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

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$$\begin{gathered} ={\varphi }_1-{\varphi }_2 \\ =\frac{2\pi x_1}{\lambda }-2\pi ft+\frac{\pi }{6}-\frac{2\pi x_2}{\lambda }-2\pi ft+\frac{\pi }{8} \\ =\frac{2\pi \left(x_1-x_2\right)}{\lambda }+\frac{\pi }{24} \end{gathered}$$

Now for constructive interference

$\Delta \varphi =2n\pi$

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