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Wave equation on a rectangular domain

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2-40

Consider the following wave equation:

utt = c2 uxx, 0<x<a, 0<y<b

Subject to the following boundary conditions:

u(0,y, t) = 0, u(a, y, t) = 0, 0<y<b, t>0
u(x,0, t) = 0, u(x, b, t) = 0, 0<x<a, t>0

Find an expression for the solution if the initial conditions are:

a) u(x, y, 0) = xy(a-x)(b-y), ut(x,y) = 0

b) u(x, y) = 0, ut(x,y) = sin(5x/a) sin(7y/b)

c) u(x, y, 0) = xy(a-x)(b-y), ut(x,y) = sin(5x/a) sin(7y/b)

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https://brainmass.com/math/fourier-analysis/wave-equation-rectangular-domain-626377

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The wave equation on the rectangular membrane is:
(1.1)
The boundary values are Dirichlet's boundary conditions:
(1.2)
And
(1.3)
And general initial conditions:
(1.4)
We start wit writing the wave function as a product of univariate functions:
(1.5)
This makes the boundary conditions a sngle-function conditions:

(1.6)
And:
(1.7)

We them plug (1.5) into (1.1), and the partial derivatives turn into full derivatives:

(1.8)
The left hand side is a function of t while the right hand side is a function of , and since it must hold true for any combination of both sides must be equal to a constant:
(1.9)
The right hand side of (1.8) can be deconstructed further:
(1.10)
Now the left hand side is a function of x while the right hand side is a function of y, so both sides must be equal the same constant:
(1.11)
But we could also write:
(1.12)
Therefore, we can define a different constant :
(1.13)
Note that
(1.14)
So we can say we have three independent second order ordinary differential equations.
One is t-dependent:
(1.15)
The second is x-dependent
(1.16)
With boundary conditions
And the third equation is y-dependent
(1.17)
With initial boundary conditions
Note that ...

Solution Summary

The 16-page solution shows how to find the solution for a wave equation on a rectangular domain with different initial conditions

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