Równanie falowe, - fizyka, wikipedia artykuły

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Równanie falowe
1
Równanie falowe
Równanie falowe
to matematyczne
drugiego rzędu,
opisujące ruch
.
Ogólną postacią równania falowego jest:
gdzie
oznacza zbiór liczb rzeczywistych nieujemnych. W równaniu funkcja
jest
niewiadomą opisującą wychylenie fali w punkcie
w chwili . Zadane są początkowe
położenie fali
oraz początkowy impuls . Fizycznie stała
oznacza prędkość światła.
Matematycznie zwykłe przyjmuje się .
Skrótowo można wyrazić równanie falowe używając
Rozwiązania równania falowego mają różne postaci i własności w zależności od parzystości
wymiaru przestrzeni. Najważniejsze równania falowe to przypadki
.
Równanie falowe jest ważne w
, gdyż opisuje
Równanie falowe można wyprowadzić z
Rozwiązania równania falowego
Równanie struny i wzór d'Alemberta
Jednowymiarowe ( ) równanie falowe nazywa się równaniem struny lub równaniem
fali płaskiej. Ma ono postać:
Bez uwzględnienia warunków brzegowych rozwiązaniem jest:
gdzie
są dowolnie wybrane. Przy założeniu regularności
oraz
uwzględnieniu warunku brzegowego rozwiązaniem jest:
Jest to
'wzór
'.
Równanie struny jest wówczas
 Równanie falowe
2
Równanie struny półnieskończonej
Struna półnieskończona to jednowymiarowa struna przymocowana na stałe z jednego
końca. Matematycznie odpowiada dodaniu dodatkowego warunku brzegowego:
dla dowolnego
Rozwiązaniem zagadnienia struny półnieskończonej jest:
Równanie falowe w wymiarze 3 i wzór Kirchhoffa
Równanie falowe dla
ma postać
Rozwiązanie równania można wyprowadzić za pomocą
Przy założeniu
regularności
rozwiązaniem jest:
Jest to wzór
.
Równanie falowe w wymiarze 2 i wzór Poissona
Równanie falowe dla
można rozwiązać
Przy założeniu regularności
rozwiązaniem jest:
Niejednorodne równanie falowe w wymiarze 3
Niejednorodne równanie falowe to równanie postaci:
Równanie to można rozwiązać
. Wynikiem jest:
Zaburzenia fali rozchodzą się więc po 4-wymiarowym stożku
.
 Równanie falowe
3
Zasada Huygensa
Zasada
opisuje pewną własność rozwiązania równania falowego, w zależności od
parzystości wymiaru przestrzeni. Podamy ją na przykładzie
oraz
.
Załóżmy, że funkcje f, g mają
.
Niech . Ze wzoru Kirchhoffa wynika wówczas, że
tylko w pewnym
skończonym czasie . Zatem falę, np. dźwiękową, zgodnie z doświadczeniem,
słychać od pewnego momentu, przez skończony czas.
Inaczej dzieje się dla
. Ze wzoru Poissona wynika, iż fala zaczyna brzmieć i nigdy nie
przestaje, choć jej
maleje jak .
Referencje

,
Równania różniczkowe cząstkowe
, PWN,
, Warszawa
 Równanie falowe
4
Źródła i autorzy artykułu
Równanie falowe
 
Source
:
 
Contributors
:
4C, Awmarcz, BartekChom, Ekr, Kakaz, Loxley,
Luka.stuka, Milek80, RManka, Reytan, Sfu, Stepa, Stok, W2023, WarX, 4 anonymous edits
 Równanie falowe
5
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