i. Consider a transverse progressive wave whose particle position is described by x and displacement from equilibrium position is described by y.

Such a sinusoidal wave can be written as follows:

∴ y (x, t) = a sin (kx – ωt + ø) ……… (1)

where a, k, ω and ø are constants,

y (x, t) = displacement as a function of position (x) and time (t)

a = amplitude of the wave,

ω = angular frequency of the wave

(kx₀ – ωt + ø) = argument of the sinusoidal wave and is the phase of the particle at x at time t.

ii. At a particular instant, t = t₀,

y (x, t₀) = a sin (kx – ωt₀ + ø)

= a sin (kx + constant)

Thus at t = t₀, shape of wave as a function of x is a sine wave.

iii. At a fixed location x = x₀

y(x₀, t) = a sin (kx₀ – ωt + ø)

= a sin (constant – ωt)

Hence the displacement y, at x = x₀ varies as a sine function.

iv. This means that the particles of the medium, through which the wave travels, execute simple harmonic motion around their equilibrium position.

v. For (kx – ωt + ø) to remain constant, x must increase in the positive direction as time t increases. Thus, the equation (1) represents a wave travelling along the positive x axis.

vi. Similarly, a wave travelling in the direction of the negative x axis is represented by,

y(x, t) = a sin (kx + ωt +ø) …….(2)