Question

What is the phase difference between the displacement of mediums of particles and pressure in case of sound waves?
A. Zero.
B. $45^\circ$.
C. $90^\circ$.
D. $180^\circ$.

Answer 1

The solution of the given question can be determined if we know about the formulae of sine function and cosine functions and know about the wave nature and its way of representation. The sine function and cosine function are perpendicular to each other i.e. 90$^0$ and displacement of particles is in sine function and pressure wave will be in cosine.

Complete answer:
Step 1:
First of all we'll know about the sine and cosine functions.
The Value of sine starts from zero and keeps on changing with time from zero to 1 then zero and again to -1 and completes one cycle.
Whereas, the cosine function is opposite to it. It starts from value 1 then zero then -1 and again zero then to 1 and completes one cycle.
They are perpendicular to each other and makes an angle of $90^\circ$ with each other
Step 2:
Displacement of a particle is given by X=A Sin($\left( {\omega t + \phi } \right)$ where, X is the displacement , A is the amplitude, $\omega$ is the angular velocity with time and $\phi$ is the phase difference between two waves. Here we are given pressure wave
So, writing the equation of particle, which is travelling in +x direction then,
$\left( {x,t} \right)$ =Sm S in $\left( {\omega t - kx} \right)$…….(1)
Here, Sm is the amplitude and –kx is the phase difference.
For pressure wave we can write:
$\Delta P$ =$\Delta Pm$ Cos$\left( {\omega t - kx} \right)$ ……(2)
From eq1 and eq2 we can firmly say that both sine and cos function will be at a phase difference of $90^\circ$ (stated above)
Hence, the phase difference will be $90^\circ$.

Option C. is correct

Note:
Sound waves: Sound is a mechanical wave that results from the back and forth vibration of the particles of the medium through which the sound wave is moving. The motion of the particles is parallel (and anti-parallel) to the direction of the energy transport. This is what characterizes sound waves in air as longitudinal waves.
Pressure waves: Mechanical longitudinal waves are also called compression or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.