Handout_6 - Classical Dynamics and Fluids ELASTIC WAVES P...

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Classical Dynamics and Fluids P 117 E LASTIC W AVES The vector wave equation for elastic waves: ρ 2 X ∂t 2 = ( B + 1 3 G ) ( · X )+ G 2 X We can Fnd wave solutions (normal modes) e.g. a wave travelling the in x -direction X = ( X 0 ,Y 0 ,Z 0 ) e i ( ωt - kx ) . The divergence of X is ikX 0 e i ( ωt - kx ) . The gradient of the divergence ( ( · X ) ) is ( - k 2 X 0 , 0 , 0) e i ( ωt - kx ) . The term 2 X is ( - k 2 X 0 , - k 2 Y 0 , - k 2 Z 0 ) e i ( ωt - kx ) . The time derivative is 2 X ∂t 2 is ( - ω 2 X 0 , - ω 2 Y 0 , - ω 2 Z 0 ) e i ( ωt - kx ) . y x If the displacement is in the y -direction it is a transverse shear, or S-wave : we Fnd ρω 2 = Gk 2 , so velocity ω 2 S = G/ρ . It’s same if the displacement is in the z -direction. x If the displacement is in the x -direction it is a longitudinal compressional , or P-wave . Both terms on the RHS now contribute so ρω 2 = ( B + 4 3 G ) k 2 . The velocity of a longitudinal wave is ω 2 P = ( B + 4 3 G ) . ±or any k -vector we can similarly Fnd waves X = X 0 e i ( ωt - k · x ) : there are three modes; two transverse ( ω 2 S = G/ρ ) and one longitudinal ( ω 2 P = ( B + 4 3 G ) ). Classical Dynamics and Fluids P 118 E LASTIC W AVES II These S-waves and P-wave modes describe waves propagating in a bulk medium (i.e. size ± wavelength). Waves near the surface of a medium have many interesting properties. d S t n Boundary conditions at a surface: 1. free boundary. The normal component of the stress vanishes: n · τ · d S = 0 , where d S = n | d S | . 2. ±ixed boundary: normal component of the displacement vanishes: n · X = 0 . Example 1 : longitudinal waves in x -direction on a thin rod (thickness ² wavelength). We have τ yy = τ zz = 0 , so Ee xx = τ xx . The wave speed on a thin rod is v 2 = E/ρ . y
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This note was uploaded on 05/09/2009 for the course DAMTP NST 1B Phy taught by Professor Sfgull during the Spring '07 term at Cambridge.

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Handout_6 - Classical Dynamics and Fluids ELASTIC WAVES P...

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