3.3 Equation of a plane progressive harmonic wave

In case of a plane progressive harmomic wave, the displacement of successive particles of the medium is given by a sine or cosine function of position.

Let a disturbance is propagating along positive $X$-axis, then the equation of a harmonic wave can be written as, $$y = A\sin k(x - vt)\quad ...(i)$$

Also, $$y = A\sin (kx - \omega t)\quad (As,\omega = kv)$$

Since the waveform in the above eqaution is based on sine function. As we know that the sine fucntion is a periodic of $$2\pi $$. Mathematically, $$x = \frac{{2\pi }}{k}$$

Also, we know that the wavelength is defined as the minimum distance between any two points in same phase having a wave motion. Hence, $$\lambda = \frac{{2\pi }}{k}\quad {\text{or}}\quad k = \frac{{2\pi }}{\lambda }\quad ...(ii)$$

From equation $(i)$ & $(ii)$ we get, $$y = A\sin \left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]$$

At time $t=0$, $$y = A\sin \frac{{2\pi }}{\lambda }x$$

Hence, the general equation of a plane progressive harmonic wave is given by, $$y(x,t) = A\sin (kx \pm \omega t + \phi )$$ or $$y(x,t) = A\cos (kx \pm \omega t + \phi )$$

where,

$y(x,t)$: Displacement

$A$: Amplitude

$(kx \pm \omega t + \phi )$: Phase angle

$t$: Time

$k$: Wave number

$\omega $: Angular frequency

$x$: Position of the particle

$\phi $: Phase difference