how to verify a solution to the 3D wave equation

Question

Is $E = (A\sin(k(x-ct)),0,0)$ a solution to the wave equation $c^2 \nabla^2E=\frac{\partial^2E}{\partial t^2}$? What is $\frac{\partial E}{\partial t}$?

Answer 1

$\partial E / \partial t = (A(-kc)cos(k(x-ct)),0,0)$ and $\partial^2 E / \partial t^2 = (-Ak^2c^2sin(k(x-ct)),0,0)$.

$\nabla E = (Akcos(k(x-ct)),0,0)$ and $\nabla^2 E = (-Ak^2 sin(k(x-ct)),0,0)$.

The equation is verified, you have a wave equation.