18.1.2 Interference of Sound Waves

What is Interference of Sound Waves?

Interference of sound waves occurs when two sound waves travelling through the same medium overlap and combine. The resultant wave at any point is determined by the Principle of Superposition:

$y=y_1+y_2$

where $y_1$ and $y_2$ are the individual displacements of the two waves at that point.

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Types of Interference

Constructive Interference

Constructive interference occurs when two waves arrive in phase (phase difference = $0$, $2\pi$, $4\pi$, … radians). The crests of one wave align with the crests of the other, producing a resultant wave with maximum amplitude.

Condition (path difference): $\Delta L=n\lambda (n=0,1,2,3,\dots )$

Since intensity is proportional to the square of amplitude ($I\propto A^2$), if each wave has amplitude $A$, the resultant amplitude is $2A$ and the resultant intensity is $4I_0$ — four times the intensity of one wave.

Destructive Interference

Destructive interference occurs when two waves arrive out of phase by $\pi$ radians (180°). The crest of one wave aligns with the trough of the other, producing a resultant wave with minimum (zero) amplitude.

Condition (path difference): $\Delta L=\left(n+\frac{1}{2}\right)\lambda (n=0,1,2,3,\dots )$

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Demonstrating Sound Interference

Two-Speaker Experiment

A classic demonstration of sound interference uses two loudspeakers connected to the same signal generator (ensuring they emit sound of the same frequency and in phase):

1. Place two speakers a short distance apart, both facing the same direction.
2. Walk slowly along a line parallel to the line joining the speakers.
3. You will hear alternating loud regions (constructive interference) and quiet regions (destructive interference).

At any point P, the path difference $\Delta L=|d_1-d_2|$ (where $d_1$ and $d_2$ are the distances from each speaker to P) determines the type of interference:

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Conditions for a Stable Interference Pattern

For a clear and stable interference pattern to be observed, the two sound sources must satisfy the following conditions:

1. Coherent sources: The sources must emit waves of the same frequency and maintain a constant phase difference over time. In the two-speaker experiment, this is achieved by driving both speakers from the same signal generator.

2. Monochromatic (single frequency): The sources should emit a single frequency (pure tone) rather than a mixture of frequencies.

3. Comparable amplitudes: The amplitudes of the two waves should be nearly equal so that destructive interference produces a clear minimum (ideally zero intensity) and the contrast between loud and quiet regions is maximised.

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Worked Example

Problem: Two coherent speakers emit sound of wavelength $\lambda =0.50\text{ m}$. Point P is $3.0\text{ m}$ from speaker 1 and $4.25\text{ m}$ from speaker 2. What type of interference occurs at P?

Solution: $\Delta L=4.25-3.0=1.25\text{ m}$ $\frac{\Delta L}{\lambda }=\frac{1.25}{0.50}=2.5=\frac{5}{2}$

Since $\Delta L=2.5\lambda =\left(2+\frac{1}{2}\right)\lambda$, this satisfies the destructive interference condition. Point P is a quiet region.