Question

When we clap our hands, the sound produced is best described by, (here p denotes the change in pressure from the equilibrium value)
A. $p={{p}_{0}}\sin \left( kx-\omega t \right)$
B. $p={{p}_{0}}\sin kx\cos \omega t$
C. $p={{p}_{0}}\cos kx\sin \omega t$
D. $p=\sum{{{p}_{0n}}}\sin \left( {{k}_{n}}x-{{\omega }_{n}}t \right)$

Answer 1

Hint: Recall how the propagation of sound takes place through a medium. You may realize that pressure variation does that. Now recall how the pressure varies with time. Also, you could think of the situation when we clap our hands, is it just one wave that is being produced? Analyzing all these points find the correct representation of required pressure variation.

Complete step by step answer:
In the question, we are asked to find the expression that best describes the change in pressure from the equilibrium value.
We know that sound waves consist of repeating patterns of high-pressure and low-pressure regions moving through a medium and thus sound waves could be otherwise called a pressure wave. We could use a detector to detect pressure variations of sound waves. Doing so, we would find that the fluctuations in pressure as detected by the detector occur at periodic and regular intervals of time. When we plot the pressure versus time graph you will get a sine curve as the result.
So the pressure wave for a single sound wave (or pressure wave) can be given by,
$p={{p}_{0}}\sin \left( kx-\omega t \right)$
Or
$p={{p}_{0}}\sin kx\cos \omega t$
Or
$p={{p}_{0}}\cos kx\sin \omega t$
However, when we clap hands we are not producing a single sound wave but we are producing a number of waves that may or may not be uniform. Since this variation in pressure cannot be uniform every time, we have to find the summation of pressure variations of n number of waves produced to get the representation of pressure variation of sound caused by the waves produced on clapping our hands.
So, when we clap our hands, the sound produced is best described by,
$p=\sum{{{p}_{0n}}}\sin \left( {{k}_{n}}x-{{\omega }_{n}}t \right)$
Which is the expression representing the summation of all n number of waves produced.

Hence option D is the right answer.

Note:
Now let us give a brief description on the plot of pressure versus time. The peak points on the sine curve shows the compressions, low points show rarefactions and the zero points shows the pressure that the air might have in the absence of any disturbance. Important point to be kept in mind is that, just the pressure-time fluctuations have sinusoidal nature, never conclude that sound is a transverse wave with crests and troughs. They are longitudinal waves that have compressions and rarefactions.