By Martin Fritz Mueller
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Extra info for Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates
If a SH-wave impinges on an interface, only reﬂected and refracted SH-waves occur, but no P- or SV-waves. If, on the other hand, a P − (SV −)wave interacts with an interface, reﬂected and refracted SV −(P −)waves occur, but no SH -waves occur. These statements hold, in general, only for the case of an interface between two solid half-spaces. In liquids, neither SH- nor SV-waves propagate; in a vacuum a rigid half-space, or no waves propagate at all. Correspondingly, the situation is even more simple if such half-spaces are involved.
The wavefronts move parallel with respect to themselves with the phase − → velocity c. Vector k multiplied by the wavenumber ω/α or ω/β, is called the wavenumber vector. 6. REFLECTION AND REFRACTION OF PLANE WAVES ... 26) − → → → → (rot(f · − n) = f · ∇ × − n −− n × ∇ f ) is transversal (perpendicular to k ). 26), it follows, that for Ψ , without loss of generality, the additional − → → → condition of orthogonality of k and − n can be introduced. (Separation of − n in − → components parallel and perpendicular to k ).
6: Additional Cartesian coordinate system (x, y, z). The x-axis of of this coordinate system is identical to the line OP. The z-axis is in the plane deﬁned by the zandζ-axis and the line OP. Then the z, ζ-axis has then in the x − y − z-system the direction of the unit vector − → n = (cos γ, 0, sin γ) = z z2 , 0, 1 − 2 r r 1 2 . 52 CHAPTER 3. BODY WAVES − → Vector r′′ from O to Q can be written in the same system as − → r′′ = (r − r′ cos ϑ, r′ sin ϑ sin λ, r′ sin ϑ cos λ). These two vectors are needed later.