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Thermodynamics

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Simple Harmonic Wave

The equation y = a sin ωt represents the displacement of a single particle vibrating simple harmonically. Let O, A, B, C etc. be different particles in the medium. Let the distances of the particles A, B, C etc. from the particle O be x₁, x₂, x₃ etc. Let t₁, t₂, t₃ etc. be the time intervals taken by the wave to travel from point O to the points A, B, C etc. The displacement of the particle O at any instant is given by

also, etc. where v is the velocity of the wave. Thus displacement of the particles A, B etc. will be given by the equations

[∴ v = n λ = λ/T ∴ vT = λ]

In equation (iii) is called the phase and is the phase difference between the vibrating particles at O and A. The distance travelled by the disturbance in time T is λ and in time t₁ is x₁.

Thus, equation (iii) can also be written as

If the distance x₁ = λ then the phase difference = 2nλ/λ = 2π,

i.e. the phase difference between the particles O and A will be zero or the two particle vibrate in phase. Similarly, all the particles distant 2λ, 3λ etc from O will be vibrating in phase.