A spherical wave has the form . The velocity of the wave (in m/s) is
A) 0.005 88.
B) 16.0
C) 340.
D) 1 360.
E) 2 720.

Question

A spherical wave has the form  . The velocity of the wave (in m/s) is
A) 0.005 88. 
B) 16.0 
C) 340. 
D) 1 360. 
E) 2 720.

Quizplus · Accepted Answer

The given wave has the form $A\cos(kr-\omega t+\phi)$ where $k$ is the wave number, $r$ is the distance from the source, $\omega$ is the angular frequency, $t$ is time, $\phi$ is the phase, and $A$ is the amplitude. The velocity of the wave is given by $v=\frac{\omega}{k}$.

In this case, we need to determine the wave number $k$ and the angular frequency $\omega$ from the given wave form. The expression $kr-\omega t$ in the argument of the cosine function should correspond to the phase of the wave. Since the wave is spherical, this phase depends on the distance $r$ according to $kr=2\pi r/\lambda$, where $\lambda$ is the wavelength of the wave. Therefore, we can write:

$$kr-\omega t+\phi = \frac{2\pi r}{\lambda}-\frac{2\pi f}{c}t+\phi$$

where $f$ is the frequency of the wave and $c$ is the speed of light. Comparing this with the given wave form, we can identify $k=2\pi/\lambda$ and $\omega=2\pi f$, so that

$$v=\frac{\omega}{k}=\frac{2\pi f}{2\pi/\lambda}=\lambda f=\frac{c}{f}=340\,	ext{m/s}$$

Therefore, the correct answer is (C).

A Spherical Wave Has the Form