Unit 2 Waves and Optics Oscillatory motion - Forced and damped oscillation differential equation and its solution - Doppler effect - Lasers : Population of energy levels, Einstein’s A and B coefficients derivation - resonant cavity, optical amplification (qualitative) - Nd.YAG laser Semiconductor lasers: homojunction and heterojunction - Fiber optics priciple, Numerical aperture and Acceptance angle - Types of optical fibres (material, refractive index, mode) - losses associated with optical fibers - fibre optic sensors: pressure and displacement. 2.1. OSCILLATORY MOTION The phenomenon of wave motion is prevalent in almost all branches of Physics. Waves have as their source a vibration. Harmonic oscillation A motion that is repeated at regular intervals of time is called periodic motion. The solutions of the equations of motion can be expressed as functions of sines and cosines. Motions described by functions of sines and cosines are referred to as harmonic oscillation. If the motion is back and forth over the same path, it is called vibratory or oscillatory. T Time required for one oscillation f Frequency (number of oscillation per unit time) A Amplitude (maximum displacement) Angular frequency Fig.2.1 Harmonic Oscillation 2 f 2.1
Engineering Physics 2.2 2.2. SIMPLE HARMONIC MOTION (SHM) The motion of a particle is said to be simple harmonic if it oscillates from an equilibrium position under the influence of a force that is proportional to the distance of the particle from the equilibrium position. Also, the force is such that it directs the particle back to its equilibrium position. Equation of SHM A familiar example of a particle executing SHM is the motion of a mass m attached to an elastic spring moving on a horizontal frictionless table. When the spring is stretched a distance x from the unextended position, the restoring force F acting on the mass is given by F = – kx .....(1) where k is the proportionality constant, but in the present case it is spring constant k of the spring. The negative sign indicates that the force is directed against the motion, which is towards the equilibrium position. d 2x The acceleration a of the mass is dt 2 . Applying Newton’s second law, we get kx m m d 2x dt 2 d 2x kx 0 dt 2 .....(2) Equ. (2) is the differential equation of motion of a particle executing SHM. Solution of Equation of SHM We can rewrite Equ. (2) as d 2x k x 2 dt m d 2x ω 2 x 2 dt .....(3)
Waves and Optics 2.3 2 where k m ..... (4) dx Multiplying both sides of Equ. (3) by 2 , we get dt 2 dx d 2 x dx 2 2 x 2 dt dt dt Integrating with respect to t 2 dx 2 2 x C dt where C is a constant. dx At the maximum displacement position x = A and velocity = 0. dt Hence, C 2 A 2 and dx dt A 2 x2 ...... (5) It may be noted from Equ. (3) and Equ. (5) that in harmonic motion, neither the acceleration nor the velocity of the particle is constant. Rewriting Equ. (5) as dx A2 x2 dt and integrating sin 1 x t A where is a constant x A sin t ...... (6) where A is the amplitude and is a constant.
Engineering Physics 2.4 2.3. DAMPED OSCILLATION When a body executing simple harmonic vibrations is left to itself, the amplitude of vibrations gradually decreases. After some time, the vibrations completely die out. This is because the motion of the body is resisted by various frictional effects. Retarding forces are called into play due to the viscosity or internal friction of the body and the resistance of the air. These forces reduce the amplitude and thus damp the oscillations. Such oscillations of a body are called damped vibrations. Example : In actual practice, when a simple pendulum vibrates in air medium, there are frictional forces (resistance of air). So energy is dissipated in each vibration. The amplitude of swing decreases continuously with time. Finally the oscillations die out. Such vibrations are called free damped vibrations. The dissipated energy appears as heat either within the system itself or in the surrounding medium. Differential Equation of a Damped Vibration Consider body of mass m. Let x be the displacement of the body from the equilibrium position at any instant. dx dt When a body is oscillating in a resisting medium, two forces are acting on the body. The instantaneous velocity is 1. A restoring force directly proportional to the displacement x but acting in the opposite direction. It may be written as – Kx Here, K is the restoring force per unit displacement. 2. A frictional (or damping) force proportional to the velocity, but opposite to the direction of motion. It may be written as dx dt Here, is a positive constant depending upon the force of resistance. Therefore, the differential equation in the case of free-damped vibrations is, m d2x dx Kx 2 dt dt