The displacement of a string carrying a traveling sinusoidal wave is given by At time t = 0 the point at x = 0 has a displacement of 0 and is moving in the positive y direction. The phase constant $\phi$ is:
A) 45 $\degree$
B) 90 $\degree$
C) 135 $\degree$
D) 180 $\degree$
E) 270 $\degree$

Question

The displacement of a string carrying a traveling sinusoidal wave is given by  At time t = 0 the point at x = 0 has a displacement of 0 and is moving in the positive y direction. The phase constant $\phi$ is:
A) 45 $\degree$ 
B) 90 $\degree$ 
C) 135 $\degree$ 
D) 180 $\degree$ 
E) 270 $\degree$

Quizplus · Accepted Answer

Given function is y(x,t) = A sin(kx - $\omega$t + $\phi$)
At time t = 0,
y(x,0) = A sin(kx - 0 + $\phi$) = 0
As given initially the point is moving in the positive y direction, this means the wave is moving in the negative x direction.
Therefore, sin(kx + $\phi$) = 0 when kx + $\phi$= n$\pi$ where n is any integer.
At x = 0, kx + $\phi$= n$\pi$
As the point is moving in a positive y direction, $\phi$ should be equal to $\pi$.
Therefore, kx + $\pi$ = n$\pi$
kx= (n-1)$\pi$
At x = $\frac{\lambda}{2}$, kx= $\frac{(2n-1)\pi}{2}$= $\frac{\pi}{2}$
Given $\lambda$= 4 m
k= $\frac{2\pi}{\lambda}$ = $\frac{\pi}{2}$
kx=$\frac{\pi}{2}$ at x = 1m
Therefore, $\phi$= kx + $\pi$ = $\frac{\pi}{2}$ + $\pi$= $\frac{3\pi}{2}$ = 180$^\circ$  (in degrees).

Hence, the answer is D) 180$^\circ$.

The Displacement of a String Carrying a Traveling Sinusoidal Wave ϕ\phiϕ

The Displacement of a String Carrying a Traveling Sinusoidal Wave $\phi$