The Wind

A young woman stands before a high flat concrete wall on a blustery day. She directly faces it, at a distance of L meters away. The wind sweeps down past the wall at the constant velocity v and blows directly perpendicular from the wall to her face. She feels compelled to shout at the wall in some way, but she takes the stopwatch from her pocket and decides upon the experiment that she shall perform (akin to the Michelson-Morley aether wind experiment).

The formula for a sound wave to echo back from a hard reflective surface fixed to the earth, when the air is still (medium at rest) is:

♦ 2L = cT

However, I speculate that this is not the formula when there is a wind blowing at the constant velocity v in the direction directly opposite to the sound wave emission source. The velocity of the air molecules (medium) will have a measurable impact on the velocity of the sound wave as it travels from the source to the wall, and then back.


Our home planet hurtles through interstellar space at a tremendous speed, 30 km/second, but the atmosphere does not get swept away, off into the cosmos. Fortunately for us, the molecular bonds of attraction and repulsion, and the force of gravity, hold a thin layer of atmosphere snugly against the earth’s surface. Though terrestrial winds may surpass 120 km/hour, most of the air molecules we depend upon to fill our lungs cannot attain enough velocity to escape the earth’s embrace. This balancing of hydrostatic pressure and gravity thus bestows upon us, the breath of life. So the earth makes its yearly orbital journey with a thin layer of atmosphere grasping tenuously to it; tornadoes and hurricanes may blow, but we shall breathe.

Girl playing by the wall

So, when the air appears still, it is actually moving at the velocity of the earth. It is supposed that we cannot detect this motion by any mechanical experiment in the reference frame of the earth; however, it is worth exploring the scenario when the air molecules are disengaged, such as by a wind, from the rapidly moving surface of the land and sea. She stands, with her mouth and ears at the ready, opposite the high and hard reflective surface, forming a moving tandem with it; that any shout she might make would come back to her some moments later. If she is standing at a reasonable distance from this reflective surface on a windless day, then the first formula applies; but if a wind is blowing as I have described before, then the maths are different.

The hydrostatic pressure casts a cloak of invisibility over the motion of the stationary earthbound tandem, and the stationary air molecules trapped near the surface of the earth. The earth, the tandem, the earthbound air molecules, are all traveling through the galaxy at the same velocity v locked together in their motion. That is, when the air is still, but a wind will cleave this triumvirate.

Despite the tandem being fixed to the earth’s surface during its daily gyre, there is no Doppler effect upon the sound wave traveling from the girl to the wall because the wave crests are squeezed together near the source (girl), but pulled apart near the receiver (wall) by an equal amount; and vice versa on the reflection’s trip, so she would observe no change in the wavelength or frequency of the wave. In the presence of a wind, the Doppler change in frequency vanishes, but the Doppler wind formula remains present and measurable:

♦ c` = c ± w, where c` is the Dopplerian speed of the sound wave in the presence of the wind.

Because of the wind’s speed and direction, the new wave speed is c` = c – w as the sound wave travels away from the emitter (her mouth); but it is c`= c + w when the wave is reflected back towards its original source (her ear). That is, when compared to speed of sound in still air, the wind slows down (subtract from) the sound wave speed as it travels in one direction; but speeds up (adds to) the sound wave speed when it travels in the opposite direction. Thusly, the total trip time interval for the sound wave is:

♦not, T = [2L/c]

♦but, T` = [L / (c + w)] + [L / (c – w)] = [2Lc] / (c² – w²) = [2L / c] [1 / (1 – [w²/c²])

♦time = distance / speed.

With the pen and pad from her other pocket, she begins to make her calculations. The given variable values for her experiment are: c = speed of sound in still air, 340 meters/second; w = speed of wind, 50 m/s; L = 100 meters.

So, in the first echo scenario (no wind):

♦T = [2L] / c; T = [2 × 100m] / [340m/s] = 0.588s

And, in the second echo scenario (wind):

♦T ` = [2Lc] / (c<sup>2</sup> – w<sup>2</sup>); T ` = [2 × 100m × 340m/s] / [(340m/s)²] – [(50m/s)²] = 0.601s

She makes note of these differing measured time values. This leads her to ponder her two scenarios of air motion: molecules at rest in a stationary reference frame, and molecules passing unencumbered through the porous walls of an apparently stationary reference frame. There is a measureable difference between an enclosed compartment and a reference frame. The “conceptual walls” of the reference frame do not compel the air molecules within it to go at the velocity of the reference frame. These free-spirit airy particles are not possessed by the earthbound reference frame. But it is difficult to say to which reference frame they belong; they belong to no reference frame, and are in all earthly reference frames. This alternative echo formula is only a close approximation.

The moving air/medium has been disengaged from the stationary earthbound girl-wall tandem in a mathematically revelatory way. This has profound implications for the motion of any material object, when that motion is investigated by sound waves. At the slow speeds of the wind, the measured time interval does not suffer the Special Relativistic effects of time dilation and length contraction; the gamma factor value is negligible at this speed. Thus, the passage of time is nearly absolute, on the scale of her everyday life behind the wall. 

The Mechanical Event

According to Galileo, Newton, and Einstein, the classical principle of relativity states that no mechanical experiment can be done to detect absolute motion, or motion of a material object relative to an everywhere stationary medium (similar to Michelson-Morley). A common reformulation of this principle states that:

♦The velocity of any motion has different values for two observers moving relative to each other.

The following thought experiment proposes to investigate this principle, that is, to find if these two values are measurably different, or measurably the same. It seeks to find the particular values of the motion of material object as seen by two observers in separate references frames, one moving and the other at rest, relative to one another. It will measure the time interval between two mechanical events occurring in the moving reference frame. This time interval is measured by two observers each possessing one of two distantly separated clocks. This thought experiment will use sound waves to determine a closely approximate measurement for the time interval between these two mechanical events.

On a windless night (air molecules at rest relative to the earth), a train of length L is traveling at the constant velocity v along a flat, straight section of train track. There is an observer in the caboose (train reference frame) as well as another observer on the station platform near the track (earth reference frame). They each have identical clocks with which to conduct the following thought experiment: they will attempt to detect absolute motion, or at least test the mentioned common reformulation of the classical principle of relativity. That is, to show that two observers can measure the same value for the velocity v of the train using the same formula, without a Galilean transformation, although these two references frames are moving relative to one another.

Placing each observer in separate reference frames which are moving relative to one another then by the Galilean transformation (addition of velocities) a material object will manifest as its velocity:

♦ u` = u – v

where u` is the velocity of the train in the reference fame attached to the train (this is usually zero); u is the velocity of the train in the reference frame attached to the platform; and v is the velocity of the train reference frame. In the reference frame attached to the train, the train itself has a velocity of zero; and in the reference attached to the platform, it will simply have the velocity of the reference frame attached to the train. The velocity of the train is represented by this formula, as well as the velocity of any material object moving within that train; each velocity is seen by the observer sharing the motion of the train, and the observer at rest on the nearby earthbound platform.

The usual method from classical physics for finding the train’s constant velocity v is to measure the time interval that it takes for it to travel between two stationary landmarks a known distance apart (velocity = distance/time). But this method would not work in the darkness of night. If an observer on the train has no access to external landmarks then that observer would have no clues to indicate that the train is in motion. For, any mechanical experiment done when traveling at a constant velocity, such as dropping a ball to the floor, or tossing a ball to a friend in another seat, would proceed as if the train were at rest. That is, these material objects would follow a trajectory through space that does not hint at the train’s motion. The following thought experiment could find the velocity of the train, even at night, without any visible external landmarks on the stationary earth as seen through a window.

If the experiment is conducted in the moving reference frame of the train, then the arm of the human ball thrower transfers a certain amount of momentum onto the ball from the train, thus increasing or decreasing the velocity of the ball. The observer sharing the motion of the reference frame attached to the train would not and cannot measure this momentum exchange, but the observer on the platform will notice this change of the velocity and trajectory of the ball.

However, a sound wave does not mechanically behave in this manner. The mechanical event of a sound wave emission at some point in space, and then the receipt of that wave at some other point in space, will progress in one of two ways. If the experiment were conducted in a closed compartment then the air molecules/medium would have the velocity of the compartment and the moving air will act to increase or decrease the velocity of the emitted wave. This increase or decrease will depend on the direction of the sound wave relative to the compartment, and the emitter’s state of motion or rest within the compartment. But if the experiment were carried out in the open still air then the air molecules will have a velocity of zero which will allow them to pass freely through the “conceptual walls” of the reference frames. Thus, the air will have no effect on the velocity of the emitted sound wave no matter if the emitter is in motion or at rest.

The new method presented by this thought experiment for finding the velocity of the train (material object) relative to the still air (medium at rest – Michelson-Morley), the observer in the caboose has a light source with which she will send a signal to the engineer at the front of the train. He will then blow the whistle, sending out sound waves which the caboose observer will be able to hear. At the moment she sends the light signal from the open caboose window, she starts the single clock that she has. The platform observer will also see this nearly instantaneous signal (the velocity of light is too fast to be measured by a normal clock) and he will start his single clock at the same moment. Thus, their mechanically identical clocks will essentially be synchronized.

Over this short distance, the light signal that the engineer and the platform observer see is approximately instantaneous so that the time t she measures is essentially the time for the sound wave to travel the length L to reach her ear. When she hears the whistle sound she stops her clock and immediately once again flashes her light. The platform observer also stops his clock upon seeing this second flash. Disregarding reaction times, both observers should measure the same interval of time t. Since the speed of the sound wave and the speed of the train are so much slower than the speed of light, the Special Relativistic (STR) effects of time dilation and length contraction are negligible, and will thus have little to no impact on the time interval measurement. The gamma factor cannot dilate the time interval enough or contract the length enough to create the illusion of a resting reference frame in the presence of a sound wave event.

The caboose moves forward to meet the rearward travelling sound wave, so the sound wave will travel a distance that is less than L, the length of the train at rest. The speed of the sound wave does not change, but the motion of the material object (train) is disengaged from the medium (still air). This should lead to, approximately, identical time interval measurements by the observers in each reference frame. This disengagement mechanically permits the air molecules to freely flow between the reference frames which are moving relative to one another. These air molecules easily pass through the “conceptual walls“ of the reference frames, like the ghostly spirits in a haunted house.

The sound wave and the caboose begin their journeys at the endpoints of L. The caboose has the constant velocity v, and the sound wave has the constant velocity c. As the train engine and the caboose move through space they form a tandem, with each car remaining at a fixed distance apart no matter whether the train is in motion, or at rest. An important premise of this experiment is that the sound wave travels between these two endpoints of the train, or alternatively, along the length L. Each observer takes the length L from the train specifications, it is measured when the train is at rest by the traditional units of measurement. Additionally, each observer knows the accepted speed of sound in still air. So, all the variable values are available to the observer within each reference frame. To reflect the conditions under which the caboose and sound wave will meet somewhere between the endpoints of L then the following equation can be set up:

♦ L = ct + vt

Certainly, they have measured the same interval of time t in both reference frames. The STR does not account for any time dilation or length contraction at the slow speeds involved here. The observer in each reference frame retrieves the length L from the train specifications. The speed of sound c is assumed to be constant or the same for both observers. Thusly, the formula can be solved for v the velocity of the train as seen from either reference frame:

♦ L = t(c + v)

♦ v = [L / t] – c

A similar argument can be made for the case when the train is moving in the reverse direction, the sound wave is then overtaking the caboose, that is, the sound wave will catch up to the caboose somewhere beyond the caboose’s initial position:

♦ L + vt = ct

♦ L = ct – vt

♦ v = c – [L / t]



The departure and arrival events of the sound wave occur at the same places and at the same times (invariance of coincidence) in space, and is mathematically observable in each reference frame. The above expression contradicts the Newtonian and Einsteinian principle of relativity in that although the two reference frames are moving relative to each other, this experiment allows each observer to use one and the same formula to find the velocity of the train as seen from either reference frame. This results in not needing the addition of velocities from the Galilean transformation between references frames when sound waves are used to investigate the motion of material objects. So, I hypothesize that a new form of motion unveils its mysteries, it is neither absolute motion, nor absolute rest. An intermediary motion of material objects can now be defined, as they travel through the space of our everyday realities. 

One Velocity, Two Reference Frames

According to Newton and Einstein, the principle of relativity states that no mechanical experiment can be done to detect absolute motion, or motion of a material object relative to a stationary medium (similar to Michelson-Morley).   Common reformulations of this principle state that:

1) The velocity of a material object takes on the simplest formula, as seen by an observer at rest in a reference frame, no matter whether the reference frame is at rest, or moving with constant velocity, v.

2) The same formula is not used for the constant velocity, v, of a material object as seen by an observer in a reference frame in which the object is viewed as being at rest; or as seen by an observer in a reference frame in which the object is viewed as being in motion (Galilean addition of velocities).

On a windless evening at dusk (air molecules at rest relative to the earth), a train of length, L, is traveling at the constant velocity, v, along a flat, straight section of train track.  There is an observer in the caboose (train reference frame) as well as another observer on the station platform near the track (earth reference frame).  They each have identical clocks with which to conduct the following thought experiment.  They will attempt to detect absolute motion, or at least test a common reformulation of the classical principle of relativity.  That is, to show that two observers can measure the same value for the velocity, v, of the train using the same formula, without a Galilean transformation, although these two references frames are moving relative to one another.  Also, this will not be the simplest form for the velocity of the train:

♦ v = [d / t]

To find the absolute motion of the train (material object) relative to the still air (medium at rest – Michelson-Morley), the observer in the caboose has a light source with which she will send a signal to the engineer at the front of the train.  He will then blow the whistle, sending out sound waves which the caboose observer will be able to hear.  At the moment she sends the light signal she starts the single clock that she has.  The platform observer will also see this signal and he will start his single clock at the same moment.

Over this short distance the light signal is effectively instantaneous, so that the time, t, she measures is essentially the time for the sound wave to travel the length, L, to her ear.  When she hears the whistle sound she stops her clock and then once again flashes her light.  The platform observer also stops his clock upon seeing this second flash.

Disregarding reaction times, both observers should measure the same interval of time, t.  Since the sound wave and the speed of the train are so much slower than the speed of light, the relativistic effects of time dilation and length contraction are negligible.  The caboose moves forward to meet the rearward travelling sound wave, so the sound wave will travel a distance that is less than, L, the length of the train at rest.  The speed of the sound wave does not change, but the motion of the material object (train) is disconnected from the medium (still air).  This should lead to, approximately, identical time interval measurements by the observer in each reference frame.  The air molecules freely flowing between the reference frames moving relative to one another make this supposition mechanically plausible. They easily pass through the “conceptual walls” of the reference frames, like the ghostly spirits of a haunted house.

The sound wave and the caboose begin their journeys at the endpoints of L.  The caboose has the constant velocity, v, and the sound wave has the constant velocity, c.  To reflect the conditions under which they will meet, then the following equation can be set up:

♦ L = ct + vt

If they have measured the same interval of time in both reference frames, then this formula can be solved for, v, the velocity of the train as seen by each reference frame:

♦ v = [L / t] - c

This is obviously not the simplest formula for the velocity of the train in either reference frame.  This expression contradicts the Newtonian and Einsteinian principle of relativity in that although the two reference frames are moving relative to each other they can each use one and the same formula to find the velocity of the train as seen from either reference frame.  This results in discarding the need for the addition of velocities from the Galilean transformation between references frames.

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.