The physical phenomena associated with the production, transmission of sound and its effects is called acoustic phenomena.

Pressure amplitude

It is maximum fluctuation in the pressure at a point in a medium when a longitudinal wave travels through the medium.

 Pressure at certain point is given by

P(x,t)  = Pmaxcos(wt – kx)

Where, P­max = Kka =KawvKawv is called pressure amplitude.

Musical Sound and Noise              

The sound which gives pleasing effect to our ears is called musical sound. Music is pleasing to ear. It is due to the periodic and regular vibration. Wave form does not show sudden change in wavelength and amplitude. Frequency is generally high. It is for long duration.

Characteristics of Musical sound

Pitch: The pitch of the sound depends upon the frequency of vibration of air and relative motion between the sounding body and the listener. The sound with low frequency is identified as low pitch and high frequency is identified as high pitch. The high pitch sound is called shrill and low pitch sound is called grave of flat.

Loudness or intensity: The intensity I of the sound is defined as the amount of energy crossing per unit area per second and the amount of intensity determines the loudness of the sound. Factors determining the loudness of the sound:

Loudness of the sound depends upon:

 Amplitude of sounding body: Greater the amplitude larger the intensity of sound.

Surface area of the vibrating source: Greater the surface area loudness of sound is also larger. eg, sound from the large drum is of larger intensity.

Presence of other bodies: If there is sound reflecting object near to the source the loudness of the sound increases. That why we hear much louder sound inside the room than outside.

Frequency of sound: The loudness of the sound of vibrating body is directly proportional to the square of the frequency of the vibrating body.

Distance between the source and listener: The loudness of the sound decreases as distance increases

Density of the medium: The loudness of the sound is directly proportional to the density of medium through which it passes.

Quality or timber: It is the factor that determines the characteristics of the sound having same pitch and loudness. It depends on the number and intensity of the harmonics.

Threshold of hearing

It is the lowest intensity of sound that can be heard by our ears is called threshold of hearing.

Intensity and intensity level

Intensity of the sound is defined as the rate of transfer of sound through one square meter area held normally at the place where the sound travels.

Let I be the intensity of the sound, E be the energy and A be the area and t be the time then the intensity is given by

Intensity (I)= energypassingarea∗time=EA∗tenergypassingarea∗time=EA∗t

Intensity level: it is define as the logarithm of the ratio of intensity of the sound to an arbitrarily chosen intensity i.e. L= log10 (I/I0)

Where I be the intensity of the wave and I0 be the arbitrary intensity taken as 10-12Wm-2 and the minimum intensity of the sound that can be heard by our ear is called threshold hearing. It is denoted by I0 and the value is equal to 10-12Wm-2.

Relation between I and L

The loudness of sound is defined as the degree of sensation of sound produced in the ear. The loudness of sound depends on its intensity but the relationship is not linear. It states that – the magnitude of any sensation is proportional to the logarithm of the physical factor which produces it. This is called the Weber-Fechner law.. Thus, according to this law,

I ∝ log I or

L = K log I ———– 1 Where L is the loudness, I is the intensity and K is a constant of proportionality. The minimum intensity of sound is denoted by I.whose value is chosen to be 10-12Wm-2. Let the loudness corresponding to this threshold of audibility be l0. Then equation (1) can be written as

L0 = K log I0 ———- 2

Subtracting equation 2 equation 1 we get

L – L0= K log I – K log I0

= K (log I – log I0)

=K log (I/I0)

(L – L0) indicates as to how much the loudness of a given sound is above the minimum value for hearing. Therefore, it is called the intensity level L.

L=K log …………………………..3

If the value of the constant K is chosen to be 1, then L is measured in a unit called bel

if I = 

then, L =Log Or L = 1 bel ………….4

Hence, the intensity level of a sound is said to be 1 bel, if its intensity is ten times the threshold intensity.

If the value of the constant K is chosen to be 10, then L is measured in decibel.

Thus L(in db)=10 log(110)(110)……………..5

Bel and decibel both are the unit of loudness and the loudness of the sound is said to be one bel if its intensity (I) is 10 times the intensity of the sound at the threshold of hearing i.e. L=1 bel if I=10I0 .we have I=log10(10)=1bel. Bel is the biggest unit of loudness and the smallest unit of the loudness is decibel and it is defined as 1/10th of bel. i.e.1decibel=bel/10

Inverse square law

Inverse square law
Inverse square law

It states that “the intensity of the sound at a point is directly proportional to the square of the distance of the point from the source of the sound “.i.e. 

 Beats

Consider two waves from two different sources of nearly same frequency travelling in the same direction. At any point let’s say A the two waves arrives in phase and intensity becomes maximum at that point. The phase difference between the two waves increases until the compression from one of the sources and rarefaction from the other arrives at the same time. At this point B, the intensity becomes minimum and no sound is produced. Each rise and fall in the intensity constitutes one beat and the corresponding time interval is called beat frequency.

Graphical representation of beat

Beat
Beat

When two sound waves of nearly the same frequency are travelling in the same medium in the same direction, the intensity of the resultant sound rises and falls regularly a number of times a second. These periodic variations in the intensity of sound due to superposition of two sound waves of slightly different frequencies are called beats and hence in this way they are formed. The number of beats heard per second is called beat frequency

Analytical method for the formation of beat

Suppose two waves of the frequency f1 and fand each of amplitude are traveling in a medium in the same direction. The equation of the waves are:

Y1=asin(ωt – k1x)…………..1

And from the superposition principle, the resultant displacement at that point is given by

= asin(ω1t – k1x)+ asin(ω2t – k2x)

When two point are superimpose at a point x=0 then we have

  1. asin(ω1+ω22)tcos(ω1−ω22)tasin(ω1+ω22)tcos⁡(ω1−ω22)t

=Where ω1=2πf1 and ω2=2πf2

= 2acos2 π f1−f22f1−f22t. sin2 π f1−f22f1−f22t

Let A=2a cos2πf1−f22f1−f22tThen the above equation become

Condition for maximum.

If the resultant amplitude A become maximum when become maxium

i.e.cos2 π f1−f22f1−f22t=±1

or,cos2 π f1−f22f1−f22t=cosnπ where n=0, 1, 2, 3, ……..

or,2π f1−f22f1−f22t=nπ

or, t=nf1−f2nf1−f2 cos2πf1−f2sf1−f2stor, putting the value of n we get

t=0,1f1−f21f1−f2, 2f1−f22f1−f2, 3f1−f23f1−f2, ………..

or, T = (1f1−f2)−0,T=(3f1−f2)−(2f1−f2)=(1f1−f2)(1f1−f2)−0,T=(3f1−f2)−(2f1−f2)=(1f1−f2)

or, T = (1f1−f2)(1f1−f2)

Condition of minima.

If the resultant amplitude A become minima whencos2π(f1−f22)cos2π(f1−f22)t is minima. i.e.

cos2π(f1−f22)t=0cos2π(f1−f22)t=0

Or,cos2π(f1−f22)t=cos(2n+1)π2Or,cos2π(f1−f22)t=cos⁡(2n+1)π2 

where n=0, 1, 2, 3, …………

Or, t = 12(2n+1f1−f2)12(2n+1f1−f2)

On putting the value of the n we get

t=12(1f1−f2),12(3f1−f2),12(5f1−f2)t=12(1f1−f2),12(3f1−f2),12(5f1−f2)

or, T = 12(3f1−f2)−12(1f1−f2)=(1f1−f2)12(3f1−f2)−12(1f1−f2)=(1f1−f2)

Therefore for both maxima and minima we have T= 1f1−f21f1−f2. Therefore the beat frequencyis equal to the frequency to the difference of the frequencies of sound waves

Doppler effects

The apparent change in frequency of the sound heard by observer is due to the relative motion of the source or observer or both are called Doppler effect of sound.

Apparent frequency is calculated by f’  = V+VrV+VsV+VrV+Vs * fs

Relative motion of the receiver

If a source is stationary, it will emit sound waves that propagate out from the source as shown below.

As the receiver moves towards the source, it will detect the sound coming from the source but each successive sound wave will be detected earlier than it would have if the receiver were stationary, due to the motion of the receiver .Thus the frequency that each successive wave front would be detectedwould be changed by this relative motion where:

Δf=VrλoΔf=Vrλo

λo is the original wavelength of the source

ΔfΔf is the change in observed frequency

Vr is the velocity of the receiver

Since the original frequency of the source can be expressed in terms of the wavelength where

fo= vλovλo, the observed frequency becomes:

f’=fo +ΔfΔf

f’=fo(v+Vr)vfo(v+Vr)v this equation works when Vr is moving towards the source.

Note that this equation only works if the relative velocity of the receiver, Vr is towards the source. If the motion is away from the source, the relative velocity would be in the opposite direction and the equation would become:

f’=fo(v−Vr)vfo(v−Vr)v

Relative motion of the source

If the source is moving towards the receiver, the spacing between the successive wave fronts would be less as seen in the figure below. This would be expressed as:

ΔΔλ=Vs/fo

‘Vs’ is the relative velocity of the source.

Observed frequency f’=vλ+λovλ+λo

Or, f’=fovv−Vsfovv−Vs

This is only when the source is moving towards the receiver. If the source is moving away it becomes f’=fovv+Vsfovv+Vs

Limitations of Doppler effect in sound

The Doppler Effect is not applicable in following conditions:

a. If the velocity of sound of the source is greater than that of the sound because the wave gets distorted due to which no change in frequency will be observed.

b. If the velocity of the sound of the observer is greater than that of the sound.

 Applications of Doppler effects:

Infrasonic and ultrasonic sound

The sound that is produced below the audible frequency is called infrasonic .It range is below 20KHz. It is produce by large object eg.earthquake waves.

The sound wave of frequency above the audible (i.e. above 20 KHz) are called ultrasonic. It is produce by small objects. Eg. quartz crystal. Ultrasonic are the longitudinal mechanical waves of frequency greater than 20 KHz and the wavelength very short. Supersonics are those objects which travel faster than sound i.e. faster than 332 m/s. Jet planes or waves are supersonics.

 Noise pollution

An unwanted sound which causes unpleasant effect in our ears is called noise pollution.

Causes of sound pollution

Transportation systems – including cars, trains and airplanes – are one of the most common sources of noise pollution since they can be particularly loud and unrelenting in certain areas. In general, people who live in urban centers are more likely to be exposed to noise pollution due to population density and the increased presence of the transportation systems described above. Living in an urban city may also mean that people are more exposed to the noises of construction, which is another major source of noise pollution.

While those in larger cities are more likely to experience noise pollution, those in rural settings may also experience this problem as well. A key example of rural noise pollution is farming, which may include a variety of machines that produce harsh or loud noises.

Effects of noise pollution

1.It may cause deafness.

2.People may suffer from headaches and migraines.

Controlling noise pollution

It decreases the working efficiency.

Rooms and wall should be covered with sound absorbing materials.

Loud speaker, radio and other music system should be played at low volume.

Industries that produces nose should be build far from the residential areas.