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.