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12 December 2003

Sound Amplitude and Loudness

It is important in our lives to be able to hear faint sounds, and also to extract information from very loud sounds. One measure of loudness is the sound intensity, or the energy per second that reaches our ear. We handle sounds over a range of 10 billion in intensity.

Physiologists are still unraveling the secrets of how we create such a wide range of sensitivity. As amateurs we can do a simple experiment with "loudness" that illustrates the kind of problem which must be solved.

Roughly speaking, vibrations with larger displacement are louder. The maximum displacement of a vibration is called the amplitude. We can seek a relation between the amplitude of a sound vibration and the loudness.

For this experiment we use the sound generating capability of the GoldWave software. From the menu bar, choose new (monaural) file, then choose tools and expression evaluator. The figure shows the result of choosing waves, and sine. (I multiplied the expression by .1 by typing in the expression window.)

In this expression, 0.1 is the amplitude of the signal that will generate the sound. It will work for this experiment to say that the sound itself has amplitude 0.1. The figures below shows a plot of signal versus time in which the amplitude is varied. It always returns to 0.1, but can go as high as 0.4. The expanded figure shows the transition between those extreme amplitudes. The frequency is 500 Hz.

Click images to enlarge

If you create these tones yourself and listen to them with earphones, you can replicate a remarkable social/physiological/physical experiment. The listener is told to pay attention to the change from low amplitude to higher amplitude sound.

The listener’s task is to identify the case in which the second sound is "twice as loud" as the original sound. The social aspect is that people can be persuaded to try the experiment even though you have no way of explaining to them what you mean by "twice as loud." They are willing to have a stab at identifying "twice as loud."

The remarkable physiological result is that everyone tends to come up with the same answer. For just about everyone, one sound is "twice as loud" as the other when it delivers between 4 and 10 times the intensity.

As far as I know, the reason for this agreement is a matter of speculation. I will pass along my personal favorite, but first I have a request: Please down load the two files, heartone.wav and hearnoise.wav and listen to them.

heartone.wav has almost the same amplitudes as in the figure above, except that the order is changed. Each louder segment is preceded by a standard tone. Start counting with the first transition, and use the form below to indicate the number of the transition for which the second tone is "twice as loud" as the first.

hearnoise.wav is a similar test, except that the sound is white noise. Please include your result for the noise sound in the same email. Include also any details about the nature of the experiment, such as earphones or speakers, number of participants, etc. If I receive enough responses to allow analysis, I will report on the results in a future column.

A nice discussion of loudness perception and hearing is found in The Science of Sound, third edition, by Rossing, Moore, and Wheeler, published by Addison Wesley, 2002. Now for some speculation:

The sense of hearing is mediated by "hair cells" which are roughly akin to the cells which give us our sense of touch. It is hard to see how our bodies would be sensitive to a factor of 10, unless the fact that we have 10 fingers is somehow related to hearing. Factors of 2 and 4 are easier to imagine.

If our hearing measures the amplitude (maximum displacement) of vibration of the hair cells, it might be sensitive to a doubling of the amplitude. In simple harmonic oscillators (see for example http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html) doubling the amplitude raises the energy by a factor of 4.

Thus, if we identified "twice as loud" as "twice the amplitude" in our ear, we would predict that twice as loud corresponds to an increase by a factor of 4 in intensity. (Remember that intensity measures the energy per second delivered to the ear.)

Still, it is hard to see how we can sense the amplitude of vibration with that kind of precision. Dr. Cooney in our department passed on another suggestion to me, based on the behavior of an ideal point source of sound waves:

For a point source, sound spreads evenly away from the source in all directions. At a given distance, R from the source, the energy emitted is spread out over a sphere of radius R. The area of such a sphere is 4pR² . The fraction of energy that our ear detects is the emitted energy multiplied by the ratio of our ear’s area (perhaps a square centimeter) to the area of that sphere.

This means that the intensity (energy delivered to our ear each second) is proportional to 1/ R² . If we double the distance between our ear and the sound source, then we double R, and the intensity falls by a factor of 2, squared: a factor of 4!

This to me is the most plausible story: Forced to invent the meaning of "twice as loud" people choose to identify "twice as loud" with "twice as close" to the source. That, in turn, means that "twice as loud" corresponds to 4 times the intensity of sound.

This argument is discussed in Musical Acoustics, by Donald Hall, page 102 of the third edition, published by Brooks/Cole in 2002.

Editor's Note: I'd like to thank Joseph DiVerdi for setting up the CGI script used to run the survey. -SG