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Harmonic Waves

The simplest type of a travelling disturbance is a ‘harmonic wave’ having the form

ƒ (x, t) = A cos (kx – ω t + Ø)

where A, k, ω, Ø are constants. Note that above function can be written in standard form as

ƒ (x, t) = A cos k (x – vt + Ø’)

where v = ω/k is wave velocity. Such a wave is produced by a source which performs SHM of frequency ω at x = 0:

ƒ (x = 0, t) = A cos (– ω t + Ø) = A cos (ω t + Ø)

where Ø is the phase constant; its value is fixed by initial conditions. For simplicity of expression, let us take Ø = 0; A is amplitude of oscillations.

The oscillations of the source point causes disturbance in the neighbourhood. From particle to particle, the disturbance propagates in space resulting in a progressive harmonic wave.

At any point in space, say x = x0, the displacement as a function of time t, during the time when disturbance passes through that point, is a simple harmonic motion:

ƒ (t) = A cos (– ω t + c) = A cos (ω t – c)

where c = kx₀ is constant. Thus, each particle of the medium performs simple harmonic oscillations with same amplitude A and frequency ω about its equilibrium position.

On the other hand, at any instant of time, say t = t₀, the displacement of particles in space (i.e. as a function of x) represents a cosine curve:

ƒ (x) = A cos (kx + c’) c’ = – ω t₀ is constant

The constant k is called wave number of the wave. It is related to wavelength λ of the wave as

Note that wavelength λ is the distance after which sine (or cosine) curve repeats itself, that is

ƒ (x + λ) = A cos (k(x + λ) + c’) = A cos (kx + c’ + 2 π)

A cos (kx + c) = ƒ (x).

The harmonic wave function can be written in various ways as

ƒ (x, t) = A cos (kx – ω t)

Note that T is time period SHM; v = ω/2 π = 1/T is the linear frequency of oscillations. The velocity v = ω/k = λ v, in case of harmonic waves, is also called phase velocity of the wave.