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The Effects of Air Velocity on Sound Waves


Geryllax Vu

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In the 1632 book entitled Dialogue Concerning the Two Chief World Systems by Galileo Galilei (translated by Stillman Drake), he presented his historically important ship thought experiment:

“Shut yourself up with some friend in the main cabin below decks on some large ship and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin……When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell whether the ship was moving or standing still …..The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.”

This paper will attempt to make a mathematical statement that expresses the differences in time measurements that would result from conducting this thought experiment in the two scenarios presented by Galileo.

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Consider two observers aboard a Great Lakes tanker traveling in a straight line on an inland portion of a placid river. The ship proceeds at the constant velocity, v, relative to the nearby riverbank. It is a windless day, so that the air/medium is also at rest relative to the moving tanker. On a line parallel to the ship’s direction of travel one observer sits at the rearward end of a lower cabin of length L, and the other observer sits at the frontward end of the same cabin. These two observers thus form a tandem at the fixed distance L apart which they maintain whether the ship is in motion or at rest. The windows and doors of this cabin below decks are closed, so that the air molecules contained within it share in the motion of the ship.

The rearward observer is holding a heavy-duty flashlight and a clock, the forward observer has only a sailor’s whistle. They sit facing each other, then, she begins their thought experiment. She flashes the light towards (horizontally and parallel to the ship’s direction of motion) the other observer, and starts her clock at the same moment. When the other ship observer sees the flash of light he blows his whistle back towards her. When the sound wave of his high-pitched whistle reaches her, she stops her clock. The light signal is effectively instantaneous over this short distance, so the duration of time she will measure is for the sound wave to travel at the constant velocity c along the length L to her ear.

I will use the symbol c as the velocity of sound, as it is often given in many scientific reference texts. Although this symbol is more often associated with the speed of light, the symbol represents a shared characteristic of waves in that both wave velocities are independent of the velocity of the source of the wave. All observers will see the wave traveling at the same speed, although the air molecules may alter that speed after the sound wave has begun its flight.

The speed of sound c is constant in that the emitter does not alter the velocity of the sound wave as a consequence of the emitter‘s motion. She would then calculate the speed of the sound wave, following a fundamental equation of motion, velocity = distance / time, as:

ce = L / te

The value, te, is the time measured for the sound wave to reach her ear while she is inside the enclosed compartment. In the enclosed cabin the apparent distance the sound wave travels is equal to L. The air / medium matches the velocity of the ship.

Next, the two ship observers clamber up to the broad flat main deck, and take their same positions, oriented similarly on a line parallel to the ship‘s motion. They are also seated the same distance L apart. She begins the same experiment that they performed earlier, but now they are exposed to the stationary outside air with the ship moving through the air molecules at the constant velocity, v. She once again flashes the light signal, and at the same moment starts her clock. He blows his whistle once again when he sees the signal, then she measures the time for the sound wave to reach her ear. Once again, using a fundamental equation of motion, she calculates the speed of the sound wave as:

cm = L / tm,

The value, tm , is the time measured for the sound wave to reach her ear as she sits on the main deck of the ship. On the main deck, the apparent distance the sound wave travels once again is L. The air / medium is at rest relative to the ship.

I make the proposition, following Galileo, that ce does not equal cm due to the differing natures of the locations where these two experiments are to take place. In both cases the ship travels forward to meet the sound wave as the wave makes its rearward flight once it is emitted from the source. The velocity for the air / medium is different for the two cases. She might be led to conclude that she has measured two different values for the speed of sound, c. Actually, however, there is a difference between measuring the time within the enclosed compartment where the air molecules have the velocity, v, of the ship relative to the riverbank; and measuring the time on the main deck where the air molecules have a velocity of zero with respect to the ship and riverbank.

In the enclosed cabin case, due to the forward velocity of the air molecules matching the ship‘s velocity, the velocity of the air molecules have slowed the sound wave so that it will cover a shortened distance at a lower speed. The time value measured should as a result be as if the ship were at rest. Alternatively, in the open air main deck case, the distance the sound wave travels is also less than L due to the ship’s forward motion. The stationary air molecules, in contrast, have zero velocity as the ship plows through them at the velocity v. The time value measured should be less than if the ship were at rest.

Thus the times should be measurably different for the two cases I have presented here. This difference results from the idea that the air molecules in the enclosed cabin have a velocity that is in the opposite direction of the sound wave, the wave is slowed by the contrary motion of this conceptual wind (akin to a Doppler wind):

[c - v] = [L - vt] / t

c = [(L - vt) / t] + [v / t]

c = [L / t]

te = [L / c]

The sound wave flight time will appear to the ship observer as though the medium, and the ship, were at rest.

In the open still air case, the air molecules do not add or subtract from the velocity of the sound wave. The air molecules have been unlinked from the motion of the ship; they are at rest relative to the velocity of the ship:

c = [(L - vt) / t]

ct + vt = L

tm = [L / (c + v)]

In this second case the sound wave velocity is unaffected by the motion of the ship, the ship simply moves toward the emitted sound wave, with the wave speed unaltered by the zero air molecules speed. As a consequence, the ship observer measures a shorter time of travel for the sound wave than if the ship were at rest.

Although the two thought experiments take place on a single ship traveling at a single velocity, the sound wave passing through the air molecules along the same distance L manifests two different results. In the cabin below decks the air molecules have the velocity of the ship; they are at rest in the reference frame attached to the ship, but in motion in the reference frame attached to the riverbank. On the main deck, in the open still air, the air molecules have a velocity of zero; they are at rest in the reference frame attached to the riverbank, but are in motion in the reference frame attached to the ship. This, I posit, leads to the different results that would be measured.

Thus, mathematical distinctions can arise and be measured in terms of the time for a sound wave to travel through air, a distance L on a moving ship. There is a difference between doing this measurement in an enclosed cabin below decks, as opposed to doing this same measurement on the main deck in the open air. Once the sound wave is in flight, its velocity can be altered by the velocity of the air / medium freely-flowing between reference frames. These alterations can be approximately calculated by this thought experiment.

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