Question

Write the expression for y as a function of x and t for a sinusoidal wave travelling along a rope in the negative x direction with the following characteristics: $A=8.00$cm, $\lambda =80.0$cm, $f=3.00$ Hz, and $y(0,t)=0$ at $t=0$. (b) Write the expression for y as a function of x and t for the wave in part (a) assuming that $y(x,0)=0$ at the point $x=10.0$cm.

Answer 1

$\left(a\right)$

$K=\frac{2\pi }{\lambda }=\frac{2\pi }{0.800}=7.85m^{-1}w=2\pi f=2\pi \times 3=6\pi rad/sec\therefore y(x,t)=A\sin (wt+kx)$

as wave is moving in x- direction

$=\left(\frac{8}{1000}m\right)\sin (6\pi t+7.85x)=0.08\sin (6\pi t+7.85x)$

$\left(b\right)$

as $y=0$ at $x=10cm=0.1m$

we know that we must shift the wave so that $y(0.1,0)=0$

Let

$y(x,t)=0.08\sin (6\pi t+7.85x+\varphi )$

now

$0=0.08\sin (6\pi t+7.85x+\varphi )0=\sin (0.785+\varphi )\therefore \varphi =-0.785\therefore y(x,t)=0.08\sin (6\pi t+7.85x+0.785)$