Question

Two coherent waves, coming from sources at different locations, move along the x axis. Their wave functions are $E_1=860\sin [\frac{2\pi x_1}{650}-924\pi t+\frac{\pi }{6}]$
and $E_1=860\sin [\frac{2\pi x_1}{650}-924\pi t+\frac{\pi }{6}]$ where $E_1$ and $E_2$ are in volts per meter, $x_1$ and $x_2$ are in nanometers, and t is in picoseconds. When the two waves are superposed, determine the relationship between $x_1$ and $x_2$ that produces constructive interference.

Answer 1

Constructive interference occurs where the phases of the waves differ by integral multiples m of $2\pi$:
$(\frac{2\pi x_1}{650}-924\pi t+\frac{\pi }{6})-(\frac{2\pi x_2}{650}-924\pi t+\frac{\pi }{8})=2\pi m$
which becomes
$\frac{2\pi (x_1-x_2)}{650}+(\frac{\pi }{6}-\frac{\pi }{8})=2\pi m$
$\frac{(x_1-x_2)}{650}+\frac{1}{12}-\frac{1}{16}=m$
$x_1-x_2=(m-\frac{1}{48})650$, where $x_1$ and $x_2$ are in nanometers and $m=0,1,-1,2,-2,3,-3,....$