Solve Wave Equation by Laplace Transform
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Reference: Wave Motion in Elastic Solid, Karl F. Graff.
\[T\frac{\partial^2y}{\partial x^2}-\rho\frac{\partial^2y}{\partial t^2}=-q(x,t)\]Solution by Laplace transform
Consider the case of a harmonically varying force acting at a single point $x=\xi$
\[q(x,t)=P\delta(x-\xi) e^{-i\omega t}\] \[\frac{\partial^2 y}{\partial x^2}-\frac{1}{c_0^2}\frac{\partial^2 y}{\partial t^2}=\delta(x-\xi) e^{-i\omega t}\]where P=-T has again been selected. Let $y(x,t)=\phi(x) e^{-i\omega t}$ so that
\[\phi''(x)+\beta^2\phi(x)=\delta(x-\xi) e^{-i\omega t}\]
