Interference of sound waves, practice problems, FAQs

When you play two coherent sound sources and place them at some distance. You will find that at some places the sound is much louder and at some places there is no sound. This is due to interference of sound waves.When two waves are in phase then constructive interference is produced and sound is louder and when opposite in phase then destructive interference is produced and sound is minimum or zero.

Table of content

Interference of sound wave
Constructive interference
Destructive interference
Practice problem
FAQs

Interference of sound wave

Consider two tuning forks, S1 and S2, acting as the source of sound waves and also consider these two sources to be coherent sources (i.e., either there is no phase difference or there is a constant phase difference between the waves that originate from them).

Suppose point O is situated at a distance x from source S1 and x - Δx from source S2. Thus, S1O = x and S2O = x - Δx and therefore, the path difference between them is |S1O - S2O| = Δx

Let the equation of the sound waves from S1 and S2 be as follows:

Equation (ii) can also be rewritten as follows:

Where ϕ is the phase difference between the waves, and this can be formulated as follows:

⇒ Phase difference Path difference ………………..(v)

Therefore, any path difference between the waves leads to a phase difference.

From equations (i) and (iii), it can be said that they are sinusoidal waves that have the same frequency and the same direction of propagation but different amplitudes. The resultant wave will also be a sinusoidal wave given by p = p0 sin(ωt - kx + δ) and they can be represented in a phasor diagram as follows:

Where p0 and δ are the amplitude and phase angle of the resultant wave and they are defined as follows:

Now, along with the phase difference originating due to the path difference, if there is an initial phase difference (i.e., the epoch, ϕ0) between the waves, then the total phase difference will be

Then, equation (vi) will reconstruct itself as

Therefore, the magnitude of the resultant wave depends upon the total phase difference between the waves.

Constructive interference

With reference to equation (vii), it can be said that constructive interference will take place if the following condition is satisfied.

cosΔϕ = 1 that is maximum resultant amplitude.

⇒ Δϕ = 2nπ

Where n = 0, 1, 2, 3,.....

Hence, from equation (viii), we get p0 = p1 + p2

Now, and we know that for constructive interference, Δϕ = 2nπ

If the epoch becomes zero, then we get

Therefore, for constructive interference to take place, the necessary and sufficient

condition is Δϕ = 2nπ, and if the initial phase difference between the interfering waves becomes zero, then Δϕ = 2nπ and Δx = nλ are both true. If path difference is integral multiple of wavelength then interference is constructive.

Destructive interference

Destructive interference will take place if the following condition is satisfied.

cosΔϕ = - 1 that is resultant amplitude is minimum.

⇒ Δϕ = (2n -1)π

Where n = 1, 2, 3,.....

Hence from equation (viii), we get, p0 = |p1 - p2|

Now, and we know that for destructive interference, Δϕ = (2n - 1)π. If the epoch becomes zero, then we get

Therefore, for destructive interference to take place, the necessary and sufficient

condition is Δϕ = (2n +-1)π and and are both true. It means if the path difference is an odd multiple of half of the wavelength then interference is destructive.

Practice problem

Q. The figure shows a tube structure in which a sound signal is sent from one end and is received at the other end. The circular portion has a radius of 20.0 cm. The frequency of the sound source varies from 1,000 Hz and 4,000 Hz. At what frequencies the maxima of intensity are detected. Take The speed of sound in air as 340 ms^-1.

A.There are two possible routes for the soundwaves to take from the source to the ear of the listener.

(i) The soundwave goes straight to the listener.

(ii) The soundwave goes to the listener via the semicircular channel.

Since there will be a difference in the path in those two routes and there is no epoch because there is only one source, we need to find the condition for constructive interference depending on that path difference.

The required path difference between the two routes is Δx = πr - 2r

Therefore, for constructive interference to take place, the following condition has to be satisfied.

This is equation for maximum intensity

By puting

, and

Since the source has the frequency range to so the required frequency at which intensity will be maximum are and .

Q.Two speakers, S1 and S2, which are driven by the same amplifier, are placed at y = 1.0 m and y = − 1.0 m as shown in the figure. The speakers vibrate in phase with the frequency 600 Hz. There is A man on the X-axis at point P which is far away from the origin. Now man starts moving parallel to the Y-axis. The speed of sound in air is 330 ms-1. At what angle (θ) will the intensity of the sound drop to a minimum for the first time?

A. wavelength of sound

Now, consider any point (Q) as shown in the figure, and consider that the point makes an angle of θ at point O, which is at the midpoint between the two sources.

the path difference at point Q : Δx = S2M

Now, from the figure, it can be observed that

Thus, Δx = S2M = d sin θ

For destructive interference to take place, the following condition should be satisfied.

For first minima,

Q. Let two sound waves having wavelengths 5m and 6m. If these two waves propagate in a gase medium with speed 300 m/s. What happens when they interfere?

A. Given and

The frequency corresponding to wavelength

When two wave of nearly same frequency interfere , Beats are produce and frequency of beat is

beat per sec.

Q.Consider the figure in which two coherent sources, S1 and S2, emit sound of wavelength λ in phase. The separation between the sources is 3λ. A circular wire of a large radius is placed in such a way that S1S2 lies in its plane and the midpoint of S1S2 is at the center of the wire. Find the angular positions θ (in first quadrant) on the wire for which constructive interference takes place./

A. Consider any point (P) on the circumference of the circle.

Therefore, the path difference at point P between the sound waves originating from sources S1 and S2 is Δx = |S1P - S2P| = S1M (assuming that d = 3λ is smaller than the radius of the circle).

Now, S1M = d cos θ

Therefore, if θ1 and θ2 are the two positions of the maxima in the first quadrant,

From equestion we get,

Therefore, the angular position on which constructive interference take place are and