17 December 2004

Distance Makes the Sound Grow Fainter
John W. Dooley, Physics Department
Millersville University

Sound become fainter as the listener moves farther away from the source. This familiar effect was invoked to understand the results of the "half as loud" experiment. Underlying the effect are two distinct processes; one conserves energy, and the other does not. Here we consider the process in which energy is conserved.

Dispersion is important in loudspeaker design. To fill a room with sound, the speaker should disperse sound waves with equal intensity in all directions. The downside of dispersion is that when the sound is dispersed, it becomes fainter. You can measure this effect using a Radio Shack piezoelectric "buzzer," GoldWave software, a microphone, and a ruler as shown in Fig. 1.

As shown in Fig. 1, the hand-held buzzer is moved along a yard stick placed between the buzzer and the microphone. The experiment is done outdoors to reduce reflected sound. The experiment is begun by starting GoldWave on your computer to record the sound and placing the buzzer two inches from the microphone for about two seconds. The buzzer is then moved two inches along the scale and again held in place for two seconds.

The GoldWave graph in Fig. 2 shows decreasing steps in the amplitude of the sound as the buzzer is moved away from the microphone. The average amplitude of each step, measured from the graph, is plotted in the simpler line graph in Fig. 2, "Sound Diminishing Due to Dispersion." (Unwanted reflected sound adding to or canceling the direct sound may be the cause of the "wiggles" at large distances in the data.)

In the line graph in Fig. 2, the dots are measured data points, and the line through the points is the graph of a simple formula. The formula for the line is 40 divided by the distance in inches between the microphone and the buzzer. To understand how such a simple formula imitates the data so closely, we need to think about the energy in a sound wave.

First, remember water ripples spreading across a basin of water under a dripping faucet. Each time a drop hits the water, a circular ripple spreads away from the impact point. This uniform dispersion in all directions is the kind of dispersion desired for loudspeakers. To the extent that a speaker is "small" like the drop of water, sound waves will also radiate uniformly in all directions.

It turns out that "small" means that the speaker diameter must be smaller than the wavelength of sound. This is the reason that woofers easily fill a room with long wavelength, low pitched sound, while tweeters, which must try to disperse short wavelength, high pitched, sound, tend to project most of their sound straight ahead.

The Radio Shack piezoelectric buzzer shown in Fig. 3 qualifies as "small." It produces a 1,400 Hz tone having a wavelength of about 25 cm. The sound comes from a hole about 0.25 cm in diameter, which is 100 times smaller than the wavelength. We may expect it to radiate sound uniformly in all forward directions, producing nearly hemispherical sound wave fronts.

If the crest of the sound wave takes a spherical form, then the energy in the crest spreads uniformly over that spherical shape. (Picture a soap bubble expanding outwards, puffed out by the buzzer at the center of the bubble.) This is our picture of sound radiating (or dispersing) without losing any energy. Now we must think about how we detect that sound.

When we hear, we listen with an ear whose opening has an area of perhaps 1 square centimeter. The fraction of total sound energy that our ear captures is equal to the ratio of our ear "window" area to the area of the spherical wave crest.

If we listen farther from the source, our 1 square centimeter "window" captures a smaller fraction, because the sphere is larger, and the energy is spread out over a larger area. The way that the area of the sphere depends on distance determines how the loudness of the sound that we perceive depends on distance.

It turns out that the area of a sphere is 4 pi times the square of the radius of the sphere. This means that the fraction of energy captured by our ear is proportional to the reciprocal of the square of the distance to the speaker (1/distance squared). But the experiment shows the amplitude is proportional to 1/distance, not 1/distance squared. To understand this, we need to understand the connection between sound energy and sound amplitude.

When you compress a gas (by clapping hands for example), you do work in the gas. The energy in the "clap" sound wave is equal to the work that you did on the gas. The work is proportional to two things: The force exerted by your hands and the distance that your hands moved. The nature of the gas itself produces a complication. The further you compress a gas, the more forcefully it pushes back. That means that you have to increase the force you apply every time you want to compress the gas a little more.

The work done on the gas is equal to the energy stored in the gas. It can spring back when you release it, and give back the work you put in. The upshot of this result is that the energy stored is proportional to the square of the compression. Since the amplitude of a sound wave indicates the compression of the air in the wave, the energy in a sound wave is proportional to the square of the amplitude. Detailed discussions are found here and here.

If the energy is proportional to the square of the amplitude, and also to the reciprocal of the square of distance from source, then, the amplitude should be proportional to the reciprocal of the distance from the source. With this result in hand, we look again at the graph of amplitude versus distance, which is shown expanded in Fig. 4.

The purple line in the graph in Fig. 4 is a plot of the function, purple = 14/(distance from source) versus distance.

14 was chosen so that the purple line would agree with the first experimental data point. We see that the purple line agrees nicely with the measured amplitude, confirming the "1/(distance from source)" result.

This result also agrees with our understanding of the "half as loud" experiment. When forced to decide, people seem to judge that sounds originating twice as far away are half as loud.

Many acoustics practitioners use intensity, rather than amplitude to judge the loudness of sound. The intensity is the energy per second through a standard window (such as the ear). Because sound intensity levels can vary over a large range, it is attractive to use the logarithm of the intensity, so that the numbers don't get so big. (Technically, we must use the ratio of our intensity to a standard intensity to calculate our logarithm correctly.) The intensity measured this way is given the unit "Bel," after Alexander Graham Bell

When this is done, the numbers get a little too small, so the logarithm is multiplied by 10, and the unit given to this measure of sound intensity is the "decibel," abbreviated dB. A plot of intensity in dB versus distance for this experiment is shown in Fig. 5. This graph is difficult to use as a test of the 1/(square of distance from source) assertion based on conservation of energy.

The graph in Fig. 6 is more easily interpreted. Figure 6 is a plot of intensity in dB versus the logarithm of the source-to-microphone distance. The slope of this graph is -20. After removing the factor of 10 by which we multiplied to convert the log of intensity to dB, we have a slope of -2 for a log-log plot of intensity versus distance. The slope of a log-log plot is easy to interpret if the vertical axis is equal to the x value raised to some power, n. In that case, the slope is equal to the power, n. In this case, the slope of -2 indicates that the intensity varies as 1/(square of distance).