Derive an expression for a one - dimensional simple harmonic progressive wave travelling in the direction of the positive x - axis. Express it in terms of `A, lambda`, v , t and x .

Explanation

Consider a simple harmonic progressive wave travelling with a speed v along the positive x-axis. Let y denote the displacement of a particle of the medium from its mean position. We assume that the initial phase of the oscillatory motion is zero . At time t, the displacement of the particle of the medium situated at the origin is
` y = A sin omega t` ...(1)
where A is the amplitude of the wave and `omega` is related to the frequency n of the wave motion by the equation `omega = 2 pi n`.

Simple harmonic progressive wave
Consider a particle of the medium situated at point P at a distance x from the origin O as shown in the figure. This particle also performs SHM with the same amplitude A and the frequency n. However, the disturbance at O reaches P only after some time , i.e., the particle at P displacement of the particle at P is
` y = A sin ( omega t - alpha)` ...(2)
Since a path difference of `lambda` (wavelength) corresponds to a phase defference of ` 2pi ` radians , a path difference of x corresponds to a phase difference of ` alpha` such that ` lambda/(2pi) = x/alpha . :. alpha = (2 pi)/lambda x`.
`:. y = A sin (omegat - (2pi)/lambda x)` ....(3)
If n is the frequency of vibration ,
`omega = 2 pi n = (2 pi v)/lambda` ...(4)
where v is the speed of the wave,
` :. y = A sin 2 pi (nt - x/lambda) ` ...(5)
Equation (5) is the equation of a simple harmonic progressive wave travelling in the positive direction of the x - axis.
Also from Eq. (4),
`y = A sin 2 pi (v/lambda t - x/lambda)`
` :. y = A sin (2 pi)/lambda (vt - x)` ....(6)
in terms of v and `lambda` , as required.