Question 428: Waves

Question

The equation of simple harmonic progressive wave is given by $Y=a \sin 2 \pi(b t-c x)$.
The maximum particle velocity will be twice the wave velocity if

Accepted Answer

The equation for a simple harmonic progressive wave is given by $ Y = a \sin 2 \pi(b t - c x) $.
To find the condition under which the maximum particle velocity is twice the wave velocity, we start with the wave equation $ y = a \sin 2 \pi(b t - c x) $.
Comparing this with the general equation of a progressive wave, $ y = A_0 \sin 2 \pi\left(f t - \frac{x}{\lambda}ight) $, we identify the components:
Frequency, $ f = b $
Wavelength, $ \lambda = \frac{1}{c} $
Amplitude, $ A_0 = a $
The maximum particle velocity ($ v_{\max} $) can be expressed as:
$ v_{\max} = A_0 \omega = a 	imes 2 \pi b \quad 	ext{.... (i)} $
The wave velocity ($ v_{	ext{wave}} $) is given by:
$ v_{	ext{wave}} = f \lambda = \frac{b}{c} \quad 	ext{.... (ii)} $
Given that the maximum particle velocity is twice the wave velocity, we have:
$ v_{\max} = 2 v_{	ext{wave}} $
Substituting values from equations (i) and (ii) into this relation, we obtain:
$ a 	imes 2 \pi b = 2 	imes \frac{b}{c} $
From this equation, we derive:
$ c = \frac{1}{a \pi} $

Answer

$c=\pi a$

Answer

$c=\frac{1}{2 \pi a}$

Answer

$c=\frac{1}{\pi a}$

Answer

$c=2 \pi a$

Waves

The equation of simple harmonic progressive wave is given by Y=a \sin 2 \pi(b t-c x). The maximum particle velocity will be twice the wave velocity if

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