Propagation of Sound: When you make a call, you may hear your friend’s voice even if he or she is a long distance away. The medium is the means through which your sound is transferred. The waves are made up of particles from the medium in which they move. A sound is a sort of energy that is conveyed as sound waves. When things vibrate, the air around them vibrates as well, and sound waves are transmitted. Vibrations in an object will not pass through it if there is no intermediary. This process of this mechanism is known as sound propagation.

What is Sound?

A sound is a form of energy that gives us the sensation of hearing. Sound is a type of mechanical wave which moves by oscillating the particles of the medium in which it is travelling. We use our ears to receive sound signals, but there is a limitation to it. Not all the vibration can be perceived by our ears. Similarly, there is a limitation to what our vocal cords can produce as sound waves.

Types of Waves

Waves are categorized based on the requirement of the medium of propagation as:

1. Mechanical waves: These types of waves require a medium to propagate. Example: sound waves. Medium for the wave must have inertia and should be elastic.
2. Electromagnetic waves: These types of waves do not require to propagate. Example: light waves, \((X)-\)rays.

Waves can be categorized into two categories on the basis of the direction of oscillation.

1. Transverse wave: In this type of wave, the medium particle oscillates in a perpendicular direction—for example, Wave in water, Waves in string.
2. Longitudinal wave: In this type of wave, the medium particles oscillate in a direction parallel to the direction of propagation of the wave—for example, Sound waves.

Important Terms Related to Wave

The sound wave follows the relation for wave,
\(v = f\lambda \)
Where,
\(v\) is the velocity of sound.
\(\lambda \) is the wavelength.
\(f\) is the frequency.

1. Wavelength
The wavelength of a wave is defined as the distance between the medium particles which are in the same phase \(\left( \lambda \right).\) In a transverse wave, the wavelength is be defined as the distance between two successive crests or troughs. In a longitudinal wave, the wavelength \(\left( \lambda \right)\) is equal to the distance from the centre of one compression or refraction to another.

2. Amplitude
The amplitude of a wave can be defined as the maximum displacement of the particles of the medium from their mean position. It can be in the same direction for the longitudinal wave or in the direction perpendicular to the direction of propagation of wave as in the case of a transverse wave.
3. Frequency
The number of vibrations made by a particle in one second is called frequency. It is represented by \(v.\) Its S.I unit is hertz \(\left({Hz} \right).\) It is given as the inverse of the time period.
\(v = \frac{1}{T}.\)

4. Time-Period
The time taken by a particle to complete one vibration or the time taken by the particle to complete one cycle is called the time period.
It is the time taken by the wave to cover a distance equal to its wavelength.
\(T = \frac{1}{v},\) it is S.I. unit is seconds.

5. Wave Velocity
It is the velocity of the wave that is the velocity at which the wave propagates. The wave velocity of a wave depends on the medium in which it is travelling. It is given by,
Wave velocity \(\left( u \right) = \) frequency \(\left( v \right) \times \) wavelength \(\left( \lambda \right)\)

Representation of Wave

A wave can be represented by the parameter that it changes while moving through the medium.

For example, a sound wave moves through the air by the oscillation of the particles of the air, which in turn changes the density and the pressure at that point; thus, we can represent sound waves in terms of pressure and density.

In terms of displacement,
\(S = {S_0}\sin \left({\omega t – Kx} \right)\)
Where,
\({S_0}\) is the amplitude of the medium particles when a sound wave passes through it.
\(\omega \) is the angular frequency of the wave.
\(k = \frac{\omega }{v}\) is known as wave number
In terms of pressure,
\(\Delta P = \Delta {P_0}\sin \left({\omega t – Kx} \right)\)
\(\Delta {P_0}\) is the amplitude of the pressure.
In terms of density,
\(\Delta P = \Delta {P_0}\sin \left({\omega t – Kx} \right)\)
\(\Delta {P_0}\) is the amplitude of density.

Relation Between Displacement and Pressure Wave

When the sound waves move, the air molecules oscillate about their mean position forming compression and faction zones. In the compression zone, the pressure and density are more than the normal, while in the rarefaction zone, the density and the pressure are less than the normal value. Thus sound wave moves by creating alternate compression and rarefaction zones and makes the pressure and density graph against distance oscillatory like a sin wave.

Velocity of Sound Wave

Sound wave requires a medium to propagate, and it has different values for its speed in a different medium. Usually, the sound travels faster through the denser materials. Thus, the velocity of the sound will be most in solid and least in air or gases, while liquids will have intermediate values for the velocity of the sound.
Also, many velocities have been termed in comparison with the speed of the sound. For Example: If a body is moving with a speed greater than that of sound in the air, then that type of speed is categorised as supersonic speed, which is sometimes referred to as Mach \(1.\)
The velocity of the sound wave is given by,
\(v = \sqrt {\frac{B}{\rho }} \)
Where,
\(B\) is the bulk modulus of the gas.
\(\rho \) is the density of the gas.
Propagation of the sound wave in the gaseous medium is ab adiabatic process, thus,
\(v = \sqrt {\frac{{\gamma RT}}{{{M_0}}}} \)
Where,
\({M_0}\) is the molecular mass of the gas,
\(\gamma \) is the polytropic index of the gas for the adiabatic process.
\(T\) is the temperature.
\(R\) is the universal gas constant.

Power Transfer in Sound Wave

The wave carries momentum and energy, so when the wave moves from one place to another, it transfers energy and momentum. If the medium is massless, then the momentum and energy transfer cannot take place, and due to this reason, the medium needs to have some mass.

Average power transferred by the sound wave is,

\({P_{{\text{avg}}}} = \frac{1}{2}\rho {\omega ^2}{S_0}^2Sv\)
Where,
\(\rho \) is the density.
\(\omega \) angular frequency of the wave.
\({S_0}\) is the amplitude of sound wave when represented in the displacement equation.
\(S\) is the area through which the power transfer is to be calculated.
\(v\) is the velocity of the sound wave in the medium.
On further simplification,
\({P_{{\text{avg}}}} = \frac{1}{2}\frac{{\Delta P_0^2S}}{{\rho v}}\)
Where,
\(\Delta {P_0}\) is the amplitude of the sound wave when represented in terms of pressure.
The intensity of the wave is given by,
\(I = \frac{P}{{4\pi {r^2}}}\)
Where,
\(p\) is the power dissipated by the source.
\(r\) is the distance from the source at which we are calculating the intensity.

Standing Wave and Organ Pipe

Standing waves are formed when two waves coming from opposite directions superimpose each other. In standing, transfer of the energy doesn’t take place. Instead, all the particles of the medium perform simple harmonic motion with an equal time period but with different amplitude in the same phase. It is important to note that, for a wave, the medium particle also performs simple harmonic motion, but they have both the amplitude and the time period equal for all particles, but each particle is not necessarily in the same phase.

Organ pipes are cylindrical columns with either one end open and one end closed or both ends open.

In the organ pipe, the sound wave travels from one end the gets reflected from the second end, and comes back, which superimposes the initial wave to form a standing wave.

The minimum length of the closed organ pipe required to form a standing wave,

\({l_{\min }} = \frac{\lambda }{4}\)

The minimum length of the open organ pipe required to form a standing wave,

\({l_{\min }} = \frac{\lambda }{2}\)

Application

When a body crosses, the speed of the sound sonic boom is produced.

SONAR is widely used by submarines and aircraft.

Ultrasounds are used for imaging organs and babies inside of the human body.

An important application of sound waves is in the speed gun using the doppler’s effect.

Solved Examples on Propagation of Sound

Q.1.Two sound waves passing through air half their wavelength in ratio \(4∶5\) their frequencies are in the ratio?
Ans: For a wave, the relation between the speed of wave \(v,\) frequency \(n,\) and its wavelength \(\lambda \) is given by,
\(v = n\lambda \)
As the sound is passing through the air in both cases of wavelengths, the speed will remain constant. So we can write,
\( \Rightarrow {n_1}{\lambda _1} = {n_2}{\lambda _2}\)
\( \Rightarrow \frac{{{n_1}}}{{{n_2}}} = \frac{{{\lambda _2}}}{{{\lambda _1}}} = \frac{5}{4} = 5:4\)

Q.2. A sound wave has a frequency of \(2\,\rm{kHz}\) and a wavelength of \(35\,{\text{cm}}.\) If an observer is \(1.4\,{\text{km}}\) away from the source, then after what time interval could the observer hear the sound?
Ans: According to the question, given frequency,
The wavelength, \(\lambda = 35~{\text{cm}} = 35 \times {10^{ – 2}}~{\text{m}}\)
Distance of observer, \(d = 1.4~{\text{km}} = 1.4 \times {10^3}~{\text{m}}\)
Now, according to the speed and frequency relationship of sound,
\(v = f\lambda \)
\( \Rightarrow 2 \times {10^3} \times 35 \times {10^{ – 2}} = 700~{\text{m}}\;{{\text{s}}^{ – 1}}\)
Now,
\(t = \frac{d}{v}\)
Now, on substituting all the values,
\( \Rightarrow \frac{{1.4 \times {{10}^3}}}{{700}} = 2\;\text{s}\)
Therefore, the required time is \(2\;\text{s}.\)

Summary

The sound wave is a longitudinal wave. A sound wave can be represented in terms of displacement, pressure, or density. Speed of sound usually increases with an increase in the density of the medium. The speed of sound in air at \({0^ \circ }\,{\text{C}}\) is observed to be \(331\,{\text{m}}/{\text{s}}.\) The speed of sound increases with an increase in temperature. A sound wave can also transfer energy. Organ pipe works on the principle of the standing wave.

The following are some of the elements that influence sound propagation:

1. Atmospheric Turbulence: Sound waves disperse owing to velocity changes in the medium if the atmosphere in which they travel is turbulent.
2. Wind Gradient: Sound that travels with the wind bends downwards, whereas sound that travels against the wind bends upwards.
3. Temperature Gradient: In a heated environment near the earth’s surface, sound waves travel quicker. Sound waves are refracting upwards in this area. The refraction would be downwards if the temperature dropped at greater elevations.

Frequently Asked Questions (FAQs) on Propagation of Sound

Q.1. What do you mean by supersonic speed?
Ans: A body is moving with a speed greater than that of sound in the air, then that type of speed is categorized as supersonic speed, which is sometimes referred to as Mach 1.

Q.2. Can we hear sound in outer space?
Ans: No, we can not hear sound in outer space because the sound wave requires a medium to propagate, and in outer space, there is no medium.

Q.3. What are the properties of the medium?
Ans: For wave propagation, the medium should have mass to allow the transfer of momentum and energy and must be elastic to allow the movement of the medium particles.

Q.4. On what principle does the organ pipe works?
Ans: The organ pipe works on the principle of the standing wave. The wave travelling from one side gets reflected from one side, and the reflected wave superimpose the incident wave to form a standing wave.

Q.5. Which has the greater speed for sound, air, or water?
Ans: The speed of sound is greater in the water than in the air.