State the mathematical expression for a transverse progressive wave travelling along the positive and negative x-axis.
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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)