Question

The equation of a simple harmonic progressive wave is given by $Y=a\sin 2\pi (bt-cx)$. The maximum particle velocity will be twice the wave velocity if
$\begin{aligned} & \text{A}\text{. c=}\pi \text{a} \\\ & \text{B}\text{. c=}\dfrac{1}{2\pi a} \\\ & \text{C}\text{. c=}\dfrac{1}{\pi a} \\\ & \text{D}\text{. c=2}\pi \text{a} \\\ \end{aligned}$

Answer 1

Hint: Waves which travel continuously in the same direction in the given medium without change of the form are called progressive waves. It travels in a given direction without change of form and particles of the medium perform simple harmonic motion about its means position. Compare the given equation with the standard wave equation. Write formulae for particle velocity and wave velocity. Using these formulae deduce the required condition for the statement in the question.

Complete step-by-step solution -
Given that, $Y=a\sin 2\pi (bt-cx)$
Comparing with $Y=A\sin 2\pi \left( \dfrac{t}{T}-\dfrac{x}{\lambda } \right)$
we get $A=a,\text{ T=}\dfrac{1}{b},\text{ }\lambda \text{=}\dfrac{1}{c}$
Maximum particle velocity is given by
${{v}_{p(\max )}}=A\omega =\dfrac{2\pi A}{T}$
Therefore, ${{v}_{p(\max )}}=2\pi ab$
Wave velocity is given by
$v=\dfrac{\lambda }{T}$
Therefore, $v=\dfrac{b}{c}$
Given $\begin{aligned} & {{v}_{p(\max )}}=2v \\\ & 2\pi ab=2\times \dfrac{b}{c} \\\ & \therefore \text{ c=}\dfrac{1}{\pi a} \\\ \end{aligned}$
Hence, option (C) is correct.

Additional Information:
Progressive Wave:
A progressive wave is defined as the vibrations of a body travelling in an elastic medium in forward direction from one particle to the successive particle.
An equation can be derived to represent the displacement of a vibrating particle in an elastic medium through which a wave passes. Thus each particle of a progressive wave performs a simple harmonic motion with the same period and amplitude but differing in phase from each other.
This equation can be written as $Y=A\sin 2\pi \left( \dfrac{t}{T}-\dfrac{x}{\lambda } \right)$

Characteristics of progressive wave:
(a) Each particle of the medium vibrates about its mean position with the same amplitude about their mean positions.
(b) All particles vibrate in the same manner with phase varying from 0 to 2π i.e. different particles attain different positions at different times.
(c) Transverse progressive waves have crests and troughs while longitudinal waves have compressions and rarefactions.
(d) Energy and momentum are transported along the direction of propagation of progressive waves.

Note: Remember that wave velocity and particle velocity are two different things. Each particle in the medium performs a simple harmonic motion with velocity periodically varying from 0 to maximum whereas the wave velocity will remain constant unless there is any variation in the affecting parameters.