Chapter 15, Problem 44P

(a)

To determine

The proof that the resultant of two waves $D_1=A\sin \left(kx-\omega t\right)$ and $D_2=A\sin \left(kx-\omega t+\varphi \right)$ is $D=\left(2A\cos \frac{\varphi }{2}\right)\sin \left(kx-\omega t+\frac{\varphi }{2}\right)$.

(b)

To determine

The amplitude of the resultant wave.

(c)

To determine

The constructive interference occurs if $\varphi =0,2\pi ,4\pi \mathrm{....}$ and destructive interference occurs if $\varphi =\pi ,3\pi ,5\pi \mathrm{.....}$.

(d)

To determine

The description for the resultant wave in terms of equation and in words when $\varphi =\frac{\pi }{2}$.