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Thermodynamics

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Motion in Waves

Before proceeding to study the various optical phenomena on the basis of Huygens wave theory, the characteristics of simple harmonic motion (the simplest form of wave motion) and the composition or superposition of two or more simple harmonic motions are discussed. The propagation of a simple harmonic wave through a medium can be transverse or longitudinal. In a transverse wave, the particles of the medium vibrate perpendicular to the direction of propagation and in a longitudinal wave, the particles of the medium vibrate parallel to the direction of propagation. When a stone is dropped on the surface of still water, transverse waves are produced. Propagation of sound through atmospheric air is in the form of longitudinal waves. When a wave is propagated through a medium, the particles of the medium are displaced from their mean positions of rest and restoring forces come into play. These restoring forces are due to the elasticity of the medium, gravity and surface tension. Due to the periodic motion of the particles of the medium a wave motion is produced. At any instant, the contour of all the particles of the medium constitutes a wave.

Let P be a particle moving on the circumference of a circle of radius a with a uniform velocity v. Let ω be the uniform angular velocity of the particle (v = aω). The circle along which P moves is called the circle of reference. As the particle P moves round the circle continuously with uniform velocity the foot of the perpendicular M, vibrates along the diameter YY’ or (XX’). If the motion of P is uniform, then the motion of M is periodic i.e. it takes the same time to vibrate once between the points Y and Y’. At any instant, the distance of M from the centre O of the circle is called the displacement. If the particle moves from X to P in time t, then ∠ POX = ∠ MPO = θ = ωt.

From the Δ MPO,

sin θ = sin ωt = OM/a

or, OM = y = a sin ωt

OM is called the displacement of the vibrating particle. The displacement of a vibrating particle at any instant can be defined as its distance from the mean position of rest. The maximum displacement of a vibrating particle is called its amplitude.

∴ Displacement = y = a sin ωt               (i)

The rate of change of displacement is called velocity of the vibrating particle.

∴ Velocity = dy/dt = + aω cos ωt                (ii)

The rate of change of velocity of a vibrating particle is called its acceleration.

∴ Acceleration = Rate of change of velocity

= –ω² (a sin ωt) = –ω²y                       (iii)

Thus, the velocity of the vibrating particle is maximum (in the direction OY or OY’) at the mean position of rest and zero at the maximum position of vibration. The acceleration of the vibrating particle is zero at the mean position of rest and maximum at the maximum position of vibration. The acceleration is always directed towards the mean position of rest and is directly proportional to the displacement of the vibrating particle. This type of motion, where the acceleration is directed towards a fixed position (the mean position of rest) and is proportional to the displacement of the vibrating particle, is called simple harmonic motion.

Further,

= –ω² × displacement

Thus, in general, the time period of a particle vibrating simple harmonically is given by T = 2π √K where K is the displacement per unit acceleration.

If the particle P revolves round the circle, n times per second, then the angular velocity ω is given by

= 2πn = 2π/T                 (∵ π = 1/T where T is the time period)

∴ y = a sin 2πnt = a sin 2π (1/T)

On the other hand, if the time is counted from the instant P is at S (∠ SOX = ) then the displacement

y = a sin (ωt + )

= a sin (2πt/T+ )

If the time is counted from the instant P is at S’, then

y = a sin (ωt – )

= a sin (2πt/T- )