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Waves yck2F6/2008 Kinds of waves Pulse / Periodic waves Transverse / Longitudinal Progressive (traveling) / Stationary (standing) Frequency (f): Number of oscillations per second Period (T): Time for one oscillation (unit is s) Amplitude (A): The largest magnitude of oscillation (unit depends on type of wave) Wavelength (): The distance traveled by wave in one period of time (m) Wave velocity (v): The distance traveled by wave per second ( m s1) Intensity (I): Energy transported through a unit perpendicular area per second Usually, (W m2) Intensity Amplitude2 Mathematical representation of wave: y A cost kx where, y is the magnitude of oscillation of wave from equilibrium state; A is the amplitude of wave; is the angular frequency of wave; = 2 f (unit is s1) t is the time ; (s) x is the position on line along which the wave propagates; (m) ` k is the wave number; (m1) is the initial phase (starting phase) For a certain moment, say t = 0 : (x=) k corresponds to the phase of 2 k 2 so, k 2 For a certain position on the propagation line, say x = 0 : T corresponds to the phase of 2 T 2 so, T 2 1 (t=T) Graphical representations of waves: y-t graph (one point, all time) y-x graph (one moment, all positions) There are crests and troughs for transverse waves; compressions and rarefactions for longitudinal waves. Wave velocity depends on the medium, not on frequency of oscillations. Doppler Effect (for sound and light) v vr f f v vs o where fo is the original frequency of wave f is the apparent frequency received by the receiver v is the wave velocity vr is the velocity of the receiver (+ approaching, moving away) vs is the velocity of wave source ( approaching, + moving away) Examples of Doppler effect Electromagnetic spectrum Wave speed is speed of light. The approximate ranges () of different e.m. waves: Radio waves 104 m -- 102 m Microwaves 102 m -- 103 m Infra red 103 m -- 107 m Visible light 7107 m -- 4107 m Ultra-violet rays 107 m -- 109 m X-rays 109 m -- 1011 m Gamma rays 1011 m -- shorter Examples and applications Huygens' Principle Every point on a wavefront behaves as a new source (sourcelet), sending out secondary wavelets. A smooth line (tangent) joining the most forward part of the secondary wavelets forms the new wavefront. 2 Reflection of waves There is phase change of when a wave is reflected from a denser medium. Laws of reflection are obeyed. Refraction of waves Snell's Law is obeyed. Refractive index of a medium (n) depends on the speed of wave in that medium. c sin v n v sin m where or, c = speed of wave in air / vacuum v = speed of wave in medium v = angle in air / vacuum m = angle in medium n1 sin 2 for the relationship between two media, 1 and 2. n 2 sin 1 Selective Polarization of waves (only for transverse waves) A polarizer and an analyser are media that only allow oscillations of a certain direction to pass through. When a unpolarized wave passes through a polarizer, its intensity drops to half. When a polarized wave passes through an analyser, then its intensity I is I I o cos 2 where is the angle between the directions of polarization of the polarizer and analyser. Polarization of waves by reflection Reflected light is completely polarized when the transmitted and reflected rays are perpendicular at Brewster's angle of incidence, b , where n tan b 2 n1 (n1 and n2 are refractive indice of the two media, and light is incident from medium 1 to medium 2) Polarization by scattering Scattered light is completely polarized when it is normal to the incident direction. 3 Principle of Superposition When two waves of the same nature meet at a point, the resultant magnitude of oscillation at that point is the algebraic sum of oscillations caused by the individual waves. Interference of waves When two waves of the same nature, same polarization direction (for transverse waves), same frequency, constant phase difference and almost the same amplitude meet, a steady interference pattern may be resulted. For two coherent waves, (waves of same phase) Constructive interference occurs at a point when path difference from the two sources = n (or phase difference = n (2) ) Destructive interference occurs at a point when Path difference from the two sources = n (n = 0, 1, 2, ...) 1 2 (or phase difference = (2n + 1) ) Young's double-slit experiment produces a interference pattern of uniform separation (detail description and calculations are needed) Multiple-slits interference Diffraction grating (for light) Constructive interference occurs at angle when a sin m (a : grating spacing; m : order of maximum) Destructive interference occurs at angle ' when 1 a sin ' m ( N : no. of slits involved ) N Angular half-width, ' Examples and applications of interference 4 Beat: occurs when two waves of slightly different frequencies superposition. Frequency of the resultant wave form = f f' 2 Beat frequency = f f ' Diffraction of light For wave passing through a single slit of width a ( a > ) There will be a central maximum (which is wider and brighter) 2m 1 2 Maximum occurs at when a sin Minimum occurs at ' when a sin ' m 1 (m = 0, 1, 2, ...) Stationary waves Stationary wave is a result of superposition of a progressive wave (of certain wavelengths) and its reflected wave. Nodes and antinodes are found in a stationary wave. At node, oscillation and kinetic energy are the minimum; at antinode, oscillation and kinetic energy are the maximum. Resonance in air columns and strings In order to obtain resonance, the length of column or string should be in integral multiple of half the wavelength of the wave. The end conditions (fixed or not) of the two ends should be considered. Harmonics and overtones Fundamental frequency, fo (not commonly known as first harmonic) Frequency of second harmonic = 2 fo Frequency of third harmonic = 3 fo Finding velocity of sound by experiments: 1. tuning fork and air column finding length of air column for resonance (error due to end correction) 2. Kundt's tube measuring distance between two leaps of powder (foam chips) 5 Intensity Level of sound ( h , unit is dB) Decibel : using 1012 W m2 as reference intensity. I h 10 log 10 Io Threshold of hearing at 1000 Hz is 0 dB. Noise and musical note Loudness depends on intensity Pitch depends on frequency Quality depends on wave form Geometric Optics Systems of lenses and mirrors Optical instruments (magnifying glass and telescope) Angular magnification and angle subtended by object and image Optical spectrometer 6