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 use propagating sound for the measurement of the time intervals between two events. These events would occur with respect to two inertial reference frames which could 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 atop one stanchion and received at the receiver/clock 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 through the air 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 through the air from the emitter to the receiver in the time: ∆t = L / (c + v). In this formula, the emitted pulse velocity would be added to the wind velocity since the wind flow passes the emitter before reaching the receiver (Morin 2008, 505).

CONCLUSION

The observer in this experiment would measure time intervals such that the measured values would incorporate the constant velocity v of the rest frame of the air relative to the rest frame of the observer. Since the observer would always be at rest relative to the clock during the experiment then these measured values would contradict the Special Relativistic definition of proper time.

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.