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 = kx_{0} 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 = t_{0}, 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 t_{0} 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.
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