# Nonhomogeneous wave equation with vanishing initial conditions

1. Jul 9, 2010

### Mmmm

1. The problem statement, all variables and given/known data

Let u(x,t) be the solution of the following initial value problem for the nonhomogeneous wave equation,

$$u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=f(x_1,x_2,x_3,t)$$

$$u(x,0)=0$$ and $$u_t(x,0)=0$$

$$x\in\Re^3 , t>0$$

Use Duhamel's Principle and Kirchoff's formula to show that

$$u(x,t)=-\frac{1}{4\pi}\int_{\overline{B}(x,t)}\frac{f(x',t-r)}{r}dx'_1dx'_2dx'_3$$

where

$$r=\left|x-x'\right|=[(x_1-x'_1)^2+(x_2-x'_2)^2+(x_3-x'_3)^2]^\frac{1}{2}$$

and $$\overline{B}(x,t)[/itex] is the ball in $\Re^3$ with center at x and radius t. 2. Relevant equations Duhamel's Principle Let $v(x,t;\tau)$ be the solution of the associated (to the above initial value problem) "pulse problem" [tex]v_{x_1x_1}+v_{x_2x_2}+v_{x_3x_3}-v_{tt}=0$$

$$v(x,\tau;\tau)=0$$ and $$v_t(x,\tau;\tau)=-f(x,\tau)$$
$$x\in\Re^3 , t>\tau$$

then
$$u(x,t)=\int^t_0v(x,t;\tau)d\tau$$

Kirchoff's Formula

Suppose $p\in C^k(\Re^3)$ where k is any integer $\geq$2 Then the solution of

$$u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=0$$

$$u(x,0)=0$$ and $$u_t(x,0)=p(x)$$

$$x\in\Re^3 , t>0$$

is given by

$$\frac{1}{4\pi t}\int_{S(x,t)}p(x')d\sigma_t$$

where S(x,t) is the surface of the sphere with radius t and centre at the point x. $d\sigma_t$ is the element of surface on S and x' is the variable point of integration.

3. The attempt at a solution

I should split this into two parts, one for each formula.
first I need to use Kirchoff's formula to find v and then Duhamel's principle to find u.

The problem with using Kirchoff is that the initial conditions are given at t=0 whereas our initial conditions for v are at $t=\tau$

So a transformation into kirchoff's formula to give the required initial conditions, $t'=t-\tau$

so

$$v(x,\tau;\tau)=0$$ and $$v_t(x,\tau;\tau)=-f(x,\tau)$$

becomes

$$v(x,0;\tau)=?$$ and $$v_t(x,0;\tau)=-f(x,?)$$

which doesn't give me anything because of that parameter in v. The t value and the parameter must be equal to give a known value.

Can anyone give me a clue as to how to get Kirchoff's formula to work with this? Or am I going in completely the wrong direction?
I'm sure that this is only the first of many sticking points in this question but I thought I'd ask one question at a time...:)