Równanie falowe.pdf

Równanie falowe

Równanie falowe to matematyczne równanie różniczkowe cząstkowe drugiego rzędu,

opisujące ruch falowy .

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 operatora d'Alemberta:

Rozwiązania równania falowego mają różne postaci i własności w zależności od parzystości

wymiaru przestrzeni. Najważniejsze równania falowe to przypadki

Równanie falowe jest ważne w mechanice kwantowej , gdyż opisuje falę de Broglie'a:

Równanie falowe można wyprowadzić z równań Maxwella.

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 d'Alemberta '. Równanie struny jest wówczas poprawnie postawione.

Równanie falowe

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 średnich sferycznych. Przy założeniu

regularności

rozwiązaniem jest:

Jest to wzór Kirchhoffa .

Równanie falowe w wymiarze 2 i wzór Poissona

Równanie falowe dla

można rozwiązać metodą spadku. Przy założeniu regularności

rozwiązaniem jest:

Niejednorodne równanie falowe w wymiarze 3

Niejednorodne równanie falowe to równanie postaci:

Równanie to można rozwiązać metodą całek Duhamela . Wynikiem jest:

Zaburzenia fali rozchodzą się więc po 4-wymiarowym stożku

Równanie falowe

Zasada Huygensa

Zasada Huygensa opisuje pewną własność rozwiązania równania falowego, w zależności od

parzystości wymiaru przestrzeni. Podamy ją na przykładzie

oraz

Załóżmy, że funkcje f, g mają zwarte nośniki

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 amplituda maleje jak .

Referencje

• Lawrence C. Evans , Równania różniczkowe cząstkowe , PWN, 2002 , Warszawa

Równanie falowe

Źródła i autorzy artykułu

Równanie falowe Source : http://pl.wikipedia.org/w/index.php?oldid=16714839 Contributors : 4C, Awmarcz, BartekChom, Ekr, Kakaz, Loxley,

Luka.stuka, Milek80, RManka, Reytan, Sfu, Stepa, Stok, W2023, WarX, 4 anonymous edits

Równanie falowe

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