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Homework Help: Nonhomogeneous wave equation with vanishing initial conditions

  1. Jul 9, 2010 #1
    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,

    [tex]u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=f(x_1,x_2,x_3,t)[/tex]

    [tex]u(x,0)=0[/tex] and [tex]u_t(x,0)=0[/tex]

    [tex]x\in\Re^3 , t>0[/tex]

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

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

    where

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

    and [tex]\overline{B}(x,t)[/itex] is the ball in [itex]\Re^3[/itex] with center at x and radius t.
    2. Relevant equations

    Duhamel's Principle
    Let [itex]v(x,t;\tau)[/itex] 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[/tex]

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

    then
    [tex]u(x,t)=\int^t_0v(x,t;\tau)d\tau[/tex]


    Kirchoff's Formula

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

    [tex]u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=0[/tex]

    [tex]u(x,0)=0[/tex] and [tex]u_t(x,0)=p(x)[/tex]

    [tex]x\in\Re^3 , t>0[/tex]

    is given by

    [tex]\frac{1}{4\pi t}\int_{S(x,t)}p(x')d\sigma_t[/tex]

    where S(x,t) is the surface of the sphere with radius t and centre at the point x. [itex]d\sigma_t[/itex] 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 [itex]t=\tau[/itex]

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

    so

    [tex]v(x,\tau;\tau)=0[/tex] and [tex]v_t(x,\tau;\tau)=-f(x,\tau)[/tex]

    becomes

    [tex]v(x,0;\tau)=?[/tex] and [tex]v_t(x,0;\tau)=-f(x,?)[/tex]

    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...:)
     
  2. jcsd
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