3 - i1 1 , 1 , 1 2 , 2 , 2 i2 Observe that when waves...

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Observe that when waves  travel from a solid of one  velocity to different  velocity, the direction  changes. This has a large impact on  the nature of seismic  waves, since the Earth is  highly variable. 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2
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Boundary conditions:  “welded contact” This implies that  displacement is continuous  across the boundary. Also, that stress is  continuous across the  boundary. 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2
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Envision a wave of frequency  f .  It cannot change frequency  at the boundary. Wavefronts are drawn  perpendicular to direction of  wave travel. Note how angle of incidence,  i 1   and  i 2   are defined. 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2 1 2 1 2 λ wavefront 1 2 i 1 i 2 wavefront A B 2 2 1 1 = = f
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Line segment AB is  common to two right  triangles.   The geometry leads to  Snell’s Law: 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2 1 2 1 2 λ wavefront 1 2 i 1 i 2 wavefront A B 1 1 sin i AB = 2 2 sin i AB = 2 2 1 1 = = f 2 2 1 1 sin sin 1 i i AB = = 2 2 1 1 sin sin i i = 2 2 1 1 sin sin i i =
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This way of drawing is  consistent with horizontal  layers in the Earth.   Lower velocities near the  surface imply wave  propagation direction is  bent towards the vertical as  the waves near the surface. 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2 2 2 1 1 sin sin i i =
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Example of a 3-component ground motion record. Note how the S-wave is dominantly showing up on the horizontal components, and the P-wave is strongest on the vertical component. P S
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In addition to the “refraction”  of energy into the second  medium, some energy is  reflected back. The angle of reflection is equal  to the angle of incidence. This brings up the issue: how is  the energy partitioned at the  interface? 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2 i 2 Incoming SH
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The energy partitioning is  determined by “reflection” and  “transmission” coefficients.  The coefficients are determined  by matching boundary  conditions For incoming SH waves, the  form is relatively simple. 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2 i 2 A T R A T Transmission coefficient Reflection coefficient A R Incoming SH
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These coefficients are not a  function of frequency. At most, the transmitted  wave has an amplitude of 2  x the amplitude of the  incoming wave. Going from a stiffer to a  softer material, the  transmission coefficient is  never less than 1.0. 1 1 1 , , β μ ρ 2 2 2 , , i 1 i 2 i 2 A T R 1 1 2 2 2 2 2 + = A T Incoming SH 1 1 2 2 1 1 2 2 + - = A R For Vertical Incidence at the boundary the refection and transmission coefficients are:
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Going from a softer to a  stiffer material, the  transmission coefficient  is never more than 1.0.
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3 - i1 1 , 1 , 1 2 , 2 , 2 i2 Observe that when waves...

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