08 August 2004

The Song of the Stadium

John W. Dooley, Physics Department
Millersville University

Calling this gift to me from Conrad Miziumski a "song" may be overstating the case. I personally have heard only one note. Nonetheless, that note is the signature tone of the stadium.

This experiment is a no-pain, no-gain experience; you will clap your hands until they are sore. If you do so while facing the (empty) stadium seats from the middle of the (empty) football field, you should hear a tone instead of a simple echo of the clap. This tone can be understood in terms of the wave nature of sound and an extremely useful mathematical technique called Fourier analysis.

In my experience, the tone that the stadium returns in response to a loud clap sounds rather like a struck chime. There is a story to go with that perception.

Before reading further, find a stadium and clap. You may want to print this article to show to stadium security before you go out on the field.

Best is a concrete stadium, because its levels reflect more sound than do the steps of the more temporary aluminum frame structures. Stand in the middle of the field, clap and listen. Do not make a "small clap" as in the figure, below, left. Make a "big clap" as in the figure below, right. (Note the redness of the palms.)

A small clap. Click image to enlarge.

A big clap. Click image to enlarge.

Thinking of a clap as a wave, is a little strange, since a clap is mostly just a single traveling region of high or low pressure. I think of the small clap as squeezing air out from between the hands. That rush of air makes a high pressure region, which propagates away and is detected as a pulse of sound.

I think of the big clap differently. There (I imagine), my hands scoop up air, capture it, and compress it. Removing that air leaves a low pressure region outside the hands. Air from further away falls into the region, and the low pressure region propagates away. It, too, is detected as a pulse of sound.

The figure shows a sketch of such a sound pulse, as it travels. The surprising conclusion of Fourier analysis is a dramatic extension of the idea of superposition. In the previous installment, two traveling waves superposed to make a standing wave with an appearance much different from either of the component waves. Fourier analysis asserts that any wave shape, even a pulse, can be created as a superposition of many many simple "pure tone" (sinusoidal) waves.

A detailed discussion is fairly sophisticated. In the simple-minded version, a large number of component waves, with wavelengths up to about the width of the pulse in space, contribute significantly to the superposition which will create that pulse. Is this just a mathematical trick or are those wavelengths "really there?" The tone that the stadium creates in response to a clap suggests that the waves are real enough to hear.

Whether they are "really there" in the traveling pulse or only occur when the sound interacts with something is not a question that is answered here. Experiments using a medium whose wave velocity is different for different wavelengths suggest that it is best to think of the component waves as always present. (In those experiments the faster waves run away from the slow ones, and the pulse changes shape.)

In the sketch of the stadium steps, we imagine the reflection of just one of the waves that contribute to the creation of the pulse. Each higher step reflects the pulse a little bit later, because it is farther from the clapping hands. This delay means that the reflection from the higher step lags behind the reflection from the step below it.

In the figure, the higher step lags behind by just one wavelength from the step below. (The step depth is half a wavelength, and by the time the wave has traveled to the upper step and back, it has fallen behind the wave from the lower step by a full wavelength.) Waves like this will add constructively and produce a loud sound.

Contributions with other wavelengths will fall behind by the same distance, because the travel time is determined by the speed of sound alone. This means that contributions with other wavelengths will not lag behind by exactly one wavelength. Most of those waves will fail to produce a loud sound.

However, there are some other waves that do manage to produce a loud sound. There is, for example, a reflected wave whose wavelength is just half that shown in the figure. For this wave, reflections from neighboring steps will add constructively, again giving a loud sound. I believe that this family of tones ("harmonics") produces the sound that I perceive as being like a chime.

There is another phenomenon, which I have not observed, but which should be heard in the proper stadium. If you are able to vary the angle at which the sound approaches the steps, you should be able to vary the pitch of the echo tone.

The second stadium sketch shows sound arriving and reflecting along a tilted path. On the lower part, the wavelength and the step size are the same as in the first sketch; only the direction has changed. Now, the reflected wave from a higher step tends to cancel with the wave from the step below. The reflected sound for this wave is almost inaudible.

To make a loud sound for this situation we can make the sound wavelength shorter, as in the sketch for the upper two steps in the stadium diagram. For this tilted path, sound with that shorter wavelength is loud. To hear this effect, you need to perch high above the field, something that I have not been able to do.

This prediction of a change in pitch of the reflected tone might be tested in another way. If you stand close to the seats, then the angle from your hands to the seats, increases for higher seats, as in the sketch.

Along the upper path, loud reflected sound will have a longer wavelength, and lower pitch, than along the lower path. In addition, sound along the upper path takes longer to return to your ear, since it has farther to travel (and all sound travels at about 345 m/sec in air). The later parts of the returned sound should have a lower pitch than the earlier parts. This kind of shift is called a "chirp."

It is possible to stand too close to the seats. When I do that I just hear the ordinary clap echo, with no tone. The most convincing evidence for a chirp echo came when I was out in the field about 30 meters from the seats.