Question

(II) The two sources of sound in Fig. 12–16 face each other and emit sounds of equal amplitude and equal frequency (305 Hz) but 180° out of phase. For what minimum separation of the two speakers will there be some point at which (a) complete constructive interference occurs and (b) complete destructive interference occurs. (Assume T = 20°C.)

Short answer

1. The minimum separation between two speakers for constructive interference is \(0.562\;{\rm{m}}\).
2. The minimum separation between two speakers for destructive interference is zero.

Step-by-step solution

Determination of the minimum separation between two speakers

The two sources of a sound wave are 180^° out of phase. So, for constructive interference, the source of the sound wave should move a distance equal to half of the wavelength of sound waves.

Given information

Given data:

The frequency of sound waves is \(f = 305\;{\rm{Hz}}\).

The temperature is \(T = 20^\circ {\rm{C}}\).

Evaluation of the minimum separation between two speakers for constructive interference

(a)

When the temperature is given, the speed of the sound wave can be calculated as:

\(\begin{aligned}{}v &= 331.3 + 0.6T\\v &= 331.3 + \left( {0.6 \times 20^\circ {\rm{C}}} \right)\\v &= 343.3\;{\rm{m/s}}\end{aligned}\)

For the constructive interference, the minimum separation of the two speakers can be calculated as:

\(\begin{aligned}{}d &= \frac{\lambda }{2}\\d &= \frac{{\left( {\frac{v}{f}} \right)}}{2}\\d &= \frac{{\left( {\frac{{343.3\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}}}{{305\;{\rm{Hz}}}}} \right)}}{2}\\d &= 0.562\;{\rm{m}}\end{aligned}\)

Thus, the minimum separation between two speakers for constructive interference is \(0.562\;{\rm{m}}\).

Evaluation of the minimum separation between two speakers for destructive interference

(b)

As both the sound waves are out of phase by \(180^\circ \), the phase shift between the waves must be \(180^\circ \) out of phase. So, the distance between the two speakers should be zero.

Hence, the minimum separation between two speakers for destructive interference is zero.