A sinusoidal progressive wave on a stretched string is described by the equation y(x,t)=Asin(kx−ωt+ϕ).
(i) Define the phase of a point on the wave situated at coordinate x at time t. State its physical significance.
(ii) Two points P and Q on the string are separated by a distance Δx and are observed at the same instant. Starting from the given equation, derive an expression for their phase difference in terms of Δx and the wavelength λ. Clearly mention every step in your working.
(iii) Using the result of part (ii), explain the precise condition under which P and Q vibrate (a) in phase and (b) out of phase. Illustrate your answer with one numerical example for each case, choosing any convenient value of λ.