Question

A transverse sinusoidal wave of amplitude a, wavelength $\lambda$ and frequency f is travelling on a stretched string. The maximum speed of any point on the string is $\dfrac{v}{10}$, where v is the speed of propagation of the wave. If $a = {10^{ - 3}}$m and v = 10 m/s, then $\lambda$ and f are given by
(A) $\lambda = 2\pi \times 10^{ - 2}$ m
(B) $\lambda = 10^{ - 3}$ m
(C) f = $10^{3} 2\pi$ Hz
(D) f = $10^{4}$Hz

Answer 1

In a transverse wave, the two types of velocities defined are for the case of (1) simple harmonic motion of particles of the wave on their position, (2) velocity of the propagation of the entire wave or the velocity with which one crest (or trough) moves.
Formula Used:
In simple harmonic motion, the maximum velocity is given as:
${{\rm{v}}_{{\rm{max}}}} = a\omega$,
where a is the amplitude.
The frequency and wavelength are related to wave velocity as:
$v = f\lambda$

Complete answer:
First, let us obtain the frequency of the simple harmonic motion that each particle of the wave undergoes:
\eqalign{ & \frac{{{{\text{v}}_{{\text{max}}}}}}{a} = \omega = 2 \times \pi \times f \cr & \Rightarrow \dfrac{{\frac{{\text{v}}}{{{\text{10}}}}}}{{2\pi a}} = f \cr & \Rightarrow \dfrac{{\frac{{10}}{{{\text{10}}}}}}{{2\pi ({{10}^{ - 3}})}}Hz = f \cr & \Rightarrow f = \dfrac{{{{10}^3}}}{{2\pi }}Hz \cr}
Now, the frequency obtained here is also the frequency of the wave as the wave consists of nothing more but particles performing simple harmonic motion.
The velocity of the propagation of the wave is given as 10 m/s. Therefore the wavelength of the wave is:
\eqalign{ & \dfrac{v}{f} = \lambda \cr & \Rightarrow \dfrac{{10{\text{ m/s}}}}{{\dfrac{{{{10}^3}}}{{2\pi }}}} = \lambda \cr & \Rightarrow \lambda = \dfrac{{2\pi }}{{100}}{\text{ m}} \cr}
Therefore this is the wavelength of the given wave.
According to the answers obtained, the correct options are (A) for the wavelength and (C) for frequency.

Additional Information:
The velocity of particle at any position in simple harmonic motion is given by the formula:
$v = \omega \sqrt {{a^2} - {x^2}}$
Here, x is the displacement of the particle, a is the amplitude and $\omega$ is the angular frequency.

Note:
Here, one has to be careful about the definition of angular frequency and frequency. The key difference in the two quantities lies in the units. Frequency f is written in terms of Hz and angular frequency is written in terms of rad/s. One might easily confuse one quantity with the other in hurry and proceed with the wrong set of calculations.