Acoustic Phenomena || Physics Grade XII

Acoustic Phenomena:

The branch of physics which deals with the process of production, transmission and reception of sound 

is called acoustics.

Pressure Amplitude: 

Sound wave is a longitudinal wave. It propagates in a medium in the form of compression and 

rarefaction i.e. variation in pressure takes place in the medium during the propagation of sound. So, 

sound wave or longitudinal wave is also called a pressure wave. The maximum change in pressure in 

the medium during the propagation of longitudinal wave is called pressure amplitude.

Expression for Pressure amplitude:

Consider a wave propagating through air along positive x-direction. The displacement of the vibrating layer of

air can be expressed as;

y = asin(𝜔𝑡 − 𝑘𝑥) ……………………….( 1)

(where a = displacement amplitude, 𝜔 = angular frequency and k = wave vector or propagation constant.)

Consider an infinitely small cylindrical vibrating layer of air of length ∆𝑥 and cross-section area ‘A’. Let the left 

cross-section of the cylinder displaces through a distance ‘y1’ and right cross-section through a distance ‘y2’ 

due to propagation of wave.

Then, the change in volume of the cylinder due to propagation of sound is given by

 ∆v = A ∆y (where ∆y=│y1-y2│)

Now, fractional change in volume of the cylinder is given by

 

∆𝑣

𝑣

 = 

𝐴∆𝑦

𝐴∆𝑥

 ∴ 

∆𝑣

𝑣

 = 

∆𝑦

∆𝑥

For limit ∆x⟶ 0, 

∆𝑣

𝑣

 = ∆𝑥 ⟶ 0

∆𝑦

∆𝑥 =

𝑑𝑦

𝑑𝑥 ……… (2)

Using equation (1)

 

∆𝑣

𝑣

=

𝑑[asin(𝜔𝑡−𝑘𝑥)]

𝑑𝑥

 ∴

∆𝑣

𝑣

= −𝑎𝑘𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑥) ……………….(3)

Now Bulk modulus of elasticity of air is given by

 𝐵 =

−∆𝑃

∆𝑣

𝑣

 ⟹ ∆𝑃 = −𝐵

∆𝑣

𝑣

Using equation ( 3)

∆𝑃 = −𝐵(−𝑎𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑥))

 ∴ ∆𝑃 = 𝐵𝑎𝑘𝑐𝑜𝑠 (𝜔𝑡 − 𝑘𝑥) …………….( 4)

This is the pressure equation of longitudinal wave.

The change in pressure is maximum, when 𝑐𝑜𝑠(𝜔t-kx) = 1.

The maximum change in pressure is denoted by ∆𝑃𝑚 called pressure amplitude.

i.e. ∆𝑃𝑚 = 𝐵𝑎𝑘 …………………………(5)

This is the expression for pressure amplitude.

Then equation (4) can also be written as,

 ∆𝑃 = ∆𝑃𝑚𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑥) ……………(6)

Using v = √

𝐵

𝜌

 where, v is the velocity of sound, the pressure amplitude can also be expressed as;

∆𝑃𝑚 = 𝑣

2𝜌𝑎𝑘

Musical Sound and Noise:

The sound that produces pleasing effect on the listeners is called musical sound. Such sound is produced by 

regular and periodic vibration of source. Example: sound produced by a flute.

The sound that produces unpleasing effect on the listeners is called a noise. Such sound is produced by 

irregular and disturbed vibration of the source. Example: barking of dog, sound produced by vehicles etc.

Characteristics of Musical Sound:

There are three following characteristics of musical sound:

1. Pitch:

The sharpness or shrillness of the sound is called pitch. It depends upon the frequency of sound. The 

pitch of the sound increases with increase in frequency. For example, the pitch of sound produced by a 

child is greater than that of adults.

2. Loudness:

It is the subjective term that depends upon the sensation of listener’s ear. The loudness of a sound 

depends upon the intensity of sound. lt increases with increase in intensity of sound. The intensity of 

sound is directly proportional to the square of amplitude. So, the loudness of the sound is also directly 

proportional to the square of amplitude. The loudness of sound also increases with increase in surface 

area of vibrating body. It is affected by the presence of other bodies in the medium.

3. Quality or Timber:

The property of sound, which enables us to distinguish between the two sounds of same frequency

and loudness produced by two different sources, is called quality of sound or timber. It is the measure 

of complexity of sound. Quality of sound depends upon the number of overtones present in it. Nature 

has provided different overtones in the sounds of different persons. So, the quality of sound differs

from person to person.

Intensity of sound:

The intensity of sound at a point is defined as the amount of sound energy passing normally through unit area 

in unit time. It is denoted by 𝐼.

If ‘E’ be the sound energy passing normally through area ‘A’ in time ‘t’, the intensity of sound is given by

𝐼 = 

𝐸

𝐴𝑡

 It’s SI unit is watt/m2 or J/m2

s.

Expression for intensity of sound:

Consider a sound wave propagating through air along x-direction. The displacement of the vibrating layer of 

air is given by

𝑦 = 𝑎𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑥), where ‘a’ is amplitude, ′𝜔′ is angular frequency and ‘k’ is wave vector or 

propagation constant.

The velocity of vibrating layer of air is given by

 𝑢 =

𝑑𝑦

𝑑𝑡

 ⟹ 𝑢 =

𝑑⦋𝑎𝑠𝑖𝑛(𝜔𝑡−𝑘𝑥)⦌

𝑑𝑡

 ∴ 𝑢 = 𝑎𝜔𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑥) … … … … … … … . . (1)

Let 𝑣 be the velocity of sound in air. Then the length of air disturbed in time ‘t’ is given by

 𝑙 = 𝑣𝑡

Volume of disturbed air = Area × length

 i.e. V = 𝐴𝑣𝑡 (where, A is cross-sectional area of disturbed air)

Mass of disturbed air (m) = density × volume

 ∴ m = 𝜌 × 𝐴𝑣𝑡 …………………………(2) ( where 𝜌 is density of air )

Now, Kinetic energy of vibrating layer of air is given by

 KE =

1

2

𝑚𝑢

2

Using eqns (1) & (2),

 KE =

1

2

𝜌𝐴𝑣t𝑎

2 𝜔2

𝑐𝑜𝑠2

(𝜔𝑡 − 𝑘𝑥)

The vibrating layer of air has both KE and PE. From conservation of energy the total energy (KE +PE) of 

vibrating layer of air is equal to the maximum kinetic energy.

 i.e E = 𝐾𝐸𝑚𝑎𝑥

 ∴ E = 1

2

𝜌𝐴𝑣ta2𝜔2 ………………..(3)

Now, the intensity of sound is given by

 𝐼 =

𝐸

𝐴𝑡

⟹ 𝐼 =

1

2

𝜌𝑣𝜔2a

2

Using 𝜔 = 2𝜋f,

 𝐼 =

1

2

𝜌𝑣2

2 𝜋

2

f

2a

2

 or, 𝐼 = 2𝜋

2𝑓

2𝜌𝑣 a

2

Since 𝜌, 𝑣 and 𝜔 or f are constant,

 𝐼 ∝ a

2

i.e the intensity of sound is directly proportional to square of amplitude of sound wave.

Relation between intensity of sound and pressure amplitude:

The intensity of sound is given by

 ɪ = 

1

2

𝜌𝑣𝜔2𝑎

2

 or, ɪ = 

1

2

𝜌𝑣𝑣

2𝑘

2𝑎

2

 [∵ 𝜔 = 2πf = 2𝜋

𝜆

 fλ = vk ]

 or, ɪ = 

1

2

𝜌 √

𝐵

𝜌

×

𝐵

𝜌

𝑘

2𝑎

2

 

 or, ɪ = 

1

2√𝜌𝐵

𝐵

2𝑘

2𝑎

2

 

 or, ɪ = 

1

2√𝜌𝐵

∆𝑃𝑚

2

 [ ∆𝑃𝑚 = Bak, the pressure amplitude ]

 ∴ ɪ ∝ ∆𝑃𝑚

2

 [∵ 𝜌 and B are conatants ]

i.e. the intensity of sound is directly proportional to the square of pressure amplitude.

Threshold of Hearing:

The lowest intensity of sound that can be heard by normal human ear is called threshold of hearing. It is 

denoted by 𝐼0 . It’s value is taken to be 1× 10-12 watt/m2

.

Threshold of pain:

The maximum intensity of sound that can be heard by normal human ear without damage to the eardrum is 

called threshold of pain. Its value is taken to be 1 watt/m2.

Inverse square law:

It states that the intensity of sound at a point is inversely proportional to the square of distance from the 

source of sound.

i.e 𝐼 ∝ 1/𝑟

2

 In general, 

𝐼1

𝐼2

= 

𝑟2

2

𝑟1

2

Sound Intensity Level(𝜷):

The loudness of sound is measured in terms of a parameter called sound intensity level. It is denoted by 𝛽 . 

If 𝐼 be the intensity of sound, it’s sound intensity level is defined by the relation

 𝛽 = 𝑙𝑜𝑔 𝐼

𝐼𝑜

 {where I0 is threshold of hearing}

The SI unit of sound intensity level is Bel.

1 Bel :

If 𝐼 = 10𝐼𝑜 , then 𝛽 = 1 Bel

Thus sound intensity level is said to be 1 Bel if it’s intensity is 10 times the threshold of hearing. Bel is a larger 

unit of sound intensity level. Most widely used unit of sound intensity is decibel (1B=10dB).

In decibel unit,

 B = 10 𝑙𝑜𝑔 𝐼

𝐼𝑜

 

Difference between two sound intensity levels:

We have,

 𝛽1 - 𝛽2 = 10 𝑙𝑜𝑔 𝐼1

𝐼𝑜

 - 10 𝑙𝑜𝑔 𝐼2

𝐼𝑜

 

or, 𝛽1 - 𝛽2 = 10 log 𝐼1

𝐼2

or, 𝛽1 - 𝛽2 = 10 log 𝑟2

2

𝑟1

2

 ( 𝐼 ∝

1

𝑟

2

)

∴ 𝛽1 - 𝛽2 = 20 log 𝑟2

𝑟1