Question

Given the function:y(x,t)=A1cosk1(x+vt)-A2 sink2(x-vt). Determine whether this a solution to the wave equation

Answer 1

The wave equation has a form

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

In our case

$$y=A_1\cos k_1(x+vt)-A_2\sin k_2(x-vt)$$

We get

$$\frac{\partial^2y}{\partial t^2}=-A_1k_1^2v^2\cos k_1(x+vt)\\ +A_2k_2^2v^2\sin k_2(x-vt)$$ $$v^2\frac{\partial^2y}{\partial x^2}=v^2\left(-A_1k_1^2\cos k_1(x+vt)\right.\\ \left.+A_2k_2^2\sin k_2(x-vt)\right)$$

Hence, the function $y(x,t)$ is a solution of the wave equation.