Question

A sound wave of frequency 245 Hz travels with the speed of $300\dfrac{m}{s}$ along the positive x-axis. Each point of the wave moves to and fro through a total distance of 6 cm. What will be the mathematical expression of this travelling wave?
A. $Y = 0.03\sin \left[5.1x - \left(0.2 \times {10^3}t\right)\right]$
B. $Y = 0.06\sin \left[5.1x - \left(1.5 \times {10^3}t\right)\right]$
C. $Y = 0.06\sin \left[0.8x - \left(0.5 \times {10^3}t\right)\right]$
D. $Y = 0.03\sin \left[5.1x - \left(1.5 \times {10^3}t\right)\right]$

Answer 1

Hint: It is crucial to understand the meanings of travelling waves before moving on to the issue. A disturbance in a medium can be characterised as a wave if it propagates while transferring energy and momentum with no net motion of the medium. A travelling wave is one whose positions of maximum and minimum amplitude move through the medium.

Complete answer:
General expression for wave travelling along positive x axis is of the form
$Y = A\sin (kx - \omega t)$

Given that,
$f = 245Hz$
$v = 300\dfrac{m}{s}$
Here we have $A = \dfrac{6}{2} = 3cm$
$\omega = 2\pi f$
$ \Rightarrow \omega = 2\pi \times 245 = 1.54 \times {10^3}\dfrac{{rad}}{{\sec }}$
$k = \dfrac{\omega }{v} = \dfrac{{1.54 \times {{10}^3}}}{{300}}$
$k = 5.1{m^{ - 1}}$

So mathematical expression of travelling wave will be given by
$Y = 0.03\sin \left[5.1x - \left(1.5 \times {10^3}t\right)\right]$

Therefore, the correct answer is option (D).

Additional information:
1. The medium must, nevertheless, possess elastic characteristics.
2. The maximum distance of the disturbance from the wave's midpoint to the top of the crest or the bottom of the trough is known as the amplitude.
3. A wavelength is the maximum separation between two adjacent troughs.
4. The time now truly refers to the duration of one vibration.
5. The number of vibrations a wave makes in one second is its frequency.
6. Both frequency and duration exhibit an inverse relationship. The connection is shown below,
$T = \dfrac{1}{f}$
The speed of a wave is given by,
$v = \lambda f$
Where $\lambda$ is the wavelength.

Note: Students might mistake in the question to write amplitude 6 cm but it’s not correct. Half of the amplitude complete length is provided to us. Always keep that in mind. A crucial second-order linear partial differential equation for the description of waves is the wave equation