tí♣♦s ♣r♠t rt ♦♥t ② ♦
♣r♦♣ó♥ ♥ ♠♦ ♦t♥ ♣rtr ♣r rs♦s ♠trs
❬❪
♥ ♥r ♠t♦♦♦í ♦rr rst ♠②♦r t ♥ ♠
trs ♦♥ s ♣érs ② q ♥ s♦s s♦s s sñs ♣♥tr♥ ♥
r♥ ♣r♦♥ ♥ ♣rát t♠é♥ s ♥♥tr♥ ♥t♦r♥♦s ♥ ♦s q
sts ♦♥♦♥s s ♣érs ♥♦ s ♠♥t♥♥ ♣♦r ♠♣♦ ♥ s♦
s♦s r♦s♦s ♦ ♦♥ strrá♥ s ♣♦r ♦ q ♣♥tró♥
♥ ♦s s ♠t
♥♦ sñ ♠t ♥③ sr ♠♦♦♥s ♦♠♦ ♦♥s♥
s ♥tró♥ ♦♥ ♠♦ q s♦♥ ♣r♦t♦ st♥t♦s ♣r♦s♦s ís♦s
♦ q ♠t♦♦♦í ♦rr s s ♥ tó♥ sñs r
s ♥ s ♥trss ss♦ ♥ó♠♥♦ r①ó♥ ♦♥stt②
♠♥s♠♦ ♥tró♥ ♣r♥♣ ♥tr sñ ② ♦s ♥♦s ♥ st♦ s
♦♥sr ♠♦♦ s♠♣ ♥ q ♥ ♦♥ ♣♥ ♥ s♦r ♥ ♥trs
♣♥ r s ♣♦s ♦t♥r ♦♥s q ♥♥ s ♠♣ts
s sñs q rst♥ ♦♠♦ ♥♦♥s ♦s ♣rá♠tr♦s ♦s ♠♦s
♠♦s ♦s ♥trs ♦s ♦♥ts ♥tr s ♠♣ts r
♥♥t R ② ♥t s ♠♣ts tr♥s♠t ♥♥t T stá♥ ♦s
♣♦r s ♦♥s ② s ♠♣♦ étr♦ s ♣r♦ ♣♥♦
♣r♦♣ó♥ ♠♦♦ tr♥srs♦ ♠♥ét♦ ② ② s ♠♣♦
étr♦ s ♣r♣♥r ♣♥♦ ♣r♦♣ó♥ ♠♦♦ tr♥srs♦ étr♦
θi s á♥♦ ♥♥ ② θt s á♥♦ tr♥s♠só♥ ❬❪
RTM =v2cos(θi)− v1cos(θt)
v2cos(θi) + v1cos(θt)
TTM =2v1cos(θi)
v2cos(θi) + v1cos(θt)
tr
Pr♠t ❱♦ ♦♥t
rt ♣r♦♣ó♥
ǫr v ♠♥s σ ♠♠
r
s
r♥t♦ s♦
r♥ s
r ú♠
♦ r♥♦s♦ s♦
♦ r♥♦s♦ ú♠♦
♦ r♦s♦ s♦
♦ r♦s♦ ú♠♦
♦ ♦♥♦
③ str
strt♦ tr ♦
❱♦rs tí♣♦s ♣r♠t rt ② ♦♥t
rs♦s ♠trs ♦ ♣r♦♣ó♥ s
♣rtr t③♥♦ ♦s ♦rs ǫr ❬❪
r ♥ ♥♥t s♦r ♥ ♥trs ♣♥ ♣r ♦s
s♦s ♥ ♦s q ♠♣♦ étr♦ ~E s ♣r♦ ♣♥♦
♥♥ ♠♦♦ tr♥srs♦ ♠♥ét♦ ② ♣r♣♥r
♣♥♦ ♥♥ ♠♦♦ tr♥srs♦ étr♦
RTE =v1cos(θi)− v2cos(θt)
v1cos(θi) + v2cos(θt)
TTE =2v1cos(θi)
v1cos(θi) + v2cos(θt)
♥♦ ♠♥♦r s ♦♥trst q ♣rs♥t♥ s ♦s ♣r♦♣ó♥
♥ ♥trs sñ r s ♠ás é ② tr♥s♠t ♠ás ♥t♥s
P♦r st r③ó♥ s ♥ ♥ s♣r s♦♥t♥ ♥♦ ② ♦♥trst
s♥t s ♠② ♣♦s q sst♠ ♦rr ♥♦ tt sñ r
② ♥ ♦♥s♥ q ♥♦ s ♦r ♥tr ♥trs ♦rrs♣♦♥♥t
♠♥trs q s ♦♥trst s r♥ sst♠ ♦rr ttr
♦♥ ♥trs ♥q ♣♦s♠♥t ♥♦ ② s♥t ♥t♥s
tr♥s♠t ♦♠♦ ♣r ttr ♦trs ♥trss ♠ás ♣r♦♥s
♠②♦rí ♦s q♣♦s ♦rr s sñ♥ ♣r ♠tr ♣s♦s ♦♥
①t♥s♦♥s t♠♣♦rs ♣qñs ♥ ró♥ ♣r♣♥r s♣r
s♦ ② ♠♦♦ q ♦s ♠s♠♦s ♥♦ s ①t♥♥ ♥ s♣♦ ♠ás á ♥
ró♥ ♦r♠ ó♥ ♦♥ ♥ á♥♦ ♣rtr 90♦ ♣r♦①♠♠♥t
♠ q ♣s♦ s ♣r♦♣ ♥ s♦ ♠s♠♦ ♠♥ árs s
③ ♦♥ ♠♥♦r ♥t♥s ② q ♥s ♥rí ♣♦r ♥
s♣r s♠♥② ♠♥tr s♣r t♦t r♥t ♦♥s q s
♣r♦♣ ♦♠♦ ♦♥s♥ r♥ ♦♠étr ♠♣♦
♥♦ ♥ sñ s r ♥ ♥ ♦t♦ ♣r♦♥♦ rrs r♣t♦r ♦♥
♠♥♦r ♥rí ♣♦r ♥ ár q s s r ♥ ♥ ♦t♦ é♥t♦ ♣r♦
♠ás s♣r ♦t♦ ♥ q s ♣r♦ r①ó♥ s s♥
t♠♥t ♣r♦♥♦ sñ ♣♦rí ♥♦ sr tt ♣♦r sst♠ r♣t♦r
♦ s ♥t♥s ♣♦r ♥ ár ♥♦♠♥ ♣r♦♥
♣♥tró♥ ♠á①♠ ♣r♦♥ q s ♣ ttr ♥ rt♦r
st ♣♥ t♥t♦ r♥ ♦♠étr sñ ♦♠♦ ♦♥trst
♥ s sss s♦♥t♥s ♠♥♦♥♦ ♥tr♦r♠♥t
sñ sr ♠ás ♥ ♣ér ♥rí ♠ q s ♣r♦♣
trés ♠♦ ♠tr s♦ró♥ ♦♠♦ ♦♥s♥
tr♥s♦r♠ó♥ ♣rt s ♥rí tr♦♠♥ét ♥ ♦r tr♠♥
♣r♥♣♠♥t ♣♦r ♦♥t étr ♠♦ ♥♦ ♠②♦r
st♥ r♦rr sñ ♠②♦r s s♠♥ó♥ s ♥t♥s ♦
s♦ró♥ r♦ s♦ró♥ ♠♥t ♣r r♥s ♠②♦rs s
ñ ② ♣♥ s rtrísts ♠♦ ♥ ♣rtr ♠
♠②♦r r♦ ♠ ♠②♦r s♦ró♥
s sñs ♦rr ♥♦r♠♠♥t s tr♥s♠t♥ trés ♠♦s
♦♠♣♦s q ♣rs♥t♥ t♦♥s ♦ tr♦♥s ♥ ss ♣r♦♣s
tr♦♠♥éts sts s♦♥t♥s s♣rs♥ ♥rí ♥r♥♦ s
ñs s♥rs ♥♦ ss ② ♦♥tr②♥♦ t♥ó♥ ② ♦r♠ó♥
sñ t♦t ♦♠♦ s sñs s♥rs ♣r♥ ♥ ♦s rstr♦s s♣r
♣sts s sñs ♥trés ♥ ♠♦s s♦s s rt♠♥t ♠♣♦s
st♥r s sñs ♥trés ♦ s ró♥ ♠♣t ♦♥
♥t♦r♥♦ ♦ s ♦♥t♥
t♦♦♦í ♦rtr s♠♣
♥ s ♣♦♥s ♠ás ♦♠♥s ♦rr s ♥t♥s s trs♥
♥ts s♦r s♣r s♦ ♦ r♦ í♥ s♦♥♦ st ♠
t♦♦♦í s ♠ ♦rtr s♠♣ ② q ♣r ♣♦só♥ s♦r
í♥ s♦♥♦ s qr ♥ ú♥ tr③ ♠♣t r ♥ ♥ó♥
t♠♣♦ ♥ ♠r♦ ♥ ♣♦♥s s♣s s t③♥ ♦♥r♦♥s
♦rtr ♠út♣ ♥ s q ♣r ♣♦só♥ s♦r í♥ s♦♥♦ s
qr♥ rs tr③s ♣r st♥ts ♣♦s♦♥s ♠s♦r ② r♣t♦r ♥
r s ♣ r ♥ ♠♣♦ ó♠♦ s ♦t♥ ♥ ♣r ♦rtr
s♠♣ s ♦♥s ② ♦rrs♣♦♥♥ st ♦♥ró♥ ♦♥
s ♥t♥s t♥♥ ♦r♥tó♥ ♣r♣♥r í♥ s♦♥♦ ♠♦♦
❬❪
r t♦♦♦í ♣r ♦t♥só♥ ♥ ♣r ♦r
tr s♠♣
r s ♥ ♠♣♦ ♥ tr③ ♥ s ♣ r q ♥
♠♥t ♥tr t = 0 ② t = 2,5 ♥s s r ♥ ♣s♦ r♥ ♠♣t q s
♥♦♠♥ sñ rt stá ♦r♠♦ ♣♦r s♣r♣♦só♥ sñ ér
q s ♣r♦♣ ♥ r ♦ ③ ♥ í♦ ♣r♦①♠♠♥
t ② sñ trrstr q s ♣r♦♣ trés strt♦ ♠ás s♣r
s♦ ♦ ♣r♦♣ó♥ ♥ ♠♦ ♥ r s ♠str
♥ sq♠ ♦s ♠♥♦s r♦rr♦s ♣♦r sts sñs sñ rt t♥
♠♣t ♠② s♣r♦r s q ♣rs♥t♥ s sñs q ♣r♦♥♥ r
①♦♥s ♥ ♦s ♦t♦s ♥trr♦s ♣♦r ♦ q t♥ tó♥ ♦t♦s
♠② s♣rs ♥ s♦ r ②s r①♦♥s ♦rr♥ ♣r
t < 2,5 ♥s
♦♥♥t♦ s tr③s qrs ♦ r♦ ♥ í♥ s♦♥♦
r ♠♣♦ ♥ tr③ s ♠str♥ ♦rs ♥♦r♠
③♦s ♠♣t
r ♠♥♦s r♦rr♦s ♣♦r s sñs rt ér ②
trrstr ② ♣♦r sñ r
r ♠♣♦ ♥ rrr♠ ♣r s♦ ♥ q
s♣ró♥ ♥tr s ♥t♥s ♠s♦r ② r♣t♦r s ♥ ♦st
s♦r s♣r s s ♠ ♣r ♦ só♥ rt t♦s ♥♦ s
tr③s ♥ ♦ ♣r s r♣rs♥t♥ ♥ ♥ rá♦ ♦♥ ♦r③♦♥t
r♣rs♥t ♦♦r♥ s♣ ♠ s♦r s♣r s♦
rt r♣rs♥t t♠♣♦ tr♥srr♦ ♣rtr ♠só♥ ♣s♦ ②
♣♦r ♠♦ ♥ s ♦♦rs s r♣rs♥t ♥t♥s sñ
s ♦t♥ ♥ rá♦ ♥♦♠♥♦ rrr♠ ♥ ♠♣♦ s ♣ r ♥
r
♠t♦♦♦í ♦rtr s♠♣ rqr t♠♣♦s ♦rt♦s qsó♥
② ♣r♦s♠♥t♦ ② ♥ ♠♦s s♦s ♣r♠t ♦t♥r ♥♦r♠ó♥ s♥t
ss♦ ♥ ♠r♦ ①st♥ s♦s í tó♥ ♦♠♦ ♣♦r ♠♣♦
♥♦ ♠tr q ♦♠♣♦♥ ♠tr③ ss♦ ♣rs♥t ♥ t♥ó♥
♠② ♥♦ ró♥ sñ r♦ s ♠s♦ t ♦ ♥♦
s strtrs tts rqr♥ ♥ ♠♦r rs♦ó♥ ♥ s♦s s♦s s
♦♥♥♥t t③r ♥ ♠t♦♦♦í ♦rtr ♠út♣ ♦♠♦ s rá ♠ás
♥t
ñs ♣r♦s ♣♦r st♥t♦s t♣♦s
rt♦rs
♥♦ s ♦♥sr♥ st♥t♦s t♣♦s ♥♦s ♥trr♦s s ♦t♥♥
sñs ♦♥ rtrísts ♣rtrs q ♣♥♥ ♦♠trí ② ♣r♦
♣s ♥♦ ② s ♣r♦♣s ♠♦ ♥ q s ♥♥tr
♥ ♥r ♥♦ s♠♣r s ♦sr ♥ ♥t ♥tr ♦r♠ ♥♦ ②
s ♣♦s♦♥s ♦s t♠♣♦s ♥ rrr♠ ♦♠♦ s rá
♦♥t♥ó♥
❯t③♥♦ s ②s ó♣t ♦♠étr s ♣♦s ♦rr ♥ só♥ s♠♣
rst♦ ♥tró♥ sñ ♦♥ st♥t♦s ♥♦s ♥ s♦
♥ ♥trs ♣♥ r t♠♣♦ s ♣ ①♣rsr ♦♠♦
tv =2z0 cos(α)
v− 2 sen(α)
vxpm
♦♥ z0 s ♣r♦♥ q ♥trs ♦rt z α s á♥♦ q
♦r♠ ♥trs ♦♥ ró♥ ♦r③♦♥t v s ♦ ♣r♦♣ó♥
♥ ♠♦ q s ♥♥tr rr ♥trs P♦r s♠♣ s ♦♥sr
♥ ♦♥ró♥ ♥ q s ♣♦s♦♥s s ♥t♥s ♠s♦r ② r♣t♦r
♦♥♥ ♥ xpm
♦♠♦ s ♣ r ♥ r sñ q s ♦t♥ ♥♦
♦♥♥t♦ ♠s♦r r♣t♦r s ♥♥tr ♥ ♥ tr♠♥♦ ♣♥t♦ s♦r
s♣r ♥♦ ♦rrs♣♦♥ r①ó♥ ♥ ♣♥t♦ ♣♥♦ q stá ♦
♦♥♥t♦ ♠s♦r r♣t♦r ♦♥strr ♥ rrr♠ ♦♥ s ♠♣ts ♥
♥ó♥ t♠♣♦ s ♦t♥ ♥ r♣rs♥tó♥ q s ♥ r s♠t
♦♥ ♦r♠ s♣r rt♦r stá st♦rs♦♥ ♣s s♠♥t♦
♥ r s ♦rrs♣♦♥ ♦♥ s♠♥t♦ ♥ ② ♠♦s
stá♥ s♣③♦s ♥tr sí sñ ♦t♥ stá s♣③ ♦♥ rs♣t♦
rt♦r q ♦r♥ ② ♣rs♥t ♥ ♥♥ó♥ st♥t ♥♥ó♥
rt♦r s α ♥ rrr♠ s ♦sr ♥ sñ ♦♥ ♥♥ó♥ αp
♣♦r ó♥
tan(αp) =2 sen(α)
v
♦s t♦s s♣③♠♥t♦ ♥♥ó♥ s♦♥ ♠ás ♥♦t♦r♦s ♣r ♥♥♦♥s
r♥s P♦r ♦tr♦ ♦ s ♥♥ó♥ rt♦r s ♣qñ ♦s t♠♣♦s
tv rstr♦s ♣♦r r♣t♦r ♣r ♦s ♣s♦s r♦s s ♥♥
♦♥ s ♣r♦♥s p ♦s rt♦rs sú♥ ①♣rsó♥
p =tvv
2
t♦r 12s q t♠♣♦ q sñ tr ♥ r♦rrr st♥
♥tr sst♠ ♦rr ② rt♦r s ♠t t♠♣♦ t♦t
r ①ó♥ ♥ ♥ ♥trs ♣♥ ♦♥ ♥♥ó♥
α ② rrr♠ rtríst♦
ó♥ ♣r♠t ♦t♥r ♥ ♣r♠r st♠ó♥ ♣r♦♥
♦s rt♦rs ♣r ♣♦só♥ s♦r s♣r ♦♥♦ ♦
v
♦♥ ♠s♠♦ ♣r♦♠♥t♦ q ♣r ♥ ♥trs ♣♥ s ♣♦s s
t♠r ♦s t♠♣♦s ♣r ♥ ♦t♦ ♣qñ♦ ♥ r s
♠str ♥ sq♠ ♥ ♦t♦ ♥trr♦ ♥ ♥ ♠♦ rtr③♦ ♣♦r
♥ ♦ ♣r♦♣ó♥ v t♠♣♦ s ♣ ①♣rsr ♣♦r
♠♦ s♥t ó♥
t2v(2z0/v)2
− (x0 − xpm)2
z20= 1
♦♥ z0 ② x0 s♦♥ ♣r♦♥ ② ♣♦só♥ ♦t♦ rs♣t♠♥t
xpm s ♣♦só♥ ♦♥♥t♦ ♥t♥s ♠s♦r r♣t♦r ♥ r
s ♠str ♥ ♠♣♦ tí♣♦ sñ q ♥r ♥ ♦t♦ ♣qñ♦ ést
♣rs♥t ♥ ♦r♠ q ♥♦ r s♠t ♦♥ ♦t♦ ♥ r s
♣ ♦srr q sñ s tt ♥ ♣♦s♦♥s ♠② s ♥♦
st♦ s ♦♥s♥ q ♦s ♠♣♦s ♠t♦s ♣♦r ♥t♥ t♥♥ ♥
♣rtr ♥r r♥ ♦ q ♣r♠t ♠♥r tr♠♥t ♦t♦
♥t♥s sñ s♠♥② ♠♥tr st♥ ♥tr ♦♥♥t♦
♠s♦r r♣t♦r ② ♥♦ ♦♠♦ ♦♥s♥ ♦s st♥t♦s ♥ó♠♥♦s
♥♠r♦s ♥ só♥
t♠♥t s ♠ rt♦rs ♦s ♦t♦s q ♣r♦♥ rs♣sts
♦♠♦ q s ♠str ♥ r ♠♥trs q ♦s ♦t♦s q
r rt♦r ♥trr♦ ♥ ♥ ♠♦ ♥♦r♠ ②
rrr♠ rtríst♦
r t♦ ♥trr♦ q ♣rs♥t ♦rs ② r
rr♠ rtríst♦ s s sñ♥ ♦s érts s
♣ér♦s
♣r♦♥ rs♣sts t♣♦ q s ♠str ♥ r s ♦s ♠
rt♦rs
♥ ♣rát á♥♦ s ♣rs♥t♥ ♥ rrr♠ sñs rt♦rs
s t r③r ♥ st♠ó♥ ♦ ♣r♦♣ó♥ sñ
♥ ♠♦ t③♥♦ ó♥ Pr ♦ ♦♥♦♦s ♦s ♦rs
tv ② xpm s s♣r♣♦♥ rrr♠ ♣ér♦ ♦t♥ ♣rtr
ó♥ ② s ♥ ♦s ♦rs ♣♦só♥ ♥♦ x0 ② z0 ②
♦ ♣r♦♣ó♥ v ♠♥r ♦t♥r ♠♦r st ♣♦s
♥tr ♣ér♦ ② sñ
② q ♦♥srr q s ♥ ♥ ♣rát s trt ♦♥ ♣r♦♠s ♠ás
♦♠♣♦s q ♦s ♦♥sr♦s ♥ ♦s ♣árr♦s ♥tr♦rs s sñs ♦♥ ♦r
♠ ❱ ♥rt s♦♥ rtríst ♦t♦s ♣qñ♦s ♦♠♣r♦s ♦♥
♦♥t ♦♥ sñ st t♣♦ sñs t♠é♥ s ♦sr ♥ ♦s
s♦s q ♥♦ ♣rs♥t ♦rs ♦ s♥s r♣t♦s ♦♠♦ s ♠str ♥
r ♦♥ s ♥r♥ sñs t♣♦ ♣ér♦s ró♥ ♥
♥ s sq♥s ♦t♦ rt♥r s s sñ♥ ♦s érts
s ♣ér♦s
♠♣♦ s♦♥♦ ♦rtr s♠♣
♥ st só♥ s ♣rs♥t ♥ ♠♣♦ ás♦ ♥tr♣rtó♥ t♦s
♦rtr s♠♣ s ♠♦♥s s r③r♦♥ ♥ ♥ ③♦♥ ♥ q s s♣♦♥í
①st♥ ñ♦s P❱ q tr♥s♣♦rt♥ ♦♥ ♣r♦♣óst♦
♦③r♦s s♣sr♦♥ í♥s s♦♥♦ ♥ ró♥ ♣r♣♥r
s♣st ♣r ♦s ñ♦s ② s qrr♦♥ t♦s ♦ r♦ s í♥s ♦♥
s♣ró♥ ♦♥st♥t 0,25 ♠ ♥tr s ♥t♥s ♠s♦r ② r♣t♦r ♦st
♦♥st♥t t③ó sst♠ rr ♥s♦rs ♦tr Ps
P ② ♥t♥s ♦♥ r♥ ♥tr ♠só♥ ③
♥ r s ♠str ♥ rrr♠ ♦t♥♦ ♣rtr ♦s t♦s
♥ s í♥s s♦♥♦ t r♦♥ rstr♦s ♣♦r sst♠ ♦r
r ♦♠♦ s ♣ ♦srr ♥ r s ♣♦s ♥tr sñ rt
♣r ♦♠♦ ♥s ♦r③♦♥ts ♥ ♣rt s♣r♦r rrr♠
♥tr t = 3 ♥s ② t = 12 ♥s sñ ♦t♦ ♥trr♦ s ♣ ♦srr ♦♥
t ♦♥ ért ♥ (x; t) = (−0,1 ♠; 16 ♥s) ♣r♦①♠♠♥t s ♥
r rr♠ ♦t♥♦ ♦♥ ♥ ♦♥ró♥ ♦
rtr s♠♣ ♦♥ ♦st 0,25 ♠ ♥ts ♣r s st♥ts
té♥s ♣r♦s♠♥t♦ ♥ ó♥ ért
sñ ró♥ ♦r♥ ♥ ♦t♦ ♥trr♦
♦♥ ♥ ♥ r
Pr ♦srr ♦♥ r sñ ♦t♦ ♥trr♦ s ♥sr♦ ♣r
st♥ts té♥s ♣r♦s♠♥t♦ ♦s t♦s ♠♥r ♠♦rr
♠♥ ésts té♥s s sr♥ ♥ só♥ ♥ r s
♠str rrr♠ ♦t♥♦ ♦ ♣r ♦ ♦rró♥ ♦r♥
t♠♣♦ ♠♥ó♥ sñ rt ② ♥♥ ♦s t♦s r
♣ ♦srr ♥ sñ tí♣ ró♥ ♥r ♣♦r ♦t♦ ♦♥
ért ♥ (x; t) = (−0,1 ♠; 12 ♥s) ♣r♦①♠♠♥t ♥ ③♦♥ ♥tr t = 0
♥s ② t = 10 ♥s s ♣♦s ♦srr ♥ ♥t sñs s♣r♣sts ②
q s trt ♥ s♦ r♥♦ ♠② ♥tr♥♦ ♥ q s ♣ ♥♦♥trr
♥ r♥ ♥t ♦t♦s ♣qñ♦s
Pr st♠r ♣r♦♥ ñ♦ ② ♦ ♣r♦♣ó♥ ♥
♠♦ s s♣r♣♦♥ ♥ rrr♠ rá♦ ♥ ♣ér♦ r
② s ♥ ss ♣rá♠tr♦s ♠♥r ♦t♥r ♠♦r st ♣♦s
r rr♠ ♦t♥♦ ♦♥ ♥ ♦♥ró♥ ♦
rtr s♠♣ ♦♥ ♦st 0,25 ♠ s st ♥ ♣ér♦
sñ ♣r♦ ♣♦r ♦t♦ ♥trr♦
r ♦t♦ ③♦♥ s♦♥♦ ♦ ①r ♣r
①♣♦♥r ♦t♦ ♥trr♦
♥tr r ② sñ ♦♠♦ s ①♣ó ♥ só♥ ♦♠♦ rst♦ s
♦t♥ ♥ ♣r♦♥ (0,38±0,12) ♠ ② ♥ ♦ ♣r♦♣ó♥
(0,07± 0,01) ♠♥s st ♦r ♦ s rtríst♦ s♦s ♦♥ t♦
r♦ ♠ r t só♥ ♦♠♦ s ♣ ♦srr
♥ r ♣r♦♥ ♣♥tró♥ s ♠t ② q ♥♦ s ♦t♥♥
sñs ♥t♥s ♣r ♣r t♠♣♦s ♠②♦rs q 16 ♥s
♥♠♥t s ♣ r ♥ r ♦t♦ ♦ sr ①♣st♦
st s ♥♦♥tr ♥ ♣r♦♥ 0,35 ♠ t♠é♥ s ♣♦s ♦srr
♣rs♥ ♥ r♥ ♥t ♦t♦s ♣qñ♦s ♥trr♦s t ♦♠♦
r ♣r♦ ♣rtr ♥áss ♦s t♦s ♦rr
Pr♦s♠♥t♦ ás♦ ♣r t♦s ♦r
tr s♠♣
♦♠♦ s ♦sr ♥ r só♥ ♥tr♦r s sñs ♦s
♦t♦s ♥trr♦s ♥♦ s ♣♥ st♥r ♦♥ r ♥ ♦s t♦s qr♦s
♦ st♥t♦s t♦rs ♦♠♦ ♣rs♥ sñs s♥rs ② ♣ér
♥t♥s q sr sñ ♣r♦♣rs ♥ ♠♦ P♦r st r③ó♥ ♦
qrr ♦s t♦s ② ♦r♥r♦s ♥ ♥ rrr♠ s ♥sr♦ ♣r rt♦s
♠ét♦♦s ♣r♦s♠♥t♦ ♦♥ ♣r♦♣óst♦ ♠♦rr s sñs ♥trés ②
s♠♥r ♦ ♠♥r s sñs s♥rs ♣r sí ♣♦r ♦t♥r ♥♦r♠ó♥
r♥t ♥tr♣rtr ♦rrt♠♥t ♦s t♦s t♣♦ ♣r♦s♠♥t♦
q s r ♣♥rá ♣♦r ♠♣♦ ♦s t♦s ♦r♥s ②
♦s rqr♠♥t♦s ♣r ♥tr♣rtó♥
♦♠♦ ♠♣♦ s rt♦♠♥ ♦s t♦s sr♣t♦s ♥ só♥ ♥ r
s r♣t r q ♠str ♥ rrr♠ ♦t♥♦ ♣rtr
♦s t♦s t ♦♠♦ s♦♥ rstr♦s ♣♦r sst♠ ♦rr ♥ st r
♦ ú♥♦ q s ♦sr ♦♥ r s sñ rt ♣r ♦♠♦
♥s ♦r③♦♥ts ♦ r♦ t♦♦ rrr♠ ♥ r s
♠str rá♦ tr③ q ♦rrs♣♦♥ x = 0 ♠ ♥ sñ
rt ♦♣ ♥tr♦ t = (3, 12) ♥s ♣r♦①♠♠♥t
t♠♥t ♥ ♠t♦♦♦í ♦rr ♠s♦r ② r♣t♦r s
♥ ♦rt st♥ r ♠s♦r ♦s ♠♣♦s t♥♥ ♦♠♣♦♥♥ts
r♥ s♦s ♦♥ ♠♣♦s tr♦stát♦s ♥t♦s q s
♦sr ♥ ♦s t♦s ♦♠♦ ♥ ró♥ t♠♣♦r ♥t ♥ ♠♦
sñ st t♦ s ♦♥♦ ♦♠♦ ♦ ② s ♣ r ♥ r ♦♠♦
♥ s♣③♠♥t♦ ♥ ♠♦ sñ ♣ s♣r♠r rst♥♦
ss♦s s♠♥t♦s tr③ ss ♦rs ♠♦s ♦ ♣♥♦ ♥ tr♦
t♠♣♦r ♣s t♦s sñ ♣r♦s♦ q s ♦♥♦ ♦♠♦ ♦ ❬❪
rst♦ ♣r st ♣r♦♠♥t♦ ♦s t♦s r s
r t♦s ♦t♥♦s ♥ts ♣r s st♥ts té
♥s ♣r♦s♠♥t♦ rrr♠ ② tr③ ♦rrs♣♦♥♥t
x = 0 ♠
r ♣ ♦ ♦s t♦s ♦t♥♦s rrr
♠ ② tr③ ♦rrs♣♦♥♥t x = 0 ♠
♠str ♥ r ♦♠♣r♥♦ s rs ② s ♣ r
♦rró♥ q s ♦r ♥ ♥ ♠♦ sñ
s♥t ♣s♦ ♦♥sst ♥ ♥r ♦r♥ t♠♣♦ t♦s s tr③s
② r♦ ♥ t = 0 ♥s ♦r♥ t♠♣♦ ♣ rr ♥ tr③ ♦tr
♣♦r rss r③♦♥s ♦♠♦ ♣♦r ♠♣♦ t♦♥s ♥ s♣r♦ ♣s♦
♠t♦♦♦í tí♣ ♦♥sst ♥ sr ♣r♠r ♣♦ ♠á①♠♦ ♦ ♠í♥♠♦
♣r♠r tr③ rrr♠ st s ♦r♥ ♥ sñ
rt ♥t♥ r♣t♦r ♦ s s♣③♥ t♦s s tr③s ♥ ♦r♠
♦♥♥t ♠♥r q ♣♦ s♦♥♦ ♥ ♣r♠r tr③ ♦♥ ♦♥
t = 0 ♥s r rs ② ♥ ♥s st♦♥s ♣ ♣rs♥trs
♥ s♣③♠♥t♦ ♦r♥ t♠♣♦s q rí ♦ r♦ rrr♠s
♥ s♦s s♦s ♥ ♥sr♦ ♣r ♣r♦♠♥t♦ sr♣t♦ ♥
s tr③s ♣♦r s♣r♦
♥ ♦s rrr♠s s♥ ♣r♦sr ♥♦r♠♠♥t s ♣rs♥t♥ ♥s ♦r
r r③ ♦rró♥ ♦r♥ t♠♣♦s
rrr♠ ② tr③ ♦rrs♣♦♥♥t x = 0 ♠
③♦♥ts t ♥t♥s q t♠♥t ♥♦ ♣r♠t♥ r s sñs
♥trés ésts s ♣♥ ♦srr ♥ r ♥tr t = −5 ♥s ② t = 5
♥s ♣r♦①♠♠♥t sts sñs s♦♥ ♦♥s♥ ♣r♥♣♠♥t
r♣ó♥ sñ rt ♣r♦♥♥t ♠s♦r ♣r♦ t♠é♥ s ♣♥
♦r♥r ♥ r①♦♥s ♣r♦s ♥ ♦t♦s ♦ ♣rs♦♥s s ♣♦r ♥
♠ s♦ ♣♦r ♠♣♦ ♥ t♦ ♦ ♦♣r♦r sst♠ ♦rr
t♠♥t ♣r r♠♦r sts ♥s ♦r③♦♥ts s ♦t♥ ♥ tr③
♣r♦♠♦ ♣rtr t♦s s tr③s q ♦♠♣♦♥♥ rrr
♠ ② s rst ♥ s tr③s rrr♠ st ♣r♦♠♥t♦
♠♥ t♦s s sñs ♦r③♦♥ts ♦ s ♦r③♦♥ts s r ♦s ♥t♦s
q stá♥ ♣rs♥ts ♥ ♥ r♥ ♥t tr③s ♥②♥♦ s sñs
rt♦rs ♥trr♦s ❯♥ tr♥t ♦♥sst ♥ r tr③ ♣r♦♠♦
♦♥sr♥♦ só♦ ♥ ♥t♥ ♥tr♦ rrr♠ ♦♥ r♥♦s ♣♦só♥
② t♠♣♦s ♣r♦♣♦s ♥ ♠♣♦ s r♠ sñ rt ♥tr♦ ♥
♥t♥ ♥ s t = −5 ♥s st t = 5 ♥s q s ①t♥ s x = −2
♠ st x = 2 ♠ st♦ ♣r♠t ♦♥srr s sñs s ♦r③♦♥ts q s
♣r♥ ♣rs♥tr ♥ t♠♣♦s ♠②♦rs t = 5 ♥s rst♦ s ♣ r ♥
r
s sñs ♣r♦♥♥ts ♥♦s ♣r♦♥♦s t♥♥ ♥ ♠♣t ♠♦
♠♥♦r q s q ♣r♦♥♥ ♥♦s s♣rs ② q ésts s t♥ú♥
rá♣♠♥t ♣r♦♣rs ♥ ss♦ r só♥ Pr s③r
s♠tá♥♠♥t s sñs ♣r♦♥♥ts st♥ts ♣r♦♥s s ♥
sr♦ r s ♠♣ts ♣♥♦ ♥ ♥ó♥ ♥♥ r♥t ♥
t♠♣♦ ♣r♦s♦ ♦♥sst ♥ ♠♥tr st♠♥t ♥t♥s ♦s
t♦s ♠♥r t q ♣r ♦s st♥t♦s t♠♣♦s ♥t♥s
r ♠♥ sñ rt rrr♠ ② tr③
♦rrs♣♦♥♥t x = 0 ♠
r st♦ ♣r ♥♥ rrr♠ ②
tr③ ♦rrs♣♦♥♥t x = 0 ♠
s s♠r Pr st♦ s ♣ ♥ ♥ó♥ ♠♣ó♥ ♣rtr♠♥
♥ ♥ t♦ ♥t♥ t♠♣♦r ♦♠♦ ♣ó♥ ♥♥ s ♥
♣r♦s♦ ♥♦ ♥ ♦tr♦s ♣r♦s♦s tr♦ ♥♦ ♣r♦♥ ♠s♠♦ rst♦
sr ♣♦s ♥ts ♦ s♣és ♣r ♥♥ rrr♠ rst♥
t ♣r ♥♥ ♦s t♦s r s ♠str ♥ r
sñ ró♥ ♦t♦ ♥trr♦ s ♣ r ♦♥ ért ♥
(x; t) = (−0,1 ♠; 12 ♥s
♥ ♥♦s s♦s s ♣♥ té♥s ♠ás ♦♠♣s ♣r♦s♠♥t♦ ♦♠♦
♣♦r ♠♣♦ st♥t♦s t♣♦s tr♦s s♣s ② t♠♣♦rs ♦♥♦ó♥
② ♦rró♥ t♦♣♦rí ♥ ♦s s♦s ♥ ♦s q sñ ♣rs♥t
ró♥ sñ r♦ s ♥sr♦ ♣r ♠t♦♦♦ís ♣r ♠♦rr ♠♥
♦♠♦ s rá ♠ás ♥t s♦♥s ② ② ♣ít♦
t♦♦♦í ♠ró♥ sñs
s sñs ♦r♥s ♥ ♦t♦s ♥trr♦s ♣r♥ ♥ rrr♠
♦♥ st♥t ♦r♠ ♣♦só♥ ♥♥ó♥ q ♦s ♦t♦s rs só♥
♦ q t ♥tr♣rtó♥ ♣r♥♣♠♥t ♥ s♦s q ♣rs♥t♥ r♥
s ♥ts sñs s♣r♣sts ♥tr sí ♦ ♣♥♥ts ♣r♦♥♥s
♥tr♣rtó♥ s ♣ tr s s ♦rr rrr♠ ♣r ♦t♥r ♥
♠♥ ♥♦ ♦s rt♦rs ♦rrt♠♥t ♦s ♥ s♣♦ st
♣r♦s♦ s ♥♦♠♥ ♠ró♥ ② ♦♠♣r♥ ♥♦q ② ♣♦s♦♥♠♥t♦
s sñs r①ó♥ ♥♦q ♦♣s s r♦♥s ♠①♠③♥♦
♠♣t ♠♥trs q ♣♦s♦♥♠♥t♦ ♦③ ♦s ♥t♦s ♦rrt♠♥t ♦
q rst r♥t ♥♦ ①st♥ rt♦rs ♦♥ ♣♥♥ts ♣r♦♥♥s ②
r♦♥s trs s♥ts ♥ ♦ ♣r♦♣ó♥ ♦♠♦ rs
t♦ s ♦t♥ ♥ ♠♥ ♠ás rst ♦s ♦t♦s ♥trr♦s ❬❪ ①st♥
rss ♠t♦♦♦ís ♠ró♥ ts ♦♠♦ ♠ró♥ r♦ s♠
♣ró ♠ró♥ ♥rs ♥ t♠♣♦ rrs t♠ ♠rt♦♥ t
❬❪
Pr ♦t♥r s♦ó♥ ♥♠ér ♠♥t ♠ró♥ s s ♦♥srr
♥ ♠♦♦ q ♦♥sst ♥ r♠♣③r rt♦r ♣♦r ♥ sr ♥ts
rts s q ♥ t = 0 ♠t♥ sñ q s ♣r♦♣ st s♣r
♦♥ s ♥ ♦s r♣t♦rs ❬❪ s s♠ ♥ ♦♥ró♥ ♥ s
♣♦s♦♥s s ♥t♥s ♠s♦r ② r♣t♦r ♦♥♥ ♥rí rstr
♥ ♣♥t♦ s♦r s♣r ♣r♦♥ ♠♥♦s ♠s♦r rt♦r ②
rt♦r r♣t♦r q ♦♥♥ ♥ s s♦ t♠♣♦ t♦t
sñ s ♦s s t♠♣♦ ♥tr ♠s♦r ② rt♦r P♦r
st r③ó♥ ♥ st ♠♦♦ ♦♥srrs ♥ ♦ q♥t q
♦rrs♣♦♥ ♠t ♦ r ♣r♦♣ó♥ ❬❪
♣rtr st ♠♦♦ s ♣ r q ♠t♦♦♦í ♠ró♥
♥rs ♥ t♠♣♦ ♦♥sst ♥ r♣rr ♠♣♦ t = 0 ♣rtr ♦s
t♦s qr♦s ♥ s♣r st ♠♥r s ♣♦s ♦t♥r ♥ ♦r♠
♥rt ♦③ó♥ s ♥ts rts s♦s ♦♥ ♦s rt♦rs
q ②♥ ♣r♦♦ ♦s ♠♣♦s Pr ①tr♣♦r ♦s t♦s trás ♥
t♠♣♦ s ♣ t③r ó♥ ♦♥s sr ♥ ♦s ♠♥s♦♥s
❬❪∂2E
∂x2+
∂2E
∂z2=
1
v2(x, z)
∂2E
∂t2
♦♥ E s ♠♣♦ sr q s ♣r♦♣ x s ♦♦r♥ s♣
♦r③♦♥t z s ♦♦r♥ s♣ rt v(x, z) s ♦ ♣r♦
♣ó♥ ② t s t♠♣♦ ♥ st ó♥ ♥♦ s ♥ s♣♦s♦♥s s♦r
♣r♦♠ ís♦ q s rs ♣♦r st r③ó♥ s ♣ qr ♦♥
q s ♣r♦♣ q s ♣ ♣r♦①♠r ♣♦r ♥ rs♣st sr ♦♥ ♥
stró♥ ♦ ♦♥ ró♥ s♣ v(x, z)
❯♥ ♣♦s ♣r rs♦r ♥♠ér♠♥t ó♥ s ♣
r ♥ r♣rs♥tó♥ r♥s ♥ts s♥♦ ♦r♥ ♥tr P♦r
♠♣♦ ♣r♠r tér♠♥♦ ó♥ s r♠♣③ ♣♦r
∂2E(x, z, t)
∂x2≈ E(xk+1, zj, ti)− 2E(xk, zj, ti) + E(xk−1, zj, ti)
h2
♦♥ h s ♥r♠♥t♦ s♣ r rs♣st E(xk, zj, ti) ♥
♣♥t♦ r (xk, zj) t♠♣♦ ti stá ♣♦r ❬❪
E(xk, zj, ti) = 2(1− 2A2)E(xk, zj, ti−1)− E(xk, zj, ti−2)+
A2[E(xk+1, zj, ti−1) + E(xk−1, zj, ti−1)+
E(xk, zj+1, ti−1) + E(xk, zj−1, ti−1)]
♦♥ v(x, z)∆t/h ∆t s ♥r♠♥t♦ t♠♣♦r
♥ts ♣r ♠ét♦♦ ♠ró♥ s r③♥ st♥t♦s ♣r♦s♠♥
t♦s ♠♥ ♦♠♦ r♠♦ó♥ ♦♥♦ ♦♠♣♦♥♥t r♥
② ♣ó♥ tr♦s s♣ ② t♠♣♦r st♦s ♣r♦♠♥t♦s st♦r
s♦♥♥ ♠♣t ♦ r♦ sñ ♣r♦ ♣r♠t♥ ♦t♥r rst♦s
♦rs ♣r ♦s ♥♦s ♠ás ♣r♦♥♦s
Pr ♠♣♠♥tr ♠ét♦♦ ♠ró♥ s srr♦ó ♥ ó♦ ♦♠♣
t♦♥ st s ♠♣♠♥tó ♥ ♥ ♠♥r tr s
♦♠♣t ♦♥ ♦tr♦s ♣r♦r♠s tr♦ ♦ s③ó♥ t♠é♥ s
rr♦♦s ♥ s ♥t♦r♥♦
♦♠♦ ♠♣♦ ♣ó♥ ♠ét♦♦ ♠ró♥ s ♦♥sr s♦
♥ ♦t♦ á♠tr♦ 0,05 ♠ q s ♦③ ♥ ♣r♦♥ 0,80 ♠
r ♦t♦ s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 5,25
② ♥ ♦♥t σ = 2,5 ♠♠ ② ♠♦ r♥♥t ♣♦r ǫr = 3,5 ②
σ = 1 ♠♠ ♠♦ s♦r ♦s s r r♥ ♥tr s ♦♥s
r ♣ó♥ ♠ét♦♦ ♠ró♥ ♠♦♦ ♦♥
♥ rt♦r rrr♠ ♥r♦ só♥ ♠r
♠ts s fc = 500 ③ ♦♥t ♦♥ λ = 0,32 ♠ ♥tr♦ s♦ ♦♥
st♦s ♣rá♠tr♦s s r③ó ♥ s♠ó♥ ♥♠ér rs♣st s♣r
♣r ♥ s♦♥♦ ♦♥ ♦st r♦ Pr ♦ s t③ó ♠ét♦♦ r♥s
♥ts ♥ ♦♠♥♦ t♠♣♦ rs♦r♦♥ ♥♠ér♠♥t s
♦♥s ① ♣r ♦t♥r ♠♣♦ Ey s♦r ♥ r ♥♦r♠
② s r♦♥ s ♦♥♦♥s ♦♥t♦r♥♦ ♠♥r tr r①♦♥s ♥
♦r r ♥ ♣é♥ s ♥ ♠ás ts ♣r♦♠♥t♦
♣r ♦t♥r ♦s t♦s s♠♦s
♥ r s ♠str ♣r ♦rtr s♠♣ ♦t♥♦ ♣r
♠♦♦ r sñ r①ó♥ rt♦r s ♣ ♦srr
♦♠♦ ♥ ♣ér♦ ♦♥ ért ♥ 11 ♥s ♣r♦①♠♠♥t ♦ s ♠♥ó
sñ rt ♦s t♦s ② s ♣ó ♠ét♦♦ ♠ró♥ t③♥♦
♥ ♦ ♦♥st♥t v = 0,16 ♠♥s ♥ r s ♣ r ♦♠♦
rst♦ q ♥rí str ♥♠♥t ♥ ♣ér♦ ró♥
♦♣s♦ ♥ ♣♦só♥ rt♦r
t♦♦♦ís ♦rtr ♠út♣ ♦♠
♠♦♥ ♣♦♥t
♥ ♥♦s s♦s t③♥♦ ♠t♦♦♦í ♦rtr s♠♣ s ♦t♥♥
sñs ♠s♦ és ② q t♥♥ sr ♦♥ss r♥t ♥tr♣rt
ó♥ ♦♠♦ ♦♥s♥ ♦ ♦♥trst q ♣rs♥t♥ s s♦♥t♥s
♥trés ♦♥ rs♣t♦ ♠♦ r♥♥t Pr ♦rr ♥ ♠♦r ♥t
ó♥ ② rtr③ó♥ st t♣♦ sñs s t t③r ♠t♦♦♦ís
♦rtr ♠út♣ ♥ s q ♣r ♣♦só♥ s♦r í♥ s♦♥♦
s qr♥ rs tr③s ♣r st♥ts ♣♦s♦♥s rts ♠s♦r ②
r♣t♦r ♠t♦♦♦í ♠ás ♥ s ♣♥t♦ ♠♦ ♦♠ú♥ ♦♠♠♦♥
♠♣♦♥t P st ♠t♦♦♦í s t③ t♥t♦ ♣r st♠r ♦
♣r♦♣ó♥ ♥ ♥ó♥ ♣r♦♥ ♦♠♦ ♣r ♦t♥r s♦♥s r
ts ♠♦rs ③♦♥ st♦
♥ ♠ét♦♦ P s t③ ♥ ♥t♥ ♠s♦r ② ♥ ♥t♥ r♣t♦r
s ♥t♥s s ♦♦♥ ♠♦s ♦s ♥ ♣♥t♦ s♦r í♥ s♦♥♦ ♦♥
♥ s♣ró♥ tr♠♥ ② s qr ♥ tr③ ♦ ♠s ♥t♥s s
♥ s♠étr♠♥t ♣♥t♦ ♠♦ ♠♥t♥♥♦ ♣♦só♥ st
♣♥t♦ ② s qr ♥ s♥ tr③ ♣r♦♠♥t♦ s r♣t st q
s qr ♥ ♦♥♥t♦ tr③s s♦s ♦♥ ♠s♠♦ ♣♥t♦ ♠♦ r
r st ♣r♦s♦ s r③ ♣r ♥♦ ♦s ♣♥t♦s í♥
s♦♥♦ ♦ qsó♥ ♦s t♦s ♥ sr ♣r♦s♦s ♠♥r
♣r♦♣ ♣r ♦t♥r ♥♠♥t ♥ ♠♦r ♥ s sñs ♠ét♦♦ P
♠♣ ♦s t♣s ♣r♦s♠♥t♦ ♥ ♥ ♣r♠r t♣ s ♣♥ rts
♦rr♦♥s ♥ s tr③s ♣r ♦♠♣♥sr s r♥s t♠♣♦s
♣r♦♣ó♥ ♦ s st♥ts st♥s ♥tr ♥t♥s ② ♥ s♥
t♣ s ♣r♦ rr♣ó♥ ♦ ♣♠♥t♦ s tr③s ♦t♥♥♦
♥ tr③ rst♥t ♣r ♣♥t♦ ♠♦
r♥t rr♣♠♥t♦ ♦ s♠ tr③s s sñs ♥trés s r
st♥ ♣♦r ♥trr♥ ♦♥strt ② s r r♦ ♦♠♦ ♦♥s♥
♥trr♥ strt ♠ ♥ú♠r♦ ♣♠♥t♦ ♥t
s♣r♦♥s ♠s♦r r♣t♦r q s ♦♥sr♥ ♥ s♠
t♠♣♦ ♥ q s r ♥ sñ ♦ rrs ♥ ♥ s♣r
s♦♥t♥ ♣♥ ♦ ♣r♦♣ó♥ ♥ ♠♦ ②
r ❯ó♥ s ♥t♥s ♠s♦r ② r♣t♦r
♥ ♠ét♦♦ P
st♥ r♦rr ♥ ♠♥♦ ♠s♦r rt♦r r♣t♦r Pr s♣r♦♥s
♠②♦rs ♥tr ♠s♦r ② r♣t♦r ♦s ♠♥♦s r♦rr♦s ♣♦r sñ s♦♥
♠②♦rs ② ♥ ♦♥s♥ t♠é♥ s♦♥ ♠②♦rs ♦s t♠♣♦s s
♥sr♦ ♦rrr st r♥ ♥ts s rr♣r s tr③s ♠♥r q
s♠ s ♠s♠s ♣r♦③ ♥ sñ ♠②♦r ♠♣t q q
s ♦sr ♥ ♥ s tr③s ♣♦r s♣r♦ st ♦rró♥ s ♦♥♦
♦♥ ♥♦♠r ♥♦r♠ ♠♦♦t ♥ r s ♣♥ r
♦s t♠♣♦s tv sñ r ♥ ♥ ♥trs ♦r③♦♥t
♥ ♥ó♥ s♣ró♥ ♠s♦r r♣t♦r x ♥ s♦ r①♦♥s
♣r♦s ♣♦r ♥ ♥trs ♣♥ ♦r③♦♥t ♦③ ♦ ♥ ♠♦
♥ ♦ ♣r♦♣ó♥ s ♥♦r♠ ♦s t♠♣♦s s
♣ ①♣rsr ♣♦r ♠♦ s♥t ó♥
t(x) =
√
t20 −x2
v2s
♦♥ t0 r♣rs♥t t♠♣♦ sñ ♥ s♦ ♥ q s ♣♦s♦♥s
s ♥t♥s ♠s♦r ② r♣t♦r ♦♥♥ ♦st r♦ ② vs s ♦
♣♠♥t♦ st ♦ s q ♣r♠t ♦t♥r ♠♦r rst♦ ♣♦s
rr♣r s tr③s ♠s♠ ♣♥ ♦ ♣r♦♣ó♥ ②
s♣s♦r t♦s s ♣s ss♦ q s ♥♥tr♥ ♣♦r ♥♠
♥ tr♠♥ ♥trs r♥ ♥ t♠♣♦ sñ ♥tr
♥ ♦♥ró♥ ♦♥ s♣ró♥ r♦ t0 ② ♥ ♦♥ s♣ró♥ x t(x) stá
♣♦r ∆t ② s ♣ ①♣rsr ♦♠♦ ❬❪
∆t = t(x)− t0
r ♠♣♦s tv sñ r ♥
♥ ♥trs ♦r③♦♥t ♥ ♥ó♥ s♣ró♥ ♠s♦r r
♣t♦r x s♥ ♣r ♦rró♥ ② ♦ ♣r
♦rró♥ ♦♥ ♥ ♦ ♠♥♦r q
② ♠②♦r q
s♦r ♦s t♦s r s r③ ♦rró♥ ó♥
s ♦t♥ rst♦ q s ♠str ♥ r ♦♥ s ♣
r q t♠♣♦ sñ ♦♥ ♣r t♦♦ x ♣r♦ ♣r ♦t♥r
st t♣♦ rst♦ ♦ t♥r ♦r ♦ ♦r
♦ s ♠s♦ ♦ r ♦ ♠s♦ t♦ r ♦s
t♠♣♦s ♥♦ stá♥ ♥♦s ② s♠r s tr③s ♥♦ s ♦r ♠♦rr
sñ ♠♥r st♦ ♥ q s ♥sr♦ r③r ♥ st
♦ ♣r ♦t♥r ♥ rst♦ ♥ ♥ q s sñs ♣rs♥t♥
♠②♦r ♠♦r ♣♦s st♦ ♣r♦♣♦r♦♥ ♥ ♠t♦♦♦í ♣r♦♣ ♣r
♦t♥r ♥ st♠ó♥ s ♦s ♣r♦♣ó♥ ♣r♦♠♦ ♣r
♣♥t♦ ♠♦ ♦♠ú♥ ② ♦ ♣r ♥r♠♥tr s ♥t♥ss s sñs
♥trés ♥ ró♥ r♦ r♥♥t ♥ s tr③s rst♥ts
♦♠♦ ♠♣♦ ♣ó♥ ♠ét♦♦ P s ♦♥sr s♦
♥ rt♦r ♣♥♦ ♦r③♦♥t ♦③♦ ♥ ♣r♦♥ 0,80 ♠ r
strt♦ s♣r s rtr③ ♣♦r ♥ ♣r♠t ǫr = 3,5 ②
♥ ♦♥t σ = 1♠♠ ♠♥trs q strt♦ ♣r♦♥♦ ♣♦r ǫr = 4
② σ = 2 ♠♠ r♥ ♠♣♦ s 500 ③ s ♣ ♥ tó♥
t♦r s ♠trs ♣r♠t ② ♦♥t ♦♥ st♦s
♣rá♠tr♦s s r③ó ♥ s♠ó♥ ♥♠ér rs♣st s♣r ♣r
r Pr ♦st 0,45 ♠ ♣r P n = 4
♦st 0,45 ♠ vst = 0,16 ♠♥s
♥ s♦♥♦ r③♦ ♦rr Pr ♦ s t③ó ♠ét♦♦ r♥s
♥ts q s sr ♥ ♣é♥
♥ r s ♠str ♥ ♣r ♦st 0,45 ♠ ♦t♥♦ ♣r
♠♦♦ r sñ r①ó♥ ♣r ♥trs s ♣
♦srr ♥ t = 11 ♥s ♣r♦①♠♠♥t ♥ r s ♠str
rst♦ ♣r P ♦♥ ♥ ♥ú♠r♦ ② ♦ ♣♠♥t♦ n = 4 ② vst =
0,16 ♠♥s rs♣t♠♥t ♦♠♣rr s rs ② s ♣ r
q ♠ét♦♦ P ♠♦r sñ q s ♦t♥ ♦♥
♣ít♦
♣ó♥ ♦s ♠ét♦♦s ss
♥ ♦rr
♥ ♥♦s s♦s ♣♦r ♠♣♦ ♥♦ ♦s ♥♦s ♣rs♥t♥ ♦ ♦♥trst
♦♥ rs♣t♦ ♠♦ r♥♥t ♦ ♦♠trís ♦♠♣s ♦s st♦s r
③♦s ♦♥ ♠t♦♦♦ís ♦rtr s♠♣ ♣r♦♥ rst♦s ♠♦s
♦ ís ♥tr♣rtr ♥ ss st♦♥s s ♣♥ st♥ts té♥s ②
s ♣r♦s♠♥t♦ s③ó♥ ♦ qsó♥ t♦s ♣r r ♥
♥tr♣rtó♥ ♠ás r ♦s rst♦s ♦♠♦ ♠♣♦ s ♣rs♥t♥ ♥ st
♣ít♦ st♦s q ♠str♥ ♣ó♥ ♦s ♠ét♦♦s ♠ró♥
sñs P ② ♠♦♦ ♣r rtr③ó♥ strtrs ♥ st♦
rq♦ó♦ P♦ ♥♦ ❬❪
t♦ rq♦ó♦ P♦ ♥♦
st♦ rq♦ó♦ P♦ ♥♦ s ♥♥tr ♥ ♠á
♥ ró♥ s♠sért ♥ ♣r♦♥ t♠r r♥t♥ r r
st st♦ stá r♦♥♦ ♦♥ ♥ s ♣r♠rs ♦♠♥s rí♦s
♣st♦rs ró♥ q s srr♦ó ♣r♥♣♠♥t r♥t Prí♦♦
♦r♠t♦ ♦♥ ♣r♦①♠♠♥t ñ♦s ♥tü
♦ s rtrísts ♦rás ③♦♥ P♦ ♥♦
stá ♦♥t♥♠♥t ①♣st ♥ ♥t♥s s♠♥tó♥ ♦♠♦ ♦♥s♥
♦s s♠♥t♦s ♥♦r♠♠♥t ♥trr♥ ♥ t♠♣♦s rt♠♥t ♦rt♦s ♦s
r ❯ó♥ st♦ rq♦ó♦ P♦ ♥♦
♣ ♦♥ ♦③ó♥ s strtrs rq♦ós st
s s♦♥ t♠s ♦♥strs ♦♥ ♣rs
s♦♥ ♥ú♦s t♦♥s ♦♥ ♣rs t♣
♦t♦s q stá♥ ♥ s♣r ♣r♦s♦ ♥tr ♦rtr ♦s ♣r♦t
♥ts ñ♥♦s ♦♠♦ ♥t♦ ♠♦s ♣ró♦s ♥ ♠ ♠♥
t ② t♠é♥ ♥tró♥ tr ♥ ♠r♦ ♦s ♦t♦s rq♦ó♦s
♣♥ qr t♠♣♦r♠♥t ①♣st♦s ♦ ♠♦♠♥t♦ ♥tr s
♠♥t♦s ♦s♦♥♦ ♣♦r ♣r♦♣♦ ♥t♦ ② ♦s ♦s ♥♦ ♣♦s
s tó♥ s sí q ♦s ♣r♠r♦s st♦s rq♦ó♦s ♥ ③♦♥ ❬❪
❬❪ rr♦♥ ♣qñs ♣♦r♦♥s ♣rs t♣ q ♣rí♥ ♥ s
♣r q ♣r♠tr♦♥ ♥tr ♥♦ ♥s rs♥s ♦♥ r♥t
♦♠♣ rqttó♥ r♦ ♦♥ s sr♣♦♥s r③s sts
strtrs ♣rs♥t♥ st♥ts ♦r♠s ② str♦♥s s♣s ♣r♦ ♥
♥r ♦♥sstí♥ ♥ trs ♦ tr♦ t♦♥s ♦r♠ rt♥r ♦♥t
s rt♠♥t ♦ ♣♦r ♠♦ árs ♣r♠♥t rts ♦ ♣t♦s ♥
♣rs♥t s♠♥tó♥ t♦ rr s strtrs q r♦♥ ♦sr
s ♣♦r ♦s rqó♦♦s ♥ ♦s ñ♦s st♦ rst ♥ ♥ s ♠②
♥s s ♦ rt♠♥t ♥ ♥♥♥ s ♥ ♠②♦rí
♦s s♦s
♥tr s strtrs st♦ ①st♥ trs ♦♥str♦♥s ♥rrs ♦♥
♦r♠ ♣r♦①♠♠♥t s♠sér sts r♦♥ srts ♣♦r ♦s
t♥ts ró♥ ♥ ♦s ñ♦s ♥♦s ñ♦s s♣és sr♠♥t♦
sts r♦ t♣s ♥♠♥t ② ár t③ ♦♠♦ ñ♦ ♣♦r s
tr♥t ñ♦s ♥t♠♥t ♥ ♠r♦ ♥ ♥stó♥ rq♦ó
♦s sts t♠s ♥ts ♦♠♦ ② r r r♦♥ ♦
③s ♣rtr rstr♦ rq♦ó♦ ①st♥t ② ①s ❬❪ sts
t♥í♥ á♠tr♦s ♣r♦♠♦ ♥tr 2,7 ♠ ② 3,3 ♠ ss ♣rs s ①t♥í♥ s
0 − 0,2 ♠ ♥ ♣r♦♥ ♠á①♠ 1,2 ♠ ② s ♥♦ st ♥
r♥♦ 0,5− 0,7 ♠ ♥ ♠r♦ trr t♠ ♥♦ ♣♦ sr ♥♦♥tr
♦♠♦ s♦♠♥t s ♦♥♦í s ó♥ ♣r♦①♠ ② ♦ q ♥st
ó♥ rq♦ó rqrí ♦③ó♥ ♣rs strtr s r③ó
♥ ♣r♦s♣ó♥ ♦♥ ♦rr ♥♦s ♦s rst♦s st ♣r♦s♣ó♥
s ♣rs♥t♥ ♥ ♣ró①♠ só♥ ♥ ♠♦strr ♥ q ♠ s ♣♥
rtr③r st t♣♦ strtrs ♣rtr t♦s ♦rtr s♠♣ ②
♣ó♥ ♠t♦♦♦í ♠ró♥
P♦r ♦tr♦ ♦ ♥ st t♣♦ st♦s rq♦ó♦s s s ♣rs♥
ss ♦tr♦s ♠♥t♦s ♥trr♦s ♥ s tó♥ ♥♦ s ♥ ♦t♦
♥tr ♦s st♦s ♦♥ ♦rr ♦ s ♣qñ♦ t♠ñ♦ rst
♠♦ ♥trés tr♠♥r t r♦♥♦r s sñs q ♥r
rí♥ st t♣♦ strtrs P♦r t ♠♦t♦ s r③r♦♥ st♦s ♦♥tr♦♦s
t♥♥ts tr♠♥r ♣♦s tó♥ ♥ ♣rtr s t③r♦♥
♠ét♦♦s ♦rtr s♠♣ ♦♥ rs t ♥s ♣r ♥ s
③ó♥ t rs♦ó♥ ss♦ ♠ás ♦rtr ♠út♣ P ②
♠♦♦ ♣r ♥ ♠♦r ♥tó♥ s sñs ♣r♠rs q ♣r♠t
♦t♥r ♥ ♥tr♣rtó♥ ♦s t♦s ♥ st ♣ít♦ s ♣rs♥
t♥ ♠♣♦s ♦s st♦s ♦s s ♠str♥ ♥s s ♣r♥♣s
rtrísts ② ♥s st♦s ♠ét♦♦s
♦③ó♥ ♥ strtr ♣rs
♦♠♦ s sró ♥ só♥ ♠ét♦♦ ♠ró♥ ♣r♠t ♦t♥r
♠á♥s ♠ás rsts ♦s ♦t♦s ♥trr♦s ② s t ♥ q♦s
s♦s ♥ q s sñs s♦♥ ♦♠♣s ♥ st só♥ s sr ♣ó♥
♠ét♦♦ ♠ró♥ ♥ st♦ rq♦ó♦ P♦ ♥♦ ♦♥ ♦t♦
r♦③r
♠♦♠♥t♦ r③r ♣r♦s♣ó♥ s í st♠♦ q ♣s
t rí♦ ♥ ③♦♥ ♣r♦♠♥t í str♦ stró♥
♥tr ♦s strt♦s ♠ás s♣rs ② ♠③♦ ss ♠trs sí só♦
s s♣r♥ sñs és ② ♣♦♦ ♦♥t♥s ♦s ♥♦s rq♦ó♦s
♦ ♦s P♦r ♦tr♦ ♦ s í♥ ♦sr♦ ♥ s♣r ♥ ♥t
♠♣♦rt♥t r♦s s♣rss ②♦ t♠ñ♦ ♥♦ r s♣r ♥ ró♥ ♦♥
s r♦s s t♠s ② P♦r ♥ ♥♦♥trrs r♦s ♥rsts
♥ s♦ ss sñs ♣♦rí♥ s♦♥r s sñs q s st♥ s♥♦
♥ t s♦ ♥áss rstrí ♠② ♦♠♣♦
♦s sts rtrísts ♥♥ q ♥tó♥ rt s
sñs s ♣rs ♣rs♥trí ts ♠♣♦rt♥ts ♦♠♦ ♥
♠♥♦ ♣♦s ♣r ♦t♥r sñs ♠ás s♥s ② rs s ♣r♦♣s♦ ♣
r ♠ró♥ ♦s t♦s st s ♣r♦s♠♥t♦ ♣rs♥t ♥ sr
♥ts ♦♠♦ s ♠♥♦♥ó ♦s ♣rs ♠r♦s ♦♥stt②♥ ♥ ♠♥ ♠ás
rst ss♦ ♣♦r ♦ q s s♠♣ ♥tr♣rtó♥ s strtrs
♦♠♣s ♥♦ ♣r♦s♦ ♥♦q q stá ♠♣ít♦ ♥ ♠ró♥
s s♣r q ♦♣s s ♣ér♦s ró♥ ② q ♣r♦③ sñs
s ♣rs ♠ás ♦♥trst♥ts ② rs♦s
♥ ♥ ♣r♠r ♠♣ñ r③ ♥ ♠r♦ st ss s ♠tó ♥
ár ♠♣ ♥ q s s♣♦♥í q s ♥♦♥tr ② s qrr♦♥ ♣rs
9m ♦♥t s♦r ♥ r r ♦♥ 1m s♣ró♥ ♥tr í♥s
♥ ♠s r♦♥s ♦s ♣rs qr♦s ♥ r♦♥s ♣r♣♥rs
♠♦r♥ ♣♦s tó♥ ♥tó♥ s ♣rs ② q s
r♦s q s ♦r♠♥ ♣♥ t♥r rt ♥ ♥ ró♥ s♦♥♦
♣r♦ s ♣r♦ q ♠r ró♥ rt ♠♥t
r③ó ♥ ♣r♦s♣ó♥ t③♥♦ ♥ ♦rr ♥♥r st♠
♠♦♥♦stát♦ s ♥t♥s ♠s♦r ② r♣t♦r s ♥ ♥ ♠s♠ ♥
♦st ♦♥ r♥ ♥tr 400 ③
❯♥ ③ qr♦s ♦s t♦s t♦♦s ♦s ♣rs ést♦s s ♣r♦sr♦♥
♠♥r ♠♦rr s③ó♥ s sñs ♥ ♥♦ ♦s ♣rs s
♣ó ♦ ♦rró♥ ♦r♥ t♠♣♦ ♠♥ó♥ sñ rt ②
♥♥ ♥ r s ♠str ♥ rrr♠ rtríst♦ ♦t♥♦ ♥
st st♦r ♥s ③♦♥s rrr♠ rrs ♥ r ♣rs♥t♥
sñs ♦♥ ♠②♦r ♥t♥s ♣r♦ ♥♦ stá♥ r③♦♥♠♥t ♥s ♣♦r ♦
q ♥♦ s ♣♥ ♥tr r♠♥t ♦♠♦ ♣♦ss ♥♦♠ís s s
♣rs s ③♦♥s ♦♣♥ ♦s ♥tr♦s x = (0,0; 1,1) ♠ x = (3,8; 4,6)
♠ x = (5,1; 5,9) ♠ ② x = (7,2; 7,8) ♠ ② st t = 10 ♥s
Pr st♠r ♦ ♠ró♥ s ♥③r♦♥ t♦♦s ♦s ♣rs ♦t
♥♦s ② s s♦♥r♦♥ t♦s s ♣ér♦s ró♥ q s ♥♦♥tr♥
♥tr♦ r♥♦ ♣r♦♥s s♣rs ♣r s ♣rs st 1,2 ♠
sú♥ ♥♦r♠ó♥ rq♦ó ♦ s stó ♥ ♣ér♦ ♥ ♥
s sñs ró♥ ♦♠♦ s ①♣ó ♥ só♥ ♥ ♠♣♦
st r③♦ s ♣ r ♥ r ♥♠♥t s r③ó ♣r♦♠♦
t♦♦s ♦s ♦rs ♦t♥♦s ♦r rst♥t v = (0,18± 0,01) ♠♥s
t③ó ♠s♠ ♦ ♠ró♥ ♣r t♦♦s ♦s ♣rs
♥ r s ♠str só♥ ♠r ♣r ♦s t♦s
r ♦sr ♥ ♠♦r ♠♦r ♥ ♥ó♥ s ♥♦♠ís
♣♦r ♠♦ ♣r♦s♦ ♠ró♥ ♦♥ ♥s ③♦♥s q ♣rs♥t♥ ♥ ♥
t♥s rt ♥t♦r♥♦ ♠②♦r q ♥ ♦s t♦s s♥ ♠rr ♣rtr
st♦s rst♦s ♦s s ③♦♥s ♠♥♦♥s ♥ ró♥ ♦♥ ♦s t♦s s♥
♠rr s ♣♥ str ♠ás r♠♠♥t ♦♠♦ ♥ó♠s r♦ s
r rr♠ ♦t♥♦ ♥ ár t♠
♦rrs♣♦♥ ♣r y = 8 ♠ r ó♥ ♠
r ó♥ ♠r q s ♣ó t♦ ♣r s
♠♣ts s♣r♦rs
r rr♠ ♦t♥♦ ♥ ár t♠
♦rrs♣♦♥ ♣r x = 2 ♠ r ♥ q s
st ♥ ♣ér♦ ♥ ♥ s sñs ró♥ q
s ♦sr♥
♥t♥s ② ♦r♠ sts r♦♥s s♦♥ s q s ①t♥♥ ♦ r♦ ♦s
r♥♦s x = (3,8; 4,6) ♠ ② x = (5,1; 5,9) ♠ ♣r ♣r♦♥s s 0,15 ♠
st 0,9 ♠ ♣r♦①♠♠♥t s s rst♥ ♦♠♣ts ♦♥ ♥♦r
♠ó♥ rq♦ó s♦r s t♠s P♦r ♦♥trr♦ ró♥ ♥ r♥♦
x = (7,2; 7,8) ♠ s ♣ srtr ♦♠♦ sñ ♦r♥ ♥ ♥ ♣r ♦
s ♠♥♦r ♥t♥s rt ② ♦r♠
Pr ♦t♥r ♥ ♠♦r s③ó♥ s ♣♦s ♥♦♠ís s t③ó ♥
t♦ s♣r♦r ♣r s ♠♣ts ♠rs s r s ó r♦ t♦♦s
♦s t♦s ♠r♦s ②♦ ♦r s♦t♦ ♠♥♦r q ♥ ♠r ♣r♥♦
P♦r ♠♣♦ ♥ r s ♠str ♠♥ q rst r
♦ ♣ó♥ t♦ s ③♦♥s ♥ó♠s q s ♠♥♦♥r♦♥ ♥
♣árr♦ ♥tr♦r s ♣♥ ♥tr r♠♥t ♥ st r trs ③♦♥s
q s ①t♥♥ ♦ 1,5 ♠ ♦♥ tr ♠♥♦r q 0,5 ♠ ♦ ♠ás s
q 0,3 ♠ s srtr♦♥ ♦♠♦ ♥♦♠ís ♣♦rq ♦♥tr♥ ♥♦r♠ó♥
rq♦ó
❯♥ r♣♦ r♦♥s ♥ó♠s ♦♥ rtrísts s♠rs s sr♣ts
♥ ♠♣♦ ♥tr♦r s ttr♦♥ ♦ r♦ ③♦♥ st♦ sts s
①t♥r♦♥ ♥ ró♥ rt ♣r♦①♠♠♥t ♥tr z = (0,15; 0,30) ♠
st z = (0,75; 1,00) ♠ ♦♥ ♥♦ ♥tr 0,45 ♠ ② 0,85 ♠ rr♦♥ ♥ ♥
♠♣ ② s ♥tr♣rtr♦♥ ♦♥sr♥♦ s rtrísts s♣rs ♣r
♥ r s ♠str ♥ ♠♣ s ♥♦♠ís ♥ ró♥ ♥ q s
♣r♦ r ♣♥t ♥ ♥tr♣rtó♥ ♦s rst♦s
♥ r s ♠str ár s♣és ①ó♥ ♥
s ♣ ♦srr t♠ r♦③ ♠ó á♠tr♦ ♠♦
r ♣ s ♥♦♠ís ♣r ② s ♥tr♣rtó♥
r ♣♥t
r ♠ ♦ ①ó♥
t♠ ①♣st s ♦t♦ ♥ r♥♦ 2,8 ♠ 3,2 ♠ s ♣rs s
①t♥r♦♥ ♥ ♣r♦♥ s 0,15− 0,30 ♠ st 1,1− 1,2 ♠ ② s ♥♦
0,5− 0,8 ♠ st♦s rst♦s s ♣ ♥♦tr q ♣r♦♥ s
♣rs ♠②♦r q st♠ 0,75 − 1,00 ♠ st ♦ ♥ q s
r①♦♥s ♥ s ♥trss ♥tr s r♦s s ② s♦ ♦ s
♥♦ r♦♥ ♥ts ♦♠♦ ♦♥s♥s s ♥t♥s ♦♥ rs♣t♦
♦trs sñs r♥♥ts P♦r ♦♥trr♦ s ♦t♦ ♥ ♥ st ♥tr
♦s ♦rs ♠♦s ② ①♣r♠♥ts ♣r ♦s ♦tr♦s ♣rá♠tr♦s
♥ s rí r tt♦ ♦s ♣rs ♥ ♥♦ ♦s ♣rs
q r③♥ t♠ só♦ ♥ s tt ♥ ♥♦ ♦s ♣rs
r r ♥ st♦s s♦s ♥ tó♥ ♣r♦♠♥t ♦r
♥ ♥ ♦♠♥ó♥ ♦s s♥ts t♦rs ♦♠trí ② s♣♦só♥
s r♦s s ♣rs rs♣t♦ s♣r ② ♣♦r③ó♥ rr
q ♦♠♥t ♣r♦♥ rt ♦♥trst ♣r♠t ♥ s
♥trss ♥tr s ♣rs ♥♦ tts ② s♦ ♥♦ ♦ s♥t♠♥t
t♦ ② ♠♦♠♥t♦ s♦ ♦ rtr tró ♦r♥
♦s ♥t♦s ♠♥r q s ♣r♦♦ ♥ ♥♦q ♠♥♦s ♥t ♠rr ♥
♦♥só♥ s ♥ ♣ó♥ té♥ ♠ró♥ ♣r♠tó ♦③
ó♥ ♦♥stró♥ s r♦ q ♥ ♠♦s s♦s s ♠♣rs♥
t③ó♥ ♠t♦♦♦ís ♦rtr ♠út♣ q t♥♥ ♥r♠♥tr
s sñs ♥trés ♦♠♦ ♣s♦ ♣r♦ ♠ró♥
tó♥ ss ♥trrs
ét♦♦ ♦rtr s♠♣
♦♥t♥ó♥ s sr ♥ st♦ r③♦ ♥ ró♥ ♦♥ tó♥
ss ♥trrs ♥ ♣rtr s ♠str♥ ♠♣♦s ♥ ♦s s s
♣♥ s ♠t♦♦♦ís ♦rtr s♠♣ ② s③ó♥ rs
t ♥s
r♥t ♥ s♥ ♠♣ñ r③ ♥ st♦ rq♦ó♦ P♦
♥♦ s ó ♦ ♥ ①♣r♠♥t♦ ♦♥tr♦♦ ♥ q ♥ s ♦♥
á♠tr♦ ♠á①♠♦ 0,2 ♠ s ♦♦ó ♥ ♣r♦♥ 1 ♠ ② s ró ♦♥
s♦ ♥tr ♦♠♣st♦ ♣r♥♣♠♥t ♣♦r r♥ ② ♥ ♠♥♦r ♣r♦♣♦ró♥
r ♥ s rs ② s ♣ r ①ó♥ ② t
s rs♣t♠♥t ♥ s rs ② s ♠str♥ ♥ só♥
rt ② ♦tr ♦r③♦♥t rs♣t♠♥t ♦♥ ♦♥ró♥ t③ ②
ó♥ s í♥s s♦♥♦ r③s s ♥t♥s ♠s♦r ② r♣t♦r
s ♦③r♦♥ ♥ ♥trs r s♦ ♦♥ s♣ró♥ 0,25 ♠ ♥tr
s ♦s s s ♥t♥s s r♦♥ ♣r♣♥rs í♥ s♦♥♦
t③ó sst♠ rr ♥s♦rs ♦tr Ps P ② ♥t♥s
500 ③ ♠tó ♥ ár 5 ♠ ① 5 ♠ ② s qró í♥s ♣rs
♦♥ ♥ s♣ró♥ 0,05 ♠ ♥tr y = −0,5 ♠ y = 0,5 ♠ s♣ró♥
♥tr í♥s s s♦♥ó ♠♥r ♦t♥r t♦s ♥s s♥t
♣r ♥rr ♥ s③ó♥ ♦♥ t rs♦ó♥ sñ ♣r♦
♣♦r s
♥ s s♦♥s qrs s ♣r♦sr♦♥ ♣r ♦t♥r ♥ s
③ó♥ s sñs ♥trés ♠♥ó sñ rt ♦
s ♣ó ♦ s ♦rró ♦r♥ t♠♣♦s ② ♥♠♥t s ♣ó
♥♥ ♥ r s ♠str rst♦ ♦t♥♦ ♣r só♥ q
♦rrs♣♦♥ y = 0 ♠ ♦♦r♥ x ♥ rá♦ s rr ♣♥t♦
♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r sñ ró♥
♦t♦ ♥trr♦ s ♣ r ♥ r ♦♥ ért ♥ (x; t) = (0 ♠; 10 ♥s)
♥ r ♠é♥ ♣r♥ rs ♣ér♦s ró♥ ♦♥s
② r①♦♥s ♣r♦①♠♠♥t ♦r③♦♥ts ♣r♦♠♥t ♣r♦s ♥ ♦s
í♠ts trs ①ó♥ ♥ r ♥ s s♦♥t♥s ♥
trs s♦ ♥ r ② ♥ í♠t ♥r♦r ①ó♥ ♥
r st♦s ♣tr♦♥s s r♣t♥ ♥ t♦s s í♥s ♦t♥s
s ♣r♦ ♥ sñ t ♥t♥s ♣♦s♠♥t ♦♠♦ ♦♥s
♥ ♣rs♥ r ♥ s ♥tr♦r q ♣rs♥t ♥ t♦ ♦♥trst
♦♥ rs♣t♦ ♠tr q r♦ ❬❪ ❬❪ st sñ s r♠♥t r♦
♥♦ ♦ s ♦r♠ P♦r ♦tr♦ ♦ ♥♦ s r♦ ♦r♥ sñ ♥
(x; t) = (0 ♠; 15 ♥s) ♣r♦①♠♠♥t s ♦♥sr ♦ ♣r♦
♣ó♥ st♠ ♣r ③♦♥ ♥ só♥ v = (0,18 ± 0,01) ♠♥s s
♦t♥ ♥ ♣r♦♥ (1,3 ± 0,1) ♠ ♣r s♦♥t♥ q ♦r♥
sñ st ♦r s ♦♠♣t ♦♥ ♣r♦♥ í♠t ♥r♦r
①ó♥
t ♥s í♥s qrs ♣r♠t ♥tr♣♦r tr♥srs♠♥t
♦s t♦s ♦♥ ♥ ♠② ♥ rs♦ó♥ ② ♦ ♥rr ♥ st sñ
r ①♣r♠♥t♦ ♦♥tr♦♦ ①ó♥ r③ ②
t s
r ♦♥ró♥ ①♣r♠♥t ♠♥s♦♥s
①ó♥ ② ♦③ó♥ s í♥s s♦♥♦
r s♣st ♣r ♣r ♦t♥♦ ♥ y = 0♠ sñ
ró♥ s sñ ♦r♥ ♥ í♠t tr
①ó♥ sñ ♥♦ ♦s strt♦s s ♦r③♦♥ts ②
sñ ♦r♥ ♥ í♠t ♥r♦r ①ó♥
r ❱s③ó♥ ♦s t♦s sñ ró♥
s sñ ♦r♥ ♥ í♠t tr ①ó♥
② sñ ♥♦ ♦s strt♦s s ♦r③♦♥ts
s ♥ r s ♠str ♥ ♠♣♦ st t♣♦ s③ó♥
♥♦♠♥tr ♦♥ q s ♥t♥ s sñs ♠ás r♥ts s ♠s♠
q ♥ r ♥ r s ♣ ♥tr sñ ♦r♥ ♥
s ♥ r sñ ♦r♥ ♥ í♠t tr ①ó♥
② s sñs ♦r♥s ♥ s s♦♥t♥s ♥trs s♦
♦♦
♥ ♦s rst♦s só♥ ♥tr♦r rs ② s ♦sr
sñ ♣r♠r s ♥trr t♠é♥ s ♣♥ r ♥s sñs
s♥rs ♦r♥s ♣♦s♠♥t ♥ ♦s í♠ts trs ♥r♦r
①ó♥ ② ♥ s s♦♥t♥s ♥trs s♦ Pr ♦♥trstr s
ts ♥tr♣rt♦♥s s ♦rr♦♥ ♠♦♦s ② s s♠r♦♥ ♥♠ér♠♥t s
rs♣sts ♦rr
st♦s ♠♦♦s s♦♥ r♣rs♥t♦♥s s♠♣s ♥ st s♦ ss♦
①♣rss ♥ tér♠♥♦s s ♣r♦♣s tr♦♠♥éts ② ♦♠
trí ♦s ♥♦s ♥trr♦s ♥ ♥r st♦s ♠♦♦s sr♥ st♦♥s
♦♠♣s ♥ s q ♣r♦♠ tr♦♠♥ét♦ ♥♦ t♥ s♦ó♥ ♥ít
♣♦r ♦ s s♠ ♥♠ér♠♥t rs♣st sst♠ ♦rr
♥ r s ♠str ♥ ♠♦♦ ♣r♦♣st♦ ♣rtr s rt
rísts ♦♥♦s ③♦♥ st♦ r♦♥ ♦rs ♣r♠t
② ♦♥t tí♣♦s ♣r ♦♠♣♦só♥ s♦ ③♦♥ st♦
♠♦♦ ♦♥t♥ trs strt♦s s♣r♦s ♣♦r ♦s ♥trss s ♦r③♦♥ts
s ♣r♦♥s 0,6 ♠ ② 0,9 ♠ strt♦ s♣r♦r s rtr③
♣♦r ǫr = 3,5 ② σ = 1 ♠♠ strt♦ ♥tr♠♦ s rtr③ ♣♦r ǫr = 4 ②
σ = 2 ♠♠ ② strt♦ ♥r♦r s rtr③ ♣♦r ǫr = 5 ② σ = 2,5 ♠♠
í♠t ♥r♦r ①ó♥ s ♥♥tr ♥ ♣r♦♥ 1,3 ♠
♠tr ♦♥ q s r ③♦♥ ① s rtr③ ♣♦r ǫr = 3
② σ = 1,5 ♠♠ ♥ st s♦ s ó ♥ ♦r ǫr ♠♥♦r ♦
♣r♦♣ó♥ ♠②♦r q ♣r ③♦♥ s♥ ①r ♣♦rq s ♦♥sró q st
♠tr st r♦ ♣♦r r s♦ r♠♦♦ r♥t ①ó♥
s s ♥♥tr ♥ ♣r♦♥ 1 ♠ s♣♦♥ q s s ♥
♥tr ♣r♠♥t ♥ r ǫr = 1 ② σ = 0 ♠♠ ♥ ♠♦♦
♠t ♥r♦r ♠s♠ s ♥♥tr ♦♣ ♣♦r ♠tr ♦♥ q
s r ③♦♥ ① s t♥ 0,2 ♠ á♠tr♦ ♣rs 0,02
♠ s♣s♦r ǫr = 6 ② σ = 2,6 ♠♠ ♥ s ♠trs ♣r♠t ②
♦♥t s ♣ ♥ tó♥ t♦r
♦♥ ♦s ♣rá♠tr♦s ♠♦♦ ♣r♦♣st♦ s r③ó ♥ s♠ó♥ ♥♠é
r rs♣st s♣r ♣r ♥ s♦♥♦ ♦rr ♥ q s ♥t♥s
♠s♦r ② r♣t♦r s♦♥ t♣♦ ♣♦♦ ♠ ♦♥ ② s ♥ ♥ ♥tr
s r s♦ ♦♥ ♦st 0,25 ♠ Pr ♦ s t③ó ♠ét♦♦ r♥s
♥ts ♥ ♦♠♥♦ t♠♣♦ ♥ ♦s ♠♥s♦♥s rs♦r♦♥
♥♠ér♠♥t s ♦♥s ① ♣r ♦t♥r ♠♣♦ Ey s♦r
♥ r ♥♦r♠ ♦♥ ♥r♠♥t♦ 0,01 ♠ ♥ ♠s r♦♥s r
♥ ♠só♥ 500 ③ ♥ ♣é♥ s ♥ ♠ás ts
♣r♦♠♥t♦ r③♦ ♣r ♦t♥r ♦s t♦s s♠♦s
❯♥ ③ st♦s ♦s ♣rá♠tr♦s ♠♦♦ s rs♣st ♣r
♠s♦r ♥ ♥ s ♣♦s♦♥s í♥ s♦♥♦ ② s rstr ♠♣♦
Ey ♥ ♣♦só♥ r♣t♦r tr♠♥ sú♥ ♦st ♦ s r♣t
♣r♦♠♥t♦ ♣r ♥ s ♣♦s♦♥s ♠s♦r s♦r í♥
s♦♥♦ ♥♠♥t s r♣♥ s tr③s ♦t♥♥♦ sí rstr♦s s♠♦s
♥á♦♦s ♦s ♦t♥♦s ①♣r♠♥t♠♥t ♣rtr st ♣♥t♦ s s
♣r♦♠♥t♦ t③♦ t♠♥t ♦s t♦s s ♦r♥♥ ♥ ♥ rrr♠
② s ♣♥ s té♥s ♣r♦s♠♥t♦ ♥srs ♣r ♠♦rr
♠♥ ♦rró♥ ♦r♥ t♠♣♦ ♠♥ó♥ sñ rt
② ♥♥
♥ r s ♠str rrr♠ ♦t♥♦ ♣r ♦s t♦s q
rst♥ ♠♦♦ r ♥ rrr♠ sñ ♦r♥
♥ s s ♣ r ♦♥ ért ♥ (x; t) = (0 ♠; 12 ♥s) ♥ r
ést s ♥t♥s t ♦♠♦ ♦♥s♥ t♦ ♦♥trst ♥tr r ♥
♥tr♦r s ② ♠♦ q r♦ s sñs ♥rs ♥ ♦s
trs ①ó♥ s ♣♥ r ♣♦r ♠♣♦ ♥ (x; t) = (−0,8 ♠; 5 ♥s)
♥ r ♠é♥ ♣rr♦♥ ♥s sñs ♦r♥s ♥ s ♥tr
ss ♥tr ♦s st♥t♦s strt♦s ♣♦r ♠♣♦ ♥ t = 8 ♥s ② ♥ t = 12 ♥s
♥ r ♥♠♥t ♥ (x; t) = (0 ♠; 16 ♥s) ♥ r s ♣
r sñ ♦r♥ ♥ í♠t ♥r♦r ①ó♥ st ♣rs♥tó ♥
s♣t♦ s♠r ♦sr ♥ ♦s rrr♠s ①♣r♠♥ts r
só♥ s sñs ♦r♥s ♥ ♦s í♠ts ①ó♥ ② ♥
s ♥trss q s♣r♥ ♦s st♥t♦s strt♦s t♥♥ ♠♥♦r ♥t♥s q
sñ q s ♦r♥ ♥ s ♦ q s ♦♥s♥ ♠♥♦r ♦♥trst
r ♦♦ s ♥trr strt♦ s♣r♦r
ǫr = 3,5 σ = 1 ♠♠ strt♦ ♥tr♠♦ ǫr = 4 σ = 2 ♠♠
strt♦ ♥r♦r ǫr = 5 σ = 2,5 ♠♠ ♠tr ♦♥ q s
♥ ①ó♥ ǫr = 3 σ = 1,5 ♠♠ s ♣r♦♥
0,9 ♠ á♠tr♦ 0,2 ♠ ♣rs 0,02 ♠ s♣s♦r ǫr = 6
σ = 2,6 ♠♠ s ♣ ♥ tó♥ t♦r st♦s
♣rá♠tr♦s rr♠ s♠♦ ♣r ♥ s♦♥♦ ♦rtr
s♠♣ ♦♥ ♦st 0,25 ♠
q ♣rs♥t♥ s s♦♥t♥s
s ♦♠♣r♥ ♦s rrr♠s ①♣r♠♥t r só♥
② s♠♦ r s ♣♦s ♦srr s♠t q ♣rs♥t♥ ♥
s ♥sr♦ ♥ ó♦ ♣r ♠♦r ♠♥t ♥ts t♣♦ ♣♦♦
② ♥♦s ♦③♦s ♦♠♦ s ♥trr s ♣ r q ó♦
t③♦ ♣r♠t r♣r♦r s ♣r♥♣s rtrísts s sñs
♦t♥s ♥ ♥ s♦♥♦ ♦♥ ♦rr
♥ st s♦ ♠♦♦ ♣r♦♣st♦ ② rrr♠ s♠♦ ♣rtr
♦ ♠♦♦ ♣r♠tr♦♥ str ró♥ ♥tr s sñs ♦srs ②
♦ ♦t♦s ♥trss q s ♥rr♦♥ ② rr♠r ♥tr♣rtó♥ r③
♣rtr ♦s t♦s ①♣r♠♥ts ♥ ♥r ♦s t♦s s♠♦s t♥♥
t♠é♥ ♦trs ♣♦♥s ♥tr s ♠ás ♠♣♦rt♥ts ♥ s t♣s ♣rs
r③r ♥ ♣r♦s♣ó♥ ♣r ♣♥r ♦s ♠ét♦♦s ② strts
qsó♥ ② r♥t ♥áss ♣♦str♦r ♥tr♣rtó♥ ♦s t♦s ♣r
♥tó♥ sñs ♣♦r ♦♠♣ró♥
ét♦♦ ♦rtr ♠út♣ P
Pr ♦♥t♥r st♦ sñ q ♥r ♥ s ♥trr s
♥③r♦♥ s ♠♦rs q ♣r♦ ♠ét♦♦ P ♥ sñ ♣r♠r
s ♠é♥ s str♦♥ s ♠♦rs ♥ s sñs s♥rs
♦r♥s ♥ s s♦♥t♥s ♥trs s♦ ② ♥ ♦s í♠ts
①ó♥ Pr ♦ s qrr♦♥ tr③s ♥ ♦♥ró♥ P ♦♥ ♥r
♠♥t♦ 0,05 ♠ ♥ s♣ró♥ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r
② ♥ s♣ró♥ ♠á①♠ 1 ♠ st♥ ♥tr ♣♥t♦s ♠♦s Ps
s♦r í♥ s♦♥♦ s ♠str ♥ r 0,05 ♠
♥ ró♥ ♦♥ r só♥ s ♠♥♦♥ó sñ ♥
strt♦ ♣r♦①♠♠♥t ♦r③♦♥t ♦ ♥ t = 11 ♥s ♥tr x = 0,5 ♠ ②
x = 2,0 ♠ st strt♦ s t③♦ qí ♣r str ♦ ♣
♠♥t♦ Pr ♦ s ♣ó ♦rró♥ sñ s♥♦ st♥ts
♦s ♣♠♥t♦ ② s s♦♥ó q q ♣r♦♦ ♠♦r st
♦r rst♥t v = (0,17 ± 0,09) ♠♥s r ♥ r s
♥♥ ♦♥ í♥s tr③♦s s sñs ♥ts ② s♣és ♣r
♥ r s ♠str rst♦ ♣♠♥t♦ ♦♥ n = 4 ♦♠
r ♦③ó♥ í♥ s♦♥♦
♣r♥♦ ♦s rst♦s ♦t♥♦s ♣r ♥ s♦♥♦ ♦rtr s♠♣ r
só♥ ② ♣r P r s ♣♦s ♦srr q
t♥t♦ s sñs ♣r♦s ♣♦r s ♥trr ♦♠♦ ♣♦r rt♦r s
♦r③♦♥t ♣rs♥t♥ ♥ ♠♦r ♣é♥♦s♦s r ♦♥ ♠②♦r r
♠s♠ ♦r♠ sñ ♦r♥ ♥ ♦♥♦ ①ó♥ s ♥ ♦♥ ♥
♥ r ♣rs♥t ♥ ♠♦r s♥t s♥t ♣r sr
♥t ♦♥ r
♥ st ♠♣♦ ♣ó♥ ♠ét♦♦ P ♣r♠tó ♠♦rr s st♥
ts sñs q s ♣rs♥t♥ ♥ rrr♠ ♥ ♠r♦ ♥ ♥♦s s♦s
s ♠♦rs q s ♦t♥♥ ♦♥ ♠ét♦♦ P ♥♦ s♦♥ s♥ts ♦ s r
trísts ♦s ♥♦s ♥trr♦s ♥♦ ♣r♠t♥ t③r ♦ ♠ét♦♦ ♦ q
srr♦r ② ♣r ♠ét♦♦s ♦rtr ♠út♣ tr♥t♦s
r ♠♣♦s tv ♥ ♥ó♥ s♣ró♥
♠s♦r r♣t♦r x s♥ ♣r ♦rró♥ ② ♦
♣r ♦rró♥ ♦♥ ♦ v = (0,17 ± 0,09)
♠♥s
r s♣st ♦t♥ ♦♥ ♠ét♦♦ P
♣ít♦
ét♦♦ rr♦s s♥tét♦s
♠s♦rs ♦rr
♠t♦♦♦í ♦rtr s♠♣ ♣r♠t rs♦r t♦t ♦ ♣r♠♥t
♠② rss st♦♥s ①♣r♠♥ts ♦♠♦ ♣♦r ♠♣♦ s ♥s ♥
♣ít♦ ♥tr♦r ♣rtr s♦ st♥ts té♥s ♣r♦s♠♥t♦
♥tr♣rtó♥ sñs ♥ ♠r♦ ② st♦♥s ♦♥ sñs ♠② és
② ♣♦r ♦ t♥t♦ ♦♥ss ♥ s q s ♥sr♦ t③r ♦trs ♠t♦♦♦ís ♦♥
♦rtr ♠út♣ ♣r ♠♦rr s s ♥ st ♣ít♦ s ♣rs♥t
♥ ♠ét♦♦ ♦rtr ♠út♣ tr♥t♦ ♥♦♠♥♦ ♠ét♦♦ rr♦s
s♥tét♦s ♠s♦rs ♦rr ②♦ ♦t♦ ♥♠♥t s ♠♦rr r
ó♥ ♥t♥s sñ ♣r♠r rs♣t♦ ♦trs sñs s♥rs
sí ♦♠♦ s ♦♥t♥ tr st ♠ét♦♦ ♦♥sr rr♦s ♥t♥s
♣r ♦t♥r ③♦♥s ♠♥s ♠ás strs ♦♥♥tr♥♦ ♠♣♦ s♦r ♦s
♥♦s ♥trés ② s♠♥②é♥♦♦ ♥ s ③♦♥s ♣rérs ♥r♦rs
sñs s♥rs ♥ss
♥ ♣r♠r tér♠♥♦ s ♦♥sr♥ rr♦s ♠s♦rs ♦s ♥ ♠♦s
♥♦r♠s ② s st ró♥ ♠♣♦ ♥♦ ♦♥ ♦t♦ strr
♠♥r ♥ q s ♣♦s ♦♥tr♦r ♣tró♥ ró♥ t♦t rr
♦ ♦♥t♥ó♥ s sr ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs
♦rr ② s st♥ s rtrísts ♦s ♠♣♦s tr♦♠♥ét♦s
♥r♦s ♣♦r st♦s rr♦s ♥ ró♥ ♠♣♦ r♥♦ ♣r ♠s♦rs
♦s ♥ s♣r s♣ró♥ r trr ♣rtr s ♥③♥
s rs♣sts q s ♦t♥♥ ♣r ♠ét♦♦ ♥ ♦s s♦s ♥♠♥ts
♦t♦ ♣qñ♦ ② rt♦r ①t♥s♦ ♥♠♥t s ♣rs♥t ♥ ♠t♦♦♦í
♣r♦s♠♥t♦ rs♣st ♦t♥ q ♣r♠t ♠♦rr ♦♥t♥
tr s sñs ♣r t♦♦ s r♥♦ ①t♥só♥ t♥♦ sí
♥tr♣rtó♥ s ♠s♠s ❬❪ ❬❪
rr♦s ♠s♦rs
♥ ♥r s ♥t♥s ♠♥ts ♠ás s♠♣s ♦♥strr ♦♠♦ s
t♣♦ ♣♦♦ ♠ ♦♥ ♣rs♥t♥ ♣tr♦♥s ró♥ ♦♥ rt
❯♥ ♣r♦♠♥t♦ q s ♣ t♠♥t ♣r ♦t♥r ♥ ♠
♦r rt ♦♥sst ♥ t③r rs ♥ts ♠♥ts strs ♥
♥ ♦♠trí ♣r♦♣ s q ♦r♠♥ ♥ rr♦ st ♠♥r s ♦r
♦♥♥trr ② rr ♥rí r ♥ ró♥ tr♠♥ ♥
s♣♦
♥ ♥ ♠♦ ♥♦r♠ ② ♥ ró♥ ♠♣♦ ♥♦ s ♣♦s ♦t♥r
①♣rs♦♥s ♥íts q sr♥ stró♥ ♥rí ♥ s♣♦
♣r♦ ♣♦r rr♦ Pr ♦ s ♦♥sr ♥ ♣♦♦ ♦ ♦♥ ♦♥
t l ♦♠♦ s ♠str ♥ r q ♥r ♠♣♦s r♠ó♥♦s
r♥ ω st ♥t s r♣rs♥t ♣♦r ♥ ♥s ♦rr♥t étr
♥♠♥s♦♥ ~Jorig(~r, t) ♦ r♦ ♣♦♦ z
~Jorig(~r, t) =
I0e−iωt sen(π
2− k|z|)δ(x)δ(y)z s |z| ≤ λ
4
0 s |z| > λ4
♦♥ I0 s ♠♣t ♦rr♥t k = 2πλ
s ♥ú♠r♦ ♦♥ λ s
♦♥t ♦♥ ♥ ♠♦ δ s t r ② sí♥ orig ♥
q ♥t s ♦③ ♥ ♦r♥ ♦♦r♥s ♥♦ s ♦♥sr ♥
♦♥t l = λ2♣r ♥t♥ stó♥ r♣rs♥t ♣r♦①♠♠♥t ♥
♣♦♦ r♥t ♠ ♦♥
♣♦t♥ ♣♦r ♥ á♥♦ só♦ dPdΩ
♦ ♥t♥s ró♥ s
♣ ①♣rsr ♦♠♦
(dP
dΩ
)
orig=
I02η
8π2
[cos(π2cos(θ))
sen(θ)
]2
♦♥ η =√
µ
ǫs ♠♣♥ ♠♦ ♦♥ µ ♣r♠ ♠♥ét
② ǫ ♣r♠t ♣♦t♥ ♣♦r ♥ á♥♦ só♦ ♥ rr♦
r ♣♦♦ ♠s♦r ② sst♠ ♦♦r♥s
♣♦♦s ♥ ró♥ ♠♣♦ ♥♦ s ♣ ♦t♥r s♣r♣♦♥♥♦ ♦s ♠
♣♦s ♦s ♣♦♦s ♥s r ó♥ ♥ ♣é♥ ♣♦t♥
♣♦r ♥ á♥♦ só♦ ♣r ♥ rr♦ N ♠♥t♦s (dPdΩ)N s ♣
①♣rsr ♦♠♦(dP
dΩ
)
N= |ΓN |2
(dP
dΩ
)
orig
♦♥ ΓN s t♦r rr♦
t♦r rr♦ s ú♥♦ q ♦♥t♥ s ♣♦s♦♥s s ♥ts~dn ② ss ♠♣ts an ② ss ϕn rts ② s q ♠♦ ♣tró♥
♠♥t♦ ♥r♦r ♠s♠♦ s ♦ ♣ ①♣rsr s♥t ♠♥r
ΓN =N−1∑
n=0
ane−iϕne−i~k·~dn
s♥♦ t♦r ó♥ s ♥s ♣♦t♥ ♠♥t♦
♥r♦r
♠♥r ♠ás s♠♣ ♥rr ♣tr♦♥s ♦♥ rtrísts s♣ís
s ♦♥sr♥♦ rr♦s rrs t ♦♠♦ r ❯♥ rr♦
rr s ♦♠♣♦♥ ♣♦r rs ♥ts q t♥♥ é♥t ♠♣t ♣rs♥t♥
♥r♠♥t♦s ♦s ♥ ss ss ② ♣♦s♥ stró♥ rr ♥ s♣♦ ♥
s s♦ t♦r rr♦ s ♣ ①♣rsr s♥t ♠♥r
|ΓN |2 =sen2(Nx(αx+kxdx)
2)
sen2( (αx+kxdx)2
)
sen2(Ny(αy+kydy)
2)
sen2( (αy+kydy)
2)
sen2(Nz(αz+kzdz)2
)
sen2( (αz+kzdz)2
)
♦♥ Ni di ② αi ♦♥ i = x, y, z r♣rs♥t♥ ♥ú♠r♦ ♥ts s♣ró♥
② s rt rs♣t♠♥t ♥ s r♦♥s x y ② z r ó♥
r rr♦ rr ♦♥ Nx = 2 Ny = 2 Nz = 4 dx =
3,0dy ② dz = 1,5dx
♥ ♣é♥ ♣♥♥ ♥r stá ♠♣ít ♥ ♦s tér♠♥♦s
ki ó♥ stá ♦♠♣st ♣♦r trs t♦rs r♦♥♦s ♦♥ ♦s
♣rá♠tr♦s r ♥ s r♥ts r♦♥s st♦ s♠♣ ♥áss
s rtrísts ♣tró♥ rst♥t ♦♠♦ ♥ó♥ ♦s ♣rá♠tr♦s
rr♦ ♦♥ ♥ s♠♣ó♥ ♦♥ ♥♦ s ♦♥sr♥ rr♦s ♥
♥ ♦ ♦s ♠♥s♦♥s ♣♦r ♠♣♦ ♣♦♦s ♥♦s ♦ ♥ rr♦ ♣♥♦
rs♣t♠♥t
♥ s♦ rr♦s rrs ♦♥ ♠♥t♦s str♦s ♦ r♦
y ó♥ q
|ΓN |2 =sen2(N(α+kd sen θ senφ)
2)
sen2( (α+kd sen θ senφ)2
)
♦♥ s s♣r♠r♦♥ ♦s sí♥s ② s r♠♣③ó ky = k sen θ senφ ♣♦
♥♥♦ q t♦♦s ♦s ♠♥t♦s t♥♥ ♠s♠ ♥t♥s ♦s ♣rá♠tr♦s q
tr♠♥♥ s rtrísts ♣tró♥ ró♥ s♦♥ ♥ú♠r♦ ♥ts
N s♣ró♥ d ② s rt α ♥tr ♦s ♠♥t♦s
Pr str ♣♥♥ ♣tró♥ ró♥ ♦♥ ♦s ♣rá♠tr♦s
rr♦ s sñó ♠♣♠♥tó ó♦ ♦♠♣t♦♥ rr② st
ó♦ srr♦♦ ♥ ♥ ♣r♠t r ♣tró♥
ró♥ rr♦s ♣♦♦s ♠ ♦♥ ♦♥ r♦♥s ♣rs ②
s ♦♥sr♥♦ ♥ ♦♥ró♥ rr ♦ rrr ♦♠♦ s ó♦
♠str rs♦s rá♦s ♥tr ♦s t♦r rr♦ Pr tr
t③ó♥ ó♦ rr② s srr♦ó ♥ ♥t♦r♥♦ ♥trt♦ q ♣r♠t
r Ptró♥ ró♥ ♣r rr♦s ♦♥ d = 0,50λ
α = 0 N = 1 N = 2 N = 4 Pr♦②ó♥ ♥ ♣♥♦
x− y ♣tró♥ ró♥ ♦s rr♦s ♥tr♦rs
rr ♦s ♣rá♠tr♦s rr♦ ② ♦srr s♠tá♥♠♥t ♣tró♥
ró♥ rst♥t ♦ ♥t♦r♥♦ ♥t ♠ás ♦♥ rss rr♠♥ts
q t♥ ♥áss ♦s rst♦s r sr♣ó♥ ó♦ ♥
♣é♥
♥ s rtrísts ♣tró♥ ró♥ ♣♥♥ ♦s ♣rá
♠tr♦s rr♦ ♥ s ♦♥♥t♦ ♥ú♠r♦ ♥ts N s♣ró♥ d ② s
rt α ♥tr ♦s ♠♥t♦s s ♣ ♦srr q ♥♦ ♥r ③
r ró♥ ♦♥ ♥ú♠r♦ ♠♥t♦s ♦♠♦ ♠♣♦ s ♦♥sr s♦
♥ ♣♦♦ ♦♥ r♥ ③ ♥ í♦ ǫ = ǫ0 µ = µ0 ② σ = 0
N = 1 r ♣tró♥ s t q ♥ ♣♥♦ ♠♦ ♣♦♦ ♣♥♦
x−y ♥rí r ♣rs♥t ♠s♠ ♥t♥s ♥ t♦s s r♦♥s
♥ s♦ ♥ rr♦ ♦♥ N = 2 d = 0,50λ ② α = 0 r ♣
tró♥ ró♥ ♣rs♥t ♦s ó♦s q ♦♥♥tr♥ t♦ ♥rí ♠t
♥ ♥r ♦♥♥tró♥ s ♠ás ♣r♦♥♥ ♥♦ s ♥r♠♥t
♥ú♠r♦ ♣♦♦s ♥♦s r
r Ptró♥ ró♥ ♣r rr♦s ♦♥ N = 4 α = 0
d = 0,25λ d = 0,50λ d = 0,90λ Pr♦②ó♥ ♥
♣♥♦ x− y ♣tró♥ ró♥ ♦s rr♦s ♥tr♦rs
tr♦ ♥ó♠♥♦ q s ♦sr ♥r♠♥tr ♥ú♠r♦ ♥ts s
♣ró♥ ó♦s s♥r♦s P♦r ♠♣♦ s N = 4 d = 0,50λ ② α = 0 s
♦t♥ ♥ ♣tró♥ ró♥ ♦♥ ♦s ó♦s ♣r♥♣s q ♦♥♥tr♥
♠②♦r ♣rt ♥rí r ② tr♦ ó♦s s♥r♦s ♠②
♥t♥s r ♥ r s ♠str♥ rá♦s ♣r♦②ó♥
♣tró♥ ró♥ ♥ ♣♥♦ x − y ♣r ♦s s♦s s rs
② ♦♥ s ♣ r t♥t♦ ♦♥♥tró♥ ♥rí ♠t
♦♠♦ ♣ró♥ ó♦s s♥r♦s
♥♦ s♣ró♥ ♥tr ♣♦♦s s ♥r♠♥t s ♦sr ♥ ♠♥t♦
♥rí q s ♣r♦♣ ♥ s r♦♥s ♦s ó♦s s♥r♦s st
♥ó♠♥♦ s ♣ ♦srr ♥ s♦ ♥ rr♦ ♦♥ N = 4 α = 0 ② ♦♥
d = 0,25λ r d = 0,50λ r ② d = 0,90λ r
♥♦ rí s rt ♥tr ♦s ♠♥t♦s rr♦ s ♦sr
♥ ♠♦ó♥ ♥ ♦r♥tó♥ ♣tró♥ ró♥ ♦♥t♥♥♦ ♦♥
♠♣♦ ♦♥ N = 4 ② d = 0,50λ s α = 0 r ♦s ó♦s ♣r♥♣s
r Ptró♥ ró♥ ♣r rr♦s ♦♥ N = 4 d =
0,50λ α = 0 α = 0,25π α = 0,50π Pr♦②ó♥ ♥
♣♥♦ x− y ♣tró♥ ró♥ ♦s rr♦s ♥tr♦rs
s ♦r♥t♥ ♥ á♥♦s 0 ② π s rt t♦♠ ♦rs α = 0,25π r
♦ α = 0,50π r s ♣ ♦srr ó♠♦ rí á♥♦
♦r♥tó♥ ♦s ó♦s q ♦r♠ ♥ ♣tró♥ ró♥
♥ rs♠♥ t♦rí rr♦s ♥ts t♣♦ ♣♦♦ ♥ ♥ ♠♦ ♥
♦r♠ ② ♥ ró♥ ♠♣♦ ♥♦ ♠str q ♦♠♥♥♦ rs ♥ts
r♥s ♥tr sí s ♣♦s ♦♥tr♦r rts rtrísts ♣tró♥ r
ó♥ ♦♥♥t♦ ♥ ♣rtr s ♣♦s ♦♥tr♦r ♦r♥tó♥ ② ♦♥
♥tró♥ ♠♣♦ ♠t♦ r♥♦ ♦s ♣rá♠tr♦s rr♦ ♥ú♠r♦
♥ts s♣ró♥ ② s rts ♥tr s
ét♦♦ rr♦s s♥tét♦s ♠s♦rs
♦rr
♦♠♦ s ♠♦stró ♥ só♥ ♥tr♦r t③r rs ♥t♥s ② str
rs ♦♥ ♥ ♦♠trí s ♣♦s ♦t♥r ♥ rt
♣r ♦♥♥t♦ ② ♦♥tr♦r ró♥ ♥ q s ♦r♥t ♥rí ♠
t st♦ s s♣♠♥t út ♥ s♦s ♥ ♦s q s ♥t♥s ♥s
t♥♥ rt ♦♠♦ ♣♦r ♠♣♦ s ♥t♥s ♦rr ss
❯♥ ♦♥s♥ rt s q ♥rí ♠t s s♣rs
♥ ♥ ♦♠♥ r♥ ♣♦r ♦ q s ♦t♥ ♥t♥s ♣r s sñs
♣r♦♥♥ts ♦s ♥♦s ♥trés ❯♥ ♠♥r s♣rr st ♣r♦♠ s
♦♥♥trr ♥rí s♣♦♥ s♦r ♦s ♥♦s ♣r ♦ s ♣r♦♣♦♥ t③r
rr♦s ♠s♦rs ♦rr
♥ ♠t♦♦♦í ♦rtr s♠♣ s t③ ♥ ú♥ ♥t♥ ♠s♦r ②
♥ ♥t♥ r♣t♦r ♥ ♠ét♦♦ ♣r♦♣st♦ s r♠♣③ ♥t♥ ♠s♦r
♣♦r ♥ rr♦ s ♥t♥s ♦ q ♦s q♣♦s ♦rr t③♦s
t♠♥t ♥t♥ ♦♥ ♥ ú♥ ♥t♥ ♠s♦r s ♣r♦♣♦♥ t③r st
ú♥ ♥t♥ ♦♦á♥♦ ss♠♥t ♥ ♥ s ♣♦s♦♥s ♦s
♠♥t♦s rr♦ ② ♦ s♣r♣♦♥r ♦s rstr♦s ♥s st
♠♥r ♠ás s t ♦♣♠♥t♦ ♥tr ♦s ♠♥t♦s ♠s♦rs st
r♥t ♠ét♦♦ ♦ ♠♠♦s ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs
s②♥tt ♠ttr rr②
♠t♦♦♦í q s sr ♦♥t♥ó♥ ♦♥sst ♥ ♦s t♣s q
♦♠♣r♥♥ ♦t♥ó♥ t♦s ② ♣r♦s♠♥t♦ ♦s ♠s♠♦s ♦s ♣
rá♠tr♦s s♦♥♦ s♦♥ ♥ú♠r♦ ♣♦s♦♥s ♠s♦r ♥ q s ♥
qrr t♦s N ② s♣ró♥ ♥tr s st♥ts ♣♦s♦♥s ♠s♦r
d P♦r s♠♣ s s r ♠s♠♦ ♦r ♣r s s♣r♦♥s
s ♣♦s♦♥s ♠s♦r ② r♣t♦r ♦ q s s♣♠♥t út ♥♦ s
tr ♥ ♦rt♦r♦ ♦ ♠♣♦
Pr ♦t♥ó♥ s tr③s s r♣t♦r ♥ ♣r♠r ♣♦só♥
í♥ s♦♥♦ ♠s♦r ♠♥♦r st♥ ♣♦s ② s ♦t♥ ♥
tr③ ♦ s s♣③ ♥t♥ ♠s♦r ♥ st♥ d s ♣♦só♥ ②
s qr ♥ s♥ tr③ ♣r♦♠♥t♦ s r♣t st ♥③r
♠á①♠ s♣ró♥ ♣rst ♥tr s ♥t♥s ♠s♦r ② r♣t♦r r r
r sq♠ ♦s ♣s♦s q rqr ♠♣♠♥tó♥
♠ét♦♦
sí s ♦t♥ ♥ ♦♥♥t♦ tr③s s♦s ♣r♠r ♣♦só♥
r♣t♦r r r
♦ s s♣③ r♣t♦r ♥ st♥ d s♥t ♣♦só♥ s♦r
í♥ s♦♥♦ s ♦♦ ♠s♦r ♠♥♦r st♥ ♣♦s ② s r♣t t♦♦
♣r♦♠♥t♦ ♦♥ ♦ s ♦t♥ ♥ s♥♦ ♦♥♥t♦ tr③s s♦s
s♥ ♣♦só♥ r♣t♦r ♦♥t♥ú♥ r③♥♦ ♦s ♠s♠♦s ♣s♦s st
rr t♦ í♥ s♦♥♦ ♦♥ ♦ q s ♥③ t♣ qsó♥
t♦s
♥ t♣ ♣r♦s♠♥t♦ s t♦♠ ♦♥♥t♦ t♦s s♦♦s
♥ s ♣♦s♦♥s r♣t♦r ② s ①tr♥ s tr③s q ♦rrs♣♦♥♥
♥ú♠r♦ ♥ts s♦♥♦ ♥ ♠♣♦ ♦♥sr♦ s ♣ ♦srr
q t♠♣♦ ♥ q sñ r♣t♦r s ♥r♠♥t ♠♥tr
s♣ró♥ ♠s♦r r♣t♦r r s♦r ♥ s tr③s s
r③ ♥ s♣③♠♥t♦ t♠♣♦r ♣r♦♣♦ s ♣♦s ♦rr q s sñs
r①ó♥ ♥ ♥ s ♥ ♥ s r♣t♦r ♠♥r q
s♠r s tr③s s ♦t♥ ♥ sñ ♠②♦r ♠♣t q s sñs
s ♥ts ♥s r ♥ s♥t ♣s♦ ♣r♦s♠♥t♦
s ♣ st t♣♦ s♣③♠♥t♦ t♠♣♦r ♥ s tr③s ②
♥♠♥t s s s♠ ♣r ♦t♥r ♥ ú♥ tr③ s♦ ♣♦só♥
r♣t♦r r s tr③s ♦t♥s ♣r ♥ s ♣♦s♦♥s
r♣t♦r s rú♥♥ ♣r ♦t♥r ♥ rrr♠ r
♦s s♣③♠♥t♦s t♠♣♦rs s ♥♥ ♠♥r q ♣r ♠és♠
♥t s♣③♠♥t♦ t♦t sté ♦ ♣♦r ∆tm = mdt ♦♥ − (N−1)2
≤ m ≤(N−1)
2s N s ♠♣r m = 0 s rr ♥t ♥tr rr♦ q
s ♣ ♥ s♣③♠♥t♦ t♠♣♦r ∆t0 = 0 ♥ ♥ s♣ró♥
♠s♦r r♣t♦r q♥t ♣r rr♦ ♦♠♦ s♣ró♥ ♥tr ♥tr♦
rr♦ ② r♣t♦r
♦♠♦ ♠t♦♦♦í ♣r s♦♥r dt s ♥r ♥ sr rrr♠s
♣r st♥t♦s ♦rs st ♣rá♠tr♦ ② ♥tr ♦s s s♦♥ q ♥
q sñ s ♦♥ ♠②♦r r
♣ó♥ ♠ét♦♦ ♣♥ só♥ ♦s ♣rá♠tr♦s
rr♦ ♥ s♦ st♦s ♥ sr ♠♥t ♦s ♣r ♦t♥r
♥ ♠♦r t ♥ rs♣st P♦r ♦ t♥t♦ s♥t ♣s♦ ♣r ♦♠
♣r♥r ♥♦♥♠♥t♦ ♠ét♦♦ ♦♥sst ♥ ♥③r s rtrísts
♦s ♠♣♦s tr♦♠♥ét♦s ♣r♦♦s ♣♦r rr♦s s♥tét♦s ♠s♦
rs ♦s s♦r s♣r r trr ② str ó♠♦ s ♠♦♥ ♦s
♠♣♦s rr ♦s ♣rá♠tr♦s rr♦
rtr③ó♥ ♠♣♦ rr♦
♠♣♦ tr♥s♠t♦ ♣♦r ♥ ♥t♥ ♦rr ♣♥ s rt
rísts ② ♦r♥tó♥ ♠s♦r ② s rtrísts ♦♥sttts ② ♦♠é
trs ♥trs r s♦ ② s ♣s ♠ás ♣r♦♥s ♥ s♦
rr♦s ♠s♦rs ♦s ♠♣♦s rst♥ts t♠é♥ ♣♥♥ ♦s ♣rá♠
tr♦s rr♦ ♥t ♠♥t♦s N ♣♦s♦♥s rts d ♠♣ts
da ② s♣③♠♥t♦s t♠♣♦rs dt st♦s ♣rá♠tr♦s s ♥ r
♠♥r ♣r rr r♥ ③ tr♥s♠t♦ ② rr
♦ s ♣♦s♦♥s st♠s ♥♦ ♦♥♥tr♥♦ st ♠♥r
♥rí s♦r é Pr r③r ♥ só♥ ♦s ♣rá♠tr♦s
rr♦ s ♣rs♦ ♦♥♦r ♠♥r ♥ q st♦s ♠♦♥ s rtrísts
♠♣♦
♥ só♥ s ♠♦stró q ♥♦ s ♦♥sr s♦ s♠♣
rr♦s ♦r♠♦s ♣♦r ♣♦♦s ♠ ♦♥ ♠t♥♦ ♥ ♥ ♠♦ ♥♦r♠
s ♣♦s ♦t♥r ①♣rs♦♥s ♥íts ♣r ♦s ♠♣♦s tr♦♠♥ét♦s
♥r♦s ♣♦r rr♦ ♥ ró♥ ♠♣♦ ♥♦
Pr♦ ♥ ♣ó♥ ♠ét♦♦ ♣r♦♠ s ♠ás ♦♠♣♦ ♦s
♠♣♦s s♦♥ ♠t♦s s ♥trs r s♦ ② s ♣r♦♣♥ trés
ss♦ ♦♥ ♥trtú♥ ♦♥ rs♦s strt♦s ♦t♦s ♥ts r
♥♦ s ③ st♦s ♥♦s ♣♥ str ♦s ♣r♦♥s q
♦rrs♣♦♥♥ t♥t♦ s r♦♥s ♠♣♦ r♥♦ ♦♠♦ s ♠♣♦
♥♦ ♦♠♦ ♥♦ ①st ♥ s♦ó♥ ♥ít ♣r ♣r♦♠ ♦♠♣t♦ s
rqr ♥ s♦ó♥ ♥♠ér
Pr ♦t♥r s rtrísts ♣r♥♣s ♠♣♦ rst♥t s ♦♥s
ró ♥ ♠♦♦ s♠♣♦ ♥ q ♦s ♠♣♦s s♦♥ ♥r♦s ♣♦r rr♦s
♣♦♦s ♠ ♦♥ ♦s ♥ ♥trs ♣♥ ♥tr ♦s ♠♦s ♥
♦r♠s q r♣rs♥t ♥trs r s♦ z = 0 ♦♠♦ s ♠str ♥
r ♦s s ♦s ♣♦♦s s♦♥ ♣r♣♥rs í♥ s♦♥
♦ ♦r♥t ♦ r♦ x st♥ ♥tr ♣♦♦s s ♥♦t ♦♥ d
r ♦♠trí rr♦ ♠s♦r st♥ ♥tr
♠♥t♦s s ♥♦t ♣♦r d
♦♥t rr♦ = (N − 1)d
Pr tr st♦ ♦s ♠♣♦s ♠t♦s ♣♦r rr♦ ♥ ♣
ó♥ ♠ét♦♦ s srr♦ó ó♦ ♦♠♣t♦♥ ♥t♦r♥♦ st
♣r♦r♠ ♣r♠t ♦t♥r ♠♣♦ ♥ rr♦ ♥ ♥ tr♠♥♦ ♠♦
t③♥♦ rs♣st ①♣r♠♥t ♦ s♠ ♣r ♥t ú♥ ♠s♠♦
♥t ♦♥ ♥ ♥t♦r♥♦ rá♦ ♠ ② ♦♥ rr♠♥ts ♣r ♥áss
♦s rst♦s r ♣é♥
Pr r③r ♦s á♦s ♥♠ér♦s ♠♣♦ ♥ st só♥ s r♦♥
♦rs ♣r♦♠♦ s♦s r♥♦s♦s ♦♥ ♥ ♣qñ♦ ♣♦r♥t r
s♠rs ♦s ♦s ♠♣♦s ♥ ♣ít♦ ♥tr♦r ♠♦ z < 0 r♣rs♥t
♥ ♣ r ② s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 1 ② ♥
♦♥t σ = 0 ♠♦ z > 0 s rtr③ó ♣♦r ǫr = 4 σ = 1
♠♠ Pr r♥ ♥tr ♠só♥ s ó fc = 500 ③ ♣rí♦♦
t♠♣♦r τ = 2 ♥s q s ♥ ♦r tí♣♦ ♦s q♣♦s ♦rr ♦♥t
♦♥ λ = 0,3 ♠ Pr ♦t♥r s♦ó♥ ♥♠ér ♣r♦♠ s ♣ó
♠ét♦♦ r♥s ♥ts ② ♠♣♠♥tó♥ s t ♥ ♣é♥
ó♦ t③♦ s ♣tó ♣r ♦t♥r ♣trs t♠♣♦ ♦♥st♥t
♦s t♦s ♠♣♦ étr♦ ♥ ♥ó♥ ♣♦só♥ (x, z)
♥ r s ♠str ♠♣♦ étr♦ ♦♠♣♦♥♥t Ey ♣r♦♦
♣♦r ♥ ú♥♦ ♣♦♦ ♦ ♥ ♦r♥ ♦♦r♥s ♣r ♥ t♠♣♦ ♦
t = 10 ♥s t = 5τ s♣és ♠só♥ ♥ r s ♣ ♦srr
q ♦♠♦ ♦♥s♥ ♣rs♥ ♥trs r s♦ ♦♥
tr♥s♠t ♥ ♠♦ ♣rs♥tó ♠♣t ♣r♦①♠♠♥t ♦♥st♥t ♥
♠②♦r ♣rt r♥♦ ♥r ♦♥ ♦s ♣qñ♦s ♠á①♠♦s ♦s ♣r á♥♦s
♣r♦♣ó♥ ♣r♦①♠♠♥t 48♦ ♦♥ rs♣t♦ rt
Pr ♦t♥r ♠♣♦ t♦t rr♦ s só ♣r♦♠♥t♦ ♣r♦♣st♦
♥ só♥ s r s s♣r♣sr♦♥ ♦s ♠♣♦s ♥ts ♥s
s ♥ ♥ s ♣♦s♦♥s rr♦ ♦s ♠♣♦s ♥r♦s
♣♦r ♥ts ♥ ♣♦s♦♥s st♥ts ♦r♥ ♦♦r♥s s ♦tr♦♥
r③♥♦ ♥ trsó♥ ♥ ró♥ x ♦s ♠♣♦s ♦t♥♦s
♣rtr s♠ó♥ ♥♠ér ♣r ♥t ♥ ♦r♥
♦♦r♥s ♥ r s ♠str ♠♣♦ ♥r♦ ♣♦r ♥ rr♦
♣♦♦s ♠t♥♦ s♠tá♥♠♥t rr♦ stá ♥tr♦ ♥ ♦r♥
st♥ ♥tr ♣♦♦s s d = 0,3λ ♥ st s♦ s ♦♥sró ♥ s♣③♠♥t♦
t♠♣♦r dt = 0τ ② ♠♣ts rts da = 1 ♣r t♦♦s ♦s ♠♥t♦s
rr♦ Pr ♥ ♦♠♣ró♥ s♠♣ í♥ ♣♥t♦s ♥ r ♥
r♥t ♦♥s ♥ ú♥ ♥t
♦♠♣r♥♦ s rs ② s ♣ ♦srr q ♠♣♦
rr♦ s ♠ás ♥t♥s♦ ② ♦♥♥tr♦ r♦r ró♥ ♥♦r♠
♥trs q ♠♣♦ ♥ s♦ ♥t Pr á♥♦s ♥ ③♦♥ ♣rér
♠♦s r♥ts ♦♥s t♥♥ rr ♦tr q t♦ ♦♥♥tró♥
s ♠ás ♥t♥s♦ ♥♦ s ♥r♠♥t N ♣♦r ♠♣♦ ♣r N = 5
r ♠♥trs q ♦s ♦s ♠á①♠♦s ♦s q s ♦sr♥ ♥ r
N = 1 s♣r♥ s ③ r♥t rst♥t rr♦ t♥
trs ♥ s ③♦♥ ♠ás ♥t♥s ♥♦ ♥ú♠r♦ ♥ts s ♥r♠♥t
s s♦♥s ♦♥ rs♣t♦ ♦♠♣♦rt♠♥t♦ ♥ ú♥ ♥t t♠
é♥ s ♥r♠♥t♥ ♥♦ ♣♥t♦ ó♥ s r rr♦ ♠♥
t♥♥♦ ♦s rst♦ ♦s ♣rá♠tr♦s ♦♠♦ ♥ ①♣ó♥ ♣r♦①♠
st ♦♠♣♦rt♠♥t♦ s ♣ ♦♥srr q rr z s ♦♥ts ♦s
♠♥♦s ♥s s ♣♦♦ ♣♥t♦ ó♥ r♥
③ ♠ás ♦♥ rs♣t♦ ♦♥t ♦♥ ♠♣♦ tr♦♠♥ét♦ q s
♣r♦♣ ♣♦rq ♦s ♠♣♦s ♥s ♥trr♥ ♠♥r ♠ás ♦♠♣
② ♦♥ rst♥t s t♦r♥ ♠ás ♦♠♣ st t♦ s ♣ ♦srr
♣♦r ♠♣♦ ♦♠♣r♥♦ s rs ② s s ♦rrs♣♦♥♥
t = 2,5τ t = 5τ ② t = 7,5τ ♦♥ N = 7 d = 0,5λ ② dt = 0τ
♠♥trN ♥♦ ③♦♥ ♥tr ♠♣♦ rr♦ s♠♥②
♥♦ ♠s♠♦ s ♥ st♥ ♥s ♣♦s ♦♥ts rr♦
r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♥ t♠♣♦ ♦
t = 5τ ♦ ♠só♥ ♣r ♥ ú♥♦ ♣♦♦ ♣r ♥
rr♦ ♣♦♦s ② ♣r ♥ rr♦ ♣♦♦s ♦♥ dt = 0τ
② d = 0,3λ ② ♠♣ts rts da = 1 s í♥s ♣♥t♦s
♥♥ ♦r♠ r♥t ♦♥s ♥ ú♥ ♥t
r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♣♦r ♥ rr♦ ♦♥
N = 7 d = 0,5λ ♥ t♠♣♦ ♦ t = 2,5τ t = 5τ ② t = 7,5τ
♦ ♠só♥ ♦♥ dt = 0τ s í♥s ♣♥t♦s ♥♥
♦r♠ r♥t ♦♥s ♥ ú♥ ♥t
♦ ♠ás ♦s ② s ♥r♠♥t ♥♦ s ú r rr♦ ♣r♠r t♦
♦rr ♣♦rq ♠♣♦ tr♥s♠t♦ ♣rs♥t ♣r♦①♠♠♥t ♦r♠ ♥
③ r♥t ♦♥ r♥ q s♠♥② ♠♥tr ♦r N
s♥♦ t♦ s ♦♥s♥ ♥r♠♥t♦ t♠ñ♦ rr♦ ♦ q
♥tr♠♥t ♣r♦ ♥ ♠♣♦ ♠ás ♥♦ r ♦s ♠♥t♦s
♥♦ s ♥③ ♣♥♥ ♦♥ d ♠♥t♥♥♦ ♦s ♦s ♦tr♦s ♣rá
♠tr♦s s ♣ ♦srr q ③ tr♥s♠t♦ s ♠♥♦s r♥t ♣r
♦rs r♥s st ♣rá♠tr♦ st ♠♥r ♠♣♦ rst♥t s
♥♦st ♥ s♣rs ó♥ s rr♦ st t♦ s ♣
♦srr ♦♠♣rr s rs ② s q ♦rrs♣♦♥♥
d = 0,1λ d = 0,3λ ② d = 0,5λ ♦♥ N = 7 ② dt = 0τ ♥ ♥ t♠♣♦ ♦
ó♥ t = 5τ ♣rtr st ♦♠♣ró♥ s ♣ ♥♦tr ♠ás q
♦s ♠♣♦s ♥s s ♥ ♥ts ♥ s ③♦♥s ♣rérs ♦
s r♥s r♥s ♥tr s ♦♥ts ♠♥♦ ♥s ♠♥trs
q ♣rt ♥tr ♠♣♦ rr♦ r ♠ás ♦♥ rs♣t♦ r♥t
♦♥s ♥ ú♥ ♥t
❯♥ s♣③♠♥t♦ t♠♣♦r dt ♥ rr♦ t♠é♥ t♥ ♥ t♦ s♦
r r♥t ♦♥s rst♥t P♦r ♠♣♦ ♥ r s ♠str
♠♣♦ ♥r♦ ♣♦r ♥ rr♦ ♦♥ N = 7 d = 0,3λ ② ♥ s♣③♠♥t♦
t♠♣♦r dt = 0,05τ ♥tr ♦s ♠♥t♦s ♣ ♦srr q t♦ ♠ás
♥t ♣r ♥ s♣③♠♥t♦ t♠♣♦r s q ♠ ró♥ ♥
q s ♣r♦♣ ③ tr♥s♠t♦ á♥♦ tr♥s♠só♥ ♥♦ s
ró♥ ♠á①♠♦ ♠♣♦ st ró♥ ♥♦r♠ ♥trs ♥
st s♦ s −9,6♦ ♠é♥ s ♣ ♥♦tr q r♥t ♦♥s tr♥s♠t♦
♣r♦①♠♠♥t ♠♥t♥ s ♦r♠ ♦r♥ ② q s ③♦♥ ♠á①♠ ♥t♥
s s s♣③ tr♠♥t s♦r r rr♥ ♥ ú♥♦ ♠s♦r
♥ r s ♣ ♦srr ♥ rá♦ á♥♦ ♦r♥tó♥
♠♣♦ ♥ ♥ó♥ s♣③♠♥t♦ t♠♣♦r dt ♥tr s ♥ts
♥ rs♠♥ ♥tr ♦s rst♦s ♦t♥♦s s ♣ str ♣♦r ♥ ♦
q ♥ó♠♥♦ ♦♥♥tró♥ s r♦♥ ♦♥ ♦s ♦rs N ② d ② q
♦r♥tó♥ ♠♣♦ s r♦♥ ♦♥ ♦r dt s ♦♥sr♥ ♦s
♠♣♦s q ♠t ♥ ♦♥♥t♦ ♥ts s ♥ ♥trs ♥tr ♦s
♠♦s ② ♥ ró♥ ♠♣♦ r♥♦ s ♦sr q s ♣♦s ♦♥tr♦r
♦♥♥tró♥ ② ♦r♥tó♥ ♠♣♦ ♠♥r s♠r ♦ q ♦rr ♥
í♦ ♣r♦ ♦♥ rts rtrísts ♣rtrs q s♦♥ ♦♥s♥
r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♣♦r ♥ rr♦
♦♥ N = 7 ♥ t♠♣♦ ♦ t = 5τ ♦ ♠só♥ ♦♥
dt = 0τ ② d = 0,1λ d = 0,3λ d = 0,5λ s í♥s
♣♥t♦s ♥♥ ♦r♠ r♥t ♦♥s ♥ ú♥ ♥t
r ♠♣♦ étr♦ Ey tr♥s♠t♦ ♣r ♥ rr♦
♦♥ N = 7 d = 0,3λ ♦♥ s♣③♠♥t♦ t♠♣♦r dt = 0,05τ
♥tr ♠♥t♦s
r ♥♦ ♦r♥tó♥ ♠♣♦ étr♦ Ey ♥
♥ó♥ s♣③♠♥t♦ t♠♣♦r ♣r ♥ rr♦ ♦♥ N = 7
d = 0,3λ ♥ ♥ t♠♣♦ ♦ t = 5τ
♣rs♥ ♥trs ② r♥í ♥tr ♣♥t♦ ó♥
♠♣♦ ② ♥t ♥ st s♦ rr♦
s♣st ♠ét♦♦ ♥ s♦s ás♦s
♦♥♦s s rtrísts ♠♣♦ ♥ ♥ó♥ ♦s ♣rá♠tr♦s
rr♦ s♥t ♣s♦ ♦♥sst ♥ ♥③r rs♣st q ♣r♦ ♠é
t♦♦ ② qé ♠♥r s ♦t♥♥ sñs ♠♦rs Pr ♦ s st♥
♠♥t s♠ó♥ ♥♠ér ♦s ♠♣♦s ♦s st♦♥s áss
♣r♠r ♥② ♥ ♦t♦ ♣qñ♦ ② s♥ ♥ rt♦r ①t♥s♦ ♦♠♦ s
rá ♠ás ♥t ♦tr♦s s♦s ♠ás ♦♠♣♦s s ♣♥ ♥tr♣rtr ♥ tér♠♥♦s
st♦s
Pr ♦t♥r rstr♦s s♠♦s ♦rr s ♦♥sró ♥ ♠♦♦ s♠
♣♦ ♥ q ♦s ♠♣♦s s♦♥ ♥r♦s ♣♦r ♣♦♦s ♠ ♦♥
♦s ♥ ♥trs ♣♥ ♥tr ♦s ♠♦s q r♣rs♥t ♥trs r
s♦ ② ♦s s ♦s ♣♦♦s s♦♥ ♣r♣♥rs í♥ s♦♥♦
t③ó ♠ét♦♦ r♥s ♥ts ② ♠♣♠♥tó♥ s t ♥
♣é♥ ó♦ s ♣tó ♣r q ♦r♠t♦ s ♦s t♦s
♦rrs♣♦♥ q♦s qr♦s ♦♥ q♣♦s ♦rr st♦ ♣r♠tó
trtr ♦s t♦s s♠♦s ♦♥ ♦s ♣r♦r♠s ♣r♦s♠♥t♦ ② s③ó♥
srr♦♦s ♣r♠♥t ♥tr♦ ♠r♦ ss
❯♥ ③ q s tr♠♥♥ ♦s ♣rá♠tr♦s ♠♦♦ s ♦♦ ♠s♦r
♥ ♥ ♣♦só♥ s♦r s♦ ② s s♠ rs♣st ♦t♥♥♦ tr③s
♣r ♥ ♥ú♠r♦ ♣♦s♦♥s r♣t♦r s ♠♥r t rr
t♦♦ r♥♦ ♦sts t③r ♦ s r♣t ♣r♦♠♥t♦ ♣r ♦trs
♣♦s♦♥s ♠s♦r ♥♠♥t s r♣♥ qs tr③s ♦♥ ♣♦s♦♥s
♦♠♥s r♣t♦r ♦t♥♥♦ sí rstr♦s s♠♦s ♥á♦♦s ♦s ♦t♥
♦s ①♣r♠♥t♠♥t ♣rtr st ♣♥t♦ s s ♣r♦♠♥t♦ ♣r
♦t♥r rs♣st sr♣t♦ ♥ só♥ s t♦♠♥ s tr③s q
♦rrs♣♦♥♥ ♥ s ♥ts rr♦ s ♣ s♣③♠♥t♦
t♠♣♦r s s♠♥ s tr③s ② ♦s rst♦s s ♦r♥♥ ♥ ♥ rrr♠
♣r ♦t♥r rs♣st t♦t
tr sñr rr♦ ♣r♦♣♦ ♣r s♦ ♠♣ str
r♦s ♣rá♠tr♦s ts ♦♠♦ st♥ ♥tr ♦s ♦♠♣♦♥♥ts rr♦
r t♦r ♦♥ 0,1 ♠ á♠tr♦ ♦③♦ ♥
♣r♦♥ 0,75 ♠ ♠♦ r♥♥t s rtr③ ♣♦r
ǫr = 4 ② σ = 1 ♠♠ ♦t♦ s rtr③ ♣♦r ǫr = 5 ②
σ = 2 ♠♠ ♣ ♥ tó♥ t♦r st♦s
♣rá♠tr♦s
② ss s♣③♠♥t♦s t♠♣♦rs rt♦s ❯s♠♥t ♠♣t rt
s t♦♠ ♦♠♦ ♦♥st♥t ♣r s♠♣r ♣r♦s♦ Pr r③r ó♥
♦s ♣rá♠tr♦s s ♥ ♦♥srr s ♦♥♦♥s s♦♥♦ ts ♦♠♦
s ♣r♦♣s ♠♦ ♣r♦♥ ♦s ♥♦s ② r♥
♠só♥ q♣♦ ♦rr Pr tr ♠♣♠♥tó♥ ♠ét♦♦
s srr♦ó ♥ ♠ó♦ ♥♦ ♥ ó♦ ♦♠♣t♦♥ ♥t♦r♥♦
st ♠♥r ó♦ ♣r♠t sñr ♦s rr♦s ♣r ♦♣t♠③r s
rtrísts ♠♣♦ s♦r ♥♦ ② ♦t♥r rs♣st ♠ét♦♦
t③♥♦ t♥t♦ t♦s ①♣r♠♥ts ♦♠♦ s♠♦s ♠ó♦ ♥t ♦♥
♥ ♥t♦r♥♦ rá♦ ♠ ② ♦♥ rr♠♥ts ♣r ♦♥tr♦r s③ó♥
♦s rst♦s q s♠♣♥ ♥♦t♠♥t ♣ó♥ ♠ét♦♦ r
♣é♥
♦♠♦ ♣r♠r s♦ ás♦ s ♥③ r①ó♥ ♥ ♥ ♦t♦ ♣qñ♦
♠♦♦ ♦♥sr♦ s ♠str ♥ r ♦t♦ t♥ 0,1 ♠ á
♠tr♦ ② s ♥♥tr ♥ ♣r♦♥ 0,75 ♠ ♠ s s♣r
st ♣rt s♣r♦r ♦t♦ ♣r♠t rt ② ♦♥t
s♦ s♦♥ ǫr = 4 ② σ = 1 ♠♠ rs♣t♠♥t ♠♥trs q ♣r
♦t♦ s♦♥ ǫr = 5 ② σ = 2 ♠♠ ♣ ♥ tó♥ t♦r
♥ ♦s ♣rá♠tr♦s ♦♥sttt♦s ♠♦ ② ♦t♦ ♦ r♦ t♦
r ♠♣♦ ♠t♦ t♥ r♥ ♥tr 500 ③ ♦♥t ♦♥
♥ ♠♦ s λ = 0,3 ♠
♦♠♦ ♠t♦♦♦í ♥r ♣r r ♠ét♦♦ rrr♠
r rr♠ ♣r ♥ ú♥ ♥t ♦♥ s♣r
ó♥ 0,5 ♠ rr♠ ♣r ♥ rr♦ ♦♥ N = 5
d = 0,1 ♠ ② dt = −0,2 ♥s dt = 0,0 ♥s dt = 0,2 ♥s
r ♠♣♦ étr♦ ♥ ♥ t♠♣♦ t = 7 ♥s ②
rs♣st ♦t♥ ♣r N = 1 ♠♣♦ étr♦ ♥ ♥ t♠♣♦
t = 7 ♥s ② rs♣st ♦t♥ ♣r ♥ rr♦ ♦♥ N = 5
♠♣♦ étr♦ ♥ ♥ t♠♣♦ t = 7 ♥s ② rs♣st ♦t♥
♣r ♥ rr♦ ♦♥ N = 10 ♥ t♦♦s ♦s s♦s d = 0,1 ♠ ②
dt = 0,0 ♥s í♥ ♣♥t♦s ♥ s rs ② r♣rs♥t
♦r♠ r♥t ♦♥s ♣r N = 1 ② ♥ s rs ②
r♣rs♥t ♦r♠ sñ ♦t♥ ♣r N = 1
rst♥t ♠s♠♦ s ♦♠♣r ♦♥ ♦t♥♦ ♣rtr ♥ ♠t♦♦♦í
♦rtr s♠♣ r ♠str ♥ rrr♠ ♣r ♥ ú♥♦ ♠s♦r
♦♥ ♦st ♦ 0,5 ♠ ♣r ♠♦♦ r ♦♦r♥ x ♥
rá♦ s rr ♣♥t♦ ♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r
♥ r s ♣ ♦srr ♥ sñ ♦♥ ♦r♠ tí♣ ❱ ♥rt ♥
sñ ró♥ q ♦rrs♣♦♥ r①ó♥ ♥ ♦t♦ ♥trr♦
r ♠str rrr♠ q rst ♥ rr♦ ♦♥ N = 5
d = 0,1♠ ② dt = −0,2 ♥s ♥ st s♦ s♣ró♥ ♥tr r♣t♦r ② ♥t
♥tr rr♦ s 0,5 ♠ s♣③♠♥t♦ t♠♣♦r st♥t♦ r♦
q ♠♣♦ s ♣r♦♣ ♦ r♦ ró♥ θ = 18 ♦♥ rs♣t♦
ró♥ ♥♦r♠ ♥trs r s♦ ♦♦r♥ x s rr ♣♥t♦
♠♦ ♥tr s ♣♦s♦♥s ♥tr♦ rr♦ ② r♣t♦r ♦♠♣r♥♦
s rs ② s ♣ ♦srr q ♥ s♦ rr♦ sñ
r①ó♥ s ♥t♥s ♥ ♥tr♦ x = (−0,8;−0,2) ♠ ② s t♥ú r
é st♦ ♦rr ♣♦rq ③ tr♥s♠t♦ ♠♥ ♥tr♠♥t r♣♦
♥ s ♥tr♦ ② só♦ ♦ ♠♥ ♠r♥♠♥t r é s r q s
♣♦ ♠♦rr ♣rt sñ ♥r♠♥tr r♦♥ ♠♣♦
tr♥s♠t♦
♥ s rs ② s ♠str rst♦ rr ♦r
dt ♠♥t♥♥♦ ♦s ♦s ♦tr♦s ♣rá♠tr♦s rr♦ st♦ s q♥t
♠r á♥♦ tr♥s♠só♥ ♦s ♦rs q s t③r♦♥ dt r♦♥ 0,0
♥s ② 0,2 ♥s ♣ r ó♠♦ s rst♥ ♣♦r♦♥s ♦♥sts sñ
ró♥ ♠r dt
♦♠♦ s ♠♦stró ♥ só♥ r♥t ♦♥s ♠♣♦ ♥r♦
♣♦r rr♦ s ♣r♦rs♠♥t r♥t ♦♥ rs♣t♦ s♦ ♥
ú♥♦ ♠s♦r ♣r ♦rs r♥s N Pr ♥③r t♦ s♦r s
ñ s ♦♠♣r rs♣st ♦t♥ ♣r ♥ rr♦ ♦♥ N = 5 s♦ ♥
q s ♦r♠♦♥s ♥ ♠♣♦ s♦♥ ♣qñs ② ♦♥ N = 10 ♥ q s
♦r♠♦♥s s♦♥ ♣rs Pr ♠♦s s♦s d = 0,1 ♠ ② dt = 0 ♥s
♥②♥ ♦s rst♦s ♣r s♦ N = 1 rs ② ♣r tr
♦♠♣ró♥ ♥ r s ♣ ♦srr q ♣r N = 5 ♠♣♦
♣rs♥t ♦r♠♦♥s ♣qñs ♥ ③♦♥ ♠②♦r ♥t♥s ♥ ♦♥s
♥ ♦r♠ tr♠♦ sñ q s rst ♦♥ ♣r♦①♠♠♥t
♦♥ ♦r♠ sñ ♣r ♥ ú♥ ♥t ♦♠♦ s ♣ r ♥ r
í♥ ♥ ♦r♠ sñ ♦t♥ ♣r N = 1 P♦r ♦tr♦ ♦
r ♦♦ ♦♥ ♥ rt♦r ①t♥s♦ ♣ s♣r
s rtr③ ♣♦r ǫr = 4 ② σ = 1 ♠♠ ② ♣ ♣r♦♥ ♣♦r
ǫr = 5 ② σ = 2 ♠♠ ♣ ♥ tó♥ t♦r
st♦s ♣rá♠tr♦s
s ♦r♠ r♥t ♦♥s rr♦ r s♥t♠♥t r♥t
♦♥s ♥ ú♥ ♥t r ③♦♥ q s rst ♥♦ ♦♥
♦♥ ♦r♠ sñ q s ♦t♥ ♣r ♥ ú♥ ♥t r ♥
♥r s ♥ ♥r♠♥t♦ N ♥t♥s sñ ♥trés ♥ ró♥
♦♥ ♦trs sñs ♣rérs ♠ás á ♥ rt♦ í♠t ♦s rrr♠s r
st♥ts t♥♥ sr ♣♦♦ ♦♠♣rs ♦♥ ♦s ♥t ú♥ ② ♥ ♥r
♣rát♠♥t ♠♣♦ss ♥tr♣rtr
♦ s♠r ♦rr ♦♥ d ②♦ ♦r ♠á①♠♦ stá ♠t♦ ♣♦r ♣ró♥
♥rí q ♥♦ s tr♥s♠t ♥ ró♥ s♣r ♦♠♦ ♥ ♣♦s
s♦ó♥ s ♣♥ t③r rr♦s ♦♥ d ♣qñ♦ ② N rt♠♥t r♥
♦ q ♣r♦ ♦s t♦s ♣r♦♠♦ ♥t♦s tr♥t♠♥t ♥
♣rát N ② d s ♥ t♠é♥ ♠t♦s ♣♦r rr♦r ♥ ♣♦só♥
♦♥t í♥ ♣r♦s♣ó♥ ② t♠♣♦ s♣♦♥ ♣r
♦♠♦ s♥♦ ♠♣♦ s ♥③ s♦ r①♦♥s ♥rs ♣♦r ♥
rt♦r ①t♥s♦ ♥ r s ♠str ♠♦♦ ♣r♦♣st♦ ♣r
st♦ ♣ s♣r♦r s rtr③ ♣♦r ǫr = 4 ② σ = 1 ♠♠ ♠♥trs
q ♠ás ♣r♦♥ ♣♦r ǫr = 5 ② σ = 2 ♠♠ rt♦r stá ♦③♦
♥ ♣r♦♥ ♣r♦♠♦ 0,9 ♠ r♥ ♠t s 500 ③
λ = 0,3 ♠ s ♣ ♥ tó♥ t♦r ♥ t♦ r
♦s ♣rá♠tr♦s ♦♥sttt♦s
r ♠str ♥ rrr♠ ♣r ♥ ú♥♦ ♠s♦r ② ♦st ♦
0,5 ♠ ♣r ♠♦♦ r q ♥ ♠♣♦ ♥tr♦r
r rr♠ ♣r ♥ ú♥♦ ♠s♦r ♦st 0,5 ♠
rr♠ ♣r ♥ rr♦ ♦♥ N = 5 d = 0,1 ♠ ② dt = 0,00
♥s dt = 0,15 ♥s dt = 0,35 ♥s ♥ ♦♥ s s ③♦♥s
rst sñ
♦♦r♥ x ♥ rá♦ s rr ♣♥t♦ ♠♦ ♥tr s ♣♦s♦♥s
♠s♦r ② r♣t♦r r ♠str rrr♠ ♣r ♥ rr♦
♦♥ N = 5 d = 0,1 ♠ ② dt = 0,00 ♥s ♦ st ♦r dt ③
tr♥s♠t♦ s ♣r♦♣ ♦ r♦ ró♥ rt ♦♠♣r♥♦ s
rs ② s ♣ ♦srr q s ♥r♠♥tó ♠♣t
♣rt sñ q stá s♦ s♠♥t♦ ♥♥♦ ③qr♦
♥trs st ③♦♥ s ♥ ♦♥ ♥ ♥ r P♦r ♦tr♦ ♦ s
♣ ♦srr q rs♣st ♣r s♠♥t♦ ♥♥♦ r s
s ♥ st ♠♣♦ ♠ét♦♦ ♥t♥só ♥t♠♥t ♣rt
③qr ♥♥ sñ ② q ③ tr♥s♠t♦ s r s♦r ♣♥♦
rs♣t♦ rt♠♥t r♣t♦r ♦ ♠② r é st ♠♥r s
♦t♥ ♠á①♠ ♠♣t ♥♠♥t ♠ét♦♦ ♥♦ s t♦ ♣r ♠♦rr
s♠♥t♦ ♥♥♦ r ② ♦s s♠♥t♦s ♦r③♦♥ts ② q ♥
st♦s s♦s só♦ ♣rt ♣rér ③ r♦ ♥③ r♣t♦r
s ♣♦s rstr ♦trs ♣♦r♦♥s sñ r r♥♦ s
♣③♠♥t♦ t♠♣♦r dt P♦r ♠♣♦ s s ♣ dt = 0,15 ♥s s ♦t♥
rst♦ q s ♠str ♥ r ♥ q s rst♥ t♦s s
♣♦r♦♥s sñ s♦s ♦♥ ♦s s♠♥t♦s ♦r③♦♥ts rt♦r
s ♣ dt = 0,35 ♥s s ♦t♥ rst♦ q s ♠str ♥ r
♦♥ s ♣ rr ó♠♦ s rst ♣rt sñ s♦
s♠♥t♦ ♥♥♦ r♦ rt♦r ♥ ♥r ♦♠♦ ♥ s rs
② s ♣ ♦srr ♥ ♦♥♥t♦ sñs s♥rs q s ♥t♥s
♥ ♣r ♠ét♦♦ sts sñs ♣r♥ ♦♠♦ ♥s
♥♥ó♥ ♣r dt ♦♥sr♦ ♥ t♦ ♠ét♦♦ ♥♦ só♦ ♥t♥s
s sñs ♦rrs♣♦♥♥ts ♦s rt♦rs ♥trés s♥♦ t♠é♥ s ♣r♦
s ♥ t♦s s ♥trss q trs ③ ♣r♦♣rs ♥ st s♦
s t♦♥s ♦♥srs ♣r ♠tr③ s♦
♥ rs♠♥ ♦s ♣rá♠tr♦s rr♦ ② s ♣r♦♣s ♠♦ tr
♠♥♥ s rtrísts ♠♣♦ ♦♠♦ á♥♦ ♥ q s ♣r♦♣
♥♦ ♥r ② ♦r♠ r♥t ♦♥s ♥ ♦s ♠♣♦s ♥tr♦rs s
♣ r q ♦s ♦s ♣rá♠tr♦s rr♦ ② s ♣r♦♣s ♠♦ s
rst♥ s sñs t♦♦s q♦s ♥♦s ② ♦♠trí s ♣r
rr ♠♣♦ ♠t♦ ♣♦r rr♦ ♣♦só♥ r♣t♦r s r
q ♦s rst♦s q s ♦t♥♥ ♣r ♠ét♦♦ ♣♥♥ t♥t♦ ♦s
♣rá♠tr♦s rr♦ ♦♠♦ ♦♠trí ② ♣r♦♣s ♥♦ ②
♠♦ r♥♥t st♦ s ♦♥srr ♣r ♠ét♦♦ ♣r ♦t♥r
♥♦s rst♦s ♥tr♣rtr♦s ♦rrt♠♥t
Pr♦s♠♥t♦ ♣r ♠♦rr sñs ♦♠
♣s
♦s ♠♣♦s ♥tr♦rs ♠str♥ q ♠ét♦♦ s út ♣r ♠♦rr
r♥ts ♣rts sñ ♣r♠r ♥ ♦t♦ ♣qñ♦ ② ♥ rt♦r
①t♥s♦ rr á♥♦ tr♥s♠só♥ Pr♦ ♥ ♥r ♦ q s s
s ♠♦rr sñ ♦♠♣t st♦ s ♠♣♦rt♥t ♥ st♦♥s rs ♦♥
♠♣t sñ ♣r♠r ② ♦①st♥ sñs ♦♥ ♠♦rr
♦s ♥s ♥t♥s ② ♦♥t♥ tr sñ ♥trés ♦r
♥ ♠♦r ♥ó♥ s ♠s♠s ② ♣♦r ♥ ♥ ♥tr♣rtó♥ ♦rrt
♠tá♥♠♥t s s ♠♥t♥r ♦r♠ sñ q s ♦t♥ ♥♦
♥♦ s ♠♥ ♦♥ r♥t ♦♥s ♥ ú♥ ♥t ② q ♥ s
s♦ s s ♦♠♦ ♥tr♣rtr ♦s rst♦s ♠ás q ést♦s s ♣♥
♣r♦sr ♦♥ ♦s ♣r♦♠♥t♦s ② ♣r♦r♠s q s t③♥ t♠♥t
r♦ st♦s rst♦s ♦s ♣s♦s ♣r♦s♠♥t♦ q s ♣r♦♣♦♥♥
♣r ♠♦rr ♥ sñ ♥ t♦ s ①t♥só♥ s♦♥ ♦s s♥ts
♥rr ♥ sr rrr♠s ♣r r♥ts ♦rs dt t③♥
♦ ♣r♦♠♥t♦ sr♣t♦ ♥ só♥
♥ ♥♦ ♦s tr♠♥r ♦ ♦s s♠♥t♦s ♥ ♦s s sñ
stá r♠♥t ♠♦r ② ♦s ♥tr♦s ♣♦s♦♥s rs♣t♦s ♥
♦s q st♦ ♦rr
♣rr s ♣♦r♦♥s rrr♠ q ♦rrs♣♦♥♥ s♦s ♥tr♦s
r♥r s ♣rts rst♥ts ♦s rrr♠s ② ♥rs
♥ r s ♠str ♥ sq♠ ♣r♦♠♥t♦ ♣r♦♣st♦ Pr
♣♦r t③r ♠t♦♦♦í ♣r♦♣st ♦♥ ♥ ② ♥ ♦r♠ ♠ s
srr♦ó ♥ ♠ó♦ s♣ ♥♦ ♥ ó♦ ♦♠♣t♦♥ ♥t♦r♥♦
st ♥t ♦♥ ♥ ♥t♦r♥♦ rá♦ q ♣r♠t ♥rsr ② ♠♦r ♦♥
t♥t♦ ♦s s♠♥t♦s ♥ ♦s q s r♥♦ ♣♦s♦♥s ♦♠♦
r sq♠ ♣r♦♠♥t♦ ♣r ♠♥tr ♦♥
t♥ tr s sñs
r rr♠ ♦t♥♦ ♣r ♥ ♦t♦ ♣qñ♦
♥♥♦ ♣♦r♦♥s ♦s rrr♠s ♦rrs♣♦♥♥ts
N = 5 d = 0,1 ♠ −0,4 ♥s ≤ dt ≤ 0,4 ♥s á♥♦s tr♥s
♠só♥ ♥tr 41♦ ② −41♦ t③r♦♥ ♥tr♦s
r s♣③♠♥t♦s t♠♣♦rs s♦♥♦s ♣r ♦
t♥r rst♦ r
♦s ♣rá♠tr♦s rr♦ ♥ ♥♦ ♦s ♠♥r ♥t ♦♥ q
s ♦t♥ rs♣st ♥t♦ ♦♥ s rr♠♥ts ♣r str s③ó♥
♠s♠ t♥ ♣r♦s♦ st ♦s ♣rá♠tr♦s rr♦ ♣r
♦t♥r ♥ rs♣st ♥ q s ♠♦r sñ ♦♠♣t r ♣é♥
s ♣ ♣r♦♠♥t♦ sr♣t♦ ♦s rst♦s ♣r ♠♦♦
r só♥ s ♦t♥ rst♦ q s ♠str ♥
r t③ó ♥ rr♦ ♦♥ N = 5 d = 0,1 ♠ ② s s♦♥r♦♥
s♣③♠♥t♦s t♠♣♦rs ♥tr −0,4 ♥s ② 0,4 ♥s ♦s q ♦rrs♣♦♥♥
á♥♦s tr♥s♠só♥ ♥tr 41♦ ② −41♦ Pr ♦t♥r rst♦ q s
♠str ♥ r ♥ q s ♠♦r sñ ♦♠♣t ró♥
r rr♠ ♦t♥♦ ♣r ♥ rt♦r ①t♥s♦
♥♥♦ ♣♦r♦♥s ♦s rrr♠s ♦rrs♣♦♥♥ts
N = 5 d = 0,1 ♠ 0,00 ♥s ≤ dt ≤ 0,30 ♥s á♥♦s tr♥s
♠só♥ ♥tr 0♦ ② −30♦ t③r♦♥ ♥tr♦s
r s♣③♠♥t♦s t♠♣♦rs s♦♥♦s ♣r ♦
t♥r rst♦ r
s t③r♦♥ ♥tr♦s ♥ r s ♠str ♥ rá♦ ♦s ♦rs
s♣③♠♥t♦ t♠♣♦r ♥ ♥♦ ♦s ♠é♥ s ♥♦tr q s
s♦♥s ♥s ♦♥st♠♥t ♠♣♠♥ ♣rt♠♥t ♥tr s st♦
♦rró ♥ ♠♣♦ ♣♦r ♥ ♦ ♣♦rq s s♦♥r♦♥ ♠♥t ♦s
♣rá♠tr♦s rr♦ N ② d s r q r♥t ♦♥s tr♥s♠t♦
t♥í ♦r♠ s♠r r♥t ♦♥s ♥ ú♥ ♥t ♥ ♣♦só♥
♥♦ P♦r ♦tr♦ ♦ s s♦♥r♦♥ ♥r♠♥t♦s dt s♥t♠♥t ♦s
♠♥r q s rstr ♥ s ♣rts sñ ② s♠♥t♦s
♦s ♣r r♦♥strr rrr♠
s♥t ♣s♦ ♦♥sst ♥ ♣r ♠ét♦♦ ♣r♦s♠♥t♦ ♣r ♠
♦rr sñ ♦♠♣t ♥ rt♦r ①t♥s♦ ♥ r s ♠str
rst♦ ♣r st ♣r♦♠♥t♦ ♠♦♦ r só♥
♥ st s♦ s t③ó ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠ s s♦♥r♦♥
s♣③♠♥t♦s t♠♣♦rs ♥tr 0,00 ♥s ② 0,30 ♥s ♦s q ♦rrs♣♦♥♥
á♥♦s tr♥s♠só♥ ♥tr 0♦ ② −30♦ ② s t③r♦♥ ♥tr♦s ♥ r
s ♠str ♥ rá♦ ♦s ♦rs s♣③♠♥t♦ t♠♣♦r ♥
♥♦ ♦s ♣ ♦srr q sñ r ♣rs♥t ♥ s♣t♦ s♠
r ♦t♥ ♣r ♥ ú♥ ♥t r só♥ ♠♥t♥♥♦
♥ ♥ ♦♥t♥ ♦r♥ ♦ ♠♦r ♣r♦ ♦♥ ♥ sñ
♦♥ ♠②♦r ♥t♥s st rst♦ str ♥ ♠ét♦♦ ♥ s♦
♥ rt♦r ①t♥s♦
♥ st♦♥s ①♣r♠♥ts s s q ♣rts s sñs ♥♦ ♣♥
sr r♠♥t ♥s ♦ ró♥ sñ r♦ ♥ ♥ só♦
s♠♥t♦ ♣ sr s♥t ♣r ♥r ♦s ♣rts s♣rs sñ ♥
♦ rt♦r ♥ ♠♦s s♦s s ♥sr♦ sr ♥ r♥ ♥ú♠r♦ ♦s
♥t ♣♥ ♦♠♣ rt♦r ❯♥ stó♥ ♦♠♣ ♦
rr ♥♦ ♥♦ stá r♦ ♦♥ ♦♥t♥ú ♥ sñ s♣és ♥ ár ♦♥s
s r ♦♥ á ♥t♦ s ♥r ♣♦r ♠♣♦ ♥♦ s s♣r♣♦♥♥
♥t♦s st♥t♦s strt♦s ♦♥ s♠rs ♥♥♦♥s ♥ st♦s s♦s ♥tr
♦s ♠s♦ ♣qñ♦s ♦ ♥r♠♥t♦s r♥s s♣③♠♥t♦ t♠♣♦r
♣♥ r rst♦s ♥♦rrt♦s ♥t♦♥s s ♦♥♥♥t s♦♥r ♥
r♥♦ ♠♣♦ ♣r dt ② ♥ ♥r♠♥t♦ t♥ ♣qñ♦ ♦♠♦ ♣rí♦♦ ♠s
tr♦ ♦ ♠♥♦r q ést ♥ s s♦ s tr③s rqr♥ ♥tr♣♦ó♥ trs
st♦♥s ♦♠♣s s♦♥ qs ♥ s q s s♣r♣♦♥♥ ♦♥ st♥t ♥
♥ó♥ ♣r♥t ♦s ♦ ♠ás sñs ♥tr♦ ♥ ♥tr♦ ♥ s♦s s♦s só♦
s ♣ ♠♦rr ♥ s sñs ♣♦r ③ ♦♥ st ♠t♦♦♦í
♣ít♦
♣ó♥ ♠ét♦♦
s♦s ♦♥ ♥♦s ♠út♣s
♥ ♣ít♦ ♥tr♦r s ♥③ó rs♣st ♦t♥ ♣r ♠ét♦
♦ ♦s s♦s s♠♣s ♥ ♦t♦ ♣qñ♦ ② ♥ rt♦r ①t♥s♦
♥ s♦s ♠♣♦s st ♣rs♥t sñ ♥ ú♥♦ ♥♦ ♦ s ♣r
s♥tó ♥ ♠t♦♦♦í ♦♠♣♠♥tr ♣r♦s♠♥t♦ rs♣st
q ♣r♠t t♠é♥ ♠♦rr ♦♥t♥ tr s sñs ② t
♥tr♣rtó♥ ♦s rst♦s
♥ st ♣ít♦ s st ♣ó♥ ♠ét♦♦ s♦s ♦♠♣♦s
♥ ♦s q stá♥ ♣rs♥ts rs sñs ♥t t♦s s♠♦s s ♠str
ó♠♦ ♣ ♣rs ♠ét♦♦ ♣r rstr ♥ s sñs ♥ ♦r♠
♥♣♥♥t ♦s rst♦s ♦t♥♦s s ♣♥ ♥ ♦s s♦s ①♣r♠♥ts
♥trés rq♦ó♦ ♦♥ t♦s qr♦s r♥t ♥ s♥ ♠♣ñ
r③ ♥ st♦ P♦ ♥♦ ♣r♠r s♦ ♦♥sst ♥ rtr③r ♥
♣r t♣ ♥trr ♣rt♥♥t ♥ s strtrs t♦♥s
tts ♥ st♦ s♥♦ s♦ ♦♠♣r♥ st♦ sñs
r①ó♥ ♣r♦s ♣♦r ss ♥trrs ❬❪ ❬❪
t♦s s♠♦s
♥ ♦s s♦s ①♣r♠♥ts s t q ♥ ③♦♥ st♦ s ♥
♥tr♥ ♣rs♥ts ♠út♣s ♥♦s ♥trés ♠ás ♦tr♦s ♦t♦s s♥
r ♦♦ ♦♥ ♦s rt♦rs á♠tr♦s 0,1 ♠ ǫr =
5 ② σ = 1,5 ♠♠ strt♦ s♣r s rtr③ ♣♦r ǫr = 4
② σ = 1 ♠♠ strt♦ ♣r♦♥♦ ♣♦r ǫr = 4,2 ② σ = 2,5 ♠♠
♣ ♥ tó♥ t♦r st♦s ♣rá♠tr♦s
r♦s Pr trtr s♦s ♥ ♦s q ①st♥ ♠út♣s sñs t
♥tr♣rtó♥ ♦s rst♦s s ♣r♦♣♦♥ ♣r ♠t♦♦♦í sr♣
t ♥ ♣ít♦ ♥tr♦r ♠♦r♥♦ ró♥ sñ ♣r♠r ♥t♦r♥♦ ②
♦♥t♥ s sñs
♠♦♦ q s ♦♥sr s q s ♠str ♥ r ést ♥②
♦s rt♦rs ♣qñ♦s ② ♥ rt♦r ①t♥s♦ ♦s ♣r♦♥s
0,5 ♠ ② 1,0 ♠ rs♣t♠♥t ♠♦s rt♦rs s rtr③♥ ♣♦r ♥
á♠tr♦ 0,1 ♠ ǫr = 5 ② σ = 1,5 ♠♠ strt♦ s♣r s rtr③
♣♦r ǫr = 4 ② σ = 1 ♠♠ ② strt♦ ♥r♦r ♣♦r ǫr = 4,2 ② σ = 2,5 ♠♠
r♥ ♠♣♦ s 500 ③ s ♣ ♥ tó♥ t♦r
s ♠trs ♣r♠t ② ♦♥t
r ♠str rrr♠ ♦t♥♦ ♣r ♥ ú♥♦ ♠s♦r ♦♥
♦st 0,5 ♠ ♥ s ♣♦s ♥tr r♠♥t s sñs ♦s r
t♦rs r①ó♥ ♥trs ♥trr s ♠② é Pr r st sñ
♠ás s s ♣ó ♠ét♦♦ ♣r ♦ s t③ó ♥ rr♦ ♦♥ N = 5
② d = 0,1 ♠ ♥ st s♦ s t③ó ♥ ♦♥♥t♦ s♣③♠♥t♦s t♠♣♦rs
♥tr 0,00 ♥s ② 0,15 ♥s ♦♥ ♥r♠♥t♦ 0,05 ♥s rrr♠ rst♥t s
♠str ♥ r ♦♥r♠ó q ♠ét♦♦ ♠♦r
sñ ♥trés ♠♦r q ♣rs♥tó sñ ♦♥sstó ♥ ♥ ♠♥
t♦ ♦♥trst ♥tr sñ ② ♥t♦r♥♦ ♣rs♥t ♥ rrr♠ ♦tr
q s sñs ♦s rt♦rs s rstr♦♥ só♦ ♥ ♥ ♣qñ ♣♦ró♥
♣r♦ q ♥ ♥r stá♥ ♠ás ss ♥ st s♥t♦ stá r♦ q
♠ét♦♦ t♥ trr s sñs ró♥ ♥♦ ést s ♣ ♣r
r rr♠ ♣r ♥ ú♥♦ ♠s♦r ♦♥ ♦st 0,5
♠ rr♠s ♣r ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠ ♦t♥♦s
♣♥♦ ♠t♦♦♦í ♣r ♠♥tr ♦♥t♥ tr
sñ ♥trs ♦r③♦♥t ② sñ rt♦r
r r♥♦ ♦rs x s ó ♥ ♠ts
② ♠t♦♦♦í s ♣ó ♣r ♠♦rr s sñs ró♥
rs♣ts
rstr sñ ♥ rt♦r ①t♥s♦
tr♥t♠♥t s ♣ ♠♦rr sñ ♥♦ ♦s rt♦rs ♣
qñ♦s P♦r ♠♣♦ ♥ r s rst sñ rt♦r
r ♣ ♦srr ♥ ♠♥t♦ ♥ ♦♥trst sñ ② ♥
♦♥t♥ tr ♥ st s♦ sñ rt♦r ①t♥s♦ s
s♣ró st♦ ♦rró s q ♣r rstr sñ ró♥
s s♦♥r♦♥ s♣③♠♥t♦s t♠♣♦rs ♥ ♦s st♥t♦s ♥tr♦s q ♥♦
r♦♥ ♦s ♣r rstr sñs rt♦rs ①t♥s♦s ♦♠♦ ♠♣♦
♥ ♥ r s rstr♦♥ ♠♦s rt♦rs Pr r st♦ s
ó r♥♦ x ♥ ♠ts ♥ ♥ s s ♣ó ♠t♦♦♦í
♣r♦s♠♥t♦ s sñs ró♥ rs♣ts ♦♥ ss♥t
s③ó♥ ♦s ♦s rst♦s s♠tá♥♠♥t
r♦ st♦s rst♦s ♠ét♦♦ ♣r ♦♠♦ ♥ rr♠♥
t ♣r ♥str ♦♠♣♦rt♠♥t♦ sñs és ♦ ♦ss ♥s♦ ♥
s♦s ♦♥ ♠út♣s ♥♦s
t♦s ①♣r♠♥ts
rtr③ó♥ ♣rs t♣
♥ st só♥ s ♠str♥ ♦s ♥♦s ♠ét♦♦ ♣♦r ♠♦
s ♣ó♥ ♥ tr♠♥ó♥ ♥♦ ② ♣r♦♥ ♣rs
t♣ ♥ st♦ rq♦ó♦ P♦ ♥♦ ♦♥♦r ♦s ♦rs s♦s
♣rá♠tr♦s s ♠② ♠♣♦rt♥t ♣r ♦s rqó♦♦s ② q ♣rtr ♦s s
r③♥ st♦s ♦♠♣rt♦s ♥áss stíst♦s t ♠ás ♦s ♦rs
♣r♦♥ s♦♥ r♥ts ♣♥r ♥ ①ó♥ ♣♦r ♠♣♦ ♣r
r ♥t s♦ q s r♠♦r
♥ s r③ó ♥ ♣r♦s♣ó♥ ♦ís ①t♥s ♥ st♦ P♦
♥♦ t③♥♦ ♠t♦♦♦í ♦rr ♦rtr s♠♣ ♥á
ss ♦s t♦s ♣r♦♦ ♠♣s s strtrs rq♦ós ♥trrs
♣rtr♠♥t rt ♥t ♣rs t♣ q ♥í♥ ♥ s
tró♥ ♦♠♣ t♦♥s ❬❪ ♥ ♣♦só♥ s ♣rs ♥
♣♥♦ ♦r③♦♥t s ♣♦ str ♦♥ ♣rsó♥ ♣♦r ♠♦ ♠t♦♦♦í
♣ ♥♠♥t só♦ s ♦tr♦♥ ♦rs ♠♣rs♦s ♥♦ s
♣rs ♦♥ ♦rs ♠② ♦s♦s ♣r♦♥ ♥ ♥s ♣qñs s
♦♥s s ♠s♠s sts ♥rt③s r♦♥ ♦♥s♥s ♥ ♣r♠r r
♦♥trst ♣r♠t ♠♦r♦ ♦ ♥ s s♣rs ♦s ♥♦s
st♦ ♦rr ♣r♥♣♠♥t ♣♦rq ♦s t♥ts ♦r♥s ♦♥str②r♦♥ s
♣rs ♦♥ ♦s ♠trs ♠♦ r♥♥t s♠s♠♦ s ss s
♣rs ♥♦r♠♠♥t ♣rs♥t♥ ♥ tr♥só♥ s ♥ ♣r♠t ♦
♦s ♣r♦s♦s ♦♠♣rsó♥ ② s♦ ♣♦r ♣♦ró♥ r♥t s ♦♥stró♥
Pr ♦t♥r ♦s t♦s s s♣s♦ ♥ í♥ s♦♥♦ 3 ♠ ♦♥
t q r③ ♥ s ♣rs tts ♥ st♦ ♣rt♥♥t
♥ú♦ t♦♥ r só♥ st♠ó ♥ ♣r♠t
ǫr = 3 ♣r ③♦♥ st♦ ♣rtr ♦ ♣r♦♣ó♥
v = (0,18±0,01) ♠♥s ♦t♥ ♣r ③♦♥ ♥ só♥ ó♥ ♣r
♠♥r ♦s ♠♣♦s s ♣r♦♥s s♣rs ♣r ♥♦ ♠♥t
♣r♦r♠ ♥t♦r♥♦ ♣é♥ ♥ó q ♣r ♥ s♣ró♥ ♥tr
♥ts d = 0,1 ♠ N ♣♦í sr ♦r♥ ♦ ♦ s qró ♦♥ ♥
s♣ró♥ ♠á①♠ q rí st r♥♦ st♥ ♠á①♠ 0,7 ♠ ♦s t♦s
s qrr♦♥ t③♥♦ sst♠ rr ♥s♦rs ♦tr Ps
P ② ♥t♥s ③ s ♥t♥s ♠s♦rs ② tr♥s♠s♦r s ♦③r♦♥
♥ ♥trs r s♦ ♦♥ ♦s s s ♥t♥s ♣r♣♥rs í♥
s♦♥♦
r ♠str ♥ rrr♠ ♣r ♥ ú♥♦ ♠s♦r ♥ st s♦
♦st s 0,5 ♠ ♣ rr q ♥ ♦s í♠ts trs ♥ s
♣r s ♣♥ str ♦♥ ♣rsó♥ ♣rtr st rrr♠ ó♦
♥ sñ ♣r♦①♠♠♥t 17 ♥s r í♥ ♣♥t♦s ♥ r
♣r ♥r s ♣r ♥q ♠s♠ s ♠s♦ é ♣r
srr♦
♦ s ♣ó ♠ét♦♦ ♣r ♠♦rr sñ r①ó♥ ♥ s
② ♣r ♥r ♠♦r s r♦♥s ♥ ♦s ♦rs ♥ r s ♠str
rrr♠ rst♥t ♣r ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠ ♥ q s
♣ó ♠t♦♦♦í ♣r♦s♠♥t♦ ♣r ♠♦rr sñ s
♣r Pr ♦t♥r st rst♦ s t③ó ♣r♦r♠ ♥t♦r♥♦ ③
tr♥s♠t♦ s ró ♦♥♦ ♣r ② s ró s♣③♠♥t♦
t♠♣♦r s −0,30 ♥s st 0,30 ♥s ♦♥ ♥r♠♥t♦s 0,05 ♥s á♥♦s
tr♥s♠só♥ ♥tr 36♦ ②−36♦ ♠♥r ♠♦rr t♦s s ♣rts r♥ts
sñ ② s♦♥r ♦s s♣③♠♥t♦s t♠♣♦rs ♠ás ♦s
r rr♠ ♣r ♥ ú♥♦ ♠s♦r ♦♥ ♦st
0,5 m rr♠s ♣r ♥ rr♦ ♦♥ N = 5 ② d = 0,1 ♠
♦t♥♦s ♣♥♦ ♠t♦♦♦í ♣r ♠♥tr ♦♥t♥
tr sñ s ♣r ② sñ
ró♥ ♦♥ ért ♥ (x; t) = (37 ♠; 8 ♥s) r♥♦ x
s ó ♥ ♠ts ② ♠t♦♦♦í s ♣ó ♣r ♠♦rr
s sñs ró♥ ♦♥ érts ♥ (x; t) = (37 ♠; 8 ♥s) ②
(x; t) = (38,5 ♠; 10 ♥s)
r r ♠ó♥ ♦t♦ ♣r ♦
sr ①♣st
♣r s♠♥t♦ ♣ ♦srr ♥ r q ♣r♦♠♥t♦
♣♦ ♠♦ró t♠♥t ♥t♦ r♦r 17 ♥s ♦♠♣t♥♦
sñ trés ♣r ♣r♦♥ q s ♥♥tr s
♣r s st♠ó ♥ (1,5 ± 0,1) ♠ ♣r ♦ s t③ó ó♥
só♥ ② ♦ ♣r♦♣ó♥ v = (0,18 ± 0,01) ♠♥s st
♠♥r ♣♦s ♠tr s ♣r s ♥ ♦♥ ♥ í♥
♣♥t♦s ♥ r
r s ♥ ♠♣♦ ó♠♦ s ♣♦s ♠♦rr ♥ sñ r
ó♥ ♥ ♥♦ ♦s ♦rs ♣r sñ q s rstó s ♣ ♥tr
r♠♥t ♥ r ♣♦só♥ s ért s (x; t) = (37 ♠; 8 ♥s) ♣r♦
①♠♠♥t s ♥ ♦♥ ♥ ♥ r ♥♠♥t ♥ r
s ♥ ♠♣♦ ♥ q s rstr♦♥ sñs rt♦rs ♥ ♦s ♦s
♦♣st♦s ♣r ♥♦ r♥♦ x ♥ ♠ts ② ♣♥♦ ♠t♦
♦♦í ♣r♦s♠♥t♦ ♥ ♥ s ♦♠♦ s ①♣ó ♣r♠♥t
sñ rst ③qr s ♠s♠ q ♥ r ♠♥trs
q sñ r t♥ ért (x; t) = (38,5 ♠; 10 ♥s) s ♥ ♦♥
♥ ♥ r ♠é♥ s ♣ ♥♦tr q s sñs ró♥
r♥♥ts ♠♦rr♦♥ ♦ q ♠♣♦ tr♥s♠t♦ t♥ ♥ ♥♦ ♥t♦
st♦s rst♦s ♣r♠tr♦♥ st♠r ♥♦ ♣r ♥ 1,5 ♠ ♦ q s
♦♥r♠ó ♦ sr ①♣st ♥ s rs ② s ♠str♥ ♦t♦s
r ♠ó♥ ② ♣r ♦ sr ①♣st rs♣t♠♥t
♣ó♥ sñs ♣r♦s ♣♦r ss
♦♠♦ s ♦ ♥tr♦r♠♥t ♥ ♦s st♦s rq♦ó♦s s ♦♠ú♥ ♥♦♥
trr ss ♦ ♠♥t♦s s♠rs ♥trr♦s ♦rrt rtr③ó♥
st t♣♦ ♥♦s ♣rs♥t ts ♦ q ♥ ♥r ♣r♦♥
sñs és ♦♠♦ ♦♥s♥ ♦ ♦♥trst q ♣rs♥t♥ ♦♥ rs♣
t♦ ♠♦ r♥♥t ♥ só♥ s ♠♦strr♦♥ ♥♦s rst♦s
r♦♥♦s ♦♥ tó♥ ss ♥trrs ♥ ♦s rrr♠s ♦t♥
♦s ♠ás sñ ♣r♠r s s ♣♥ ♦srr rss sñs
♦r♥s ♥ rt♦rs ♥trr♦s ② s♦♥t♥s ♥trs ss♦
♥ st só♥ s st♥ s ♠♦rs ♥ s sñs q s ♦t♥♥ ♠♥t
♣ó♥ ♠ét♦♦
♦s t♦s s qrr♦♥ t③♥♦ ♦♥ró♥ ①♣r♠♥t sr♣t
♥ só♥ ♥ st s♦ s s♣sr♦♥ ♦s í♥s s♦♥♦ 5 ♠
♦♥t ♠♥r q ésts ♣s♥ ♣♦r ♥♠ s r ②
♥ r ♦r ♥ s í♥s s qró ♦♥ s♣ró♥ ♥tr
♥ts 0,05 ♠ ② ♦♥ ♥ s♣ró♥ ♠á①♠ ♥tr r♣t♦r ② ♠s♦r 1
♠ s ♥t♥s ♠s♦r ② r♣t♦r s r♦♥ ♥ ♥trs r s♦ ♦♥
♦s s s ♥t♥s ♣r♣♥rs í♥ s♦♥♦ t③ó sst♠
rr ♥s♦rs ♦tr Ps P ② ♥t♥s 500 ③
♥ ♣r♠r tér♠♥♦ s ♥③r♦♥ ♦s t♦s ♦t♥♦s ♣r í♥ s♦♥♦
r ♥ r s ♠str ♥ ♣r ♦st 0,25 ♠
sñ ró♥ ♦t♦ ♥trr♦ s ♣ r ♦♥ r ♥ r
♦♥ ért ♥ (x; t) = (0 ♠; 10 ♥s) ♦♦r♥ x ♥ rá♦ s rr
♣♥t♦ ♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r ❱rs ♣ér♦s
ró♥ ♦♥s ② r①♦♥s ♣r♦①♠♠♥t ♦r③♦♥ts ♣r♦s
♥ s s♦♥t♥s ♥trs s♦ t♠é♥ ♣r♥ ♥ r
r ♦♥ró♥ ①♣r♠♥t t③ ♣r qrr
♦s t♦s
P♦r ♠♣♦ ♥ sñ ♣r♦①♠♠♥t ♦r③♦♥t ♥ strt♦ s ♣
♦srr r♦r 8 ♥s ♥tr x = 0,8 ♠ ② x = 2 ♠ ♣r♦①♠♠♥t
①tr♠♦ ③qr♦ st sñ ♦♥t♥ú ♥ ♥ sñ ró♥ ért ♥
x = 0,8 ♠ ♦r♥ ♥ ♣♥t♦ ♦♥ ①ó♥ ♦rtó strt♦
r ♠str rst♦ ♠ét♦♦ ♣r ró♥ ♥
s ♥trr ♦s ♦rs N ② d s s♦♥r♦♥ ♠♥r q
r♥t ♦♥s rst♥t ♥♦ s st♦rs♦♥ s♥t♠♥t ♥ ♣♦só♥
♥♦ ♣r ♦ s t③ó ♣r♦r♠ ♥t♦r♥♦ ② s ♦♥sró ǫr = 3 ♦s
♣rá♠tr♦s rr♦ r♦♥N = 4 d = 0,1♠ ② ♦st 0,45♠ ♠♥trs q s
s♦♥r♦♥ ♦s s♣③♠♥t♦s t♠♣♦rs ♥tr−0,6 ♥s ② 0,6 ♥s á♥♦s
tr♥s♠só♥ ♥tr 56♦ ② −56♦ ♣r ♠♦rr sñ ♥ ♥tr♦s ♦♥st♦s
t③r♦♥ ♥ t♦t ♥tr♦s ♥ r s s♦♥ó ♥ r♣♦
r♥t s♣③♠♥t♦s t♠♣♦rs ♥tr −0,5 ♥s ② 0,2 ♥s á♥♦s
tr♥s♠só♥ ♥tr 54♦ ② −22♦ ♦♥ ♣r♦♣óst♦ rstr r①ó♥
♣♥♦ ♣r♦①♠♠♥t ♦r③♦♥t ② ♣ér♦ s♦
♥♦ s ♦♠♣r♥ s rs s ♣ ♦srr q s
sñs s♦♥ ♠ás rs ♣r ♠ét♦♦ q ♣r ♠ás ♠ét♦♦
♠♦ró sñ ♦t♦ ♥trr♦ ♦ strt♦ ♦r♠ st
♥♦ s rstó sñ ró♥ ♣r♥♣ r s t♥ó
sñ ♣r♦①♠♠♥t ♦r③♦♥t ② ró♥ s♦ ② rs r
♥♠♥t s ♥③r♦♥ ♦s t♦s ♦t♥♦s ♣r í♥ s♦♥♦
r ♥ r s ♠str ♥ ♣r ♦st 0,25 ♠ ♦♠♦ ♥
r í♥ s♦♥♦ rr♠ ♣r ♥ ú♥♦
♠s♦r ♦st 0,25 ♠ rr♠ ♥ q s ♠♦r
sñ ♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ♦st 0,45 ♠
−0,6 ≤ dt ≤ 0,6 ♥s rr♠ ♥ q s ♠♦r
sñ rt♦r ♣r♦①♠♠♥t ♦r③♦♥t ♦③♦ ♥
t ≈ 8 ♥s 0,8 ♠ < x < 2 ♠ ② sñ ró♥ s♦ ♦♥
ért ♥ (x; t) = (0,8 ♠; 8 ♥s) N = 4 d = 0,1 ♠ ♦st 0,45 ♠
−0,5 ≤ dt ≤ 0,2 ♥s
s♦ ♥tr♦r sñ r①ó♥ s s ♣ ♦srr ♦♥ ért
♥ (y, t) = (0 ♠; 10 ♥s) ♦♦r♥ y ♥ rá♦ s rr ♣♥t♦
♠♦ ♥tr s ♣♦s♦♥s ♠s♦r ② r♣t♦r ♥ st s♦ t♠é♥ s
♣♦s ♦srr sñ ♣r♦①♠♠♥t ♦r③♦♥t ♥ strt♦ r♦r
8 ♥s ♥tr y = 0,2 ♠ y = 2 ♠ ♣r♦①♠♠♥t sñ s
♦♣ó ♥ st s♦ ♥ r♥♦ ♣♦s♦♥s ♠♥♦r q ♥ ♣r ♣r
♣♦s♠♥t ♦♠♦ ♦♥s♥ ♦r♠ s ② stró♥
s♠♥t♦s ♥ s ♥tr♦r
♦ s ♣ó ♠ét♦♦ ♣r ♠♦rr sñ ♣r♠r s
♥ r s ♠str rrr♠ rst♥t ♣r ♥ rr♦ ♦♥
N = 4 ② d = 0,1 ♠ s s♦♥r♦♥ ♦s s♣③♠♥t♦s t♠♣♦rs s
−0,6 ♥s st 0,6 ♥s ♦♥ ♥r♠♥t♦s 0,05 ♥s ♠♥r ♠♦rr t♦s
s ♣rts r♥ts sñ ♣ ♦srr q ♥ ③♦♥ ♥tr
−0,75 ♠ ≤ y ≤ 0,75 ♠ ♥t♥s sñ ♠♥tó ♦♥ rs♣t♦ s
sñs r♥♥t ② ♠ás q sñ s ③♦ s ♥ t♦♦ r♥♦ y
♥q é♠♥t ♥ ♥♦s tr♠♦s
♠é♥ s ♣ó ♠ét♦♦ ♣r ♠♦rr r①ó♥ ♣♥♦ ♣r♦
①♠♠♥t ♦r③♦♥t ② ♣ér♦ s♦ ♥ r s ♠str
rrr♠ rst♥t ♣r ♥ rr♦ ♦♥ N = 4 ② d = 0,1 ♠ ♥ q
s ♣ó ♠t♦♦♦í ♣r ♠♦rr ♦♥t♥ s sñs r♥♦
s♣③♠♥t♦ t♠♣♦r s −0,5 ♥s st 0,2 ♥s ♦♥ ♥r♠♥t♦s 0,05
♥s ♦♠♦ ♥ s♦ s ♣ ♦srr ♦♥ ♠②♦r r ró♥
♥ ♦r strt♦ ♠♥trs q r①ó♥ ♥ s t♥ s♣
rr ♠ás s ♠♦ró ♦♥t♥ strt♦ ♥ t♦ s ①t♥só♥
r í♥ s♦♥♦ rr♠ ♣r ♥ ú♥♦
♠s♦r ♦st 0,25 ♠ rr♠ ♥ q s ♠♦r
sñ ♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ♦st 0,45 ♠
−0,6 ≤ dt ≤ 0,6 ♥s rr♠ ♥ q s ♠♦r
sñ rt♦r ♣r♦①♠♠♥t ♦r③♦♥t ♦③♦ ♥
t ≈ 8 ♥s 0,2 ♠ < x < 2 ♠ ② sñ ró♥ s♦ ♦♥
ért ♥ (x; t) = (0,2 ♠, 8 ♥s) N = 4 d = 0,1 ♠ ♦st 0,45 ♠
−0,5 ≤ dt ≤ 0,2 ♥s
♣ít♦
ó♥ ♠ét♦♦
♥ ♦s ♣ít♦s ♥tr♦rs s ♠♦stró q ♠ét♦♦ ♣r♠t ♠♦rr
s sñs t♥t♦ ♥♦s ♦③♦s ♦♠♦ rt♦rs ①t♥s♦s ♥ t♦♦s
♦s s♦s ♣r r s ♠♦rs s ♦♠♣rr♦♥ tt♠♥t s sñs
♦t♥s ♣rtr ♦s ♠ét♦♦s ②
♥ st ♣ít♦ s ♦♠♣r♥ ♥ttt♠♥t s rs♣sts ♦t♥s
♦♥ ♦s ♠ét♦♦s ② P st út♠♦ ♠ét♦♦ ♦rtr ♠út
♣ ♠ás t③♦ s ♠♦rs ♦t♥s s ú♥ ♥t♥♦ ♥s
s rtrísts sñ q ♥②♥ ♥ s s③ó♥ ró♥
♥t♥s rs♣t♦ s sñs r♥♥ts ② s t♦♥s ♥ t♠♣♦
♥t♦ ② ♥ s ♠♣t s s s ♥♥ ♦♥ ♦♥t♥
sñ ❬❪
♥ó♥ ♥♦rs
Pr r ♥ttt♠♥t ♦s rst♦s ♦s ♠ét♦♦s
② P s ♣r♦♣♦♥♥ rts ♥ts ♦♠♦ ♥♦rs
♠♥ ♦t♥ ♦♠♦ rs♣st ♥③♥ s t♦♥s ♥ t♠♣♦ ② ♥
♠♣t sñ ② ró♥ ♥tr s ♥t♥ss sñ ② ♠♦
r♥♥t
Pr ♠r s t♦♥s t♠♣♦ ♥ q ♦rr ♥ r①ó♥
s tr♠♥ t♠♣♦ ♠á①♠♦ ♠♣t ♣r sñ ♥ tr③
tn s ♦ ♦♠♣r ♦♥ t♠♣♦ s tr③s ②♥ts tn−1 ② tn+1 ② s
♥♦r♠③ rst♦
NTD =|tn − (tn−1+tn+1)
2|
tn
st ♠♥t s ♥♦♠♥ r♥ t♠♣♦s ♥♦r♠③ ♥♦r♠③
t♠ r♥
♦♠♦ ♠♣♦ s ♣r s♦ s♠♦ r①ó♥ ♥
♥ ♥trs ♣♥ ♦r③♦♥t ♥ r s ♠str ♠♦♦
♣ s♣r♦r s rtr③ ♣♦r ǫr = 5 ② σ = 1,5 ♠♠ ♠♥trs q ♠ás
♣r♦♥ ♣♦r ǫr = 6 ② σ = 2 ♠♠ rt♦r stá ♦③♦ ♥ ♣r♦♥
1 ♠ r♥ t③ s ③ ♥ s rs ② s
♠str♥ rrr♠s ♣r ♦st ♦ 0,3 ♠ ♥♦ s ♣♥ t♦♥s
t♦rs ② rs♣t♠♥t ♥ t♦ r ♦s ♣rá♠tr♦s
♦♥sttt♦s
♥ r s ♠str♥ ♦s ♦rs ♦s ♣r s sñs
s rs ② ♥ ♥ó♥ x ♥ s rs s ♣ó ♥ ♣r♦♠
♦ s♦r ♦rs ♦♥st♦s ♣r s③rs ♦♠♦ r s♣rr ♦s
♦rs ♦t♥♦s ♣r s♦♥ ♠♥♦rs ♣r ♠♦♦ ♦♥ t
ó♥ s r ♥♦ sñ s ♣ r ♦♥ ♠②♦r r ♠♥trs q
s♠♥r sñ ♦s ♦rs ♠♥t♥ ♠é♥ s
♣ ♦srr ♥ ♠♥t♦ ♥ tó♥ ♦s ♦rs st s
♥ rtríst s♥♦ ♦r♥ q t♠é♥ s s♦ ♦♥ s♠♦r
♥ s③ó♥ s sñs
Pr ♠r s t♦♥s ♠♣t ♦rrs♣♦♥♥ts r①ó♥
♥trés s t♦♠ tr③ ② ss ♣r♠rs ♥s s s♣③ sts út♠s
♥ t♠♣♦ ♠♥r q ♦s ♠á①♠♦s ♠♣t q♥ ♥♦s ② s
s ♦rr♦♥ ♥ ♥ ♥tr♦ t♠♣♦r ♥tr♦ ♥ ♦ ♠á①♠♦ ② ♦♥
①t♥só♥ ♥ ♣rí♦♦ t♠♣♦r Pr rr rst♦ st ♣s♦
rt♠♥t s ♦ ♠ ♦rró♥ ♥ r s ♠str♥ rs
♦rró♥ ♣r ♦s ♠♦♦s r ♣ r q ♦rs ♠②♦rs
② ♠♥♦s t♥ts ♦rró♥ s s♦♥ ♥ ♠♦r s③ó♥ s
sñs
♥♠♥t s ♠ ró♥ ♥t♥s ♥tr sñ ♣r♥♣ ② s
r♥♥ts ♠♥t♦srr♦♥♥ s♥ ♥t♥st② rt♦ ♣r ♦ s
r③ ♥ ♦♥t ♥tr s ♥t♥ss ♣r♦♠♦ s sñs ♣r♥♣ ②
r ♦♦ ♥ ♥trs ♣♥ ♦r③♦♥t 1 ♠
♣r♦♥ ♣ s♣r♦r ǫr = 5 ② σ = 1,5 ♠♠ ♣
♣r♦♥ ǫr = 6 ② σ = 2 ♠♠ rr♠s ♦t♥♦s ♣r
♥ ♦♥ró♥ ♦♥ ♦st 0,3 ♠ r♥ ③ ♣r
♦s s♦s ♥ ♦s q s ♣♥ t♦♥s ②
♠tr③ ♦s ♣rá♠tr♦s ♦♥sttt♦s
r ❱♦rs ♦rró♥ ②
♦s ♣r ♥ ♠♦♦ ♦♥ ♥ ♥trs ♣♥ ♦r③♦♥t ♦♥
tó♥ ② ♥ ♠tr③ ♦s ♣rá♠tr♦s ♦♥st
tt♦s ② ♦st ♦ 0,3 ♠
r r③ ♥ q s ♠str♥ ♦s ♥tr♦s ♥ ♦s q
s ♥ s ♥t♥ss sñ ♣r♥♣ ② s sñs
r♥♥ts
r♥♥ts st♦s ♣r♦♠♦s s ♥ ♥ trs ♥tr♦s t♠♣♦ st♥t♦s
r ♣rí♦♦s ①t♥só♥ ♥♦s ♦ r♦ tr③
♣r♠r ♥tr♦ stá ♥tr♦ ♥ ♠á①♠♦ ♥t♦ ♣r♥♣ ♣r r
s ♥t♥s ♠ ♠♥trs q ♦s ♦tr♦s s s♣♦♥♥ s♠étr♠♥t r
♦r ♥t♦ ♣r♥♣ ♣r r ♦r ♠♦ ♥t♥s ♦s
♥t♦s r♥♥ts ♥ r s ♠str rst♦ ♦t♥♦ ♣r
s sñs r①ó♥ ♦s ♠♦♦s ♥tr♦rs ♣ r q ♦rs
♠②♦rs s s♦♥ ♥ ♠♦r s③ó♥ s sñs
♦♠♣ró♥ ♥tr ♦s ♠ét♦♦s ②
P
♥ st só♥ s ♣rs♥t ♥ ♦♠♣ró♥ ♦s rst♦s ♦t♥♦s
♦♥ ♦s ♠ét♦♦s ② P st♦ s r③ ♣rtr s ♦s
st♦♥s áss q s ♥③r♦♥ st ♠♦♠♥t♦ ró♥ ♥ ♥
♦t♦ ♣qñ♦ ② r①ó♥ ♥ ♥ ♥trs ①t♥s Pr ♠s st♦♥s
s st♥ t♥t♦ t♦s s♠♦s ♦♠♦ ①♣r♠♥ts
♥ ♠ét♦♦ ♥ ③ q s st♥ N ② d s♣③♠♥t♦
t♠♣♦r dt ♥tr s ♦♥s ♠ts s s♦♥ ♥ ♣♦ró♥ s♣r
♠♥r rr ③ tr♥s♠t♦ ♦ r♦ ♠♥♦ ♦
rr♦ rt♦r r♣t♦r s ♣♦r ♦ ♥♠♥t r ♦s t♦s q
r t③ ♠s♠♦ ♦♥♥t♦ t♦s ♣r ♦s ♠é
t♦♦s ② P
s ♣r♦♥ ♥ ♦s rst♦s ♠ét♦♦ r dt
♥ st ♣ít♦ s ♥③ t♠é♥ ♥♥ r♥ts ♥s
tó♥ ♥ ♠tr③ s♦ sts t♦♥s tú♥ ♦♠♦ ♦tr♦s ♦
t♦s ♥ s♦ ♦s q ♣r♦♥ r①♦♥s s♥rs q ♥trr♥ ♦♥
s r①♦♥s ♥trés ♣♦r ♦ s ♠♣♦rt♥t ♥③r á s ♥
♠ét♦♦ ♣r ♠♦rr ♠♥r t ♦s ♥s ♦♥t♥ ②
♠♣t sñ
♦♠♦ s ♦ ♠ás rr ♥ st ♣ít♦ s ♦♠♣r♥ ♦s rst♦s ♦
t♥♦s ♦♥ ♦s ♠ét♦♦s ② P ♥♦tr q st♦s ♠ét♦♦s s
♣♥ ♣r ♠s♠♦ ♦♥♥t♦ t♦s P♦r ♠♣♦ ♥ r s
q ♣rtr ♦s t♦s qr♦s ♣r ♠ét♦♦ s ♣♦s s♦♥r
qs tr③s q ♦rrs♣♦♥♥ ♥ ♠s♠♦ ♣♥t♦ ♠♦ ♦♠ú♥ ♦♥ s♣
r♦♥s r♥ts ♥tr ♠s♦r ② r♣t♦r st ♠♥r ♥ ♦ q s s
♣♥ ♦s ♠ét♦♦s ② P ♠s♠♦ ♦♥♥t♦ t♦s
t♦s s♠♦s
t♦ ♣qñ♦
♥ r s ♠str ♣r♠r ♠♦♦ ♥③♦ ❯♥ ♦t♦ ♦♥ á
♠tr♦ 0,05 ♠ s ♦③ ♥ ♣r♦♥ 0,80 ♠ ♦t♦ s rtr③
♣♦r ♥ ♣r♠t rt ǫr = 5,25 ② ♥ ♦♥t σ = 2,5 ♠♠ ②
♠♦ r♥♥t ♣♦r ǫr = 3,5 ② σ = 1 ♠♠ ♠♦ s♦r ♦s s r
♥tr♦ r s 0,01 ♠ ♥ ♠s r♦♥s ♥ ♦s ♣rá♠tr♦s
s♦ s ♣♥ t♦♥s t♦rs ♦♥ ♥ ♠♣t ♠á①♠
♦♦s ♦s ♣♦♦s s ♦③♥ ♥ ♥trs r s♦ ♥tr♦s ♥ x
ró♥ s♦♥♦ ♦♥ ♦s s ♦s ♣♦♦s ♣r♣♥rs ♠s♠♦
r♥ ♥tr s ♦♥s ♠ts s fc = 500 ③ ♦♥t ♦♥
λ = 0,32 ♠ ♥tr♦ s♦
♥ r s ♠str ♥ ♣r ♦t♥♦ ♣r ♠♦♦ r
♦♥ ♦st ♦ 0,3 ♠ ♦♦r♥ x s rr ♣♥t♦ ♠♦ ♥tr
s ♣♦s♦♥s ♠s♦r ② r♣t♦r ♥ r sñ r①ó♥
rt♦r s ♣ ♦srr ♦♥ t ♦♠♦ ♥ ♣ér♦ ♦♥ ért ♥
t = 11 ♥s ♣r♦①♠♠♥t ♦ q ♥t♥s sñ ♣r♠r
r ♦♦ ♥ ♦t♦ ♣qñ♦ ♦③♦ ♥
♣r♦♥ 0,80 ♠ ♠♦ r♥♥t s rtr③ ♣♦r
ǫr = 3,5 ② σ = 1 ♠♠ ♦t♦ s rtr③ ♣♦r ǫr = 5,25 ②
σ = 2,5 ♠♠
② ♠♦ r♥♥t s♦♥ ♦♠♣rs ② ♥trr♥
r ♠str ♥ rst♦ ♣r ♠ét♦♦ ♦t♥♦ ♣r
♠♦♦ r s st♥s rts ♥tr ♦s ♦♠♣♦♥♥ts
rr♦ s♦♥ d = 0,1 ♠ ♠♥trs q N = 4 ♦♦r♥ x s rr
♣♥t♦ ♠♦ ♥tr ♥tr♦ rr♦ ② ♣♦só♥ r♣t♦r s t③ó ♥
♦st q♥t 0,45 ♠ ♥♦ ♥tr sts ♣♦s♦♥s s♦♥r♦♥
s♣③♠♥t♦s t♠♣♦rs −0,6 ♥s ≤ dt ≤ 0,6 ♥s ♦s q ♦rrs♣♦♥♥
á♥♦s tr♥s♠só♥ ♥tr 50♦ ② −50♦ ♣r ♠♦rr sñ rt♦r ♥
♥tr♦s ♦♥st♦s x t③r♦♥ ♥ t♦t ♥tr♦s ♣r ♦t♥r
rst♦ r
♦♠♣rr s rs ② s ♣ ♦srr q sñ q
rst ♠ét♦♦ s ♠ás s q sñ ♦r♥ ♥ st ♣♥t♦
♥♦ stá r♦ s ♠♦r stá r♦♥ ♣r♥♣♠♥t ♦♥ ♠♥♦rs
t♦♥s sñ ♥ s♣♦ ② ♥ t♠♣♦ ♦ ♦♥ ♥ ♥r♠♥t♦ ♥
ró♥ ♥tr ♥t♥s sñ ♣r♠r ② s sñs r♥♥ts ♦♥
rs♣t♦ st♦ út♠♦ ♥♦ só♦ s rr③♥ s sñs rt♦r ♣r♥♣
♥ r s♥♦ t♠é♥ ♣rts sñs rt♦rs ♠♦ r♥
♥t s t♦♥s ♥ ♠tr③ s♦ s q s ♣♥ ♦srr ♥
r ♦♠♦ r♥s ♥♥ó♥ ♦♥st♥t ♣rs sñ ♣r♠r
♥tr♦ t♦♦s ♦s ♥tr♦s x ♠ás ♣r♥ sñs ♣r♣♥rs
sñ ♣r♥♣ ♥ ♦s ♥tr♦s ♠ás ①tr♠♦s x = (−2;−1) ♠ ② x = (1; 2)
♠ s s s♦♥ ♦♥s♥ ♣ró♥ ♥rí q ♥♦ s tr♥s♠t
♥ ró♥ s♣r ♣r |dt| > 0,4 ♥s ♦ q t♠é♥ ♣ tr
r Pr ♦st 0,3 ♠ Pr N = 4
d = 0,1 ♠ ♦st 0,45 ♠ −0,6 ≤ dt ≤ 0,6 ♥s Pr P
n = 4
♦♥t♥ ♥t♥s ♦s rst♦s
♥ r s ♠str rst♦ ♠ét♦♦ P ♣r ♠♦♦
r ♥ú♠r♦ ♣♠♥t♦ ♥ r s n = 4 ♦♠♦ ♦
♣♠♥t♦ s ♦♥sr ♦ ♣r♦♣ó♥ ♥ ♠♦ v = 0,16
♠♥s P♦r ♦tr♦ ♦ ♦♠♣r♥♦ s rs ② s q sñ
s ♠ás r ♣r ♠ét♦♦s q ♣r ♠ét♦♦ P st rst♦
s t♥ó♥ s sñs ♣r♦s ♣♦r rt♦rs s♥r♦s
r ③ tr♥s♠t♦ ♥ ♠ét♦♦
Pr r ♥ttt♠♥t ♦s rst♦s ♦s ♠ét♦♦s ②
P s ♥③r♦♥ ♦s ♥♦rs ♥♦s ♥ só♥ s t♦♥s
t♠♣♦ ② ♠♣t ♦ r♦ r①ó♥ ♣r♥♣ ② s ♥t♥s
♦♥ rs♣t♦ ♥t♥s ♦♥♦
♥ r s ♠str r♥ t♠♣♦s ♥♦r♠③ ♣r s
sñs ró♥ ♥ s rs ♦♠♦ ♥ó♥ x s rs
♥ r s♦♥ rst♦ ♣r♦♠r ♦s ♦rs tr③s ♦♥sts
♥ r s ♣ ♦srr q ♠ét♦♦ r ♦ r♦
t♦♦ r♥♦ x ♠ás r ♣rs♥t ♠♥♦s r♦♥s q s
♦trs ② s ①t♥ ♥ ♥ r♥♦ x ♠②♦r ♦♠♦ s ♠♥♦♥ó sts r
trísts stá♥ r♦♥s t♠é♥ ♦♥ ♥ ♠♦r s③ó♥ sñ
ró♥ ♦♠♥t ♦r ♣r♦♠♦ ♦ ♦ r♦
t♦♦ r♥♦ x s 0,60 10−2 1,61 10−2 ② 1,64 10−2 ♣r ♦s ♠ét♦♦s
② P rs♣t♠♥t ♦ q ♥ ♥ ♠♦r s♥t ♦r
♦r♥ ♦♠♦ ♦♥s♥ ♣ó♥ ♠ét♦♦ ♦tr
q ♥ st s♦ rt♦r ♣qñ♦ ♠ét♦♦ P ♥♦ ♦ ♥ rst♦
t♥ stst♦r♦ st♦ s s♣rr ♦ q ♠s♠♦ s♦ ♦r♥♠♥t
sñ♦ ♣r st♦ strt♦s ♣r♦♥♦s ♦♥ ♣qñ ♥♥ó♥ ❬❪
s rs q rstr♦♥ r ♦rró♥ ♣r sñ r
ó♥ r s ♠str♥ ♥ r Pr s③r s rs
s r③ó ♥ ♣r♦♠♦ ♦rs ♦♥st♦s ♦ r♦ s ♠s♠s
♣ ♦srr q ♠ét♦♦ ♥r♠♥tó ♦rró♥ ♥ ♣r♦①
♠♠♥t r♥♦ x ♥♦ ♥ r ♠♥♦s t♥t P♦r
♦♥trr♦ ♥ ♥tr♦ x = (0,25; 0,45) ♠ r stá ♣♦r ♦
r ♣r st s ♦♠♣♦rt♠♥t♦ ♥ó♠♦ ♦rr ♦s♦♥♠♥t
♥ r♦♥s ♥ s q ♥ sñ s♥r ♥t♥s ♠♣♦rt♥t r③
r ♦rró♥ ② ♣r sñ
ró♥
r Pr♦♠♦ ♦rró♥ ②
♥ ♥ó♥ s♣③♠♥t♦ t♠♣♦r dt ♥ ♥tr♦ x =
(−1,5; − 1,1) ♠ ①tr♠♦ ③qr♦ sñ ró♥
Pr♦♠♦ ♦rró♥ ② ♥ ♥ó♥
s♣③♠♥t♦ t♠♣♦r dt ♥ ♥tr♦ x = (−0,2; 0,2) ♠ ♥
♥tr♦ sñ
t P P
10−2 10−2 10−2
10−2 10−2 10−2
♦rró♥
♦rró♥ ② ♣r♦♠♦s ♦ r♦
t♦♦ r♥♦ x ♣r ♠♣♦ s♠♦ rt♦r
♥♥ ♦s ♥s tó♥ ♥ ♦s ♣rá♠tr♦s s♦
s út♠s ♦s ♦♠♥s s rr♥ ♦rs rt♦s sts
♥ts
sñ st ♦ ♦r♠ó♥ st út♠ ♦♥ rs♣t♦
♠ét♦♦ P s ♦sr q ♦rró♥ sñ ♦r♥ ♠♦ró ♥ s
t♥t♦s ♥tr♦s x ♣r♦ ♣♦r ♦ ♦s rst♦s ♠ét♦♦
♦rró♥ ♦ ♣r ♦s trs ♠ét♦♦s ♦ r♦ t♦♦ r♥♦
x s ♥ ♥ t
♥ r s ♠str♥ ♦♥ts ♥tr s ♥t♥ss ♣r♦♠♦
sñ ♣r♥♣ ② r♥♥ts ♣r ♦s ♠ét♦♦s ② P
♦♠♣rr s rs r s ♣ ♦srr q ♠ét♦♦
♣r♦♦ ♥ ♦r ♠②♦r ♥ ♠②♦r ♣rt r
st ♣r♠r ♠♣♦ ♠♦stró q ♠♦r ♥ sñ ♣r♦ ♣♦r
♠ét♦♦ s r♦♥ t♥t♦ ♦♥ r♠♥t♦ s t♦♥s ♦♠♦
♦♥ ♥r♠♥t♦ ♥ ró♥ ♥tr s ♥t♥ss sñ ♣r♠r ② s
r♥♥ts s ♠♥ts ♠♦r ♥ s rs ♦rró♥
② rí♥ ♦♥ ♣♦só♥ ♠é♥ s ♣s♦ rt♦ ♦♥ ♥ ♦rró♥
♥♦ ♣r ♥tr s ♣♦s♦♥s ♦s ♠í♥♠♦s ② ♠á①♠♦s ss rs ♥
♥r ♥♦ s t♦♥s ♥ ♠tr③ s♦ s♠♥②♥ s rs
s t♦r♥♥ ♠ás ss ② s ♣r♦①♠♥ ♦r ♦♥st♥t r♦ ♠♥trs
q ♠♦r rt ♥r ♣r♦ ♣♦r ♠ét♦♦ s♠♥② P♦r
♠♣♦ ♣r ♥ tó♥ ♥ ♦s ♣rá♠tr♦s s♦ ♦r
♥r ♣r♦♠♦ ♦ r♦ t♦♦ r♥♦ x ♣r ♦ ♣♦r
♦r ♥r ♣r s 0,37 stá ♦ ♦♥t rs♣t♦ ♣r
s♦ 0,46 r t ♠r♠♥t ♦r ♥r rt♦
♣r ♠ét♦♦ P ♣s 1,02 1,04 ♠♥t♥é♥♦s ♣rát♠♥t
♦♥st♥t P♦r ♦tr♦ ♦ s rs ♦rró♥ s s③♥ ② s ♣r♦①♠♥
♦r ♦♥st♥t 1 ♦♥ ♥ ♠♦r rt sñ ♣r♦ ♥♦ s
t♦♥s s♦ s r♥ Pr ♥ tó♥ ♦rró♥
♥r rt ♣r ♦s ♠ét♦♦s ② P s♦♥ 1,26 ② 1,07 ♣♦r ♥♠
♦s ♦rs rs♣t♦s ♣r s♦ 1,09 ② 0,98 P♦r ♦♥trr♦ ♦s
♦rs s♦t♦s ② rt♦s s ♥r♠♥t♥ ♥♦ ♥
tó♥ s♠♥② ♠♥trs q s rs ♣r ♦s trs ♠ét♦♦s s ♥
♠♥♦s t♥ts ② ♥ s♣r♣♦♥rs ♥tr sí Pr ♥ tó♥
♦s ♣rá♠tr♦s s♦ ♦r ♥r rt♦
♣r ♦s ♠ét♦♦s ② P s♦♥ 1,14 ② 0,96 ♦s q stá♥ ♦ ♦s
♦rs ♣r 1,26 ② 1,09 rs♣t♠♥t
Pr ♥③r ♦s t♦s só♥ ró♥ ③ tr♥s♠t♦
s♦r ♦s rst♦s ♠ét♦♦ s ♥ ♥ ♥tr♦ x ② s ♣r♦♠♥
s ♠♥ts ♦rró♥ ② ♥tr♦ é ♣r dt s♣
③♠♥t♦ t♠♣♦r r ♠str ♣r♦♠♦ ♣r
② P ♥ st s♦ ♥tr♦ ♥ q s ♣r♦♠ x = (−1,5;−1,1) ♠
s ♦③ ♥ ①tr♠♦ ③qr♦ sñ ró♥ ♣ ♦srr
q s rs ♣r ♣rs♥t♥ ♥ ♣♦ ♦♥ ♠í♥♠♦ ♥ dt = −0,5 ♥s ♦s
♦rs dt ♣♦ ♦rrs♣♦♥♥ á♥♦ tr♥s♠só♥ q ♠♦r r
♥rí ♦ r♦ ♠♥♦ ♠s♦r rt♦r r♣t♦r s rs ♦
rró♥ ② rs ② t♠é♥ ♣rs♥t♥ ♣♦s ♦♥ ♠á①♠♦s
♥ dt = −0,5 ♥s ♦tr q sts rs s ♣♥ sr ♣r s♦♥r ♥
s♣③♠♥t♦ t♠♣♦r q ♦♣t♠ ♦s rst♦s ② q st só♥
♥♦ s rít ♦ ♥♦ ♦♥sr ♦s ♣♦s P♦r ♠♣♦ ♠ét♦
♦ ♠♦r s♠tá♥♠♥t ♦s ♦rs ♦r♥s ♦rró♥
② s dt = −0,6 ♥s dt = −0,3 ♥s ♦ q ♦♥stt② ♥ ♥tr♦
st♥t ♠♣♦
s rs ② s♦♥ ♥á♦s s rs ②
rs♣t♠♥t ♣r♦ ♣r ♥ ♥tr♦ ♣r♦♠♦ ♥tr♦ ♥ ♦r♥
x = (−0,2; 0,2) ♠ s rs ♣r ♣rs♥t♥ rtrísts s♠rs
s ♥tr♦rs ①♣t♦ ♣♦r ♣♦só♥ ss ①tr♠♦s ♦s q stá♥ ♦③♦s
♥ dt = −0,1 ♥s ♥ st s♦ ♠ét♦♦ t♠é♥ ♠♦r s♠tá♥♠♥t
s ♠♥ts ♦s trs ♥♦rs ♥ ♥ ♥tr♦ ①t♥s♦ s dt = 0,0
♥s dt = 0,2 ♥s
t♦r ①t♥s♦
♥ r s ♠str s♥♦ ♠♦♦ ♥③♦ st ♦rrs♣♦♥
♥ rt♦r ♣♥♦ ♦r③♦♥t ♦③♦ ♥ ♣r♦♥ 0,80♠ strt♦
s♣r s rtr③ ♣♦r ♥ ♣r♠t ǫr = 3,5 ② ♥ ♦♥t
σ = 1 ♠♠ ♠♥trs q strt♦ ♣r♦♥♦ ♣♦r ǫr = 4 ② σ = 2 ♠♠
♥♠♥t ♥ tó♥ ♥ ♠tr③ s♦ s ♦s
♦tr♦s ♣rá♠tr♦s ♦♠♦ ♦♥ró♥ ♦s ♣♦♦s ② s rtrísts
sñ tr♥s♠t s♦♥ s ♠s♠s q ♥ ♠♦♦ ♥tr♦r
r ♦♦ ♥ rt♦r ♣♥♦ ♦r③♦♥t ♦③♦
♥ ♣r♦♥ 0,80 ♠ strt♦ s♣r s rtr③
♣♦r ǫr = 3,5 ② σ = 1 ♠♠ strt♦ ♣r♦♥♦ ♣♦r ǫr = 4 ②
σ = 2 ♠♠
♥ r s ♠str ♥ ♣r ♦st 0,45 ♠ ♦t♥♦ ♣r
♠♦♦ r sñ r①ó♥ ♣r ♥trs s ♣
♦srr r♠♥t t = 11 ♥s ♣r♦①♠♠♥t ♥ r s
♠str rst♦ rs♣t♦ ♣r ♦♠♦ ♣r ♠♦♦ ♥tr♦r ♦s
♣rá♠tr♦s rr♦ s♦♥ N = 4 ② d = 0,1 ♠ ♥ st s♦ só♦ s ♥sr♦
♥ s♣③♠♥t♦ t♠♣♦r ♦♥st♥t dt = 0,1 ♥s á♥♦ tr♥s♠só♥
−10♦ ♥ t♦♦ r♥♦ x ♦♠♣rr s rs ② s ♣
r q ♠ét♦♦ ♠♦r sñ q s ♦t♥ ♦♥ ♥ r
s ♠str rst♦ ♠ét♦♦ P ♣r ♠♦♦ r
♥ú♠r♦ ♣♠♥t♦ s n = 4 ♦♠♦ ♦ ♣♠♥t♦ s
♦♥sr ♦ ♣r♦♣ó♥ ♥ strt♦ s♣r v = 0,16 ♠♥s
♠é♥ rst♦ P ♠♦r sñ ♦r♥ ♣r♦ ♥ ♠♥♦r ♠
r Pr ♦st 0,45 ♠ Pr N = 4
d = 0,1 ♠ ♦st 0,45 ♠ dt = 0,1 ♥s Pr P n = 4
t P P
10−2 10−2 10−2
10−2 10−2 10−2
♦rró♥
♦rró♥ ② ♣r♦♠♦s ♦ r♦
t♦♦ r♥♦ x ♣r ♠♣♦ s♠♦ rt♦r
♥♥ ♦s ♥s tó♥ ♥ ♦s ♣rá♠tr♦s s♦
s út♠s ♦s ♦♠♥s s rr♥ ♦rs rt♦s sts
♥ts
q ♠ét♦♦
Pr ♦♥r♠r sts ♦sr♦♥s ♥ s rs s ♠str♥
s rs ♦rró♥ ② rs♣t♠♥t ♣r s sñs
② P s rs s ♦t♥♥ ♣♦r ♠♦ ♠s♠♦ ♣r♦♠♥t♦
t③♦ ♥ s♦ rt♦r ♣ ♦srr q ♦s ♠ét♦♦s ②
P ♠♦rr♦♥ rst♦ ♣r s trs ♠♥ts ♦♥ ♠♦rs rst♦s
♣r ♠ét♦♦ ♦s ♦rs ♥rs ♣r♦♠♦s ♥ t♦♦ r♥♦
x ♦rró♥ ② s♦♥ 0,29 10−2 1,25 10−2 ② 0,83 10−2 ♣r
♦s rst♦s ② P rs♣t♠♥t
♦♠♦ ♥ s♦ rt♦r ♠♦r ♥ s rs ② ♦rró♥
q s ♦t♥ ♦♥ ♦s ♠ét♦♦s ② P ♣r rt♦r ♣♥♦ ♦r③♦♥t
s ♥r♠♥t ♥♦ ♥ tó♥ ♥ ♠tr③ s♦ ♠♥t
r t ♣r ♦s ♦rs rt♦s sts ♠♥ts P♦r ♦♥tr
r♦ ♠♦r ♥ s ♥r♠♥t ♥♦ s t♦♥s ♥ s♦
s♠♥②♥
♥ s rs ② s ♠str♥ ♦s ♦rs ♣r♦♠♦s
♦rró♥ ② ♦♠♦ ♥ó♥ dt ♣r rt♦r ♣♥♦ ♦
r③♦♥t ♥ st s♦ s t③ r♥♦ ♦♠♣t♦ x ♥ ♣r♦♠♦
♣ ♦srr q s rs s♦♥ tt♠♥t s♠rs s rs
♦t♥s ♣r♠♥t ♣r rt♦r rs ♣rtr♠♥t
qs s ♣r♦♠♥♦ ♥ ♥ ♥tr♦ x ♥tr♦ ♥ ♣♦só♥
r ♦rró♥ ② ♣r ♥ r
t♦r ♣♥♦ ♦r③♦♥t
r Pr♦♠♦ ♦rró♥ ②
♦♠♦ ♥ó♥ dt
♦t♦ rs ② q ♣r sts ♣♦s♦♥s ♠s st♦♥s
s♦♥ ♣r♦①♠♠♥t s♠rs
t♦s ①♣r♠♥ts
♥ st só♥ s sr♥ ♦s ♠♣♦s q ♠str♥ qé ♠♥r
♠ét♦♦ ♠♦r t♦s ①♣r♠♥ts ♦♥sr♥ ♦s t♦s ♦t♥♦s
♥ st♦ ss ♥trrs ♥③♦s ♥ só♥ qr♦s
s♦r í♥ s♦♥♦ t③♥♦ ♦♥ró♥ ①♣r♠♥t sr♣t
♥ só♥ ♥ s rs s ♠str♥ ♦s rst♦s
♥ ♣r ♦♥ ♦st 0,25 ♠ r sñ ró♥ ♦t♦
♥trr♦ s ♣ r ♦♥ r ♦♥ ért ♥ t = 10 ♥s ❱rs ♣ér♦s
ró♥ ♦♥s ② r①♦♥s ♣r♦①♠♠♥t ♦r③♦♥ts ♣r♦
s ♥ s s♦♥t♥s ♥trs s♦ t♠é♥ ♣r♥ ♥ r
P♦r ♠♣♦ ♥ sñ ♣r♦①♠♠♥t ♦r③♦♥t ♥ strt♦ s ♣ ♦
srr r♦r t = 8 ♥s ♥tr x = 0,8 ♠ ② x = 2,4 ♠ ♣r♦①♠♠♥t
①tr♠♦ ③qr♦ st sñ ♦♥t♥ú ♥ ♥ sñ ró♥ ért
♥ x = 0,80 ♠ ♦r♥ ♥ ♣♥t♦ ♦♥ ①ó♥ ♦rtó strt♦
r ♠str rst♦ ♠ét♦♦ ♣r ró♥ ♥
♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ② ♦st 0,45 ♠ s s♦♥r♦♥ ♦s
s♣③♠♥t♦s t♠♣♦rs −0,6 ♥s ≤ dt ≤ 0,6 ♥s ♣r ♠♦rr sñ ♥
♥tr♦s ♦♥st♦s ♥ r s s♦♥ó ♥ r♣♦ r♥t
s♣③♠♥t♦s t♠♣♦rs−0,5 ♥s ≤ dt ≤ 0,2 ♥s ♦♥ ♣r♦♣óst♦ rstr
r①ó♥ ♣♥♦ ♣r♦①♠♠♥t ♦r③♦♥t ② ♣ér♦ s♦ P♦r
♦tr♦ ♦ r ♠str rst♦ ♣r P ♦♥ ♥ ♥ú♠r♦
♣♠♥t♦ n = 4 t③ó ♦ ♣♠♥t♦ ♦t♥ ♥ só♥
♣♥♦ ♦rró♥ sñ ♣♥♦ ♣r♦①♠♠♥t
♦r③♦♥t v = (0,17± 0,09) ♠♥s
♥♦ s ♦♠♣r♥ s rs s ♣ ♦srr q s
sñs s♦♥ ♠ás rs ♣r ♦s ♠ét♦♦s ② P q ♣r ♦♥ rs
t♦s ♠♦rs ♣r s sñs ♠ás ♦♠♦ s sró ♥ só♥
♠ét♦♦ ♠♦r sñ ♦t♦ ♥trr♦ ♦ strt♦
♦r♠ st ♥♦ s rst sñ ró♥ ♣r♥♣ r
s t♥ú sñ ♣r♦①♠♠♥t ♦r③♦♥t ② ró♥ s♦
② rs r
r Pr ♦st 0,25 ♠ Pr ♥ q
s ♠♦r sñ ♦t♦ ♥trr♦ N = 4 d = 0,1 ♠ ♦st
0,45 ♠ −0,6 ≤ dt ≤ 0,6 ♥s Pr ♥ q s ♠♦r
sñ rt♦r ♣r♦①♠♠♥t ♦r③♦♥t ♦③♦ ♥
t = 8 ♥s 0,8 ♠ < x < 2,4 ♠ ② sñ ró♥ s♦ ♦♥
ért ♥ (x, t) = (0,8 ♠; 8 ♥s) N = 4 d = 0,1 ♠ ♦st 0,45 ♠
−0,5 ≤ dt ≤ 0,2 ♥s Pr P n = 4
r ♦rró♥ ② ♣r r
ó♥ ♦t♦ ♥trr♦
r Pr♦♠♦ ♦rró♥ ②
♥ ♥ó♥ dt ♥ ♥tr♦ x = (−1,2;−0,8) ♠ ♥ ①tr♠♦
③qr♦ sñ Pr♦♠♦ ♦rró♥ ②
♥ ♥ó♥ dt ♥ ♥tr♦ x = (−0,2; 0,2) ♠ ♥
♥tr♦ sñ
r ♦rró♥ ② ♥ ♥ó♥
x ♣r rt♦r ♣r♦①♠♠♥t ♦r③♦♥t
r Pr♦♠♦ ♦rró♥ ②
♦♠♦ ♥ó♥ dt
P P
rt♦r 10−2 10−2 10−2
t♦r 10−2 10−2 10−2
♦rró♥rt♦r
t♦r
rt♦r
t♦r
♦rró♥ ② ♣r♦♠♦s ♦ r♦
t♦♦ r♥♦ x ♣r ♠♣♦ ①♣r♠♥t s út♠s
♦s ♦♠♥s s rr♥ ♦rs rt♦s sts ♥ts
Pr ♥③r s t♦♥s ② ♠♣ts rts s sñs s
r♦♥ s ♠s♠s ♠♥ts q s t③r♦♥ ♥ s s♦♥s ♥tr♦rs
s rs ② ♠str♥ s rs ♦rró♥ ②
rs♣t♠♥t ♣r ró♥ ♥ ♦t♦ ♥trr♦ ♣
♦srr q ♠ét♦♦ ♠♦ró rst♦ ♥ ♠②♦r ♣rt
r♥♦ x ② r♥♦ ♣r s ♠♥ts rs♣ts
♠ás ♠ét♦♦ P ♠♦ró s rs sñ ♦r♥ ♣r♦ ♠♥♦s
q ♠ét♦♦ ♦s ♦rs ♦s s ♠♦rs ♦rró♥
② ♦s ♦ r♦ r♥♦ ♦♠♣t♦ x s ♥♥ ♥ t
♦♠♣r♥♦ s rs ② s ♣ ♦srr
♠ás q ♦s ♠♣♦s ①♣r♠♥ts ② ♥♠ér♦s ♣rs♥t♥ ♦♠♣♦rt♠♥t♦s
s♠rs ♦ r♦ s rs
♥ s rs ② s ♠str♥ rs ♦s ♣r♦♠♦s
♦rró♥ ② ♦♠♦ ♥♦♥s dt ♣r ♥ ♥tr♦
♣r♦♠♦ x = (−1,2;−0,8) ♠ ♦③♦ ♥ ①tr♠♦ ③qr♦ s
sñs ró♥ s rs ② ♠str♥ rs s♠rs
♣r ♥tr♦ ♣r♦♠♦ x = (−0,2; 0,2) ♠ ♦③♦ ♥ ♥tr♦
s sñs ♦♠♦ ♥ ♦s ♠♣♦s s♠♦s s rs ♣rs♥t♥ ♣♦s
♣r q♦s ♦rs dt q ♠♦r r♥ ♥rí ♦ r♦ ♠♥♦
♠s♦r rt♦r r♣t♦r ♥tr♦ dt ♥ ♠ét♦♦ ♠♦r
rst♦ s ♠②♦r q 0,2 ♥s s♥t♠♥t ♥♦ ♣r r ♣♦s
♥ á só♥ st ♣rá♠tr♦
♦♠♦ út♠♦ ♠♣♦ s ♥③ r①ó♥ ♥ strt♦ ♣r♦①♠♠♥t
♦r③♦♥t q s rstó ♣♦r ♠♦ ♠ét♦♦ ♥ r s
rs ♠str♥ rs ♦rró♥ ② ♣r s
sñs ♦s ♠ét♦♦s ② P ♣ ♦srr q ♦s ♠ét♦♦s
② P ♠♦r♥ r ♥ ♠②♦r ♣rt r♥♦ x ♥
②♥♦ ♥tr♦ r♦♥♦ ró♥ s♦ x < 1 ♠ ♦
tó♥ ② ♥ ♠♣t sñ ♦r♥ ♦s ♥♦s
♠ét♦♦ s♦♥ ♠ás ♠♦r♦s ♥ st ♠♣♦ q ♥ ♦s ♥tr♦rs
♦♥ rst♦s s♠rs ♣r ② P ♥♠♥t ♥ s rs
s ♠str♥ rs ♣r♦♠♦ ♦rró♥ ② ♦♠♦
♥ó♥ dt ♣r ♥ ♥tr♦ ♣r♦♠♦ x = (1,0; 2,4) ♠ ♦③♦ ♥
♣rt ♣r♦①♠♠♥t ♦r③♦♥t sñ sts rs ♦♥r♠r♦♥ s
rtrísts ♦srs ♥ ♦s ♠♣♦s ♣r♦s
♣ít♦
♦♥s♦♥s
s♠♥ rst♦s ② ♦♥s♦♥s
♥ st ss s stó ♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦
rr ② s srr♦r♦♥ s té♥s ② rr♠♥ts áss ♣r r t
s ♠♣♠♥tó♥ ♠é♥ s r♦♥ tt ② ♥ttt♠♥t s
♠♦rs ♦t♥s ♥ s sñs ♠♥t ♣ó♥ ♠ét♦♦
♦♠♦ ♥tr♦ó♥ s ♣rs♥tr♦♥ ♦s ♥♠♥t♦s ♠ét♦♦ ♦rr
sí ♦♠♦ s té♥s ♦rtr s♠♣ ② ♠út♣ q s t③♥ t
♠♥t ♥ s ♣ó♥ ♦ s sró r♠♥t ♥ sr tr♦s
r③♦s ♥ st♦ rq♦ó♦ P♦ ♥♦ ♦s q ♥♦r♥ t
ó♥ ♣rs ♥trrs ② st♦ ♣♦s tó♥ ss
♥trrs ♥ ♠♦s s♦s s ♣rs♥t♥ sñs ♦♥ss q ♥♦ ♣♥ sr
♥tr♣rts ♠♥t ♣rtr ♦s ♣rs t♦s ♦rtr s♠♣
♥t ♣ó♥ ♠ét♦♦ ♠ró♥ ♥ s♦ ♣r ②
P ② ♠♦♦ ♥ s♦ s ss ♣♦s ♦rr ♥ ♥tr♣rt
ó♥ ♠②♦rí ss sñs ♦s tr♦s ♥ r♥ ♠ ♠♦tr♦♥
srr♦♦ sst♠át♦ ♠ét♦♦
Pr ♦♠♥③r ♦s st♦s s♦r ♠ét♦♦ s ♥③r♦♥ ♦s ♠♣♦s
tr♦♠♥ét♦s ♣r♦♦s ♣♦r rr♦s ♠s♦rs t♦rí rr♦s
♥ts t♣♦ ♣♦♦ ♥ í♦ ② ♥ ró♥ ♠♣♦ ♥♦ ♠str q
♦♠♥♥♦ rs ♥ts r♥s ♥tr sí s ♣♦s ♦♥tr♦r rts r
trísts ♣tró♥ ró♥ ♦♥♥t♦ ♥ ♣rtr r♥ ②
á♥♦ ♣r♦♣ó♥ ♦ q s ♣♦t♥♠♥t út ♣r ♦rr ♥r♠♥
t♦ ♥rí s♦r ♦s ♥♦s ♥trés ② ró♥ ♥♥
♥t♦r♥♦ s ♥ s s s s ♠ét♦♦ Pr tr ♦s st♦s
r③♦s s srr♦ó ó♦ ♦♠♣t♦♥ rr② st ó♦ ♣r♠t
r ♦s ♣tr♦♥s ró♥ qr stró♥ ♣♦♦s ② r♣r♦
r ♦s r♠s ♦rrs♣♦♥♥ts ♥t♥s rs ♠♥r st♠r s
♣♥♥ ♥r
♦ s ♣r♦♥ s ♠s♠s s ♣r ♦t♥r ♦s ♠♣♦s tr♦♠
♥ét♦s ♥r♦s ♣♦r ♥ts s ♥ ♥trs r s♦ ② s ♦♥
srr♦♥ ♦s ♠♣♦s q ♠t ♦♥♥t♦ ♥ ró♥ ♠♣♦ r♥♦
♦sró q t♠é♥ ♥ s s♦ s ♣♦s ♦♥tr♦r ♦♥♥tró♥ ②
♦r♥tó♥ ♠♣♦ ♠♥r s♠r ♦ q ♦rr ♥ í♦ ♣r♦ ♦♥
rts rtrísts ♣rtrs q s♦♥ ♦♥s♥ ♣rs♥
♥trs ♥tr ♦s ♠♦s ② r♥í ♥tr ♣♥t♦ ó♥
♠♣♦ ② ♥t ♥ st s♦ rr♦ ♥ ♣rtr s ♥③ó ó♠♦ s
♥ s♦♥r ♦s ♣rá♠tr♦s rr♦ ♣r ♦t♥r ♥ r♥t ♦♥s
q rst ♦ ♣r ♣r♦r ♥ ♠♦r t s sñs
①♠♥ó ♠ét♦♦ ♦♠♦ ♥ ♠♥r ♠♦rr s sñs
♦rr ①♣ó ó♠♦ ♥♦♥ ♠ét♦♦ rr♦s s♥tét♦s ♦♥ s
t♥t♦s t♣♦s ♥♦s ② qé ♠♥r ss rst♦s ♠♦r♥ s sñs
♦♥srr♦♥ ♦s st♦♥s ♥♠♥ts ♣rtr s s s ♣♦s
♥tr♣rtr ♦s s♦s ♠ás ♦♠♣♦s ♥ ♦♥t♥♥♦ ♥ ♦t♦ ♣qñ♦ ②
♦tr ♥②♥♦ ♥ ♥trs ①t♥s r♥ts ♣rts sñ r
♣r♦♥ sr rsts r♥♦ á♥♦ tr♥s♠só♥ t♥♥♦ s ③
♦trs ♣rts sñ ♥♦ s ♣rts ♥t♥ss ♦rrs♣♦♥r♦♥
qs ♣♦r♦♥s s ♥trss ♥trrs q rr♦♥ ♠ás ♥t♠♥
t ♥rí r♣t♦r s ♦trs ♣rts ♦rrs♣♦♥r♦♥ s♠♥t♦s
q ♥♦ r♦♥ ♠♥♦s ♦ q rr♦♥ ③ r ♣♦só♥ r
♣t♦r s s ♥r♠♥tó r♠♥t ♣r s ♣rts rsts
sñ ♥ ♥r ♣r♦♠♦ tr③s ♠♣ít♦ ♥ ♠ét♦♦ t♥ó
♥r r♦ t♦r♦ ② t r♥ s♣ ② r♦r③r r①ó♥
♣r♠r
♣♦ st ♠♥r ♠ét♦♦ ♠♦stró sr t♦ ♣r ♠♦rr
st♠♥t ♣rts sñ ♦s ♥♦s ♥q ♥♦ ♣r r③r ♥t♦s
♦♠♣t♦s ♥ ♦♥s♥ s ♠♣♠♥tó ♥ ♠t♦♦♦í q ♠♦ró
s ♦ r♦ t♦ sñ st ♦♥sst ♥ s♦♥r ♥ sr
s♣③♠♥t♦s t♠♣♦rs ♦ q♥t♠♥t á♥♦s tr♥s♠só♥
♣r ♦s s t♦s s ♣♦r♦♥s r♥ts sñ s♦♥ ♠♦rs ♦
s ♣♦r♦♥s ♠♦rs s ♦rt♥ ② s ♦♦♥ ♥ts ♥ rrr♠ ♥
st ♠t♦♦♦í s ♣ó ♥ ♠♦♦ q ♥② ♦s rt♦rs ♣qñ♦s
② ♥ ♥trs ①t♥s ♠♣♦ ♣r♠tó ♦srr ♣ ♠ét♦♦
♣r ♠♦rr ♠♥r st s sñs ♦t♦ ② trr s ♦trs
sñs ♥♠♥t s ♣ó ♥ r♥t st té♥ ♣r rstr
♠♥r s♠tá♥ ♠s sñs ró♥ ♥ rrr♠ rst♥t
♣♦rt ♠ét♦♦ ♣r str rt♦rs q ♥r♥ s♦♠♥t
sñs t♥s ♦ ♦ss s ♠♦stró ♣á♥♦♦ ♥ st♦ s♦s rs
♣r♠r♦ ♦s ♦♥sstó ♥ ♠ó♥ s♣s♦r ② ♣r♦♥
♣rs t♣ ♥ st♦ rq♦ó♦ P♦ ♥♦ ♣r s s s♦♥♦s
♣r♦s ♦♥ ♦rtr s♠♣ í♥ ♦ rst♦s ♥t♦s ♠ét♦♦
rr♦s ♣r♦ó sr ♥t ♣r s③r sñ s ♣r ②
s r♦♥s ♥ ss ♦rs st ♠♥r s♣s♦r ② ♣r♦♥
♣r ♣r♦♥ sr ♠♦s ♥ s♥♦ s♦ r♦♥♦ ♦♥ tó♥
ss ♥trrs ♠ét♦♦ ♠♦stró ♠♣♦rt♥ts ♥♦s ♥ ♥t♦
♣ trr s r ♠♦rr ♥ ♦r♠ st sñs ♦r♥s
♥ st♥t♦s ♦t♦s
srr♦ó ♣r♦r♠ ♦♠♣t♦♥ ♥t♦r♥♦ q t sñ♦
♦s rr♦s ♣r ♦♣t♠③r s rtrísts ♠♣♦ s♦r ♥♦
♦ s ①t♥ó ♣r♦r♠ ♠♥r q ♦♥stt② ♥ rr♠♥t
♣r s st♥ts t♣s ♠♣♠♥tó♥ ♠ét♦♦ st ♣r♦r♠
♣r♠t s♥tt③r ♠♣♦ ♥ rr♦ ♥ ♥ ♠♦ ♥♦r♠ ♦t♥r
rs♣st rr♦ ♣r t♦s ①♣r♠♥ts ♦ s♠♦s ② ♣r st
rs♣st ♠t♦♦♦í q ♣r♠t ♠♦rr ♦♥t♥ tr s
sñs ♥♠♥t s srr♦ó ♥ ♥t♦r♥♦ rá♦ ♠ ♦♥ rr♠♥ts
♣r ♥áss ♦s rst♦s q s♠♣♥ ♣ó♥ ♠ét♦♦
♦ s ♣rs♥tó ♥ st♦ ♥ttt♦ s ♠♦rs ♣r♦s ♥
s sñs ♣♦r ♠ét♦♦ r♦♥ s r♦♥s t♠♣♦ ♥
♦rr r①ó♥ ♦r♠ ♣s♦ r♦ ② s ♥t♥s
♦♥ rs♣t♦ r♦ r♥♥t t♦♦ ♦ ♦♠♦ ♥ó♥ ♣♦só♥
♦♠♣rr♦♥ ♦s rst♦s ♠ét♦♦ ♦♥ ♦s ♦s ♠ét♦♦s ② P
♥③r♦♥ ♦s t♦s r♥ts ♥s tó♥ ♥ ♦s ♣rá♠tr♦s
s♦ ② ó♥ s rt ♥tr ♦s ♦♠♣♦♥♥ts rr♦
r♦♥ t♥t♦ t♦s s♠♦s ♦♠♦ ①♣r♠♥ts ♣r ♦s s♦s
♦t♦ ♣qñ♦ ② rt♦r ①t♥s♦
s sñs ♦t♥s ♦♥ ♠ét♦♦ rstr♦♥ ♠ás ss q s
♦t♥s ♦♥ ♦s ♠ét♦♦s ② P ♦♠♦ ♦♥s♥ t♥ó♥
♥t s sñs s♥rs rt♦rs r ③ q s ♣r♦♣
♠♦stró q ♠♦r ♦rr s ♦♥s♥ s♥s♦s ♥ s
t♦♥s t♠♣♦ r①ó♥ ② ♠♣t ♣s♦ r♦ ② ♥
♥r♠♥t♦ ♥ ró♥ ♥t♥s sñ ♣r♥♣ rs♣t♦ s
r♥♥ts ♠♣♦rt♥ rt st♦s ♦rs ♠ ♦♥ ♣♦só♥
rr♦ ♦ r♦ í♥ s♦♥♦ s♥ ♥ ró♥ ♥t ♥tr s
rs♣ts rs ♥ ♥r s rs ♦t♥s ♣♦r ♠♦ ♠ét♦♦
♣rs♥t♥ ♠♥♦s r♦♥s q s ♠ás ♦tr rtríst r♦♥ ♦♥
♥ ♠♦r s③ó♥ s sñs
♥ ♦s st♦s r③♦s s ♦♠♣rr♦♥ rst♦s ♦t♥♦s ♥♦
♥ú♠r♦ ♥ts t③s ♥ ♠ét♦♦ ♦♥ ♦♥ ♥ú♠r♦
♣♠♥t♦ ♠ét♦♦ P s r③♥ s♦♥♦s ♥ ♦s q s ♥t♥s
s ♣♦s♦♥♥ ♠♥♠♥t s♦r s♣r s ♥sr♦ qrr ♣r♦①
♠♠♥t ♠s♠ ♥t t♦s ♥ ♠♦s ♠ét♦♦s s r q ♦s
♠ét♦♦s ② P rqr♥ s♠rs t♠♣♦s qsó♥ ♥ ♠
r♦ ♠ét♦♦ ♥ ♥r ♠♦str♦ rst♦s s♣r♦rs ♦♠♦ s
st♦ ♣rtr ♦s st♦s r③♦s ♥ st ss
♠♦strr♦♥ ♦s t♦s só♥ ró♥ ③ tr♥s♠t♦
♥♦ ♦♥ s♣③♠♥t♦ t♠♣♦r s♦r ♦s rst♦s ♠ét♦♦
♦s rst♦s ♣r ♠ét♦♦ ♣rs♥t♥ ♠♦rs q s ♠①♠③♥
♣r q♦s s♣③♠♥t♦s t♠♣♦rs q r♥ ♠ás ♥t♠♥t
♥rí ♦ r♦ ♠♥♦ ♠s♦r rt♦r r♣t♦r Pr st♦s s♣③
♠♥t♦s t♠♣♦rs s ♠♦rs ♥ ♦s rst♦s ♣r ♠ét♦♦ ①♥
s ♦rrs♣♦♥♥ts ♠ét♦♦ ♥ ♥tr♦s s♣③♠♥t♦ t♠♣♦r
♦♥ ♥ ①t♥só♥ ♥ ♦r♥ ♦ ♣♦r ♥♠ ♥ ♣rí♦♦ ♠str♦ tí
♣♦ st♦ ♠str q ♥♦ s rít ó♥ st ♣rá♠tr♦ r♥t
♠♣♠♥tó♥ ♠ét♦♦ ② q ♠s♠♦ ♣ sr ♦ ♦♥
rt ① s♠s♠♦ ♦s rst♦s ♦t♥♦s ♣r t♦s s♠♦s ②
①♣r♠♥ts rstr♦♥ tt♠♥t s♠rs ♥tr sí
♥ rs♠♥ ♥ st ss s ♦ró ♥③r sst♥♠♥t ♥ ♦♠♣r♥
só♥ ♠ét♦♦ ② ♥ s ♠♣♠♥tó♥ srr♦r♦♥ s té♥s ②
s rr♠♥ts áss ♥srs ♥ s ♣ó♥ ② s r♦♥ tt
② ♥ttt♠♥t s ♠♦rs ♥ ♥tr♣rtó♥ ♦s t♦s ♦t♥s
♦♥ t③ó♥ ♠ét♦♦ ♦♠♦ rst♦ t♦♦ ♦ ♠ét♦♦
s ♣ ♦♥srr ♥ ♥ ♦♠♣♠♥t♦ ♣r ♦s s♦♥♦s ts
♠ás ♠ét♦♦ s ♣ ♣r ♥ ♣r♦ ♦ s♣és ♦tr♦s ♠ét♦♦s
♦rtr ♠út♣ ♥♦r♠♠♥t s♥ qrr t♦s ♦♥s ♦s rst♦s
♦t♥♦s ♥ ♣rs♥t tr♦ ♠♦strr♦♥ ♥♦s s♥t♦s ♥ ♥
t♦ ♣ó♥ st ♥♦ ♠ét♦♦ ♥ ♦rr ♦ q ♦♥stt② ♥ ♣s♦
♥♠♥t ♣r ♦♥t♥r ♦♥ srr♦♦ ♠ét♦♦
trs í♥s tr♦
♦s rst♦s ♦t♥♦s ♥ st ss ♠str♥ r♥ ♣♦t♥ q ♣r
s♥t ♠ét♦♦ ♥ ♦♥s♥ s ♣r♦♣♦♥♥ s s♥ts í♥s
tr♦ ♣r ♦♥t♥r srr♦♦ ♠s♠♦
③r ♠♣♠♥tó♥ rr♦s ♥♦ ♥♦r♠s ♥ ♥t♦ s
s♣r♦♥s ss ② ♠♣ts rts ♦s st♥t♦s ♠♥t♦s
♠♥r ♦♣t♠③r ♦s ♠♣♦s ♠t♦s s♦r ♦s ♥♦s ♥trés
str t♦ s♦r rs♣st ♠ét♦♦ s st♥ts
♦r♥t♦♥s s ♥t♥s ♠s♦r ② r♣t♦r ♥tr sí ② ♦♥ rs♣t♦
í♥ s♦♥♦ ♠s♠♦ ♠♦♦ str ♦s ♥♦s ♠
♣♠♥tr rr♦s ②♦s ♠♥t♦s t♥♥ st♥t ♦r♥tó♥ s r
st♥t ♣♦r③ó♥
❱♥♦ ♦♥ st♦ ♦♠♣rr ♠ét♦♦ ♦♥ ♠t♦♦♦ís q ♣
♥ r♦tó♥ ♦r ❬❪ ♥ st ♠ét♦♦ s qr♥ rs tr③s
♥ ♣♥t♦ s♣r ♣r st♥ts ♦r♥t♦♥s rts ♥
tr s ♥t♥s ♠s♦r ② r♣t♦r ♦ q ♥ts ♥ ♥t♦
♦③ó♥ ♦t♦s r♦s ② st♠ó♥ s ♦r♥tó♥ ❬❪
♥q ♣rs♥t ts ♥♦ sñ ♣♦s ♠♣t
♥ r♦ ♥ ♠♦ s ♦ ♦ s ♣r♦♥ rr♦rs ♥ ♣♦s
♦♥♠♥t♦ s ♥t♥s ♥ ts st♦♥s ♠ét♦♦ ♣♦rí
♣r♦r ♠♦rs r♥
①t♥r ♣ó♥ ♠ét♦♦ rr♦s ♥ ♦s q ♦s
♠♥t♦s s ♥♥tr♥ str♦s t♥t♦ ♦ r♦ í♥ s♦♥♦
♦♠♦ ♥ ró♥ ♣r♣♥r ♠♥r ♦rr ♥♦str
③ ② ♦♥tr♦r s ♦r♥tó♥ ♥ ró♥ ♣r♣♥r í♥
s♦♥♦
③r st♦s ♠♣♦ ♥r♦ ♣♦r ♦s st♥t♦s rr♦s tr
és s♠♦♥s ♥ ♠♥r ♥③r ♥ ♦♠♣r♥só♥ ♠ás
♣r♦♥ ♥♦♥♠♥t♦ ♠ét♦♦ ② ♦rr ♦♥tr♦r ♥
t♠♥t r♥t ♦♥s rr♦ ♣r ♦♣t♠③r rs♣st
srr♦r ♥ sst♠ qsó♥ t♦♠át q ♣r♠t ♣r
♠ét♦♦ ♥ t♠♣♦s ♦rt♦s ♠♥r r ♣♦s s t③ó♥
♣♦r ♠♣♦ ♥ rs t ♥s ♦ ♣r r③r st♦s ♣rs
①t♥s♦s
③r st♦s s♠♦s ② ①♣r♠♥ts ♥ ró♥ ♦♥ ♦s t♦s
♦s st♥t♦s t♣♦s ♥ts sñs s♥rs q s ♣♥
♦srr ♥ ♦s t♦s ①♣r♠♥ts ♣r♦♣♦♥ str ♦s t♦s
r♦ ♦r♥t ♣r♦♦ ♣♦r ♥ts ♥ ss♦ ♦ r ② ♥♦
♦r♥t ♣♦r ♠♣♦ r♦ tró♥♦
③r st♦s sst♠át♦s t♥t♦ ①♣r♠♥ts ♦♠♦ s♠♦s ♣
r ♥③r s ♠♦rs q ♣r♦ ♠ét♦♦ ♥ rs♦ó♥
sñs ♣ró①♠s ♥t sí ② ♥ ♣r♦♥ ♣♥tró♥
♣r t♦r♠ r♣r♦ q ♥ q ♦s ♣tr♦♥s r
ó♥ ♥ ♥t♥ ♥ ♠só♥ ② r♣ó♥ s♦♥ ♦s ♠s♠♦s ❬❪ ♣r
t③r rr♦s r♣t♦rs ♥ ♦r♠ s♠r ♦ r③♦ ♥ st
ss ♦♥ ♠s♦rs
♣é♥
♠ó♥ ♥♠ér
rrr♠s
♥ s ♣♦♥s ♦♥srs ♠ét♦♦ ♦rr ♦s ♠♣♦s s♦♥
♠t♦s s ♥trs r s♦ ② s ♣r♦♣♥ trés ss♦ ♦♥
♥trtú♥ ♦♥ rs♦s ♦t♦s ♦s ♥♦s ♥trés ♣♥ str ♦s
♣r♦♥s q ♦rrs♣♦♥♥ t♥t♦ s r♦♥s ♠♣♦ r♥♦ ♦♠♦
♠♣♦ ♥♦ ♦ q s trt ♥ ♣r♦♠ ♦♠♣♦ q ♥♦ s
♣♦s ♦t♥r ♥ s♦ó♥ ♥ít s t s♠r ♥♠ér♠♥t
rs♣st sst♠
♦s rrr♠s s♠♦s ♥♠ér♠♥t ♣r♠t♥ str ró♥
♥tr s sñs ♦srs ② ♦s ♥♦s q s ♥r♥ ♦s t♦s s♠♦s
t♥♥ ♦trs ♣♦♥s ♦♠♦ ♣♦r ♠♣♦ ♥tr sñs ♦♥ ♦r♠s
rtrísts Pr ♦ s ♥r ♥ rs♣st s♠ ② ♦ s ♦♠♣r
♦♥ s sñs tts st ♠♥r s st ás s ♣♦rí♥
♦rrs♣♦♥r ♦s ♦t♦s ♥trr♦s ❬❪ tr s ♣♦♥s ♦♥sst ♥
rr ♥tr♣rtó♥ r③ ♦s t♦s ♦ ♥ ♣r♠r ♥tr♣r
tó♥ ♦s t♦s s ♣r♦♣♦♥ ♥ ♠♦♦ s r s ♣r♦♣s étrs ②
♦♠trí t♥t♦ ♠♦ ♦♠♦ ♦s ♥♦s ② s s♠ rs♣st ♦
rr ♣r ♠♦♦ ♣r♦♣st♦ ♦ s ♦♠♣r♥ ♦s rst♦s s♠♦s
♣r ♦ ♠♦♦ ♦♥ s sñs qrs s t q ♦♠♦ rst♦
♥ ♣r♠r ♥áss s ♣r♦♣♦♥ ♥ ♠♦ó♥ ♠♦♦ ♥ ♦ q
♥ ♣r♦s♦ st ♠♦♦ st r ♥tr♣rtó♥ ♥
♦s t♦s
ét♦♦ s♠ó♥
♦♦s ♦s t♦s s♠♦s ♥ st ss s ♦t♥♥ ♦♥ ♠ét♦♦
r♥s ♥ts ♥ ♦♠♥♦ t♠♣♦ ♥tr♥ t♠♦♠♥
❬❪ ♥ st ♠ét♦♦ s rs ♥♠ér♠♥t ♥ rsó♥ srt s
♦♥s ① ♦♥sr♥ s ♦♥s ❬❪
~∇× ~E = −µ∂ ~H
∂t
~∇× ~H = σ ~E + ǫ∂ ~E
∂t
♦♥ ǫ µ ② σ s♦♥ ♣r♠t ♣r♠ ♠♥ét ② ♦♥t
rs♣t♠♥t ~E s t♦r ♠♣♦ étr♦ ② ~H s t♦r ♠♣♦
♠♥ét♦ ♦ ♣r r③r s♠♦♥s s♦♥♦s r①ó♥ ♥ ♦s
q s ♥t♥s s ♥ ♥ s♣r r s♦ ♦♥ ss s ♣r♣♥
rs í♥ s♦♥♦ q s ①t♥ ♦ r♦ x s s♠
q s ♣r♦♣s íss ♠♦ s♦♥ ♥r♥ts ♥ ró♥ y
♥ s s♦ s ♦♥s ② s ♦t♥♥ ♦♥s s♦
♣s ♣r s ♦♠♣♦♥♥ts ♦s ♠♣♦s tr♦♠♥ét♦s ④Ey Hx Hz⑥
♦♠♣♦♥♥t tr♥srs♦ étr ② ④Ex Ez Hy⑥ ♦♠♣♦♥♥t tr♥srs♦
♠♥ét Pr ♦♠♣♦♥♥t s ♦♥s s ♣♥ ①♣rsr
s♥t ♠♥r∂Ey
∂z= µ
∂Hx
∂t
∂Ey
∂x= −µ
∂Hz
∂t
σEy − ǫ∂Ey
∂t=
∂Hz
∂x− ∂Hx
∂z
Pr ♦t♥r ♥ s♦ó♥ ♥♠ér s ♦♥s ② s
r ♥ ♦♥♥t♦ ♦♥s ♥ r♥s ♥ts s♦r ♥ r srt
② ♥♦r♠ ♦s ♣♥t♦s r s ♥♦t♥ ♣♦r ❬❪
(x, z) = (j∆x, k∆z)
♦♥ ∆x ② ∆z s♦♥ ♦s ♥r♠♥t♦s s♣s r s ③
♥ó♥ s♣♦ ② t♠♣♦ s sr s♥t ♠♥r
F (x, z, t) = F (j∆x, k∆z, n∆t) = F n(j, k)
♦♥ ∆t s ♥r♠♥t♦ t♠♣♦r ♥♠♥t s rs s♣s ②
t♠♣♦rs ♥ s ♦♥s s ♣r♦①♠♥ t③♥♦ ♠ét♦♦ r♥s
♥ts
♠♣♠♥tó♥ ♦♠♣t♦♥
t③ ♠♣♠♥tó♥ ♠ét♦♦ r③ ♣♦r r♥ ②
♥t ❬❪ trt ♥ ó♦ ♥ ♥ q ♣r♠t s♠
r ♥♠ér♠♥t ♦s ♠♣♦s ♥r♦s ♣♦r ♥ sst♠ ♦rr ♥ ♦s
♠♥s♦♥s ó♦ t♥ s s♥ts rtrísts
Pr s rs s♣s s t③ ♥ ♣r♦①♠ó♥ r♥s
♥ts rt♦ ♦r♥ ♠♥trs q ♣r s rs t♠♣♦rs s
t③♥ ①♣rs♦♥s s♥♦ ♦r♥
t③♥ ♦♥♦♥s ♦♥t♦r♥♦ t♣♦ ♣rt② ♠t ②r P
♠♥r tr r①♦♥s ♦s ♦rs r Pr ♦ s
r ♥ ♦♥t♦r♥♦ r rt ♥t s ♦♥ ♣r♦♣
s tr♦♠♥éts ts q ♦s ♠♣♦s q ♥ s ró♥ s♥
s♦r♦s ♣♦r ♦♠♣t♦ ♠♥r tr r①♦♥s ♦s ♠s♠♦s
♥tr♦r r
Pr ♥tr♦r ♥ ♥t ♠♣♦ étr♦ ♥ r s ♦♥sr
♣s♦ r ♣r ♠♣♦ étr♦ Ey s s♠ ♥
♣s♦ t♠♣♦r ♥ ♣♦só♥ r ♥ q s ♥♥tr
♥t st♦ s q♥t s♠r ♣s♦ ♦rrs♣♦♥♥t ♥t
♦♠♣♦♥♥t y tr♠♥♦ ♥s ♦rr♥t ♥ s ♦♥s
①
Pr s♠r ♦s r♣t♦rs s rstr ♠♣♦ étr♦ ♥ ♥ó♥
t♠♣♦ ♥ ♦s ♣♥t♦s r ♥ ♦s q s ♥♥tr♥ ♦s r♣t♦rs
♦ q s trt ♥ ó♦ s ♥ts s♦♥ ♠♥t♦s ♥s
q s ①t♥♥ ♥♥t♠♥t ♥ ró♥ ♣r♣♥r ♣♥♦
s♦♥♦ ♥ ♦♥s♥ ó♦ ♥♦ ♠♦ ♠♥t s♣rsó♥
② ♣tró♥ ró♥ ♥ts t♣♦ ♣♦♦ ♣r ♦ s ♥sr♦
♥ ó♦ ♥ ♠r♦ ó♦ ♣r♠t r♣r♦r s ♣r♥♣s
rtrísts s sñs ♦t♥s ♥ ♥ s♦♥♦ ♦♥ ♦rr
♣r♦r♠ rqr ♦♠♦ ♥tr s ♠trs ♦♥ s ♣r♦♣s
tr♦♠♥éts ♠♦♦ ǫ σ ② µ s ♣♦s♦♥s s ♥ts ②
♦s r♣t♦rs ♠ás r♥ ♥tr ♥t ♦s ♣s♦s
s♣s ② t♠♣♦r ② t♠♣♦ ♥ s♠ó♥
Pr ♥ s ♣♦s♦♥s s ♥ts ó♦ r③ ♥
sr ♦♠♣t tr♦♥s s t = 0 st t♠♣♦ ♥
s♠ó♥ st ♠♥r s ♦t♥ ♥ só♥ t♦s Ey(xr, t)
♥ ♦♥ró♥ ♠s♦r ♦♠ú♥ xr r♣rs♥t s ♣♦s♦♥s r♣t♦r
♦♠♦ s ó♦ ♥r ♥ ♦ t♦s s r ♥ ♠tr③
♦♥ trs ♥s q ♦♥t♥ t♦s s s♦♥s t♦s Ey(xr, t) ♥
♦♥ró♥ ♠s♦r ♦♠ú♥ ♥ ♣r ♣♦só♥ ♠s♦r
r Ps♦ q ♠♥t ♠♣♦ étr♦ Ey ♣r ♥
r♥ ③
♦♦♥s r③s s♦r ó♦
♣r♦r♠ s ♠♦ó ♠♥r tr s ♣ó♥ ♦♥ ♦s ♣r♦♣ó
st♦s s♣í♦s st ss ♥ ♣r♠r s ♠♦♦♥s r③s s
♣tó ♦r♠t♦ s ♦s t♦s ♠♥r q ♦rrs♣♦♥ ♦r♠t♦
q♦s t♦s qr♦s ♦♥ q♣♦s ♦rr st♦ ♣r♠tó trtr ♦s
t♦s s♠♦s ♦♥ ♦s ♣r♦r♠s ♣r♦s♠♥t♦ ② s③ó♥ srr♦
♦s ♣r♠♥t ♥tr♦ ♠r♦ st ss Pr ♦ s ♠♣♠♥tó
♥ r♠♥t♦ ó♦ q ♣r♠t rr ♥ r♦s s♣r♦s ♦s t♦s
♦rrs♣♦♥♥ts ♥ s s♦♥s ♥ ♦♥ró♥ ♠s♦r ♦♠ú♥
♣r ♠♣♦ Ey(xr, t) ♥ st♦s r♦s s ♥②♥ ♠ás ♦s t♠♣♦s ♥
♦s q s rstr ♠♣♦ t ② s ♣♦s♦♥s ♠s♦r ② ♦s r♣t♦rs
xr ♦♥tr ♦♥ st ♥♦r♠ó♥ t ♣♦str♦r trt♠♥t♦ ♦s
t♦s ♦♠♦ ♣♦r ♠♣♦ ♥♦ s ①tr♥ ♦s t♦s ♥sr♦s ♣r s♠r
♦s rstr♦s ♦t♥♦s ♦♥ ♥ sst♠ ♦rr ♥ ♦s st♥t♦s t♣♦s
s♦♥♦s
♦ q ♣r ♥ s ♣♦s♦♥s ♥t ♦s rst♦s s
rr ♥ r♦s s♣r♦s s ♠♦♦♥s r③s ♣rs♥t♥ ♥
t ♦♥ ♣r♠tr t♥r s♠ó♥ ♦ q s ♣rtr♠♥t út
♥ ♦s s♦s ♦♥ rs ①t♥ss ♣r ♦s q s rqr♥ t♠♣♦s r♦s
♦rr
Pr ♦s st♦s r③♦s ♥ st ss rstó ♥trés ♥③r
♣r♦♣ó♥ s sñs ② ♥tró♥ s ♠s♠s ♦♥ ♦s ♥♦s P♦r
st r③ó♥ ♦tr ♠♦ó♥ r③ s♦r ó♦ ♦r♥ ♦♥sstó ♥
srr♦r ♥ rsó♥ tr♥t ♦ ó♦ ♥ q s ♥♦r♣♦r ♥
r♠♥t♦ ♣r♦r♠ q ♣r♠t ①trr ♣trs ♠♣♦ ♥ ♣s♦
t♠♣♦r s♠ó♥ ♥♦ s ♦rr ♥ rsó♥ ó♦ s
♦t♥ ♦♠♦ s ♥ sr r♦s q ♦♥t♥♥ s ♠trs ♦♥ ♦s
♦rs ♠♣♦ étr♦ ♥ ♣♦só♥ Ey(x, z) ♣r ♣s♦ t♠♣♦r
s♠ó♥
♣ó♥ ó♦ s s♠♦♥s r③s
♦s ♣rá♠tr♦s t③♦s ♥ s s♠♦♥s r③s ♥ st ss
r♦♥
r♥ ♥tr ♠só♥ s ③
s♣♠♥t♦ r s 0,01 ♠
Ps♦ t♠♣♦r dt = 0,02 ♥s
Pr s♠r ♠s♦rs ② r♣t♦rs ♦s ♥ ♥trs r s♦
st♦s s ♥ ♥ ♣r♠r í♥ r ♦♥ ♦s ♣rá♠tr♦s
♠♦
♣é♥
rr♦s ♥t♥s ♥ ♥ ♠♦
♥♦r♠
♥ st só♥ s ♣rs♥t♥ s ♦♥s q sr♥ stró♥
♥rí r ♣♦r ♥ ♣♦♦ ♦s♥t ♠ ♦♥ ② s ♦rr♦♥s
♥srs ♥♦ ♣♦♦ stá s♣③♦ ♦r♥ ♦ s ♣
♣r♥♣♦ s♣r♣♦só♥ ♣r r t♦r rr♦ s♠♥♦ ♣♦r
s♠♣ q ♦s ♣♦♦s s♦♥ ♣r♦s ❬❪ ❬❪
♦♥sr ♥ ♥t♥ ♦♥ ♦♥t l ♦♠♦ s ♠str ♥
r q ♥r ♠♣♦s r♠ó♥♦s r♥ ω s ♥ts
rr♦ s r♣rs♥t♥ ♣♦r ♥ ♥s ♦rr♥t étr ♥♠♥s♦♥~Jorig(~r, t) ♦ r♦ ♥t♥ z
r ♣♦♦ ♠s♦r ② sst♠ ♦♦r♥s
~Jorig(~r, t) =
I0e−iωt sen(π
2− k|z|)δ(x)δ(y)z s |z| ≤ λ
4
0 s |z| > λ4
♦♥ I0 s ♠♣t ♦rr♥t k = 2πλ
s ♥ú♠r♦ ♦♥
λ s ♦♥t ♦♥ ♥ ♠♦ δ s ♥ó♥ t r ②
sí♥ orig ♥ q ♥t s ♦③ ♥ ♦r♥ ♦♦r♥s
♦♥sr ♥ ♦♥t l = λ2♣r ♥t♥ st stó♥ r♣rs♥t
♣r♦①♠♠♥t ♥ ♣♦♦ r♥t ♠ ♦♥ ó♥ s ♥
♥ ♣r♦①♠ó♥ ♥♦ á♠tr♦ ♥t♥ s ♣qñ♦ ♦♥ rs♣t♦
s ♦♥t ♣r♦ ♥♦ s á ♣r ♥t♥s rss ♥ s s♦ rí q
♦♥srr ♣♥♥ r ~Jorig
♠♣♦ étr♦ rst♥t r♦ ♣♦r ♣♦♦ ①♣rs♦ ♥ ♦♦r
♥s sérs r stá ♦ ♣♦r
~Eorig(~r, t) = i2Iηei(kr−ωt)
4πr
cos(π2cos(θ))
sen(θ)θ
♦♥ η =√
µ
ǫs ♠♣♥ ♠♦ ♦♥ µ ♣r♠ ♠♥ét
② ǫ ♣r♠t ♥s ♣♦t♥ dPds
♥t♥ ♣r
srr s ♣tró♥ ró♥
dP
ds= |〈~S〉| =
∣
∣
∣
1
2Re( ~E × ~H∗)
∣
∣
∣
~S ♥♦t t♦r P♦②♥t♥ ② ♦s ♦rts ♥♥ ♣r♦♠♦ t♠♣♦r ♥
♥ ♣rí♦♦ Pr ♥t ♦♥sr rst
(dP
ds
)
orig=
I02η
8π2r2
[cos(π2cos(θ))
sen(θ)
]2
♥t♦♥s ♣♦t♥ ♣♦r ♥ á♥♦ só♦ dPdΩ
♦ ♥t♥s r
ó♥ s ♣ ①♣rsr ♦♠♦
(dP
dΩ
)
orig= r2
(dP
ds
)
orig=
I02η
8π2
[cos(π2cos(θ))
sen(θ)
]2
♥♦ s ♦♥sr ♥ ♣♦♦ s♣③♦ ♦r♥ ♦ r♦ r
ó♥ ~d ♥s ♦rr♥t rs♣t ~Jdesp s ♣ srr s♥t
♠♥r~Jdesp(~r, t) = ae−i(ωt+ϕ) ~Jorig(~r − ~d)
♦♥ ~Jorig ♥ ♥s ♦rr♥t ♥♦ ♥t s ♥♥tr ♥
♦r♥ ó♥ r ♥ tér♠♥♦ s ϕ ② ♥ t♦r ♠♣t
a ♦♥ ♣r♦♣óst♦ ♦♥t③r ♣♦ss ♠♦s ♥ s♦s ♣rá♠tr♦s
♠♣♦ étr♦ rst♥t s
~Edesp(~r, t) = ae−iϕe−i~k·~d ~Eorig(~r, t)
♥♠♥t ♠♣♦ étr♦ ♥ rr♦ ♣♦♦s ♥ ró♥
♠♣♦ ♥♦ s ♣ ♦t♥r s♣r♣♦♥♥♦ ♦s ♠♣♦s ♦s ♣♦♦s ♥
s Pr ♦s ♣♦♦s ♠ ♦♥ ♦♥sr♦s ♥ rr♦ N
♠♥t♦s s sr ♣♦r s♥t ①♣rsó♥
~EN(~r, t) = ~Eorig(~r, t)(
N−1∑
n=0
ane−iϕne−i~k·~dn
)
tér♠♥♦ ♥tr ♣ré♥tss s ú♥♦ q ♦♥t♥ s ♣♦s♦♥s s ♥ts~dn ② ss ♠♣ts an ② ss ϕn rts ② s ♦ ♥ ♦♠♦ t♦r rr♦
ΓN
ΓN =N−1∑
n=0
ane−iϕne−i~k·~dn
♣♦t♥ ♣♦r ♥ ár r ♣♦r rr♦(
dPds
)
Ns ♣
①♣rsr ♦♠♦(dP
ds
)
N= |ΓN |2
(dP
ds
)
orig
♦♥(
dPds
)
origs ♥s ♣♦t♥ r ♣♦r ♥ ú♥♦ ♣♦♦ ♦
③♦ ♥ ♦r♥ ó♥ ♥á♦♠♥t ♣♦t♥ ♣♦r ♥
á♥♦ só♦ s ♣ ①♣rsr ♦♠♦
(dP
dΩ
)
N= |ΓN |2
(dP
dΩ
)
orig
♣r♠r t♦r st ó♥ t♦r rr♦ t♥ ♥ ♥t
stró♥ s♣ ♦s ♠♥t♦s ss ss ② ♠♣ts rts ♠♥trs
q s♥♦ s ♥s ♣♦t♥ ♠♥t♦ ♥r♦r ♥t♦♥s s
♣ ♣♥sr q t♦r rr♦ ♠♦ ♣tró♥ ♠♥t♦ ♥r♦r
♠♥r ♠ás s♠♣ ♥rr ♣tr♦♥s ♦♥ rtrísts s♣ís
s ♦♥sr♥♦ rr♦s rrs ❯♥ rr♦ rr s ♦♠♣♦♥ ♣♦r rs
♥ts q t♥ é♥t ♠♣t ② ♣rs♥t♥ ♥r♠♥t♦s ♦s ♥ ss ss
r rr♦ rr ♦♥ N1 = 2 N2 = 2 N3 = 4
dx = 3,0 dy ② dz = 1,5 dx
② ♦♥ stró♥ rr ♥ s♣♦ ♥ st s♦ s út ①♣rsr t♦r
rr♦ ♣♦r ♠♦ ♥♦tó♥ rst♦rá ♣♦só♥ ♠♥t♦
rr♦ ~dn s sr ♣♦r ♠♦ t♦r ❬❪
~dn = n1~b1 + n2
~b2 + n3~b3
♦♥ ~b1 ~b2 ② ~b3 s♦♥ ♥r♦rs ♥♦ ♦♣♥rs ② ni = 0, 1, . . . , (Ni − 1)
i = 1, 2, 3 ♦♥ st ♥♦tó♥ ♥ú♠r♦ t♦t ♣♦♦s N s r♦♥ ♦♥
Ni ♣♦r ♠♦ N = N1N2N3 ♠♥trs q ♦♥t t♦t r ♥
ró♥ ~bi s (Ni − 1)|~bi| P♦r s♠♣ ♥ ♦ q s s s♠ q ~b1~b2 ② ~b3 s♦♥ ♣r♦s x y ② z rs♣t♠♥t s r q s ♥
~b1 = dxx,~b2 = dyy,~b3 = dz z
② ♥♠♥t ó♥ s ♣ ①♣rsr ♦♠♦ s
~dn = n1dxx+ n2dyy + n3dz z
♦♥ dx dy ② dz ♥♦t♥ s ♦♥st♥ts r ♦ r♦ ♦s trs s
rts♥♦s
r ♠str ♥ ♠♣♦ ♦♥ N1 = 2 N2 = 2 N3 = 4 dx = 3,0dy ②
dz = 1,5dx s ♠♥t♦ ϕn s ♣ ①♣rsr ♠♥r s♠r
♦♠♦
ϕn = n1α1 + n2α2 + n3α3
♦♥ α1 α2 ② α3 ♦s ♥r♠♥t♦s ♥ s r♦♥s~b1 ~b2 ②~b3 rs♣t♠♥t
♦♥sr♥♦ s ♦♥s ② ② s♠♥♦ q s ♥ts
t♥♥ é♥t ♠♣t ai = 1, ∀i t♦r rr♦ ó♥ s
♣ srr ♦♠♦
ΓN =(
N1−1∑
n1=0
e−in1(α1+kxdz))(
N2−1∑
n2=0
e−in2(α2+kydy))(
N3−1∑
n3=0
e−in3(α3+kzdz))
s ♦♥sr q
N−1∑
m=0
e−imγ =sen(Nγ
2)
sen(γ2)e−i(N−1) γ
2
♥t♦♥s s ♦t♥ s♥t ①♣rsó♥
|ΓN |2 =sen2(N1(α1+kxdx)
2)
sen2( (α1+kxdx)2
)
sen2(N2(α2+kydy)
2)
sen2( (α2+kydy)
2)
sen2(N3(α3+kzdz)2
)
sen2( (α3+kzdz)2
)
♣♥♥ ♥r stá ♠♣ít ♥ ♦s tér♠♥♦s ki ó♥
stá ♦♠♣st ♣♦r trs t♦rs r♦♥♦s ♦♥ ♦s ♣rá♠tr♦s r ♥
r♥ts r♦♥s st♦ s♠♣ ♥áss s rtrísts ♣tró♥
rst♥t ♦♠♦ ♥ó♥ ♦s ♣rá♠tr♦s r ♦♥ ♥ s♠♣ó♥
♦♥ ♥♦ s ♦♥sr♥ rr♦s ♥ ♥ ♦ ♦s r♦♥s ♣♦r ♠♣♦
♣♦♦s ♥♦s ♦ ♥ rr♦ ♣♥♦ rs♣t♠♥t
♣é♥
Pr♦r♠ rr②
ó♦ rr② srr♦♦ ♥ ♥ ♣r♠t r
♣tró♥ ró♥ rr♦s ♣♦♦s ♠ ♦♥ ♦♥ r♦♥s
♣rs ② s ♦♥sr♥♦ ♥ ♦♥ró♥ rr ♦ rrr ♥ ♣r♠r
tér♠♥♦ ó♦ ♣tró♥ ró♥ ♥ ♣♦♦ ♠ ♦♥
♦③♦ ♥ ♦r♥ ó♥ ② q ést s ♠♥t♦ ♣r♠r♦
q ♦♥stt② rr♦ ♦ s s rtrísts rr♦ ♥ú♠r♦
♣♦♦s ♣♦só♥ t s tr♠♥ t♦r rr♦ ó♥
♣r rr♦s rrs ♦ ó♥ ♣r rr♦s rrrs ♥♠♥t
♣tró♥ ró♥ rr♦ s ♦t♥♥ ó♥
♣r♦r♠ ♠str ♦s ♦rs ♥s ♥ts q ♣r♠t♥ s
rr s rtrísts ♣r♥♣s ♣tró♥ ró♥ ♥t♥ ♥
st s♦ rr♦ ❬❪ ❯♥ ♥t ♠♣♦rt♥t s P♦t♥
♦t PT ♠t ♣♦r ♥t♥ q s ♥ ♦♠♦
PT =
∫ π
0
∫ 2π
0
dP
dΩdΩ
♣ ♥t♥ ♣r rr st ♣♦t♥ ♥ ♥ ró♥
(θ, φ) s♠♥t s ♠ ♣♦r ♠♦ rt D(θ, φ) q s ♥
♦♠♦ ♣♦t♥ ♣♦r ♥ á♥♦ só♦ ♥♦r♠③ ♣♦r t♦r PT/4π
D(θ, φ) =4π
PT
dP
dΩ
t♦r ♥♦r♠③ó♥ PT/4π r♣rs♥t ♣♦t♥ r ♣♦r ♥
á♥♦ só♦ ♥♦ s ♦♥sr ♥ ♥t♥ ♥♦ r♦♥ s♥ ♣♥♥
♥r st s ♣ ♥tr♣rtr ♠ás ♦♠♦ ♥s ♣♦t♥
♣r♦♠♦ ♣♦r ♥ á♥♦ só♦ ♥t♦♥s ♣r ♣tr♦♥s ♥♦r♠s s
D = 1 ♥ t♦s r♦♥s P♦r ♦♥trr♦ ♣r ♣tr♦♥s r♦♥s s
D > 1 ♣r q♦s á♥♦s ♥ ♦s s s ♣rs♥t♥ ó♦s ♠♥trs q
s D < 1 ♥ ♦s s q ♣rs♥t ♣tró♥ ❯♥ ♥t r♦♥ s
♥♥ ♥t♥ G(θ, φ) ♥ st s♦ ♣♦t♥ t♦t ♣r♦st ♣♦r
♥t PS t♠♥t ♠ P♦t♥ ♥tr r♠♣③ ♣♦t♥
♥t PT ♥ ó♥
G(θ, φ) =4π
PS
dP
dΩ
s ♣érs q s ♣r♦♥ r♥t ♣r♦s♦ ♠só♥ s♠♥t
s r♣rs♥t♥ ♣♦r t♦r ♥ e q s ♥ ♦♠♦ ♦♥t
PT/PS t♦r e t♦♠ ♦rs ♥tr ② ♦rrs♣♦♥ ♥ s♣♦st♦
s♥ ♣érs ♠♥trs q ♦s ♦rs ♠ás ♦s ♦rrs♣♦♥♥ sst♠s
♠② ♥♥ts t♦r e r♦♥ ♠ás s ♥♦♥s G ② D
G(θ, φ) = eD(θ, φ)
út♠ ♥t q s ♦♥sr s á♥♦ só♦ ③
∆Ω =
∫ π
0
∫ 2π
0
D(θ, φ)
Dmax
dΩ
♦♥ Dmax s ♠á①♠♦ ♦r D(θ, φ) Pr ♥t♥s t♠♥t rts
s q ♦♥♥tr♥ ♠②♦r ♣rt s ♣♦t♥ r ♥ ♥ ♣qñ♦ á♥♦
só♦ ∆Ω s r♥♦ s ♠í♥♠♦ ♦r P♦r ♦♥trr♦ ♣r ♥t♥s
sótr♦♣s ∆Ω t♦♠ s ♠á①♠♦ ♦r 4π ♥t ∆Ω r♣rs♥t á♥♦
só♦ trés t♦ ♣♦t♥ r s ♥srí s ♣♦t♥
♣♦r ♥ á♥♦ só♦ dP/dΩ r ♦♥st♥t s♦r st ró♥ ♥r
♣r♦r♠ t♥ ♥ ♥trs rá q ♣r♠t ♥ s♦ ♠ ♣♦r
♠♦ ♥ rr ♠♥ú ② r♥ts ♦♥tr♦s rá♦s ❯t③♥♦ st ♥
trs s ♣♥ rr á♠♥t s rtrísts rr♦ ♥ r
s ♠str st ♥trs ♣r rr♦s ♦♥ ♦♥ró♥ rr ♦♣ó♥
r ♥ ♠♥ú rr② ♥ st♦r s♣r♦r ③qr♦ st ♥t♥
♣r♥ ♦s ♣rá♠tr♦s rr♦ rr② ♣r♠trs Nx Ny ② Nz ♥ú♠r♦
♣♦♦s ♥ ♦s s x y ② z rs♣t♠♥t di/λ i = x, y, z ss ♦rrs
♣♦♥♥ts s♣r♦♥s di ♥ ♥s ♦♥t ♦♥ λ ② ♥♠♥t
φi/π s ss φi ♥♦r♠③s π ❯♥ ③ q s ♥ st♦s ♣rá♠tr♦s
s ♥ ♦s ♦rs P♦t♥ ♦t ♦t rt ♣♦r
ó♥ ①♣rs ♥ ❲ts ② ♥♦ ó♦ ③ ♠ s♦
♥ ó♥ ♥ r♥s st♦s ♦rs s ♠str♥ ♥ st♦r
Pttr♥ rtrsts r ♥ ♣r♦ ♣r♥ tr♦ rá♦s
♥ r♥ts ♥t♥s ♦♠♦ s ♠str ♥ r ♥t♥
♣♦ rt♦♥ ♣ttr♥ q ♦rrs♣♦♥ ♣tró♥ ró♥ ♥
♣♦♦ ♠♥t ♥ ♦r♥ (dPdΩ)orig ♥t♥ rr② q ♠str
stró♥ ♥ s♣♦ ♦s ♣♦♦s t♦r rr♦ rr② t♦r
|ΓN |2 ② ♣tró♥ ró♥ rr♦ rr② rt♦♥ ♣ttr♥ (dPdΩ)N
♥♦ s rí♥ ♦s ♣rá♠tr♦s ♥tr s ♥ts ♥ st♦r Pttr♥
rtrsts ② ♦s rá♦s s t③♥ t♦♠át♠♥t st ♠♥r
s á r ó♠♦ r♦♥s ♥ rr♦ ♠♦♥ ♣tró♥ ró♥
♥t♥
♥ sst♦r t♦♥ ♥ ♣♦♥t st♦r Pttr♥ rtrsts s
♠str♥ ♥s rtrísts ♣tró♥ ró♥ q s ♥ ♥
ró♥ ♥ ♣♦r ♦s á♥♦s sér♦s θ ② φ ♥trsó♥ ♥tr sts
r♦♥s ② ♣tró♥ ró♥ s ♠str ♦♠♦ ♥ ♣♥t♦ ♥ ♣tró♥
ró♥ rr♦ ♥t♥ rr② rt♦♥ ♣ttr♥ r r s
♥ts s ♥ st♦r t♦♥ ♥ ♣♦♥t s♦♥ ♥t♥s
ró♥ ó♥ ♥s ♣♦t♥ ó♥
♥♥ ó♥ ② rt ó♥ ♦tr q ♣r
r rt s ♥sr♦ s♥r ♥ ♦r t♦r ♥ e
st ♦r ♥t♦ ♦♥ ♥t♥s I0 ♣♦♦ ♠♥t s ♣♥ str
s♦♥♥♦ ♣t♦♥s ♥ rr ♠♥ú
♠é♥ s ♣♦s ♠r t♣♦ rá♦ s♦♥♥♦ rá♦s t♣♦
s♣r r ♦ r r ♥ st♦r P♦t st ♦r♠ s ♣
♦t♥r ♥ ♦ ♣tró♥ t③♥♦ ♥ rá♦ s♣r ② ♦
s ♣♥ ♦srr ♦s ts t③♥♦ r♥ts ♦rts ♥♦ sr♦
r stá ♠r♦ s ♠str♥ trs rs ♥ ♣r θ ♦♥st♥t ② s ♦trs
♣r ♦s ♦rs r♥ts φ st♦s ♦rs s ♣♥ rr t③♥♦ ♦s
♦♥tr♦s s③s
rr ♠♥ú t♥ ♦♣♦♥s ♣r rr ♥ s♦ ♦♥ró♥
rr♦ ② ♦s rst♦s ② ♣r rr♦s ♥♠♥t P♦r ♦tr♦ ♦
♣r♦r♠ ♥② ♦s ♦♥r♦♥s ♣rtr♠♥s ♥ s ♣r
♥t♥ ♥ s t♦♥í r ② ♦tr ♣r ♥ ♥t♥ ♣♦♥s
r ♥trs rá ♣r♦r♠ rr② ♦♥r♦
♣r rr♦s rrs ♥ st♦r s♣r♦r ③qr♦ s ♣♥
♥tr♦r ♦s ♣rá♠tr♦s rr♦ st♦r Pttr♥ r
trsts ♠str ♥s ♥ts q rtr③♥ ♣tró♥
ró♥ st♦r P♦t ♣r♠t s♦♥r ♦r♠t♦
s③ó♥ s♣r ②♦ rs ♥ tr♠♥s ♣♦s♦♥s ♥
rs ♥♠♥t ♥ rt♦ st♦r s t③ ♥ ♣r♦s♦
st
r st♦s rá♦s ♠str♥ s③ó♥
♣tró♥ ró♥ ♥ ♣♦♦ ♠ ♦♥ stró♥
♦♠étr ♣♦♦s ♥ rr♦ t♦r rr♦ ②
♣tró♥ ró♥ rr♦
♦rr
♦♠♦ ♥ ♠♣♦ ♥♦♥♠♥t♦ ♣r♦r♠ rr② s ♦♥sr
♥ rr♦ rr ♦r♠♦ ♣♦r tr♦ ♣♦♦s é♥t♦s ♣r♦s ♥♦s ♦♥
y r ♦s ♣rá♠tr♦s ♦♠♣t♦s st ♠♣♦ s♦ s♦♥
♦s s♥ts(Nx, Ny, Nz) = (1; 4; 1)
(dx, dy, dz)/λ = (0; 0,50; 0)
(φx, φy, φz)/π = (0; 0; 0)
♥ r s ♠str ♣tró♥ ró♥ rst♥t st ♦♥
ró♥ ♣r♦ ♥ ♣tró♥ t♦ ♦♥ ♦s ó♦s ♣r♥♣s q ♦♥♥tr♥
♠②♦r ♣rt ♥rí ♠t ② tr♦ ó♦s s♥r♦s ♠② ♣q
ñ♦s ♣♦só♥ ♦s tr♦ ♣♦♦s sr♣t♦s ♥ ♠♣♦ ♠ ♥
stró♥ rt♥r s♦
(Nx, Ny, Nz) = (2; 2; 1)
(dx, dy, dz)/λ = (0,25; 0,50; 0)
(φx, φy, φz)/π = (0; 0; 0)
♥ ♦♥ró♥ t♠é♥ ♣r♦ ♦s ó♦s ♣r♥♣s r
♣r♦ ♦♥ ♥ ♣♦t♥ r 162 W ♠②♦r q ♦r 115 W ♣r♦
♥ s♦ tr♦ ♣rá♠tr♦ q s ♣ rr ♣♦r ♠♦ ♥trs
rá s s rt ♥tr ♦s ♣♦♦s P♦r ♠♣♦ s s rt φi/π
♠ 0 0,5 ♣tró♥ ró♥ rst♥t r s ♠②
r♥t ♥ ó♦ s♣r ② ♥rí r s ♦♥♥tr ♥ ♥ ú♥
ró♥
♥♦ s trt rr♦s ♥♦ rrs s ♥t ♦♥ ♠②♦r rt ♥
ó♥ ♦s ♣rá♠tr♦s ♥♦ s ♥③ó s♦ r s ♠♦stró
q ♥t♦ ♦♥ ♦s ó♦s ♣r♥♣s ♣r♥ tr♦ ó♦s ♠ás ♣qñ♦s
st♦ s♥ q ♣rt ♥rí s♣♦♥ s ♣r♦♣ ♥ ♥ ró♥
st♥t s ♣r♥♣s ♥t♦♥s s s ♥t♥tr ♠♥♠③r ♦s ó♦s s
♥r♦s ❯♥ ♠♥r r③r st♦ s ♠r ♥t♥s rt ♦s
♣♦♦s Pr♠r♦ s ♥sr♦ ♠r ♥t♥ rrr rr② r
s♦♥♥♦ ♦♣ó♥ rr② ♥ rr ♠♥ú ② ♦ s♦♥r
rrr ♥ st ♥t♥ s ♣ r ♥ú♠r♦ t♦t ♣♦♦s N ②
♠♥r ♥♣♥♥t rr s rtrísts ♠♥t♦ s ♠♣t
s a s ♦♦r♥s ♥ s q s ♥♦ ♦s ♣♦♦s x/λ y/λ
r rr♦ rr tr♦ ♣♦♦s ♦s ♥
② ♦♥ dy = 0,5λ s♦ ② ♣tró♥ ró♥
♦rrs♣♦♥♥t
r rr♦ rr ♦r♠♦ ♣♦r ♦s s ♦s
♣♦♦s ♦♥ dx = 0,25 λ ② dy = 0,5 λ s♦ ② ♣tró♥
ró♥ ♦rrs♣♦♥♥t Ptró♥ ró♥ ♠s♠ s
tró♥ ♣r♦ ♥tr♦♥♦ ♥ ♦rr♠♥t♦ s φx = 0,5 π
♥tr ♦s ♣♦♦s
② z/λ ② s ss ♦rrs♣♦♥♥ts φ/π ♦s ♦rs q s ♠str♥ ♥
r ♦rrs♣♦♥♥ s♦ ①♣t♦ ♣♦r ♥ ♠♦ ♥ s ♥t♥ss
♦s ♣♦♦s ♥ ♣tró♥ ró♥ rst♥t r s ♣ r
q ♦s ó♦s s♥r♦s s♣r♥
♥ ♠♦s s♦s ♣rát♦s sr♣ó♥ té♥ ♥ ♥t♥ ♥②
♣tró♥ ró♥ ♣r♦ ♥♦ s ♦♠trí ♥ s stró♥ ♦rr♥t ♥
♦♥s♥ s t♥ ♥ r♥ ♠tó♥ r③r ①♣r♠♥t♦s ♥♠ér♦s
♣r ♦s s s ♥sr♦ ♦♥♦r ♦s ♠♣♦s tr♦♠♥ét♦s ♥r♦s
♣♦r ♥t♥ P♦r ♠♣♦ ♥♦ s trt r ♣♥tró♥ ♦s
♠♣♦s tr♦♠♥ét♦s ♥ s♦ ♦ sñ ♣r♦ ♣♦r ♥ ♦t♦ ♥
♣rtr ❯♥ ♠♥r s♣rr st ♠tó♥ s ♦t♥r ♥ ♥t q
♥t ♦ ♥ sst♠ ♥ts q ♣r♦③ ♥ ♣tró♥ ró♥ s♠r
♣tró♥ ♦♥♦♦ ♣r ♥ tr♠♥ ♥t♥ ♦s ♠♣♦s tr♦♠♥ét♦s
s ♣♥ r ♦ ♣r ♥t q♥t
♦♠♦ ♠♣♦ s ♦♥sr ♣tró♥ ró♥ ♥ ♥t♥ ♦
rr q s ♠str ♥ r ❬❪ ② s s ♥ rr♦ ♣♦♦s
②♦ ♣tró♥ s st ♦r♥ Pr ♦ s t③ ♦♣ó♥ tt♥ ♠
♥ú ♣r♥♣ ♣r♦r♠ rr② r ♥♠♥t s ♣r♦♣♦♥ ♥
rr♦ ♦♠♣st♦ ♣♦r ♦s ♣♦♦s ♦s ♦ r♦ ① ② s rí s
♣♦só♥ ② ss rts t③♥♦ ♠ét♦♦ ♣r ② rr♦r ♦♠♦ rr♦r
st ♣r♦r♠ só♥ stá♥r Pr ♦♥ró♥
♦s ♣♦♦s s ♦t♥ ♥ ♥ st ♣r dx/λ = 0,17 ② φx/π = 0,5 r
rr♦r st ♣r st ♦♥ró♥ s Pr ♠♦rr st
s r ♦tr♦ ♣♦♦ t♠é♥ ♦ ♥ x ♦ s ♦ s♣③ ♥
♥ ♣♦só♥ ♥ x ♠♥r q rst ♥ rr♦ ♥♦ rr
♠♦r st s ♦t♥ ♣r ♣♦♦ ♥ st♥ 0,31 λ ♦r♥ ② s
rt r♦ rr♦r r ♠str ♣tró♥ ró♥ r
st♥t r♠♥t st rst♦ s ♠♦r ② ♥♦ s ♦ ♣♦rí r ♦t♥♦
t③♥♦ ♥ rr♦ rr
♦s ♠♣♦s ♥tr♦rs ♠str♥ r♠♥t ó♠♦ s ♣♦s str s
rtrísts ♣tró♥ ró♥ ② s ♣♥♥ ♦s ♣rá♠tr♦s
rr♦ ♥ú♠r♦ ♣♦♦s ♣♦s♦♥s ♠♣ts ② ss ♦♥♦♠♥t♦
r♦ ①♣r♠♥tr r♥ts ss rr♦s s ♠♣♦rt♥t ♣r sñr
rr♦s ♦♥ str♦♥s s♣ís ♥rí r
r ♥trs rá ♣r♦r♠ rr② ♦♥
r♦ ♣r rr♦s ♥♦ rrs ♥ st♦r rr② Pr♠trs N
s ♥ú♠r♦ t♦t ♣♦♦s a s ♠♣ts ♦rrs♣♦♥♥ts
x/λ y/λ ② z/λ s ♦♥s ♣♦♦ ② phi/π ss
♦rrs♣♦♥♥ts Ptró♥ ró♥ ♦rrs♣♦♥♥t ♦s ♣
rá♠tr♦s N = 4 a = (1 3 3 1) x/λ = (0,0 0,0 0,0 0,0) y/λ =
(0,0 0,5 1,0 1,5) z/λ = (0,0 0,0 0,0 0,0) ② φ/π = (0,0 0,0 0,0 0,0)
r Ptró♥ ró♥ ♦rrs♣♦♥♥t ♥ ♥
t♥ ♦rr Ptró♥ ró♥ ♦t♥♦ st♥♦ ♦s
♣rá♠tr♦s ♥ rr♦ rr ♦♠♣st♦ ♦s ♣♦♦s rá
♦ s♣r ② ♣tró♥ ♦r♥ í♥ ♣♥t♦s Ptró♥
ró♥ ♦t♥♦ st♥♦ ♦s ♣rá♠tr♦s ♥ rr♦ ♥♦
rr ♦♠♣st♦ ♣♦r trs ♣♦♦s
♣é♥
Pr♦r♠ ♥t♦r♥♦
♠ét♦♦ rr♦s s♥tét♦s ♠s♦rs ♦rr ♣r♠t ♠♥
tr rt ♦s ♠♣♦s tr♥s♠t♦s ♣♦r ♦s q♣♦s ② ♦♥♥trr
♥rí s♣♦♥ s♦r ♦s ♥♦s ♥trés tr sñr rr♦
♣r♦♣♦ ♣r s♦ ♠♣ str r♦s ♣rá♠tr♦s ts ♦♠♦ s
t♥ ♥tr ♦s ♦♠♣♦♥♥ts rr♦ ② ss ss rts Pr r③r
ó♥ ♦s ♣rá♠tr♦s ♦s s ♥ ♦♥srr s ♣r♦♣s
♠♦ ♣r♦♥ ♦s ♥♦s ② r♥ ♠só♥ q♣♦
♦rr
♥ ♠r♦ st ss s srr♦ó ♥ ó♦ ♦♠♣t♦♥ ♣r
♠♣♠♥tó♥ s st♥ts t♣s ♠ét♦♦ ♣r♦r♠ s
rr♦♦ ♥ ♥ rsó♥ stá ♦ ♥ trs ♠ó♦s
rr② t♦ rr♠ ♠ Pr♦ss♥ st♦s ♣r♠t♥ ♦
t♥r rá♦ ♠♣♦ ♥ rr♦ ♥ ♥ ♠♦ ♥♦r♠ ♦t♥r
rs♣st rr♦ ♣r t♦s ①♣r♠♥ts ♦ s♥tét♦s ② ♣r ♥ ♠
t♦♦♦í ♣r♦s♠♥t♦ ♣r ♠♥tr ♦♥t♥ tr s sñs
ó♦ t sñ♦ ♦s rr♦s ♣r ♦♣t♠③r s rtrísts
♠♣♦ s♦r ♥♦ srr♦ó ♠ás ♥ ♥t♦r♥♦ rá♦ ♠ ♦♥
rr♠♥ts ♣r ♥áss ♦s rst♦s ♣r s♠♣r ♣ó♥
♠s♠♦
♥t♦r♥♦ ♦♥sst ♥ ♦s ♥t♥s ♥ s s ♦♥t♥ ♥trs
rá ② ♦tr ♦♥t♥ r ♦s rst♦s ♥trs rá ♣r♠t
♥ s♦ s♥♦ ♣r♦r♠ trés ♥ rr ♠♥ú ② r♥ts
♦♥tr♦s stá ♥ trs ♣♥s q ♣r♠t♥ rr ♦s t♦s ♥rr
r ❱♥t♥ ♣rs♥tó♥ ♣r♦r♠ ♥t♦r♥♦
rs♣st rr♦ ② ♠♦r ♠♥ ♦t♥
♥r ♣r♦r♠ s ♠str ♥ ♥t♥ ♣rs♥tó♥ r
♦♥ ♥♦♠r ♣r♦r♠ ② ♦t♦♥s q ♣r♠t♥ ♥rsr ♥♦ ♦s
♠ó♦s rr② t♦ rr♠ ♠ Pr♦ss♥ ♠é♥
s ♠str♥ ♦t♦♥s q ♣r♠t♥ rr ♥t♥ ② ♣ ♦ sr
♣r♦r♠ ①t
♠ó♦ rr② ♣r♠t s♠r ♦♥stró♥ rr♦s ♠
s♦rs ♦rr ♣rtr ♥ts t♣♦ ♣♦♦ ♣rs ♥tr sí ② ♥ ♥
♦♥ró♥ rr s♥tt③r ♠♣♦ rr♦ ② ♠♦strr ♥ rá♦
♠s♠♦ ♠ó♦ t♦ rr♠ ♣r♠t ♦t♥r rrr♠ ♦
rrs♣♦♥♥t ♥ rr♦ ② ♠ó♦ ♠ Pr♦ss♥ ♣r♠t ♦t♥r ♥
rrr♠ ♦♠♣st♦ ♣♦r s♠♥t♦s rrr♠s ♥r♦s ♦♥ st♥t♦
s♣③♠♥t♦ t♠♣♦r
♦♦s ♦s ♠ó♦s stá♥ ♦s ♥ trs ♣♥s ♥♣t t rr② ♣r
♠trs ♠ r ❯♥ ③ q s ♥rs qr ♦s trs
♠ó♦s s ♣ r ♥ ♣rt s♣r♦r ♥t♥ ♥ rr ♠♥ú ♦♥
♦♣♦♥s ♣r ♦♣rr ♦♥ r♦s ♣r ♥rsr ♦s st♥t♦s ♠ó♦s
♣r♦r♠ rr② t♦ rr♠ ♠ Pr♦ss♥ ♦
♥t♥ ② ♣
♦♣ó♥ s♣ ♥ ♠♥ú ♦♥ s ♦♣♦♥s ♣r♠trs r
♥ ♥ r♦ ♦s ♦rs t♦♦s ♦s ♦♥tr♦s q s ♠str♥ ♥ ♥t♥
♦ ♣r♠trs r s ♥ r♦ ♦s ♦rs t♦♦s ♦s ♦♥tr♦s
q s ♠str♥ ♥ ♥t♥ ①♣♦rt t r ♥ ♥ r♦ s ♦s
♦rs rá♦ q s ♠str ♥ r ①♣♦rt r r ♥ ♥
r♦ t ♣ rá♦ q s ♠str ♥ r st ♠♣ ♦s
♦rs q s ♠str♥ ♥ ♦s ♦♥tr♦s ② ①t ♣r♠t sr ♣r♦r♠
ó♦ rr②
♠ó♦ rr② r stá ♦ ♥ trs ♣♥s ♥♣t t
rr② ♣r♠trs ♠ ♣♥ ♥♣t t ♣r♠t ♥rsr ó♥
♦s r♦ q ♦♥t♥♥ ♦s t♦s Pt ♦s r♦s ♥tr ♠ó♦
rr② ♥ ♦♥t♥r ♦s t♦s ♦rrs♣♦♥♥ts ♦s ♦rs ♠♣♦
♥ ♥ó♥ ♣♦só♥ ♣r ♥ ♠♦ ♥♦r♠ ♦ ♣s ♥ ♥r
♦♥ s♠trí trsó♥ ♥ ró♥ ♦r③♦♥t ♦s t♦s ♥tr ♥
♥♦♥trrs ♥ ♥ ♠tr③ rt♥r ♥ ♦r♠t♦ s ② ♦♥ s♣♦ ♦♠♦
s♣r♦r ♥tr♦ ♠tr③ rt♥r ♦s t♦s ♥ str ♦r♥♦s
s♥t ♠♥r ♥ ♣r♠r ♠♥t♦ ♣r♠r ② ♣r♠r ♦♠♥
t♠♣♦ q ♦rrs♣♦♥♥ ♦s t♦s ♦♥t♥♦s ♥ ♠tr③ ♣r♠r
♣♦s♦♥s ♦r③♦♥ts x ♣r♠r ♦♠♥ ♣♦s♦♥s rts z ♥
rst♦ ♦s rs ♦s ♦rs ♠♣♦ ♥ ♦s ♣♥t♦s (x, z) r r
♦♦s ♦s r♦s t♥♥ q t♥r ♠s♠ ♥t s ② ♦♠♥s
♣r♦r♠ r t♦♦s ♦s r♦s q s ♥♥tr♥ ♥ ó♥
♥ ② ♠str ♦s ♦rs ♥ ♥ ② ♣s♦ ♣r t♠♣♦ ② s
♣♦s♦♥s x ② z ♦s t♦s ♦s ♣rá♠tr♦s rr♦ r sr ♦♥sst♥ts
♦♥ st♦s ♦rs
♣♥ rr② ♣r♠trs r ♣r♠t ♥tr♦r ♦s ♣rá♠tr♦s
rr♦ ♥ú♠r♦ ♥ts N s♣ró♥ d ② s♣③♠♥t♦ t♠♣♦r
r ó♦ rr②
rt♦ dt ♥tr ♥ts ② t♠♣♦ ♥ q s qr ♦t♥r ♠♣♦
♣♥ ♠ r ♣r♠t ♠♦strr r♥ts ♠♥t♦s s♦r
♠♥ q s ♣♥ t③r ♥ ♥áss ♠s♠ P♦r ♠♦
♦♣ó♥ ♦♥t♦r ♦ t r♦♥t ♦ s♥ tr♥s♠ttr s ♣♦s s♣r♣♦♥r
r ♥ r ♣r sr t③ ♦♠♦ rr♥ ♣r ♦♠♣rr ♦r♠
♦s r♥ts ♦♥s ♥ ú♥ ♥t ② rr♦ r s ♥ r♦
♣s ♣r♦r♠ ♣r♠t ♥rsr ♦s ♦rs ♦s s♠s rt ②
♦r③♦♥t ♣r♦♠♥t♦ ♦♥sst ♥ ♥rr rá♦ ♠♣♦ ♥
♥t ♣r ♥ tr♠♥♦ t♠♣♦ str r r♥t ♦♥s ② ♦
♥rr rá♦ ♠♣♦ rr♦ ♣r ♠s♠♦ t♠♣♦ st ♦r♠ s
♣♦s ♦srr ♣♦r ♠♣♦ á s r♥♦ ♦s ♣rá♠tr♦s ♣r ♦s q
r♥t ♦♥s ♥♦ s ♦r♠ ♦♥ rs♣t♦ r♥t ♥ ♥t ♦♣ó♥
♥ ♦ t ♠①♠♥ ♠str s♦r r ♥ rt ♥tr ♦r♥
♦♦r♥s ② ♣♦só♥ ♠á①♠♦ ♥t♥s ♠♣♦ ♠str ♥
x1 x2 x3
z1 E11 E12
z2 E21
z3
r ♦r♠t♦ ♦s t♦s ♥tr ♣r ♠ó♦
rr②
♥t♥ ♦r ♥♠ér♦ á♥♦ ♦♣ó♥ ♥r ♣rtr ♠str
s♦r r ♦s í♥s q ♥♥ ♦s á♥♦s ①tr♠♦s ♥ ♦s q ♠♣♦
t♥ ♥t♥s ♥ ♠ó♦ s♣r♦r ♥ ♥t♥s ♠r ♥t♥s
♠r IT s ♥ ♥ ó♥ ♦♥ IMax s ♥t♥s ♠á①♠
♠♣♦ ② δ s ♥ ♦r ♠♥♦r ♦ 1
IT = IMax · δ
s ♥t♥ rá s ♣ rr ♦r δ ♠♣t rt♦♥
♣r ♦srr s ③♦♥s st♥t ♥t♥s r♥t ♦♥s t♠é♥
s ♠str ♦r ♥♠ér♦ ♣rtr ♥r ♥♠♥t ♣♦só♥
♦ rr② ♠str s ♣♦s♦♥s rr♦ s♦r r ② s ♣♦s♦♥s
trt♦♥ ② ①s ♣r♠t♥ ♦♥tr♦r s ♦♦rs ② s s♣
rá♦
♦♠♦ ♠♣♦ ♣ó♥ ♣r♦r♠ s ♦♥sr ♥áss ♠♦
♦ q s ♠str ♥ r Pr ♥rr ♦s t♦s s♠♦s s t③
♥ r♥ ♥tr ♠só♥ fc = 500 ③ ♠♦ ♣r z < 0 r
♣rs♥t ♥ ♣ r ② s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 1
② ♥ ♦♥t σ = 0 s♦ ♣r z > 0 s rtr③ ♣♦r ǫr = 3,5
σ = 1 ♠♠ Pr r③r s♠ó♥ ♥♠ér s t③ ♠ét♦♦
♣é♥ ♥tr♦ r s 0,01 ♠ ♥ ♠s r♦♥s
rá♦ q ♥r ♦♠♦ rs♣st ♣r♦r♠ s ♠str ♥ r
♦♥ ♦s ♣rá♠tr♦s s♦♥♦s N = 5 d = 0,1 ♠ dt = 0,2 ♥s ② t = 8
♥s s ♦t♥ ♥ ♠♣♦ ♦♥♥tr♦ ② ♦r♥t♦ ♥ r t♠é♥ s
♠str♥ ♦s st♥t♦s ♠♥t♦s q ♦♦rr ♥ ♥áss rst♦
r rr♥ q ♣r♠t ♦♠♣rr ♦♥ ♦r♠ r♥t ♦♥s
♥ ú♥ ♥t ♦r♥tó♥ ② ♣rtr ♥r ③ ② s ♣♦s♦♥s
s ♥ts rr♦
r ♦♦ ♥ s♠s♣♦ ♥♦r♠ ♦♥ ǫr = 3,5 ②
σ = 1 ♠♠
ó♦ t♦ rr♠
♠ó♦ t♦ rr♠ r stá ♦ ♥ trs ♣♥
s ♥♣t t rr② ♣r♠trs ♠ ♣♥ ♥♣t t ♣r♠t ♥rsr
ó♥ ♦s r♦ q ♦♥t♥♥ s tr③s Pt
♣r♦r♠ ♠t ♦♠♦ ♥tr t♦s qr♦s ♥ ♦♥ró♥
r♣t♦r ♦♠ú♥ s r q r♣t♦r s ♦♦ ♥ ♥ ♣♥t♦ í♥
s♦♥♦ ② ♦ s qr♥ tr③s ♣r st♥ts ♣♦s♦♥s ♠s♦r s♦r
♠s♠ í♥ Pr ♥ s ♣♦s♦♥s r♣t♦r s♦r í♥
s♦♥♦ s ♦t♥ ♥ ♦♥♥t♦ t♦s q t♠♥t s r ♥ ♥
r♦ s♣r♦ ♥ ♥ ♦♥tr♦ sst♠ ♦rr ♣r♦r♠
r t♦♦s ♦s r♦s q s ♥♥tr♥ ♥ ó♥ ♦s r♦s
♥ ♦♥t♥r ♦s ♦rs ♠♣♦ ♥ ♥ó♥ t♠♣♦ ♣r ♥
s ♣♦s♦♥s s ♣♦s♦♥s ♥ str s♦r ♥ ♠s♠ í♥ ② ♥
s tr③s ♦rrs♣♦♥r ♣♦s♦♥s st♥ts s ♥ts rr♦
♦s t♦s ♥ ♥♦♥trrs ♥ ♥ ♠tr③ rt♥r ♥ ♦r♠t♦ s ② ♦♥
s♣♦ ♦♠♦ s♣r♦r r ♦♦s ♦s r♦s t♥♥ q t♥r
♠s♠ ♥t s ② ♦♠♥s ♥tr♦ ♠tr③ rt♥r ♦s
t♦s ♥ str ♦r♥♦s s♥t ♦r♠ ♥ ♣r♠r ♠♥t♦
♣r♠r ② ♣r♠r ♦♠♥ ♣♦só♥ r♣t♦r xR ♥
♣r♠r s ♣♦s♦♥s ♠s♦r xE ♥ ♣r♠r ♦♠♥ ♦s t♠♣♦s
② ♥ rst♦ s ♣♦s♦♥s ♠tr③ ♦s ♦rs ♠♣♦ ♥ ♥ó♥
♣♦só♥ ② t♠♣♦
♣♥ ♥♣t t ♣r♠t ♣r rt♦s ♣r♦s♦s ♦s t♦s ♦
♠♥ ♦♠♣♦♥♥t r♥ ♦r♠s ♠♣t ♥♦r♠③
r rá♦ ♠♣♦ ♣r N = 5 d = 0,1 ♠ dt = 0,2
♥s ② t = 8 ♥s ♦t♥♦ ♦♥ ♠ó♦ rr②
♠♣t ♥ s tr③s ♣♦r s♣r♦ ② ♦rrt t♠ ♦r♥ ♦rr
♦r♥ t♠♣♦ ♣♥♦ ♦♣ó♥ ♠ rs♠♣♥ s ♣ ♥tr♣♦r
♥ t♠♣♦ ♥ s tr③s ♣♦r s♣r♦ ♥♦tr q ♣s♦ t♠♣♦r
♦s t♦s s ♦rrs♣♦♥ ♦♥ ♠í♥♠ ró♥ q s ♣ ♥tr♦r
♥ s♣③♠♥t♦ t♠♣♦r ♥tr s ♥ts rr♦ ② ♥ ♦♥s♥
♥ á♥♦ ♦r♥tó♥
♣♥ rr② ♣r♠trs ♣r♠t ♥tr♦r ♦s ♣rá♠tr♦s rr♦
♥ú♠r♦ ♥ts N st♥ d ② s♣③♠♥t♦ t♠♣♦r dt rt♦
♥tr ♥ts ② ♦r♥ ♦s ♠♥t♦s r♣t♦r r ♦ ♠s♦r
r
♣♥ ♠ ♣r♠t ♠♦r s♣t♦ ♠♥ ♥t ♦♥ s
♦♣♦♥s ♥ ♣ ♥♥ r P♦st♦♥ rs♠♣♥ ♥tr♣♦
♥ s ♣♦s♦♥s ♠♦ rt rst sñ rt r
trt♦♥ ♣r♠t rr stró♥ s ♦♦rs rá♦ ②
①s ♣r♠t rr ♦s s r
♦♠♦ ♠♣♦ s ♦♥sr♥ ♦s t♦s ♥♠ér♦s ♦t♥♦s ♣r ♠♦♦
r ❯♥ ♦t♦ ♦♥ á♠tr♦ 0,05 ♠ s ♦③ ♥ ♣r♦♥
r ó♦ t♦ rr♠
xR xE1 xE2 xE3
t1 S11 S12
t2 S21
t3
r ♦r♠t♦ ♦s t♦s ♥tr ♣r ♦s ♠ó♦s
t♦ rr♠ ♠ Pr♦ss♥
r ♦♦ ♥ rt♦r ♣qñ♦ ♦♥ á♠tr♦ 0,05♠
♥ ♣r♦♥ 0,80 ♠ ♦♥ ♣r♠t rt ǫr = 5,25
② ♦♥t σ = 2,5 ♠♠ ♥ ♥ ♠♦ ♦♥ ǫr = 3,5 ② σ = 1
♠♠
0,80 ♠ ♦t♦ s rtr③ ♣♦r ♥ ♣r♠t rt ǫr = 5,25 ②
♥ ♦♥t σ = 2,5 ♠♠ ② ♠♦ r♥♥t ♣♦r ǫr = 3,5 ② σ = 1
♠♠ ♠♦ s♦r ♦s s r ♥tr♦ r s 0,01 ♠ ♥ ♠s
r♦♥s ♥ ♦s ♣rá♠tr♦s s♦ s ♣♥ t♦♥s t♦rs
r♥ ♥tr s ♦♥s ♠ts s fc = 500 ③ ♥ r
s ♠str rs♣st ♦t♥ ♣r N = 5 d = 0,1 ♠ ② dt = 0,2 ♥s
P♦r ♠♦ ♣♥ ♠ s st ♠♦♦ s③ó♥ ♣r ♦srr
♥ ♦r♠ rst♦
ó♦ ♠ Pr♦ss♥
♠ó♦ ♠ Pr♦ss♥ stá ♦ ♥ trs ♣♥s ♥♣t t rr②
♣r♠trs ♠ r ♦s ♣♥s ♥♣t t ♠ ♦♥♥
♦♥ ♦s q ♣r♥ ♥ ♠ó♦ t♦ rr♠
♣♥ rr② ♣r♠trs ♣r♠t ♥tr♦r ♦s ♣rá♠tr♦s rr♦
♥ú♠r♦ ♥ts N st♥ rt ♥tr ♥ts d ② ♦r♥ ♦s
♠♥t♦s r♣t♦r r ♦ ♠s♦r r ♥ st ♣♥ s ♠s
tr♥ ♠ás trs ♦♠♥s ♠♥t s s s ♣♦s ♥rsr ♦s í♠ts
♦s ♥tr♦s ♥t ♣♦st♦♥ ② ♥ ♣♦st♦♥ ② s♣③♠♥t♦ t♠♣♦r
♥ ♥♦ ♦s t t♠ st t♥ s♦rs ♥ r
s ♠str rs♣st ♦t♥ ♣r ♠♦♦ r ♦♥ N = 5
d = 0,1 ♠ ② dt s♦♥♦ ♠♥r rstr sñ rt♦r ♥
♥tr♦
r rá♦ ♠♣♦ ♣r N = 5 d = 0,1 ♠ ② dt = 0,2
♥s ♦t♥♦ ♦♥ ♠ó♦ t♦ rr♠
♣♥ ♠ ♥t ♦♥ ♦♣ó♥ ♦ t ♠ts ♦ t ♥trs q
♠str ♦s í♠ts ♦s ♥tr♦s ♥ ♦s q s ♣♥ ♦s st♥t♦s ♦rs
dt
rtrísts ♥rs ♣r♦r♠
♥ t♦s s ♠♦s ♣r♦r♠ ♥♦ s ♠♦ ♥♦
♦s ♣rá♠tr♦s ♥tr ♦♥tr♦ ♠ ♦♦r ♥♥♦ q ♦r q
s ♠str ♥♦ ♦rrs♣♦♥ rá♦ ❯♥ ③ q s t③ rá♦
♣♦r ♠♦ ♦s ♦t♦♥s ♦ ♦ ♣♣② ♦s ♦♥tr♦s rrs♥ s ♦♦r ♥
♣r♦r♠ ♥t ♦♥ ♥♦s ♠♥t♦s q t♥ s s♦ ♣♦r ♠♣♦ s s
trt ♥rr r s♥ r r♦ ♦s t♦s ♣r♠♥t s ♠str
♥ s♦ ♦ ♠s♠♦ ♦rr s s ♥t♥t ♣r ú♥ t♣♦ ♣r♦s♠♥t♦
♥ts r ♥r♦ r
♣r♦r♠ t♥ rss ♣♦♥s ♣♦r ♠♣♦ s ♦ ♣ t③r
♣r sñr rr♦ ♥ts r③r ♥ st♦ ♠♣♦ ♣r st♠r
♥t ♥ts ② s♣ró♥ ♥tr s q ♣r♠t♥ ♦t♥r
t♦s ♦s ♣r ♣r ♠ét♦♦ ♠é♥ ♣r♠t tr♠♥r ♦s
♣rá♠tr♦s rr♦ ♣r rstr sñs ♥ tr♠♥s ③♦♥s rr
r ó♦ ♠ Pr♦ss♥
r♠ ♣♦r ♠♣♦ st♥ts ♣r♦♥s á♥♦ s t③ ♠ó♦
t♦ rr♠ ② s ♦♥ t♦s s♠♦s ♦ ①♣r♠♥ts ♣r♦
r♠ ♣r♠t str s st♥ts sñs ② rr ♦s ♣rá♠tr♦s rr♦
st r③r ♥ st ♥♦ ♦s ♣rá♠tr♦s st♠♦s ♦♥ ♠ó♦ rr②
♥♠♥t ♠ó♦ ♠ Pr♦ss♥ s ♣ t③r ♥ s♦ ♥
q s sñs ♣rs♥t♥ r♦♥s ♥♥ó♥ r♥s ② q ♥ ú♥
s ♥♦ ♣r♠t rstrs ♥ s t♦t ♥ s♦ ♦srr q sñ
♥trés ♥♦ s rst ♥ t♦ s ①t♥só♥ ♥♦ s ♦sr rs♣st
♠ét♦♦ ♠ó♦ t♦ rr♠ ♠t♦♦♦í ♦♥sst
♥ ♣sr ♠ó♦ ♠ Pr♦ss♥ ② s♦♥r s ♣r♦♣ ♣r ♦s
st♥t♦s ♥tr♦s ♣♦só♥ ♣r ♠♥tr sí ♦♥t♥ tr s
sñs ♥trés
r rá♦ ♠♣♦ ♣r N = 5 d = 0,1 ♠ ② dt
s♦♥♦ ♠♥r rstr sñ rt♦r ♥
♥tr♦ ♦t♥♦ ♦♥ ♠ó♦ ♠ Pr♦ss♥
r♠♥t♦s
r♦ rr ♥ ② ést♦r ♣♦r r♠ ♦ r♦ st ss ♦♥
t♥t ♣♥ ② ó♥
r♦ rr t♦s s ♣rs♦♥s q ♦r♠♥ r♣♦ ♦ís
♣ ② ♠♥t ♣♦r s t♦ ♥ t♦♦s ♦s ♠♦♠♥t♦s ♦♠♣rt♦s
r♦ rr ♠ ♠ ♣♦r ♣♦②r♠ s♠♣r
♦rí
❬❪ ♦s Prs st ♥ ❱ ♥ ♥
♠♦t s♥s♥ ♥ P sts ♦ t♦ t ts ❲str♥ ♥t
P♥s ♥ ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ ♦♠③ ♥ ♠♦♦ s♣♥ss ♥ ❲ss♠r P
♦♥s ♦ ♣♦st♦♥ ♦♥♣t ♠♦ r♦♠ P
sr② ♦♥ ♣②r♦st♦ ♣♦sts t r♣ ❱♦♥♦ ♥♦♥s ♦
♠♦r♣♦♦②
❬❪ Ptt♥ ♥ ♦♠r ♥ ♥♥♥ P P
♥ ♦♠str② ♠sr♠♥ts ♦ ♥ t s ♥t t♦ st②
♥rsr s♠rt♦♥ ♣t②s ♦♣②ss
❬❪ ♦♥♦♠♦ r♥ s ♥ tt♦ P ♣r♦s♣t♥
♥ ♣rs♣♥ ❲ r♥t♥ ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ r rr r t♥ss ♦
r♦♥♣♥trt♥ rr sr②s ♥ t ♦t♦♥ ♦ ♥♠r r sts
♥ ♠♦r♥ ♠trs ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ Pér③r ❱ ss ♣s s♦r♦ rtí♥③ ♥
♥s ♥trt ♥rsr ♦♣②s sr② ♦ t tr
♦ ♦r ♦r♥ ♦ r♦♦ ♥
❬❪ r♥♦ ♦ ❯s♥ ♠t♦♠♣♦♥♥t P t♦ ♠♦♥t♦r rs
♥ st♦r ♥ ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ P♦rs♥ ♥ ❲ r♦♥♣♥trt♥ rr ♣r♦s ♦r
♠t♣ st t♥s rtt r♠♦ tr♦ t t ♣r♦ss♥
♦♣②ss
❬❪ ♥s♠t ♥ ♦r♦♣♦♦s ♥s♣t♦♥ ♦ rt
♥♥ s s♥ P ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ r♦♦ Prs♦ ♦♦r ♥ ss② r②st
♥ ♣♣s P♠♦♥t♦r♥ ♠r♦ t♦♠♦r♣ ♥rs♦♥ ♦r
♥ ♦ ♣♣ ♦♣②ss
❬❪ s♠ ❨ ❩♦ t♦ st♠t♦♥ ♦ r♦♥tr
② P ♥ ♥ r t ♠t♣ ♠♦s rt♦♥s ♦r♥ ♦ ♣♣
♦♣②ss
❬❪ ♦♦r Prs♦ ♥ Prs♦ ♣♣t♦♥ ♦ ♠r♦
t♦♠♦r♣② ♥ ②r♦♦♣②ss ♦♠ ①♠♣s ❱♦s ❩♦♥ ♦r♥
❬❪ ♦♥ rr② ♦r③ ♥ ❩♥ ♥trt
♦♣②s ♥ ♦♦ ♥stt♦♥ ♦ tr♦♥♦s qr
♥ ♦♠s ssss♣♣ ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ s ♦♦ ③③♦ ♥ ♣♥♥ ❱ ♦♣②s
rtrst♦♥ ♦ r♦s ❱ st r♦t♦♥ t② ♦r♥ ♦ ♣♣
♦♣②ss
❬❪ ♥ r ❯s ♦ r♦♥ ♣♥trt♥ rr t♦ ♠♣
ssr r♦♦ trs ♥ ♥ r♥ r ♦r♥ ♦ r♦
♦ ♥
❬❪ r♥s ♥③ ♥ ♦r rs♦t♦♥ ♠t
♥♥ rr ♥stt♦♥ ♦ ♦♠♥ ♥ ♦rtr♥ t② ♦r♥
♦ ♣♣ ♦♣②ss
❬❪ rs♠ ❲r ♥ ♦rst♠②r rs♦t♦♥
P ♠♥ ♦♣②ss
❬❪ ♦ ❨ ♥ ♥ Ptt♥ ♥ st ♥
♠sr♠♥ts ♦ rt♦♥ ♣ttr♥s ♦ ♣♦ P ♥t♥♥s
♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ ③s ♥ Ptrs r ♥s r
♣♦ rt♦♥ ②♥♠s tr♦ ♠♦♥ ♦r♥ ♦ ♣♣
♦♣②ss
❬❪ rr ♥ ♦ t♦st r♦♥ ♣♥trt♥ rr
t ♦r ♠♣r♦ ♠♥ ♥ rs ♦ tr ♦♠♣①t② ♣♣t♦♥ t
t ♠r♥ st ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ P♣♥ r♦ ♦rt Pr③③♦♥ ♥ ♥tt
♥ ♣r♦ss♥ ♥ ♥tr♣rtt♦♥ ♦ ♠t♦ r♦♥ ♣♥trt♥
rr t s st♦r② r♦♠ ♥ r♦♦ st ♦r♥ ♦ ♣♣
♦♣②ss
❬❪ r Pr♥ ♠♥ ♦♥ rt♦ P ♥ ♥rs
sts ♦ t P r♦♥ ♠t♦ ♦r st♠t♥ s♦
tr ♦♥t♥t r♥ rrt♦♥ ♥ r♥ ♦r♥ ♦ ②r♦♦②
❬❪ r ♣♣②♥ ❱ ♥②ss t♦ P t ♦♣②s
sr ttrs
❬❪ ♦r♥ r ♥♥ ssr ❯s♥ ♠♣t
rt♦♥ t ♦st ♥ ♥♦r♠③ rs ♣♦r③t♦♥ ♥②ss ♦ r♦♥
♣♥trt♥ rr t t♦ r♥tt ♥ P rs r♦♠ strtr♣
♥s ♦r♥ ♦ ♣♣ ♦♣②ss
❬❪ r♦♥ ♦t♦ s ♥ ❱
rs♥ rt♦♥ ♦♥ts ♦r P❱ ♥②ss ♥ tt♦♥ ♦ s
tr ♥ P ♦♥t♠♥♥ts r r ♦♣②ss
❬❪ r♥ r r♥s ts P s♦♥t ♠♥
rs ♥s♥ ♥ rst♦ ♥t rs ♥
rr♦♥ tt② ♠♣♣♥ s♥ ♥ ♠t♥♥ r♦♥♣♥trt♥
♠♥ s②st♠ Pr♦♥s ♦ ♥ ♥♥ ♦♥r♥ ♦♥ t ♣♣
t♦♥ ♦ ♦♣②s ♥ t♦♦♦s t♦ r♥s♣♦rtt♦♥ ts
♥ ♥rstrtr ♦s ♥s ❯ P❨
❬❪ t♦ ♠ ❨ ♥ ❳ ♦♥ ❩♥ ❩ ♥ ♥
P s♥ ♥ rr② ♥t♥♥ ♦r ♥♠♥ tt♦♥ r r
♦♣②ss
❬❪ ♥ ♥ ♦♥ P♥ ♠rt♦♥ ♥ tt ♦♦r♥ts
♦♣②ss
❬❪ t♦ P ♥ ♦ Pst♥ ♥ ♦♠
P♥ ♣t ♠rt♦♥ ♦♣②ss
❬❪ r♦♥r ♥ ♦♥♦♥r ♥stt♥ tt♥
t♦♥ sttr♥ ♣s ♥tr ♥ t♦t t s♥ s♠t ♥trr♦
♠tr ♠s ♦ ♦rst rs r♥st♦♥s ♦♥ ♦s♥s
♥ ♠♦t ♥s♥
❬❪ ❲♥ ❲ ♣♣r♦ ♦ ♣t s②♥r♦♥③t♦♥ ♦r stt
rt♠ ♠♥ r♥st♦♥s ♦♥ ♦s♥s ♥ ♠♦t ♥
s♥
❬❪ s♥ r♥ ♥ rr P ♦♣♠♥t ♦ ♦♠
♣t P rr s ♦♥ ♠ts♥s♦rs t♥♦♦② Pr♦♥s ♦
t ♥ ♥tr♥t♦♥ ❲♦rs♦♣ ♦♥ ♥ r♦♥ P♥trt♥
r tr♥s
❬❪ s ❯ ♦r ♥ ♥ r♥♥ Ps rr② t♥♦♦②
♦r P ♥t♥♥ s♥ ♦r ♥r ssr ①♣♦rt♦♥ Pr♦♥s ♦
t ♥ ♥tr♥t♦♥ ❲♦rs♦♣ ♦♥ ♥ r♦♥ P♥trt♥ r
tr♥s
❬❪ ♥ ❩ ③♥ ♥ ♥ ♦①♥ ❳ sr ♦♥ t
♦♠♥t♦♥ ♦r Psrr② r♦♥ P♥trt♥ r ❲♥ ❯♥r
st② ♦r♥ ♦ tr ♥s
❬❪ ② ❨r♦♦② ♥ trt ❲t ♥r ♦
s♥ ♥ ♥ rr②s P ♦ ♥
❬❪ t③ P ♥ Prr♦ Psrr② tr♥s♠ttrs ♦r P sr
②s ♦♣②s ♥
❬❪ s♦♥ ss tr♦②♥♠s ❲② ❨♦r
❬❪ trtt♦♥ tr♦♠♥t ♦r② r ❨♦r
♥ ♦♥♦♥
❬❪ ♥ s ② ♦sé s ♥t ♦♠♣♦rs rq♦♦ís
❯♥ ♠t♦♦♦í ♥trs♣♥r ♣r ①♣♦rr ♣s♦ ♥ó♥
st♦r tr ❯♥rs ♠ó♥s s s
❬❪ t ♣ts s♦♥ ❲s② ♥ r♥s♦
❬❪ ♦ t♦r r♦♥ P♥trt♥ r ♦r② ♥ ♣♣
t♦♥s sr tr♥s
❬❪ ❨♠③ s♠ t ♥②ss ♦t② ♦ ①♣♦rt♦♥ ♦♣②
ss
❬❪ sr ♥ ♥♥♥ P ♦s② ①♠♣s ♦
rrst♠ ♠rt♦♥ ♦ s♥♥♥ r♦♥♣♥trt♥ rr ♣r♦
s ♦♣②ss
❬❪ ♠♣é ♦♥tró♥ rq♦♦í á♥
ss ♦t♦r ❯♥rs ♦♥ Pt ♥♦s rs
r♥t♥
❬❪ ♠♣é s trs r♦rrs ♣rs♣á♥s
á♥ ♥♦stt♠r ♦♥s ♦ r♥t♥
♥tr♦♣♦♦í
❬❪ tt♦ ② ♥ s ♦①st♥ sñ♦s t♥♦
stíst♦s ♥ Prí♦♦ rí♦ ♣r♥♦ s♦ ♥trr♦ ♥ r♥
é r♦② ♥♦st t♠r r♥t♥ st ♥tr
s♦♥s ♥ ♥tr♦♣♦♦í
❬❪ r♦♥ r♦♥ rr s♠t♦♥ ♦r r♦♦ ♣♣
t♦♥s ♦♣②s Pr♦s♣t♥
❬❪ r♦♥ r♦♥rr ♥♠r ♠♦♥ ♣♣ t♦ ♥
♥r♥ ♣r♦♠s r♦♣♥ ♦r♥ ♦ ♥r♦♥♠♥t ♥ ♥♥r♥
♦♣②ss
❬❪ r♥ ♦♥♦♠♦ s ♥ ♣♣t♦♥ ♦ t s②♥t
t ♠ttrrr② ♠t♦ t♦ ♠♣r♦ P s♥s ♦r♥ ♦ ♣♣
♦♣②ss
❬❪ r♥ ♦♥♦♠♦ s Ps♥ ♠♣r♦♠♥t tr♦
t s②♥tt ♠ttr rr② ♠t♦ st② ♦ ts rtrsts ♦r♥
♦ ♣♣ ♦♣②ss ♥í♦ ♥♦♠r ♥s
r♣t ♥♠r PP
❬❪ rt♥♦ ♦♥♦♠♦ s♥♦ s tt♦ tr
♥ P ♣r♦s♣t♥ t P♦ ♥♦ r♦ st ♥♦rtstr♥
r♥t♥ ♦♣②ss
❬❪ ❨ ♠r s♦t♦♥ ♦ ♥t ♦♥r② ♣r♦♠s ♥
♦♥ ①s qt♦♥s ♥ s♦tr♦♣ ♠ r♥st♦♥s ♦♥
♥t♥♥s ♥ Pr♦♣t♦♥ P
❬❪ ❱♥ st ♥ t♦ P ♣♣t♦♥ ♦ ♦r r♦tt♦♥ t♦
r♦♥♣♥trt♥ rr t ♦♣②ss
❬❪ ♥s ♥t♥♥ ♦r② ♥②ss ♥ s♥ ♦♥ ❲② ♥
♦♥s ♥ ❨♦r
❬❪ r♥ ♥ ♥t ♠r ♠♦♥ ♦ r♦♥♣♥trt♥
rr ♥ s♥ ♦♠♣trs ♦s♥s
❬❪ ♥s ♥t♥♥ ♦r② Pr♦♥s ♦ t
❬❪ sr♦t ❲ ♥ r♠♥ ♦ tt P②ss ♥rs
♦ ♦r ❲♦rt
❬❪ r♦♥ t♦♥ Pttr♥s ♦r P ♦rr ♦♥
♦♣②ss