Consider a plane progressive wave travelling along the +x–direction.
(a) Starting from the analytical expression of the wave, derive expressions that relate the phase difference Δϕ to (i) the path difference Δx between two points, and (ii) the time difference Δt between two instants for the same particle.
(b) Hence show that, for the same phase to be observed at a single position at times differing by Δt, the corresponding path difference that the wave has advanced in that interval is Δx=vΔt, where v is the speed of the wave.
(c) A sound wave of frequency 512Hz travels through air with a speed of 340m s−1. At a fixed point, two observations are made at instants separated by 0.00075s. Calculate the path difference represented by the wave train that has passed the point in this time, expressing your answer (i) in metres and (ii) as a multiple of the wavelength.