In a closed room the sound is repeatedly reflected from the walls and it is even heard long after the sound source ceases to function. The residual sound remaining in an enclosure and the phenomenon of multiple reflections of sound is called reverberation.

An echo is a repetition of sound produced by the reflection of sound waves from a wall, mountain or other obstructing surfaces.

Hurricane produces a shock wave which has a speed greater than the speed of sound. If travels with a supersonic sound that can be easily sensed by animals, using Doppler effect, The multiple reflections of sound can be easily sensed by animals.

Transverse waves travel in the form of crests and troughs and so involve change in shape. Transverse waves can be produced in a medium which has elasticity in shape. As gas has no elasticity of shape, hence transverse wave cannot be produced in a gas. Transverse waves can be produced in solids and on the surface of liquids.

An echo is a repetition of sound produced by the reflection of sound waves fiom a wall, mountain or other obstructing surfaces. The speed of sound in air at 20De is 344 ms^-1. If we shout at a wall which is at 344 m away, then the sound will take 1 second to reach the wall. After reflection, the sound will take one more second to reach us. Therefore, we hear the echo after two seconds. Scientists have estimated that we can hear two sounds properly if the time gap or time interval between each sound is \((\frac{1}{10})^{th}\) of a second (persistence of hearing) i.e., 0.1 s. Then,
Velocity = \(\frac{Distance\ travelled}{time\ taken}=\frac{2d}{r}\)
2d = 344\(\times\)0.1 = 34.4 m
d = 17.2 m
The minimum distance from a sound reflecting wall to hear an echo at 2^oCe is 17.2 meter.

When the source and the ob.server are in relative motion with respect to each other and to the medium in which sound propagates, the frequency of the sound wave observed is different from the frequency of the source. This phenomenon is called Doppler effect.

For transverse waves, the distance between two neighbouring crests or troughs is known as the wavelength. For longitudinal waves, the distance between two neighbouring compressions or rarefactions is known as the wavelength. The SI unit of wavelength is meter.

The disturbance which carries energy and momentum from one point in space to another point in space without the transfer of the medium is known as a wave.

This implies, when increasing the value of a, the line shifts towards right side. For a = vt, y = x − vt satisfies the differential equation. Though this function satisfies the differential equation, it is not finite for all values of x and t. Hence, it does not represent a wave.

\(v=\sqrt { \frac { y }{ \rho } } =\sqrt { \frac { 2\times { 10 }^{ 11 } }{ 7800 } } =\sqrt { 0.2564\times { 10 }^{ 8 } } \)
= 0.506\(\times\)10⁴ms^-1= 5\(\times\)10³ms^-1
Therefore, longitudinal waves travel faster in a solid than in a liquid or a gas. Now you may understand why a shepherd checks before crossing railway track by keeping his ears on the rails to safeguard his cattle.

This means that the speed of the wave along a string is a constant. Higher the frequency, shorter the wavelength and vice versa, and their product is velocity which remains the same.

\({ \lambda }_{ 1 }=\frac { v }{ { f }_{ 1 } } =\frac { 340 }{ 20 } =17m\)
\({ \lambda }_{ 2 }=\frac { v }{ { f }_{ 2 } } =\frac { 340 }{ 20\times { 10 }^{ 3 } } =0.017m\)
Therefore, the audible wavelength region is from 0.017 m to 17 m when the velocity of sound in that region is 340 m s ^-1.

According to Weber-Fechner's law, "loudness (L) is proportional to the logarithm of the actual intensity (1) measured with an accurate nonhuman instrument". This means that
L ∝ 1n I|
L = k  1n I
where k is a constant, which depends on the unit of measurement. The difference between two loudnesses, L₁ and L_o measures the relative loudness between two precisely measured intensities and is called as sound intensity level. Mathematically, sound intensity level is
ΔL = L₁-L_o = k In I₁- k In I₀ = k In \(\left[I_1\over I_0\right]\)
If k = 1, then sound intensity level is measured in bel, Therefore,
\(ΔL=In\)\(\left[I_1\over I_0\right]\)bel
However, this to express smaller unit, decibel. Thus, [decibel = \(1\over10\)bel] by multiplying and dividing by 10
\(ΔL=10\left(In\left[I_1\over I_0\right]\right){1\over 10}bel\)
\(ΔL=10In\left[I_1\over I_0\right]\)decibel with k = 10
For practical purposes,
\(ΔL=10log_{10}\left[I_1\over I_0\right]\)decibel.

It works on the principle of multiple reflections.
It consists of three main parts:
(i) Chest piece
(ii) Ear piece
(iii) Rubber tube
(i) Chest piece: It consists of a small disc-shaped resonator (diaphragm) which is very sensitive to sound and amplifies the sound it detects.
(ii) Ear piece: It is made up of metal tubes which are used to hear sounds detected by the chest piece.
(iii) Rubber tube: This tube connects both chest piece and ear piece. It is used to transmit the sound signal detected by the diaphragm, to the ear piece. The sound of heart beats (or lungs) or any sound produced by internal organs can be detected, and it reaches the ear piece through this tube by multiple reflections.

(i) For the propagation of the waves, the medium must possess both inertia and elasticity, which decide the velocity of the wave in that medium.
(ii) In a given medium, the velocity of a wave is a constant whereas the constituent particles in that medium move with different velocities at different positions. Velocity is maximum at their mean position and zero at extreme positions.
(iii) Waves undergo reflections, refraction, interference, diffraction and polarization.

The first minimum occurs when the two waves reaching the point B are 180° (out of phase). The path difference \(\Delta x=\frac { \lambda }{ 2 } \) . In order to calculate the path difference, we have to find the path lengths x₁ and x₂. In a right triangle BDC,

The given equation is similar to
\(\mathrm{y} =\mathrm{A} \sin \left(\omega t-\frac{x}{\lambda}\right) \)
\(\omega=300, \ \mathrm{f} =\frac{300}{2 \pi}=\frac{150}{\pi} \)
\(\text { Speed } \mathrm{v} =\mathrm{f} \lambda \)
\(=\frac{150}{\pi} \times \pi \)
\(=150 \mathrm{~ms}^{-1} \)

Frequency = 5000 Hz
Speed of sound in air = 332 m/s
Speed of sound in water = 1450 m/s
Wavelength of sound in air  \(=\frac{332}{5000}\)
\(=66.4 \times 10^{-3}\)
\(\text { Wavelength of sound water } \lambda_{\text {water }}\)
\(=\frac{1450}{5000} =290 \times 10^{-3} \mathrm{~m} \)
\(\therefore \frac{\lambda_{\text {water }}}{\lambda_{\text {air }}} =\frac{290 \times 10^{3}}{66.4 \times 10^{-3}} =4.367 \)

\(\mathrm{f}_{\mathrm{o}}=150 \mathrm{~Hz}\)
\(\text { From the series of frequencies is }\)
\(f_{1} =2 f_{0} \)
\(f_{2} =3 f_{0} \)
\(=3 \times 150=450 \mathrm{~Hz} \)

f = 120Hz
Frequency of the 4th string is obtained from harmonic series
120,119, 118 and 117 Hz
(1)  (2)    (3)           (4)
\(\therefore\) f₄ = 117 Hz

Sono means sound related, and sonometer implies sound-related measurements. It is a device for demonstrating the relationship between the frequency of the sound produced in the transverse standing wave in a string, and the tension, length and mass per unit length of the string. Therefore, using this device, we can determine the following quantities:
(a) the frequency of the tuning fork or frequency of alternating current
(b) the tension in the string
(c) the unknown hanging mass

Construction:
The sonometer is made up of a hollow box which is one meter long with a uniform metallic thin string attached to it. One end of the string is connected to a hook and the other end is connected to a weight hanger through a pulley as shown in Figure. Since only one string is used, it is also known as monochord. The weights are added to the free end of the wire to increase the tension of the wire. Two adjustable wooden knives are put over the board, and their positions are adjusted to change the vibrating length of the stretched wire.
Working:
A transverse stationary or standing wave is produced and hence, at the knife edges P and Q, nodes are formed. In between the knife edges, anti-nodes are formed.
If the length of the vibrating element is I then
\(l={\lambda\over2}\Rightarrow\lambda=2l\)
Let f be the frequency of the vibrating element, T the tension of in the string and μ the mass per unit length of the string. Then using equation
\(v={\sqrt{T\over \mu}}\), we get
\(f={v\over \lambda}={1\over 2l}\sqrt{T\over \mu}\) in Hertz ....(1)
Let ρ be the density of the material of the string and d be the diameter of the string. Then the mass per unit length μ,
μ = Area\(\times\)density = πr²ρ = \(π\rho d^2\over4\)
frequency \(f={v\over \lambda}={1\over 2l}\sqrt{T\over {\pi d^2\rho\over4}}\)
\(∴ f={1\over ld}\sqrt{T\over \pi \rho}\)   .....(2)

When a jerk is given to a stretched string which is tied at one end, a wave pulse is produced and the pulse travels along the string. Suppose two persons holding the stretched string on either side give a jerk simultaneously, then these two wave pulses move towards each other, meet at some point and move away from each other with their original identity. Their behaviour is very different only at the crossing/meeting points; this behaviour depends on whether the two pulses have the same or different shape as shown in Figure.

When the pulses have the same shape, at the crossing, the total displacement is the algebraic sum of their individual displacements and hence its net amplitude is higher than the amplitudes of the individual pulses. Whereas, if the two pulses have same amplitude but shapes are 1800 out of phase at the crossing point, the net amplitude vanishes at that point and the pulses will recover their identities after crossing. Only waves can possess such a peculiar property and It is called superposition of waves. This means that the principle of superposition explains the net behaviour of the waves when they overlap. Generalizing to any number of waves i.e, if two or more waves in a medium move simultaneously, when they overlap, their total displacement is the vector sum of the individual displacements.
To express mathematically, consider two functions which characterize the displacement of the waves, for example,
Y₁ = A₁ sin(kx - \(\omega t\))
and
Y₂ = A₂ cos(kx - \(\omega t\))
Since, both Y₁ and Y₂ satisfy the wave equation (solutions of wave equation) then their algebraic sum
Y = Y₁ + Y₂
also satisfies the wave equation. This means, the displacements are additive. Suppose we multiply Y₁ and y₂ with some constant then their amplitude is scaled by that constant Further, if C₁ and C₂ are used to multiply the displacernents y₁ andY₂ respectively, then, their net displacement Y is
Y = C₁Y₁ + C₂Y₂
This can be generalized to any number of waves. In the case of n such waves in more than one dimension the displacements are written using vector notation.
Here, the net displacement \(\vec y\) is
\(\vec { y } =\overset { n }{ \underset { i=1 }{ \Sigma } } { C }_{ i }\vec { { y }_{ i } } \)
The principle of superposition can explain the following:
(a) Space (or spatial) Interference (also known as Interference)
(b) Time (or Temporal) Interference (also known as Beats)
(c) Concept of stationary waves
Waves that obey principle of superposition are called linear waves (amplitude is much smaller than their wavelengths). In general, if the amplitude of the wave is not small then they are called non-linear waves.