A plane P wave travelling in a Poisson solid (

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Fluids & Continuum Physics · Superposition, boundaries and interference

Bending, Ringing, Breaking and Flowing, Ch Elastic waves in a solid and at its surface

Problem

A plane P wave travelling in a Poisson solid (ν=0.25\nu=0.25, so cL=3 cTc_L=\sqrt3\,c_T) strikes the material's free surface at an angle of incidence θP=45∘\theta_P=45^\circ from the normal. ((a)) Find the angle θS\theta_S of the reflected SV wave, using phase matching along the surface. ((b)) Solving the two traction conditions on the free surface for a single incident P wave with reflected P and SV waves gives the reflected-P amplitude ratio RPP=b−aa+b,a=(1cT2−2p2)2,b=4p2cos⁡θPcLcos⁡θScT,R_{PP}=\frac{b-a}{a+b},\qquad a=\left(\frac{1}{c_T^2}-2p^2\right)^2,\quad b=4p^2\frac{\cos\theta_P}{c_L}\frac{\cos\theta_S}{c_T}, with p=sin⁡θP/cL=sin⁡θS/cTp=\sin\theta_P/c_L=\sin\theta_S/c_T. The fraction of the incident energy returning as P is RPP2R_{PP}^2. Compute RPPR_{PP} and the fraction returning as P. ((c)) The same working gives the reflected SV amplitude ratio RPS=4(cL/cT) p (cos⁡θP/cL)(cT−2−2p2)/(a+b)R_{PS}=4(c_L/c_T)\,p\,(\cos\theta_P/c_L)\left(c_T^{-2}-2p^2\right)/(a+b), and the fraction of the energy returning as SV is RPS2R_{PS}^2 times the obliquity factor (cTcos⁡θS)/(cLcos⁡θP)(c_T\cos\theta_S)/(c_L\cos\theta_P). Compute that fraction, and check that the two energy fractions sum to one.

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