John W. Dooley, Physics Department

Millersville University

The basic parameters required to estimate the speed and altitude of an aircraft. Click image to enlarge.

This experiment is necessarily done outdoors, because it requires observation, with sight and sound, of an airplane flying overhead.

If you have ever noticed the sound of an airplane seeming to come from a place behind the plane, you have made most of the observations necessary to determine the speed, and even the altitude, of the plane.

Before analyzing a moving sound source like an airplane, consider a simpler case: the stationary sound source caused by a lightning bolt. As indicated in the adjacent figure, two signals from the lightning bolt travel the distance (h) to you:

A light signal traveling at speed c = 3E8 m/sec, and a sound signal traveling at speed s = 3.45E2 m/sec. The light travels a million times faster than the sound. Even though the two signals left at the same time, the light signal in the figure has already traveled much farther than the sound signal.

The travel time for the sound is called t_S, and the travel time for the light is called t_L. Since both signals travel at constant speed, the distance to you is given by

h = c t_L

and also by

h = s t_S

The distance traveled is the same, but the sound takes a million times longer than the light to make the trip:

t_S ~ 1E6 t_L.

A little algebra shows that
.
The quantity (t_S- t_L) is the time between seeing the light flash and hearing the thunder. This is the time interval which an observer can measure. Since the speed of light is so large, the speed of sound in the denominator can be ignored, giving,

h ~ (t_S- t_L) s

Plugging in the speed of sound gives h of about 1.6 km (1 mile) when the time interval is 5 seconds, the usual result.

The sound of an overhead plane is heard behind the plane. Click image to enlarge.

Now we are ready for the special situation when the plane is seen to be overhead, with the sound heard to be coming from behind it, as in the figure on the left. The figure shows the plane flying on a level path, and we will assume that the plane flies at constant speed, v.

The sound arrives (finally), having traveled the distance (z) from where it was created. The dotted image of the plane represents the plane's location at the time when the sound was created. For simplicity, we follow the lightning argument, and assume that the time for light to travel from plane to eye is negligible; the instant we see the plane overhead is the same instant that it was overhead.

X is the distance traveled by the plane while its sound is on the way to an observer. Click image to enlarge.

In the figure above, h is the altitude of the plane, z is the distance that the sound traveled on its way to the ear, and x is the distance the plane traveled while the sound was making its way to the ear. The figure on the right is cleaned up to emphasize the geometry of the situation. We see a right triangle, with sides h, x, and z, z being the hypotenuse.

To quantify the observations and find the speed of the plane, you need to find the ratio x/z. This ratio is the sine of the angle A.
In the figure, A is about 30 degrees, so that x/z = 0.5.

One way to find that ratio is to point your left arm towards the sound, and your right arm straight up, towards the plane. Have a friend measure the horizontal distance between your left finger tips and your right arm. The ratio of that distance to the length of your left arm is the same as x/z.

To finish the analysis, let us call the time it takes the sound to travel from plane to ear by the name T. During that time, the sound travels a distance z. During that time, the plane travels a distance x. In terms of the speed of sound (s) and the speed of the plane (v),

z = s T

and

x = v T .

The ratio x/z is given by x/z = v/s, because the T cancels. Thus, the ratio x/z is the ratio of the plane speed to the speed of sound:
v = (x/z) s. When the angle A is 30 degrees, the plane is traveling at half the speed of sound.

The observations can be extended to find the altitude of the plane by adding a time measurement. Start measuring time when the plane is overhead, and determine how long it takes before the sound of the plane comes from directly overhead. Call this interval TD. TD is the time it took for the sound to travel down to your ear, after leaving the plane when it was directly overhead. Thus,
h = (s)(TD).

Using the speed of sound as 345 m/sec, you can estimate the altitude of the plane in meters. Note that this is exactly the same formula as the one used to find the distance to a lightning bolt. If TD is 5 seconds, the plane is 1.6 km (1 mile) above the earth.

You can check your earlier speed measurements by measuring the angle the plane makes with the vertical when the sound comes from directly overhead. Details are left to the interested experimenter, with this hint: Consider the tangent of the angle instead of the sine. (As a check, note that when this new angle is 45 degrees, the plane is moving at the speed of sound.)

In a check, a large plane displayed an angle (A) of about 30 degrees or less, and the overhead sound took about 30 seconds to reach my ear. The plane was estimated to be flying at about half the speed of sound (perhaps 300 mph) at an altitude of about 10 km (about 30,000 feet or about 6 miles). Finally, I noted that I could hide the plane behind my little finger, when my arm was stretched above my head. Similar triangles let me estimate the wingspan of the plane at about 40 meters.