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 could be used by both observers in the same experiment. This would contradict the Special Relativistic effect of time dilation.

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 could be used by both observers in the same experiment. This would contradict the Special Relativistic effect of length contraction.

 

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.