Chapter 16, Problem 147CP

Consider two wave functions $y_1\left(x,t\right)=A\sin \left(kx-\omega t\right)$ and $y_2\left(x,t\right)=A\sin \left(kx+\omega t+\varphi \right)$ . What is the wave function resulting from the interference of the two wave? (Hint: $\sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$ and $\varphi =\frac{\varphi }{2}+\frac{\varphi }{2}$ .)