Consider a harmonic wave travelling along the +x–direction and described by
y(x,t)=Asin(kx−ωt).
Answer the following, taking A, k and ω as constants.
(i) Starting from the given equation, derive expressions for the instantaneous velocity and acceleration of a particle of the medium in terms of x and t.
(ii) For a fixed position x, show that the particle velocity becomes zero twice during each complete oscillation and that the time interval between two consecutive instants of zero velocity is 2T, where T is the time period of the wave.
(iii) Eliminate time t between the expressions for displacement and acceleration to obtain a direct relationship between them, and hence prove that the acceleration of a particle is proportional to its displacement and directed towards the mean position.