Questionnaire 1.2.7 — The Classical Wave Equation

Which of the following waves are possible solutions to the classical wave equation for a wave on an infinitely long string, assuming there are no boundary conditions. The wave equation is given by

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

where $v$ is the magnitude of the velocity of the wave. SIX of the following choices are possible solutions. Choose ALL of the options that are correct for at least some value of the wave velocity (possibly different in each case).

a) $\phi(z,t) = 6\left(\sin(z - 4t) + \sin(z + 4t)\right)$ b) $\phi(z,t) = \cos(0.1t)\,\sin(11.1z)$ c) $\phi(z,t) = i\cos(z - vt^2)$ d) $\phi(z,t) = \exp(i(3z - 5t)) + \exp(i(2z + 4t))$ e) $\phi(z,t) = \exp(2z - t) + \exp(2z + t)$ f) $\phi(z,t) = 4z - 4vt - 5\exp(z + vt)$ g) $\phi(z,t) = \cos(z - 2t)\exp(z - 2t)$ h) $\phi(z,t) = \sin(z - vt)\cos(z + vt)$ i) $\phi(z,t) = (2z + 5t)^2$ j) $\phi(z,t) = (az)^2 - (bt)^2$ where $a$ and $b$ are real constants with appropriate physical dimensions