# Wave equation on a rectangular domain

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(5x/a) sin(7y/b)

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

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#### Solution Preview

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