AE02.pdf

In this case the expressions for the reflected waves

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In this case, the expressions for the reflected waves are as follows: (75) (76) (77) (78) The terms α 1 and α 2 in Eqs. 75 and 76 are the direction cosines of the reflected transverse waves and are given by the expressions α 1 = sin θ i and α 2 = cos θ i. The terms β 1 and ζ in Eqs. 77 and 78 are determined from β 1 = κ sin θ i , β 2 = j ζ and ζ = ( κ 2 sin 2 θ i – 1). With the angles of incidence θ i > θ c , the amplitude of the reflected transverse wave is equal to the amplitude of the incident transverse wave but the phase of the reflected transverse wave is shifted by 2 α . The other term (Eqs. 77 and 78) was a reflected longitudinal wave for θ i < θ c . However, for θ i > θ c , it is no longer a plane wave traveling into the medium but rather is a heterogeneous disturbance propagating along the surface at the same speed as the projection of the transverse wave along the surface. Plots of Eqs. 69 and 70 (or for θ i > θ c , Eqs. 71 and 72) are shown in Fig. 35. Note that, as the incident angle increases from zero, the magnitude of the reflected transverse wave decreases whereas the amplitude of the reflected longitudinal wave increases. In this region, the incident transverse wave is increasingly converted to a longitudinal wave as the incident angle increases up to about 0.5 rad (30 deg). At this incident angle, the amplitude of the reflected transverse wave begins to increase. The discontinuity in Fig. 35 occurs at the critical angle, 0.564 rad (32.3 deg) for Poisson’s ratio of 0.3. For angles greater than the critical, the amplitude of the reflected transverse wave is equal to the amplitude of the incident transverse wave but the phase of the reflected wave changes as discussed above. The amplitude of the reflected longitudinal u S e e x j k ß x t 2 2 2 2 1 1 rl = ( ) β ζ ω α u S e e x j k ß x t 1 1 2 2 1 1 rl = ( ) β ζ ω α u e j k x x t 2 1 2 2 1 1 2 2 rt = ( ) [ ] α α α ω α u e j k x x t 1 2 2 2 1 1 2 2 rt = ( ) [ ] α α α ω α tan sin sin sin cos α κ θ θ θ κ θ = ( ) × ( ) ( ) ] ÷ ( ) 2 1 2 2 2 2 i i i i S = ( ) ÷ ( ) { + ( ) [ ] × ( ) ( ) } κ θ κ θ κ θ θ θ sin cos sin sin sin 4 2 4 1 2 2 4 2 2 2 2 2 i i i i i A Se j rl = α A e j rt = 2 α A rt i i rl i = ( ) ( ) ( ) ( ) κ θ θ θ κ θ sin sin sin cos 4 2 2 2 2 2 A rt i rl i i rl i = ( ) ( ) ( ) ( ) ( ) + ( ) sin sin cos sin sin cos 2 2 2 2 2 2 2 2 2 2 θ θ κ θ θ θ κ θ θ θ i c arc = = sin c c 2 1 84 Acoustic Emission Testing
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wave for θ i > θ c is really the amplitude of the disturbance propagating parallel to the surface and this disturbance amplitude in turn depends on the incident angle. Incident Longitudinal Wave As shown in Fig. 36, a plane longitudinal wave incident on a free surface at angle θ i produces a reflected longitudinal and a reflected transverse wave. The angle the reflected transverse wave makes with the normal to the surface is determined by Snell’s law: (79) The amplitudes of the reflected waves are given by Eqs. 80 and 81: (80) (81) where κ = c 1 · c 2 –1 as before.
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  • Fall '19
  • Nondestructive testing, Acoustic Emission, Acoustic Emission Testing

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