Question

The maximum particle velocity is $3$ times the wave velocity of a progressive wave. If $A$ is the amplitude of an oscillating particle, find the phase difference between two particles of separation $x$.

Answer 1

Hint: We have to find the phase difference between the particles which are separated by a distance $x$. To find the phase difference we need to find the wavelength of the wave, which in turn depends on the velocity of the wave. Using the given relation, we can find the velocity of the wave.

Formula used:
$v_{p}=A\omega$,$\phi=\dfrac{2\pi}{\lambda}\times x$, $\lambda\times f=v$ and $f=\dfrac{\omega}{2\pi}$

Complete answer:
The wave equation of a wave is given as $y(x,t)=Asin(kx\pm\omega t+\phi)$ where, $x$ is the position of the wave at time $t$, $t$ is the time taken, $A$ is the amplitude , $k$ is the wavenumber , $\omega$ is the angular frequency of the wave and the phase difference $\phi$.
Let us consider a wave with amplitude $A$ , angular velocity $\omega$.Then the velocity of the particle will be the velocity of the wave $v_{p}=A\omega$.
Given that particle velocity is $3$ times the wave velocity of a progressive wave, which can be written as $v_{p}=3\times v$
Let us consider that wavelength $\lambda$ and frequency $f$ travel at a velocity $v$. Then we also know that $\lambda\times f=v$ and the frequency of the wave can be written as$f=\dfrac{\omega}{2\pi}$
Then the velocity of the wave is given as, $v=\lambda \times\dfrac{\omega}{2\pi}$
Substituting the values in $v_{p}=3\times v$, we get $3\lambda\times\dfrac{\omega}{2\pi}=A\omega$
Or, $\dfrac{3\lambda}{2\pi}=A$
Or, $\lambda=\dfrac{2A\pi}{3}$
If the separation of the particle is $x$, then the phase difference $\phi=\dfrac{2\pi}{\lambda}\times x$
Substituting the value of $\lambda$, we get $\phi=\dfrac{2\pi}{\dfrac{2A\pi}{3}}\times x$
Reducing we get $\phi=\dfrac{3x}{A}$
Hence, the phase difference between two particles of separation $x$ is $\phi=\dfrac{3x}{A}$

Note:
The particle velocity is the oscillation of a particle along its mean position, whereas the propagation velocity of the wave is the velocity at which the wave travels in any medium. Clearly both are different, the maximum particle velocity is given by $v_{p}=A\omega$ and we know that the velocity of the wave is, $v=\lambda \times\dfrac{\omega}{2\pi}$.