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,
and
Use Duhamel's Principle and Kirchoff's formula to show that
where
and is the ball in with center at x and radius t.
2. Relevant equations
Duhamel's Principle
Let be the solution of the associated (to the above initial value problem) "pulse problem"
and
then
Kirchoff's Formula
Suppose where k is any integer 2 Then the solution of
and
is given by
where S(x,t) is the surface of the sphere with radius t and centre at the point x. 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
So a transformation into kirchoff's formula to give the required initial conditions,
so
and
becomes
and
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 as it doesn't affect kirchoffs formula?
and
but I can't really see how I would get to the required answer with this, so
and
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...