Question

Determine if the function y = A sinkx cosωt is a solution to the wave equation.

Answer 1

Given:

$y(x,t)=A\sin kx\cos \omega t$

The wave equation

$$\frac{\partial^2y}{\partial t^2}-v^2\frac{\partial^2y}{\partial x^2}=0$$

The speed of the wave $v=\omega/k$

We have

$$\frac{\partial^2y}{\partial t^2}=-\omega^2A\sin kx\cos \omega t$$

$$\frac{\partial^2y}{\partial x^2}=-k^2A\sin kx\cos \omega t$$

So

$$-\omega^2A\sin kx\cos \omega t-v^2(-k^2A\sin kx\cos \omega t)=$$

$$-k^2v^2A\sin kx\cos \omega t+k^2v^2A\sin kx\cos \omega t=0$$