Nonlinear Acoustic Wedge Waves

Abstract

Among the various types of guided acoustic waves, acoustic wedge waves are non-diffractive and non-dispersive. Both properties make them susceptible to nonlinear effects. Investigations have recently been focused on effects of second-order nonlinearity in connection with anisotropy. The current status of these investigations is reviewed in the context of earlier work on nonlinear properties of two-dimensional guided acoustic waves, in particular surface waves. The role of weak dispersion, leading to solitary waves, is also discussed. For anti-symmetric flexural wedge waves propagating in isotropic media or in anisotropic media with reflection symmetry with respect to the wedge’s mid-plane, an evolution equation is derived that accounts for an effective third-order nonlinearity of acoustic wedge waves. For the kernel functions occurring in the nonlinear terms of this equation, expressions in terms of overlap integrals with Laguerre functions are provided, which allow for their quantitative numerical evaluation. First numerical results for the efficiency of third-harmonic generation of flexural wedge waves are presented.

Appendices

Appendix A

We define the operator

$$ D_{\alpha } (q) = \left\{ {\begin{array}{*{20}c} {iq\,\,\,\,\,\,{\text{for}}\,\,\,\alpha = 1} \\ {{\partial \mathord{\left/ {\vphantom {\partial {\partial x_{\alpha } \,\,{\text{for}}\,\,\,\alpha = 2,3}}} \right. \kern-0pt} {\partial x_{\alpha } \,\,{\text{for}}\,\,\,\alpha = 2,3}},} \\ \end{array} } \right. $$

(A.1)

and the quantities

$$ g_{I}^{(\alpha )} (x_{2} ,\,x_{3} ;\,q) = D_{\alpha } (q)\hat{f}_{I} (| q | x_{2} ,| q | x_{3} )\,, $$

(A.2)

which possesses the scaling property for q > 0

$$ g_{I}^{(\alpha )} (x_{2} ,\,x_{3} ;\,q) = Xg_{I}^{(\alpha )} (Xx_{2} ,\,Xx_{3} ;\,{q \mathord{\left/ {\vphantom {q X}} \right. \kern-0pt} X}) $$

(A.3)

and, if the function f_I is real,

$$ g_{I}^{(\alpha )} (x_{2} ,\,x_{3} ;\, - q) = g_{I}^{(\alpha )*} (x_{2} ,\,x_{3} ;\,q). $$

(A.4)

The elements of the matrix M in (8.2.7) may then be expressed in the form

$$ M_{IJ}^{(\alpha \beta )} (q) = {\iint\limits_{A}} {g_{I}^{(\mu )*} (x_{2} ,\,x_{3} ;\,q)C_{\alpha \mu \,\beta \nu } g_{J}^{(\nu )} (x_{2} ,\,x_{3} ;\,q)\,dx_{2} \,dx_{3} \,}. $$

(A.5)

The right-hand side of (A.2) implies that M does not depend on |q|, that

$$ M_{IJ}^{(\alpha \beta )} (q) = M_{JI}^{(\beta \alpha )*} (q) $$

(A.6)

and, if the functions f_I are all real, which is the case with the choice (8.2.6), that

$$ M_{IJ}^{(\alpha \beta )} ( - q) = M_{IJ}^{(\alpha \beta )*} (q). $$

(A.7)

For the quantities V and W we obtain expressions analogous to (A.5),

$$ \begin{aligned} V_{IJK}^{(\alpha \beta \gamma )} ( - q,k,q - k) = & \iint\limits_{A} {S_{\alpha \mu \,\beta \nu \,\gamma \lambda } \,\,g_{I}^{(\mu )} (x_{2} ,\,x_{3} ;\, - q)} \\ & \quad \times \,g_{J}^{(\nu )} (x_{2} ,\,x_{3} ;\,k)\,g_{K}^{(\lambda )} (x_{2} ,\,x_{3} ;\,q - k)dx_{2} \,dx_{3} , \\ \end{aligned} $$

(A.8)

$$ \begin{aligned} W_{IJKL}^{(\alpha \beta \gamma \delta )} ( - q,k,k',q - k - k') = & \iint\limits_{A} {S_{\alpha \mu \,\beta \nu \,\gamma \lambda \,\delta \kappa } \,\,g_{I}^{(\mu )} (x_{2} ,\,x_{3} ;\, - q)} \\ & \quad \times \,g_{J}^{(\nu )} (x_{2} ,\,x_{3} ;\,k)\,g_{K}^{(\lambda )} (x_{2} ,\,x_{3} ;\,k')g_{L}^{(\kappa )} (x_{2} ,\,x_{3} ;\,q - k - k')dx_{2} \,dx_{3} \\ \end{aligned} $$

(A.9)

with scaling properties that follow from (A.3).

Appendix B

The kernel function in the effective second-order nonlinearity of the evolution Eq. (8.4.12) may be expressed in the form

$$ \begin{aligned} K_{2} ( - q,k,q - k) = & \frac{ - iq}{{\hat{N}\,k(q - k)}} \\ & \quad \times \sum\limits_{I,J,K} {\left\{ {V_{IJK}^{(1,\alpha \beta \gamma )} ( - q,k,q - k)\,w_{I}^{(\alpha )} ( - q)\,w_{J}^{(\beta )} (k)\,w_{K}^{(\gamma )} (q - k)} \right.} \\ & \quad + \,V_{IJK}^{(0,\alpha \beta \gamma )} ( - q,k,q - k)\,\left[ {w_{I}^{(\alpha )} ( - q)\,w_{J}^{(\beta )} (k)\,\,\Delta w_{K}^{(\gamma )} (q - k)} \right. \\ & \quad + \,\left. {\left. {w_{I}^{(\alpha )} ( - q)\,\,\,\Delta w_{J}^{(\beta )} (k)\,w_{K}^{(\gamma )} (q - k) + \Delta w_{I}^{(\alpha )} ( - q)\,\,w_{J}^{(\beta )} (k)\,w_{K}^{(\gamma )} (q - k)} \right]} \right\}. \\ \end{aligned} $$

(B.1)

It depends only on the signs and on ratios of the 1D wavevectors q, k, q − k, and it vanishes if the reflection symmetry with respect to the mid-plane of the wedge is not broken.

The kernel function in the effective third-order nonlinearity in (8.4.12) consists of a direct contribution, which is linear in the components of the eighth-rank tensor S, and an indirect contribution, which is quadratic in the sixth-rank tensor S, K₃ = K_3,d + K_3,i, where

$$ \begin{aligned} K_{3,d} ( - q,k,k',q - k - k') = & \frac{ - q}{{3\hat{N}\,k\,k'\,(q - k - k')}} \\ & \, \times \,\sum\limits_{I,J,K,L} {\left[ {W_{IJKL}^{(0,\alpha \beta \gamma \delta )} ( - q,k,k',q - k - k')\,\,w_{I}^{(\alpha )} ( - q)\,\,w_{J}^{(\beta )} (k)\,\,w_{K}^{(\gamma )} (k')\,\,w_{L}^{(\delta )} (q - k - k')} \right]} \,,\, \\ \end{aligned} $$

(B.2)

and

$$ \begin{aligned} K_{3,i} ( - q,k,k',q - k - k') = & \frac{ - q}{{\hat{N}\,k\,k'\,(q - k - k')}} \\ & \quad \times \,\sum\limits_{I,J,K,L} {\left[ {w_{I}^{(\alpha )} ( - q)\,\,w_{J}^{(\beta )} (k)\,\,V_{IJK}^{(0,\alpha \beta \gamma )} ( - q,k,q - k)\,\,\varGamma_{KL}^{(\gamma \lambda )} (q - k)} \right.} \\ & \quad \times \,\left. {V_{LMN}^{(0,\lambda \mu \nu )} ( - q + k,k',q - k - k')w_{M}^{(\mu )} (k')\,\,w_{N}^{(\nu )} (q - k - k')} \right]\,. \\ \end{aligned} $$

(B.3)