Question

If the slope is $s$, wave velocity is ${v_w}$ and particle velocity of ${v_P}$ then
A. ${v_P} = - s{v_w}$
B. ${v_P} = \dfrac{{{v_w}}}{s}$
C. ${v_P} = s{v_w}$
D. ${v_P} = \dfrac{{ - {v_w}}}{s}$

Answer 1

Recall the basics of the particle velocity and wave velocity. Also recall the formula for slope of a particle in a wave. This formula gives the relation between the particle velocity and wave velocity. Rearrange this formula to determine the formula for particle velocity of the particle in a wave.

Complete step by step answer:
We have given that the slope is $s$, wave velocity is ${v_w}$ and particle velocity is ${v_P}$.
We know that the slope of a particle in a wave at any instant is defined as the ratio of particle velocity to wave velocity.
The mathematical expression for slope of the particle in a wave is
$s = \dfrac{{ - {v_P}}}{{{v_w}}}$
Rearrange the above equation for ${v_P}$.
${v_P} = - s{v_w}$
This is the required expression for particle velocity.

So, the correct answer is “Option A”.

Additional Information:
Let us derive the formula for particle velocity.
The displacement $y$ of wave at any instant $t$ is given by
$y\left( {x,t} \right) = A\sin \left( {\omega t - kx} \right)$
Here, $A$ is the amplitude of the wave, $\omega$ is angular frequency of the wave, $k$ is the wave number and $x$ is the displacement of the wave.
The particle velocity ${v_P}$ is given by
${v_P} = \dfrac{{dy}}{{dt}}$
Substitute $A\sin \left( {\omega t - kx} \right)$ for $y$ in the above equation,
${v_P} = \dfrac{{dA\sin \left( {\omega t - kx} \right)}}{{dt}}$
$\Rightarrow {v_P} = A\omega \cos \left( {\omega t - kx} \right)$ …… (1)
Let us also determine the value of $\dfrac{{dy}}{{dx}}$ which is the slope of the particle in a wave.
$\dfrac{{dy}}{{dx}} = \dfrac{{dA\sin \left( {\omega t - kx} \right)}}{{dx}}$
$\Rightarrow \dfrac{{dy}}{{dx}} = - kA\cos \left( {\omega t - kx} \right)$ …… (2)
From equations (1) and (2), we can also write the particle velocity as
${v_P} = - \dfrac{\omega }{k}\dfrac{{dy}}{{dx}}$
Substitute ${v_w}$ for $\dfrac{\omega }{k}$ in the above equation.
${v_P} = - {v_w}\dfrac{{dy}}{{dx}}$
This is the required expression for particle velocity.

Note:
The students may get confused between the signs in the formula for particle velocity because the slope of the particle in a wave is the ratio of particle velocity and wave velocity. It does not include negative signs. But the actual formula for particle velocity includes negative sign as shown in the above derived formula.