A sinusoidal wave having wavelength of 6 m propagates along positive x direction on a string. The displacement (y) of a particle at x = 2 m varies with time (t) as shown in the graph .If the equation of the wave is (3 mm) sin (ax-bt +c) , find the value of  c.

 

 $\mathrm{\lambda }=6\mathrm{m}$

$\therefore \mathrm{k}=\frac{2\mathrm{\pi }}{\mathrm{\lambda }}=\frac{\mathrm{\pi }}{3}$
From the graph T = 4s
$\therefore \mathrm{\omega }=\frac{2\mathrm{\pi }}{\mathrm{T}}=\frac{\mathrm{\pi }}{2}$
Let the equation of the wave be
$\mathrm{y}=\left(3\mathrm{mm}\right)\sin \left(\frac{\mathrm{\pi }}{3}\mathrm{x}-\frac{\mathrm{\pi }}{2}\mathrm{t}+\mathrm{\delta }\right)$
At x = 2; t = 0, y is zero
$\therefore \sin \left(2\frac{\mathrm{\pi }}{3}+\mathrm{\delta }\right)=0$
$\Rightarrow \mathrm{\delta }=-\frac{2\mathrm{\pi }}{3} \mathrm{or} \frac{\mathrm{\pi }}{3}$
Also, $\frac{\mathrm{\partial }\mathrm{y}}{\mathrm{\partial }\mathrm{t}}>0$ at t = 0 at x = 2 
$\frac{\mathrm{\partial }\mathrm{y}}{\mathrm{\partial }\mathrm{t}}=-3\frac{\mathrm{\pi }}{2}\cos \left(\frac{\mathrm{\pi }}{3}\mathrm{x}-\frac{\mathrm{\pi }}{3}\mathrm{t}+\mathrm{\delta }\right)$
At t = 0, x = 2
$\frac{\mathrm{\partial }\mathrm{y}}{\mathrm{\partial }\mathrm{t}}=-\frac{3\mathrm{\pi }}{2}\cos \left(\frac{2\mathrm{\pi }}{3}+\mathrm{\delta }\right)$
for this to be positive
$\begin{array}{l}\cos \left(\frac{2\mathrm{\pi }}{3}+\mathrm{\delta }\right)<0\\ \therefore \mathrm{\delta }=\frac{\mathrm{\pi }}{3}\left[\text{ and }\mathrm{\delta }\ne -\frac{2\mathrm{\pi }}{3}\right]\\ \therefore \mathrm{y}=\left(3\mathrm{mm}\right)\sin \left(\frac{\mathrm{\pi }}{3}\mathrm{x}-\frac{\mathrm{\pi }}{2}\mathrm{t}+\frac{\mathrm{\pi }}{3}\right)\end{array}$