Question

can you give me the derivation of y=Asin(wt-kx)?

Answer 1

Consider a particle O, situated at the origin. It performs a S.H.M. of amplitude A, angular frequency ω and produces a simple harmonic wave in the medium.

Suppose that the wave travels in the positive direction of X-axis and the particle O starts from the mean position. The relation between the displacement (y) of the particle and its position (x) in the medium.



y = A sin ωt

The particles on the righ of O, will recieve the disturbance after sometime and hence they will progressively lag behind the phase.

For a particle P at a distance x from O, 

y = A sin (ωt-α)    ... (1)

where α is the phase lag of P w.r.t. O.

Since a path difference of λ corresponds to a phase difference of 2Π radian, a path difference of x will correspond to a phase difference of .

Thus, the phase difference = ×(path difference)



Therefore, the displacement of P at any instant is given by,



This is the equation of a one-dimensional simple harmonic progressive wave.

The quantity  is called the propagation constant (K).

Hence the equation can be written as y = A sin (ωt - Kx).    ... (3)

If the wave is travelling towards left, in the -ve direction of X-axis, the negative sign should be changed to a +ve sign and the equation is written as,