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 lengthwise, 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 typical clock. If S′ and the rod were at rest relative to S the medium and, then: D = c∆t′ = c∆t. 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: D = c∆t′ ± v∆t′ = c∆t ± v∆t.

 

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 lengthwise, 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. 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).

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.

Not Every Reference Frame is an Enclosed Compartment.

It is a premise of the Galilean principle of relativity that every reference frame behaves mechanically like an enclosed compartment at rest.  As a consequence of this premise it is presumed to be mechanically impossible to discern the motion of any reference frame by observing experiments conducted within that reference frame.  Material objects in flight within an enclosed compartment will manifest a particular velocity that arises from momentum transfer through physical contact with the compartment walls.  Objects in flight outside of the compartment will exhibit essentially the same behavior via contact with the external physical structure of the moving compartment.    However, a sound wave in flight through an enclosed compartment where the air has no wind currents in it will manifest one particular velocity while a sound wave propagating through the still air outside the compartment will manifest some other velocity — in a moving enclosed compartment the contained air’s velocity is the same as the compartment’s velocity and would add to or subtract from the sound wave’s propagation velocity.   There is then a difference in the mechanical behaviors of material objects and sound waves when they are moving through any particular medium based on whether that medium is within or outside of a moving enclosed compartment.  Under certain conditions an observer in a stationary or moving reference frame may not have to apply the principle of addition of velocities from the Galilean or Lorentz transformation equations to the propagating sound wave.  Not every reference frame is an enclosed compartment.