Interference of Sound Wave

Superposition principle

When two waves travel in a medium simultaneously in such a way that each wave represents its separate motion, then the resultant displacement at any point at any time is equal to the vector sum of the individual displacements of the waves.

This principle is illustrated by means of a slinky as shown in the figure.

(a) In the figure, (i) shows that the two pulses pass each other,

(b) In the figure, (ii) shows that they are at some distance apart

(c) In the figure, (iii) shows that they overlap partly

(d) In the figure, (iv) shows that  resultant is maximum

Figure illustrates the (ii) same events but with pulses that are equal and opposite. (iii)

If Y₁   and Y₂ are the (iv) displacements at a point, then the resultant displacement is (v)

given by Y = Y₁ +Y₂ .

If |Y₁ |=|Y₂ |= a,  and if the two waves have their displacements in the same direction, then |Y| = a + a = 2a

If the two waves have their displacements in the opposite direction, then |Y | = a + (-a) = 0

The principle of superposition of waves is applied in wave phenomena such as interference, beats and stationary waves.

Interference of waves

When two waves of same frequency travelling in the same direction in a medium superpose with each other, their resultant intensity is maximum at some points and minimum at some other points. This phenomenon of superposition is called interference.

Let us consider two simple harmonic waves of same frequency travelling in the same direction. If a₁ and a₂ are the amplitudes of the waves and φ is the phase difference between them, then their instantaneous displacements are

y₁ = a₁sin ωt          …... (1)

y₂ = a₂sin (ωt + φ)     …... (2)

According to the principle of superposition, the resultant displacement is represented by

y = y₁ + y₂

= a₁ sin ωt + a₂ sin (ωt + φ)

= a₁ sin ωt + a₂ (sin ωt. cos φ + cos ωt.sin φ)

= (a₁ + a₂ cos φ) sin ωt + a₂ sin φ cos ωt           …... (3)

Put a₁ + a₂ cos φ = A cos θ               …... (4)

a₂ sin φ = A sin θ                             …... (5)

where A and θ are constants, then

y = A sin ωt. cos θ + A cos ωt. sin θ

or  y = A sin (ωt + θ)

This equation gives the resultant displacement with amplitude A. From eqn. (4) and (5),

This equation gives the resultant displacement with amplitude A. From equation (4) and (5),

A²cos²θ + A²sin²θ = (a₁+a₂cosφ)² + (a₂sinφ)²

So, A² = a₁² + a₂² + 2a₁a₂cosφ                  

Thus, A = √[a₁² + a₂² + 2a₁a₂cosφ]           …... (7)

Also tanθ = a₂ sinφ/[a₁+a₂ cosφ]               …... (8)

We know that intensity is directly proportional to the square of the amplitude

So, I ∝ A² 

Thus, I ∝ (a₁² + a₂² + 2a₁a₂ cosφ)           …... (9)

Special cases

The resultant amplitude A is maximum, when cosφ = 1 or φ = 2mπ where m is an integer (i.e) I_max ∝ (a₁+a₂)²

The resultant amplitude A is minimum when

cosφ = -1 or φ = (2m + 1)π

I_min ∝ (a₁ – a₂)²

The points at which interfering waves meet in the same phase φ = 2mπ i.e 0, 2π, 4π, ... are points of maximum intensity, where constructive interference takes place. The points at which two interfering waves meet out of phase φ = (2m + 1)π i.e π, 3π, ... are called points of minimum intensity, where destructive interference takes place.

Refer this video to know more about interference of wave

Experimental demonstration of interference of sound

The phenomenon of interference between two longitudinal waves in air can be demonstrated by Quincke’s tube shown in figure. Quincke’s tube consists of U shaped glass tubes A and B. The tube SAR has two openings at S and R. The other tube B can slide over the tube A. A sound wave from S travels along both the  paths  SAR  and  SBR  in opposite  directions  and  meet at R.

If the path difference between the two waves (i.e) SAR ~ SBR is an integral multiple of wavelength, intensity of sound will be maximum due to constructive interference.

i.e  SAR ~ SBR = mλ

The corresponding phase difference φ between the two waves is even multiples of π. (i.e) φ = m 2π where m = 0, 1, 2, 3 ....

If the tube B is gradually slided over A, a stage is reached when the intensity of sound is zero at R due to destructive interference. Then no sound will be heard at R.

If the path difference between the waves is odd multiples of λ/2 , intensity of sound will be minimum.

i.e  SAR ~ SBR = (2m + 1) λ/2

The corresponding phase difference φ between the two waves is odd multiples of π. (i.e) φ = (2m + 1)π where m = 0, 1, 2, 3 .....

Problem (JEE Main):

In a plane progressive harmonic wave particle speed is always less than the wave speed if

(a) amplitude of wave is less than  λ/2π.

(b) amplitude of wave is greater than  λ/2π.

(c) amplitude of wave is less than λ.

(d) amplitude of wave is greater than  λ/π.

(λ = wavelength)

Solution:

Let equation of a plane progressive harmonic wave be,

y = A sin (ωt – kx)

wave speed, v = ω/k

and maximum particle speed, v_p = Aω

Given that, v_p < v

Or, Aω < ω/k

Or, A < 1/k

Or, A< λ/2π

Thus, from the above observation we conclude that, option (a) is correct. 

Which of the following statements is correct:

(a) both sound and light waves in air are longitudinal.

(b) both sound and light waves in air are transverse.

(c) sound waves in air are transverse and light waves are longitudinal.

(d) sound waves in air are longitudinal and light waves are transverse.

Question 2

The reciprocal of wavelength is called_____

a) frequency

b) wave velocity

c) wave number

d) amplitude

Question 3

When the particles of a medium are vibrating at right angles to the direction of energy transport, then the wave is a ____ wave.

(a) transverse       (b) sound

(c) standing           (d) longitudinal

Question 4

Many wave properties are dependent upon other wave properties. Yet, one wave property is independent of all other wave properties. Which one of the following properties of a wave is independent of all the others?

(a) wavelength          (b) frequency

(c) period                  (d) velocity

Question 5

If the frequency of a wave is doubled and if the speed remains constant, its wavelength is ____.

(a) quartered             (b) halved

(c) unchanged           (d) doubled

Related Resources:-

• You might like to refer progressive wave. 

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