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I want to get the fundamental solution for the following 1D nonhomogeneous wave equation:\begin{align}\left\{ \begin{aligned} &\frac{\partial^2u}{\partial^2t}-\frac{\partial^2u}{\partial^2x}-au=0,x\in R^1,t>0,a\ is\ a\ constant.\\& u(x,0)=0,u_t(x,0)=\phi(x) \end{aligned} \right.\end{align} I know the fundamental solution should statisfy the following equation:\begin{align}\left\{ \begin{aligned} &\frac{\partial^2E}{\partial^2t}-\frac{\partial^2E}{\partial^2x}-aE=0,x\in R^1,t>0,a\ is\ a\ constant.\\& E(x,0)=0,E_t(x,0)=\delta(x) \end{aligned} \right.\end{align}Use the fourier transform and then: \begin{align}\left\{ \begin{aligned} &\frac{\partial^2\hat{E}}{\partial^2t}+(\xi^2-a)\hat{E}=0,\xi\in R^1,t>0,a\ is\ a\ constant.\\& \hat{E}(\xi,0)=0,\hat{E}_t(\xi,0)=1 \end{aligned} \right.\end{align} Solve the equation then we got:$$\hat{E}(\xi,t)=\frac{sin(\sqrt{\xi^2-a}t)}{\sqrt{\xi^2-a}}$$Using the inverse Fourier transform:$$E(x,t)=\frac{1}{2\pi}\int_R\frac{sin(\sqrt{\xi^2-a}t)}{\sqrt{\xi^2-a}}e^{-ix\xi}d\xi=\frac{1}{\pi}\int_{0}^{\infty}\frac{sin(\sqrt{\xi^2-a}t)}{\sqrt{\xi^2-a}}cos(x\xi)d\xi$$I don't know how to compute this integration,or maybe I shouldn't use the fourier transform to solve this problem? Appreciate anybody who answer this question first

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  • $\begingroup$ "I don't know how to compute this integration" - that may be, but it doesn't mean that the solution is wrong. Giving a solution in the form of an integral is valid, in general you can always compute the integral numerically $\endgroup$
    – Yuriy S
    Commented Apr 17, 2019 at 11:43

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If you replace $au$ and $aE$ with $k(x,t)$ then you will have the non homogeneous wave equation as its normally given.

If you then set $E(x,0)=E_t(x,0)=0$ you will have the purely non homogeneous equation with a forcing function and no initial conditions.

Then setting $E(x,0)=0, E_t(x,0)=g(x,t)$ and $k(x,t)=0$ will give you the non homogeneous wave equation with a velocity initial condition.

Finally, setting $E(x,0)=f(x,t), E_t(x,0)=0$ and $k(x,t)=0$ will give you the non homogeneous wave equation with a displacement initial condition.

So you can get three separate solutions, and the final solution is their sum.

If when used, $k,f,g$ are set to $\delta$ then the sum of the solutions will be the fundamental solution.

The homogeneous solutions can be found from D'Alemberts formula. The non homogeneous solution is a little more work--Google it.

Bottom line: I think your wave equation may be wrong, and you do not need the Fourier transform.

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