16.6 Exercise 30

Given the wave functions $y_{1} \left(x , t\right) = A \text{sin} \left(k x - \omega t\right)$ and $y_{2} \left(x , t\right) = A \text{sin} \left(k x - \omega t + \phi\right)$ with $\phi \neq \frac{\pi}{2}$, show that $y_{1} \left(x , t\right) + y_{2} \left(x , t\right)$ is a solution to the linear wave equation with a wave velocity of $v = \frac{\omega}{k} .$