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,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




and \overline{B}(x,t) 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"


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


Kirchoff's Formula

Suppose p\in C^k(\Re^3) where k is any integer \geq2 Then the solution of


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


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


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.

or can I just use anything I like for \tau as it doesn't affect kirchoffs formula?

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

but I can't really see how I would get to the required answer with this, so

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

which is a bit of a desperate attempt to match the format of the given answer.

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...