Question

Some loudspeaker horns for outdoor concerts (at which the entire audience is seated on the ground) are wider vertically than horizontally. Use diffraction ideas to explain why this is more efficient at spreading the sound uniformly over the audience than either a square speaker horn or a horn that is wider horizontally than vertically. Would this still be the case if the audience were seated at different elevations, as in an amphitheater? Why or why not?

Short answer

Some loudspeaker horns for outdoor concerts (at which the entire audience is seated on the ground) are wider vertically than horizontally since this is more efficient at spreading the sound uniformly over the audience than either a square speaker horn or a horn that is wider horizontally than vertically. Meanwhile, if the audience were seated at different elevations, as in an amphitheater then speaker should have more horizontal length.

Step-by-step solution

Define diffraction of sound

Diffraction of sound is a physical phenomenon in which sound wave bends around the edges to reach non-audible region. Necessary condition for the diffraction to happen with sound wave is, width of obstacle should be comparable to wavelength of the wave.

Diffraction angle $\theta$of central maxima is given as,

role="math" localid="1663899332758" $\mathrm{sin\theta }=\frac{\mathrm{\lambda }}{\mathrm{a}}$

Explain situation in field and amphitheater

Start from the equation,

$\mathrm{sin\theta }=\frac{\mathrm{\lambda }}{\mathrm{a}}$

Analyze it in two dimensions, vertical and horizontal. If there is audience at same altitude and spans far from the source, in the horizontal plane then to have first intensity maxima as wide as possible, length of horizontal plane should be sufficiently small. On the other hand, if audience is planar and to have very narrow maxima which will cover the entire audience equally then it is done by keeping vertical length of speaker big enough.

Meanwhile in amphitheater, distribution of audience is changed and now, audience is distributed in vertical plane only. However, in horizontal plane, audience is limited as the size of venue itself. So, horizontal length of speaker is larger than vertical length of speaker.

Thus, some loudspeaker horns for outdoor concerts (at which the entire audience is seated on the ground) are wider vertically than horizontally since this is more efficient at spreading the sound uniformly over the audience than either a square speaker horn or a horn that is wider horizontally than vertically. Meanwhile, if the audience were seated at different elevations, as in an amphitheater then speaker should have more horizontal length.