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Unformatted text preview: Chapter 14: Wave Motion Types of mechanical waves Mechanical waves are disturbances that travel through some material or substance called medium for the waves. travel through the medium by displacing particles in the medium travel in the perpendicular to or along the movement of the particles or in a combination of both transverse waves: waves in a string etc. longitudinal waves: sound waves etc. waves in water etc. Types of mechanical waves (contd) Longitudinal and transverse waves sound wave = longitudinal wave C = compression R = rarefaction air compressed air rarefied Types of mechanical waves (contd) Longitudinaltransverse waves Types of mechanical waves (contd) Periodic waves When particles of the medium in a wave undergo periodic motion as the wave propagates, the wave is called periodic. x=0 x A t=0 t=T/4 t=T period amplitude wavelength Mathematical description of a wave Wave function The wave function describes the displacement of particles in a wave as a function of time and their positions: t x y t x y y , at nt displaceme is ; ) , ( = A sinusoidal wave is described by the wave function: ) / / ( 2 cos ) / ( 2 cos )] / ( cos[ )] / ( cos[ ) , ( T t x A t v x f A t v x A v x t A t x y = = = = sinusoidal wave moving in +x direction angular frequency f 2 = velocity of wave, NOT of particles of the medium wavelength period v f = T f / 1 = )] / ( cos[ ) , ( x v t A t x y + = sinusoidal wave moving inx direction v>v phase velocity Mathematical description of a wave (contd) Wave function (contd) t=0 x=0 x t=T/4 t=T period wavelength ) / / ( 2 cos ) , ( T t x A t x y = ) , ( ) , ( T t x y t x y + = + = Mathematical description of a wave (contd) Wave number and phase velocity / 2 = k wave number: ) cos( ) , ( t kx A t x y = phase The speed of wave is the speed with which we have to move along a point of a given phase. So for a fixed phase, . const t kx = v k dt dx = = / / phase velocity )] ( cos[ ) cos( ) , ( vt x k A t kx A t x y = = Mathematical description of a wave (contd) Particle velocity and acceleration in a sinusoidal wave ) cos( ) , ( t kx A t x y = ) , ( ) cos( / ) , ( ) , ( ) sin( / ) , ( ) , ( 2 2 2 2 t x y t kx A t t x y t x a t kx A t t x y t x v y y = = = = = acceleration u in textbook velocity Also ) , ( ) cos( / ) , ( 2 2 2 2 t x y k t kx A k x t x y = = 2 2 2 2 2 2 2 2 2 / ) , ( / ) , ( ) / ( / ) , ( t v t x y t t x y k x t x y = = wave equation Mathematical description of a wave (contd) General solution to the wave equation 2 2 2 2 2 2 2 2 2 ) , ( ) , ( ) , ( t v t x y t t x y k x t x y = = wave equation ) ( ) , ( vt x f t x y = ) cos( t kx Solutions: such as The most general form of the solution: ) ( ) ( ) , ( vt x g vt x f t x y + + = Speed of a transverse wave...
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This note was uploaded on 05/28/2011 for the course PHY 126 taught by Professor Staff during the Summer '08 term at SUNY Stony Brook.
 Summer '08
 Staff
 Physics

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