The Effects of Wind Velocity on Sound Waves

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.



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 Lapart 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:

c_e = L / t_e

The value, t_e, 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:

c_m = L / t_m,

The value, t_m , 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 c_e does not equal c_m 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]
t_e = [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
t_m = [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. They easily pass through the “conceptual walls” of the reference frames, like the ghostly spirits of a haunted house.

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. 

A Material Object In Space

As a material object makes its flight through the air (molecules / medium), it is a substantially different thing from a sound wave traveling through air.  The principle of relativity holds the pair in a tension of physics contrariness.  A material object making a straight line flight at constant velocity through space (air / medium), is seen to have two different velocities, when viewed by two different observers, in two different frames. One frame is considered as being at rest, and the other frame is considered as being in motion with constant velocity.  Transfer of momentum and addition of velocities mask the velocity of the reference frame considered in motion, and the principle of relativity as a scientific concept prevents the detection of this motion.

The foundational propositions of Einstein’s Special Theory of Relativity (STR): the Lorentz transformation, time dilation, length contraction, etc., are based on a particular interpretation of the nature of the relationship between two inertial reference frames. Given two reference frames moving relatively to each other, the observer within the moving frame is considered at rest, though the reference frame is moving.  An observer in another reference frame that is at rest or stationary, views the motion of the first frame.  The observer in the first reference frame can not by any mechanical experiment detect his or her own reference frame’s motion.

Two Reference Frames iin Relative Motion

By STR, though the train is in motion, it is regarded as being at rest in the reference frame attached to the train.  In the frame attached to the platform, the observer can clearly see the train’s motion.  Nonetheless, the observer on the train is assumed to be unable to do any mechanical experiment that can detect his or her motion.  The transfer of momentum of material objects cloaks any motions that might disclose any strange forces at work; through the addition of velocities, substantial speeds are kept hidden.  The property of waves (sound, EM, etc.), to not accept this transfer of momentum from its source, leads to the violation of Galilean invariance.  In other words, the wave speed remains fixed across reference frames, regardless of their relative velocity.

For example, a ball thrown rearward from the engine of the moving train has the velocity of the train subtracted from the ball’s velocity.  This maintains the appearance of the same distance of travel, and the same time for the journey; unbeknownst to the observer within the reference frame of the thrown ball. However, a sound wave directed rearward will not have any velocity subtracted, so that the wave will appear to travel a decreased distance, over a decreased duration of time, as the train moves forward.  This would hint at a possibly deeper reality.  For a material object, the influences of forces are somewhat hidden; for a sound wave they are not, they are just dodged and evaded.

In this thought experiment, I have shown that it is possible by the properties of sound waves, to lift this veil; to pull aside the curtain from the aforementioned proposition of the STR postulate.  That is, it is possible to pass through the wall between reference frames like a subatomic particle; to measure the same velocity value of a sound wave, by observers in separate reference frames that are moving with a constant velocity relative to one another.  It may become possible to overcome the static that jams any two-way communications between reference frames.

A Flash of Light

I introduce a modification to my thought experiment; it will expand the use of the lantern as a light signal by the caboose observer. This modification will pull the platform observer into the midst of the experiment.

There is once again a train of length, L, moving at a constant velocity, v, along a level straight section of track on a windless day. Also, there is again an operator in the engine car and an observer in the caboose car. The platform observer will now take on a more significant role.

The air molecules, because it is a windless day, along with the platform and the Earth, are at rest relative to the moving train. There is a reference frame attached to the train and a reference frame attached to the Earth thus creating two coordinates systems moving relative to one another. Since the outside air / medium is disconnected from the motion of the train, the method outlined in this thought experiment makes it possible to calculate the velocity of the train. If the experiment were conducted within a single enclosed car, then the air molecules would follow the motion of the train and it would be impossible to find the velocity of the train by the method I have presented here.

The thought experiment begins with the caboose observer flashing her light signal to the engine car; at the same moment she begins her clock. The operator blows the train whistle at the moment he receives the light signal. Over this short distance the light signal is effectively instantaneous. She is prepared to measure the time, t, for the sound wave from the whistle to reach her.

When the she hears the sound wave travel the length of the train, from the engine to the caboose, she flashes her lantern once more, and stops her clock. Now this is where my modification enters the experiment. The platform observer also has a clock and he is able to see the flashes of light from the lantern. So, at the first flash he begins his clock, and at the second flash he stops his clock, thus also measuring the flight time, t, of the sound wave.

Now the physics question becomes, will both observers measure the same time, t? Since the two observers are moving at a constant rectilinear velocity relative to one another, by the principle of relativity they should find different velocities as viewed from the other’s reference frame. This applies to a material object flying through space, because they are in reference frames moving relative to each other. However, this does not apply to sound waves because of their violation of invariance, a concept known to science.

Her goal is, once again, to find the velocity of the train entirely from within the reference frame attached to the train. The principle of relativity says this not possible, but she imagines herself to be a clever science girl. She ponders upon the problem and imagines that a sound wave would be a solution to her problem; but may open a Pandora’s Box, of which she knows not the contents. Nonetheless, she proceeds.

She sets her equations as I have shown before. For the train moving forward, the caboose meets the rearward traveling sound wave within the distance L = ct - vt, with c representing the known speed of sound; the sound wave and the caboose start at the endpoints of L. If the train were to go in reverse, the sound wave from the whistle at the engine would have to overtake the rearward going caboose, so then, a similar type of formula would be applied: L - vt = ct. Both the sound wave and the caboose start at the endpoints of L, with each moving in the same direction. Each of the above formulas can be solved for the velocity of the train, v.

If her algebra is correct, what are the implications? She has found the velocity of the train, she thinks, but she is also aware that this seems to contradict the principle of relativity. The two observers have measured the same velocity for the train though each is in a reference frame moving relative to the other. The train length, L, is found from the technical specifications.The speed of sound, c, can be found in any science text.  Thus the platform observer and the caboose observer can use the same equation, L = ct - vt.  If they measure the same time, t, as implied by the formulas, then they will each find the same velocity, v, for the train.

The caboose and platform observer could also find the velocity, v, of the train by another means.  By noting the landmarks immediately opposite to the flashes, relative to the embankment, and by measuring the distance between the landmarks by some device, then v = d / t could be found.  But the landmark method cannot be extended to find a definition for simultaneity, or make use of Doppler to find a deeper interpretation of the motion of a material object through space.  Even more, only sound waves are a mechanical means that can be done from within the train’s reference frame to find v in as advantageous a way, as by sound waves.

A scenario that is similar, but not the same, is by throwing a lump of coal rearward (instead of a sound wave) from the steam engine with a strong arm. Neglecting air resistance and gravity, it should fly in a level and straight line. From this thrown material object or any other similar type of mechanical experiment, she cannot find the velocity of the train, while riding upon the train. But by a sound wave, she can perform an experiment that allows her to find the train’s velocity. She has uncovered another of reality’s many paradoxes.

If the platform observer threw a lump of coal, with the same arm strength as the train operator, to another person on the platform, then the platform observer would measure the same velocity (v = d / t) as the train observer for the velocity of the of lump of coal between the engine and caboose, though the train is moving and the platform is at rest. The outsider, by addition of velocities, measures a different velocity for the lump of coal, but he cannot communicate the illusion of the caboose observer’s measurement to her. He sees the lump of coal on the train travel a shorter distance and thus a shorter time (by Galilean transformation). But she is trapped in her illusions, with no mathematical way to clear the shadows of her blindness. She has no way to find the velocity of the train.

Until a sound wave is applied to the problem. That sound waves violate invariance is already well-known to physicists. Exploiting the phenomenon that sound waves do not gain any addition of velocity (v_x = v'_x - v₀) from transfer of momentum, then this thought experiment makes it possible to measure the same velocity, distance, and time, across reference frames moving relative to each other. The seemingly paradoxical statements can both be true with a little cleverness. That the principle of relativity reflects reality and does not reflect reality seems an inescapable trap. The observers in two reference frames moving relative to one another can both measure different velocities for an object and the same velocity for that object, namely the velocity of the train.

An intermediary motion arises from betwixt the reference frames. It is different from the transfer of momentum imposed upon a material object by its initial cause of motion. Whether the object or reference frame is already in motion, or at rest, the law describing the object’s motion will be the same simple law (v = d / t), as viewed from within the reference frame. A sound wave leaps the hedge between reference frames; its velocity is not altered by the state of motion, or by the state of rest of the source. And by lifting the veil from this intermediary motion, she has found the velocity, v, of the train.

Song from the Black Hole

I have recently learned about the concept that sound travels through Space.  At Harvard’s Chandra X-Ray Observatory they found that the collapse of a Black Hole causes sound waves that travel through interstellar space.  Empty space is not a pure vacuum; it has got stuff in it!

There is cosmic dust, high-energy particles and magnetic fields in the so-called vacuum of space; that can be detected as evidence of sound waves across thousands of light years of space by Earth instruments. This is evidence of violent space events, such as the collapse of Black Holes.  The frequency of the waves detected translate to a B-flat that registers well below the level of human hearing.  It’s more a single constant tone rather than a melodious song; but it is far more than the silence of a vacuum as we had formerly thought.

A sonic anemometer, or a Pitot tube for Lord Vader’s Death Star may be on the technological horizon.  If there is a medium, then my thought experiment becomes plausible.  For that matter, any device that is dependent on airflow measurements will become mechanically useful.

All objects that move through a medium, such as air or water, drags a thin layer of the medium along with it as it moves through the medium.  From a planet down to a golf ball, hydrostatic pressure causes this anti-aerodynamical layer of medium to stick to the surface of any object in flight.  But when the object is far away from any other large object the influence of this thin layer is minimized (such as two objects moving in tandem through space, but at some distance, L, apart).

On a day with no interstellar wind a starcruiser travels through the galaxy at some fraction of the speed of light (it is undoubtedly more than one Mach).  There are two devices attached to the outer shell of this one kilometer long spacecraft; at the front-end is a sound emitter and at the backend of the spacecraft is an ultra sensitive microphone.  Since the air above the thin hydrostatic pressure layer is disconnected from the spacecraft, then the formulas from my thought experiment can be used as a means of determining the velocity of the spacecraft anywhere in interstellar space.

Bcome a Black Hole Hunter, listen to the song  http://www.blackholehunter.org/