Question

Two coherent waves are described by the expressions.
$E_1=E_0\sin (\frac{2\pi x_1}{\lambda }-2\pi ft+\frac{\pi }{6})$
$E_2=E_0\sin (\frac{2\pi x_2}{\lambda }-2\pi ft+\frac{\pi }{8})$
Determine the relationship between $x_1$ and $x_2$ that produces constructive interference when the two waves are superposed.

Answer 1

Phase of first wave $=\frac{2\pi x_1}{\lambda }-2\pi ft+\frac{\pi }{6}$

Phase of second wave$=\frac{2\pi x_2}{\lambda }-2\pi ft+\frac{\pi }{8}$

Phase difference$={\varphi }_1-{\varphi }_2=\frac{2\pi x_1}{\lambda }-2\pi ft+\frac{\pi }{6}-\frac{2\pi x_2}{\lambda }-2\pi ft+\frac{\pi }{8}=\frac{2\pi (x_1-x_2)}{\lambda }+\frac{\pi }{24}$

Now for constructive interference

$\Delta \varphi =2n\pi$

$or,\frac{2\pi (x_1-x_2)}{\lambda }+\frac{\pi }{24}=2n\pi or,(x_1-x_2)=(2n\pi -\frac{\pi }{24})\times \frac{\lambda }{2\pi }or,x_1-x_2=(n-\frac{1}{48})\lambda wheren=(0,1,2,....)$