Sound Propagation in a Wind Tunnel: Time

INTRODUCTION

This is a proposal for an experiment which could be conducted within a university research wind tunnel. Its objective would be to measure the time intervals between two sound events which occur with respect to two inertial reference frames which may be at rest or in motion relative to each other.

METHODOLOGY

An experimental apparatus consisting of two heavy stanchions could be placed within the wind tunnel a distance L apart on a straight line which is parallel to the direction of the wind flow. A sound pulse would be emitted from the emitter/sensor atop one stanchion and received at the receiver/clock/activator atop the other stanchion. 

The clock and the emitter would be activated by an electrical signal so that the clock would start simultaneously with the pulse emission event. Then, the clock would stop simultaneously with the pulse reception event. Therefore, the clock would measure the time interval, ∆t, between the events.

In inertial reference frame S′ the air would be at rest (Tipler & Mosca 2008, 522). In inertial reference frame S an observer and the experimental apparatus would be at rest. The x′-axis of S′ and the x-axis of S would be parallel to each other and parallel to the direction of wind flow.

The pulse would propagate from the emitter to the receiver along their connecting line at the constant velocity c relative to the air. The letter c is used for both the speed of light and the speed of sound because of some of their shared wave characteristics (Born 1965, 227). One such characteristic is that both of their wave velocities are independent of the source velocity (Tipler & Llewellyn 2012, 12).

This experiment could be conducted in two stages. In the first stage, S′ could be at rest relative to S. The pulse would propagate from the emitter to the receiver in the time: ∆t = L / c. This is just a rearrangement of the classical velocity formula: time = distance / velocity.

In the second stage, S′ could be in motion at the constant velocity v (v < c) relative to S in a direction which is parallel to the x′-axis of S′, the x-axis of S, and the wind flow. The pulse would propagate from the emitter to the receiver in the time: ∆t = L / (c ± v). In this formula, the wind velocity would be added to or subtracted from the emitted pulse velocity depending upon whether the wind flow passes the emitter or the receiver first (Morin 2008, 505).

CONCLUSION

In this experiment, propagating sound would measure different times when the medium's inertial reference frame is either at rest or in motion relative to the observer's inertial reference frame. These results would contradict the definition of times from the Special Theory of Relativity.

References

Born, Max. 1965. Einstein's Theory of Relativity. New York: Dover Publications.

Morin, David. 2008. Introduction to Classical Mechanics. Cambridge: Cambridge University Press.

Tipler, Paul & Ralph Llewellyn. 2012. Modern Physics. New York: W. H. Freeman & Company.

Tipler, Paul & Gene Mosca. 2008. Physics for Scientists and Engineers. New York: W. H. Freeman & Company.

Sound Clock.

A sound clock could measure time with respect to observers at rest in inertial reference frames S′ and S. A straight rigid rod of fixed length D could be oriented parallel to the x′-axis of S′ and the x-axis of S. The rod would have a sound emitter at one end and a sound receiver at its other end. This apparatus could emit a sound pulse which would propagate at the constant velocity c relative to some medium from the emitter to the receiver. Each emission and reception event would represent a tick of the clock. The times between ticks would be ∆t′ in S′ and ∆t in S, as measured by a typical clock. If S′ and the rod were at rest relative to S and the medium, then: ∆t′ = ∆t = D /c. If S′ and the rod were in motion at the constant velocity v (v < c) relative to S and the medium, parallel to the rod's length, then: ∆t′ = ∆t = D /(c ± v).  The formulas containing v would tend to contradict the Special Relativistic interpretation of length and time intervals.

Length of a Rod.

Propagating sound could be used to measure the length of a material object with respect to observers at rest in inertial reference frames S′ and S. A straight rigid rod of fixed length D could be oriented parallel to the x′-axis of S′ and the x-axis of S. The rod would have a sound emitter at one end and a sound receiver at its other end. This apparatus could emit a sound pulse which would propagate at the constant velocity c through some medium from the emitter to the receiver. The times between the emission and reception events would be ∆t′ in S′ and ∆t in S, as measured by a common clock. If S′ and the rod were at rest relative to S and the medium, then: D = c∆t′ = c∆t. If S′ and the rod were in motion, parallel to the rod's length, at the constant velocity v (v < c) relative to S and the medium, then: D = c∆t′ ± v∆t′ = c∆t ± v∆t.  The formulas containing v would tend to contradict the Special Relativistic interpretation of length and time intervals.

Using Sound to Define Simultaneity.

Propagating sound could be used to define simultaneity with respect to observers at rest in inertial reference frames S′ and S. A straight rigid rod of fixed length 2D could be oriented lengthwise, parallel to the x′-axis of S′ and the x-axis of S. The rod would have a sound emitter at each end, with a receiver at its midpoint. This apparatus could simultaneously emit two sound pulses which would propagate with the constant velocity c through some medium towards the receiver. The times between each emission and reception event would be ∆t′, ∆τ′ in S′; and ∆t , ∆τ in S, as measured by a typical clock. If S′ and the rod were at rest relative to S and the medium, then: ∆t′ = ∆t = [D /c ] = ∆τ′ = ∆τ. The observers would agree that the reception events were simultaneous. If S′ and the rod and were moving at the constant velocity v (v < c) relative to S and the medium, parallel to the rod's length, then: ∆t′ = ∆t = [D /(c + v)]; ∆τ′ = ∆τ = [D /(c – v)]. The observers would agree that the reception events were not simultaneous. The formulas containing v could be used by both observers in the same experiment. This would contradict the Special Relativistic definition of simultaneity.