Question

A sinusoidal wave propagates along a string. In figure (a) and (b). 'y' represents displacement of particle from the mean position. 'x' & 't' have usual meanings. Find the magnitude of slope of the string at x = 2 at t =4 sec.

• A. $\frac{-3\pi }{2}\times {10}^{-3}$
• B. $\frac{-5\pi }{2}\times {10}^{-3}$
• C. $\frac{-3\pi }{2}\times {10}^{-2}$
• D. $\frac{-3\pi }{5}\times {10}^{-3}$

Answer 1

The correct option is A $\frac{-3\pi }{2}\times {10}^{-3}$

The wave has form: $y(x,t)=Asin(kx-\omega t+\varphi )$

Since from first figure, $y(x=2)=Asin(2k-\omega t+\varphi )=Asin\left(\omega t\right)$

And from second figure, $y(t=4)=Asin(kx-4\omega +\varphi )=Acos\left(kx\right)$

We have $2k+\varphi =\pi$

and $\varphi -4\omega =\frac{\pi }{2}$

$⟹2\frac{2\pi }{4}+\varphi =\pi$

$⟹\varphi =0$

Slope of wave is given by: $\frac{dy}{dx}=Akcos(kx-\omega t)$

$⟹\frac{dy}{dx}(2,4)=A\frac{2\pi }{4}cos(\frac{2\pi }{4}2-\frac{2\pi }{4}4)$$=-A\frac{\pi }{2}$$=-\frac{3\pi }{2}\times {10}^{-3}m$