Questionnaire 2.1.2 — Plane Waves & Interference

Consider the three-dimensional classical wave equation

$$\nabla^2 \phi - \frac{1}{v^2}\,\frac{\partial^2 \phi}{\partial t^2} = 0$$

where $v$ is the magnitude of the velocity of the wave. Is the following function a solution of the wave equation for some value of the wave velocity?

$$\phi(\mathbf{r}, t) = \sin(x - 0.2t) + \sin(x + 2.2t)$$

where $x$ is the component of the vector $\mathbf{r}$ in the $x$-direction.

(Note: formally the numbers $0.2$ and $2.2$ must have appropriate dimensions — here, of velocity — for this equation to make physical sense. Where a variable such as $x$ stands on its own, presume it is multiplied by a number "1" carrying the appropriate dimensions: $\sin(x - 0.2t)$ means $\sin(1x - 0.2t)$ with the "1" having dimensions of inverse length.)

a) yes b) no c) cannot be determined without more information