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Unformatted text preview: Chapter 14: Wave Motion Types of mechanical waves Â¡ Mechanical waves â€¢ are disturbances that travel through some material or substance called medium for the waves. â€¢ travel through the medium by displacing particles in the medium â€¢ 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 (contâ€™d) Â¡ Longitudinal and transverse waves sound wave = longitudinal wave C = compression R = rarefaction air compressed air rarefied Types of mechanical waves (contâ€™d) Â¡ Longitudinaltransverse waves Types of mechanical waves (contâ€™d) Â¡ Periodic waves â€¢ When particles of the medium in a wave undergo periodic motion as the wave propagates, the wave is called periodic. x=0 x Î» A t=0 t=T/4 t=T period amplitude wavelength Mathematical description of a wave Â¡ Wave function â€¢ 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 ; ) , ( = â€¢ 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 âˆ’ = âˆ’ = âˆ’ = âˆ’ = Î» Ï€ Ï€ Ï‰ Ï‰ sinusoidal wave moving in +x direction angular frequency f Ï€ Ï‰ 2 = velocity of wave, NOT of particles of the medium wavelength period v f = Î» T f / 1 = )] / ( cos[ ) , ( x v t A t x y + = Ï‰ sinusoidal wave moving inx direction v>v phase velocity Mathematical description of a wave (contâ€™d) Â¡ Wave function (contâ€™d) t=0 x=0 x Î» t=T/4 t=T period wavelength ) / / ( 2 cos ) , ( T t x A t x y âˆ’ = Î» Ï€ ) , ( ) , ( T t x y t x y + = + = Î» Mathematical description of a wave (contâ€™d) Â¡ Wave number and phase velocity Î» Ï€ / 2 = k wave number: ) cos( ) , ( t kx A t x y Ï‰ âˆ’ = 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 = âˆ’ Ï‰ v k dt dx = = / / Ï‰ phase velocity )] ( cos[ ) cos( ) , ( vt x k A t kx A t x y âˆ’ = âˆ’ = Ï‰ Mathematical description of a wave (contâ€™d) Â¡ Particle velocity and acceleration in a sinusoidal wave ) cos( ) , ( t kx A t x y Ï‰ âˆ’ = ) , ( ) 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 Ï‰ Ï‰ Ï‰ Ï‰ Ï‰ âˆ’ = âˆ’ âˆ’ = âˆ‚ âˆ‚ = âˆ’ = âˆ‚ âˆ‚ = acceleration u in textbook velocity Also ) , ( ) cos( / ) , ( 2 2 2 2 t x y k t kx A k x t x y âˆ’ = âˆ’ âˆ’ = âˆ‚ âˆ‚ Ï‰ 2 2 2 2 2 2 2 2 2 / ) , ( / ) , ( ) / ( / ) , ( t v t x y t t x y k x t x y âˆ‚ âˆ‚ = âˆ‚ âˆ‚ = âˆ‚ âˆ‚ Ï‰ wave equation Mathematical description of a wave (contâ€™d) Â¡ 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 âˆ‚ âˆ‚ = âˆ‚ âˆ‚ = âˆ‚ âˆ‚ Ï‰ wave equation ) ( ) , ( vt x f t x y Â± = ) cos( t kx Ï‰ âˆ’ Solutions: such as The most general form of the solution: ) ( ) ( ) , ( vt x g vt x f t x y + + âˆ’ = Speed of a transverse wave...
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 Summer '08
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 Physics

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