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Chapter 5 Waves 5.1 The wave equation The general form of the wave equation is: ± u =0 , or: 2 u - 1 v 2 2 u t 2 = 2 u x 2 + 2 u y 2 + 2 u z 2 - 1 v 2 2 u t 2 =0 where u is the disturbance and v the propagation velocity . In general holds: v = f λ . By deFnition holds: k λ =2 π and ω =2 π f . In principle, there are two types of waves: 1. Longitudinal waves: for these holds ± k ± ± v ± ± u . 2. Transversal waves: for these holds ± k ± ± v ± u . The phase velocity is given by v ph = ω /k . The group velocity is given by: v g = d ω dk = v ph + k dv ph dk = v ph ± 1 - k n dn dk ² where n is the refractive index of the medium. If v ph does not depend on ω holds: v ph = v g . In a dispersive medium it is possible that v g >v ph or v g <v ph , and v g · v f = c 2 . If one wants to transfer information with a wave, e.g. by modulation of an EM wave, the information travels with the velocity at with a change in the electromagnetic Feld propagates. This velocity is often almost equal to the group velocity.
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Unformatted text preview: ±or some media, the propagation velocity follows from: • Pressure waves in a liquid or gas: v = ³ κ / ² , where κ is the modulus of compression. • ±or pressure waves in a gas also holds: v = ³ γ p/ ² = ³ γ RT/M . • Pressure waves in a thin solid bar with diameter << λ : v = ³ E/ ² • waves in a string: v = ³ F span l/m • Surface waves on a liquid: v = ´ ± g λ 2 π + 2 πγ ²λ ² tanh ± 2 π h λ ² where h is the depth of the liquid and γ the surface tension. If h ² λ holds: v ≈ √ gh . 5.2 Solutions of the wave equation 5.2.1 Plane waves In n dimensions a harmonic plane wave is deFned by: u ( ± x,t ) = 2 n ˆ u cos( ω t ) n µ i =1 sin( k i x i )...
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.

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