By Martin Fritz Mueller

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**Extra resources for Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates**

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24), the movement at all times and locations is non-zero, the wavefronts can no longer be deﬁned as surfaces separating undisturbed-disturbed from disturbed regions. We, therefore, consider wave − → → fronts as surfaces of constant phase ω(t − − x k /c) with c = α or c = β. These surfaces are deﬁned by d dt t− − → − → x k c = 0. − → They are perpendicular to vector k , which also gives the direction of propagation. The wavefronts move parallel with respect to themselves with the phase − → velocity c.

29) Which of the displacement components is continuous depends on the special combination of the half-spaces. 6. REFLECTION AND REFRACTION OF PLANE WAVES ... 65 Since no connection of v with Φ and Ψ exists via the boundary conditions and, therefore, with u and w, it follows, that the S-waves, the displacement of which is only horizontal (in y-direction: SH-waves), propagate independently from the P -waves, following from Φ, and the S-waves, following from Ψ, that also have a vertical component (in z-direction: SV-waves).

Pressure P (t) acts on the top of the elastic layer. What is the movement in the layer? Examine the case P (t) = P 0 δ(t). 2 (constant pressure P1 on the surface). 4 Spherical waves from explosion point sources In the previous sections, we considered inﬁnitely extended waves. They are an idealisation, because they cannot be produced in reality since they require inﬁnitely extended sources. , waves which originate at a point (point source) and propagate in the full-space. Their wavefronts are spheres.