Abstract Generated abstract
The paper develops a ray-theoretic generalization of classical Rayleigh waves for a homogeneous elastic body bounded by an arbitrary analytic surface. Longitudinal and transverse ray solutions of the elasticity equations are combined with complex phase functions chosen to satisfy eikonal equations, decay into the body, and meet the stress-free boundary condition on the surface. The boundary conditions lead to solvability conditions and transport equations for the leading amplitude along surface geodesics. For the principal term, the resulting intensity is proportional to the inverse square root of the divergence coefficient of the surface rays, reducing in the plane case to constant intensity during propagation.
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Mathematics
V. M. BABICH
ON THE PROPAGATION OF RAYLEIGH WAVES ALONG THE SURFACE OF A HOMOGENEOUS ELASTIC BODY OF ARBITRARY SHAPE
(Presented by Academician V. I. Smirnov on 28 XI 1960)
1°. In the present paper, solutions of the equations of the dynamics of an elastic body will be constructed that generalize the well-known solutions of Rayleigh. As is known, “classical” Rayleigh waves may be regarded as the superposition of two complex plane waves—a longitudinal and a transverse one—chosen so that the stresses vanish on the surface of the half-space \((^{1})\). If the plane wave is replaced by a “ray” solution and the half-space by an arbitrary analytic surface, then we arrive at the generalization of Rayleigh waves that is considered here*.
2°. Let \(f_0(\zeta)\) be a function of the complex variable \(\zeta\), regular in the upper half-plane, and let \(f_k(\zeta)\) be successive integrals of \(f_0(\zeta)\). A longitudinal ray solution of the equations of the theory of elasticity is the series \((^{2,3})\)
\[ \mathbf{u}=\sum_{k=0}^{+\infty}\mathbf{u}_{ka}(x,y,z)\,f_k\bigl(t-\tau_a(x,y,z)\bigr), \tag{1} \]
where
\[ (\nabla \tau_a)^2=\frac{1}{a^2}\qquad \left(a^2=\frac{\lambda+2\mu}{\rho}\right); \tag{2} \]
\[ \mathbf{u}_{ka}=\mathbf{u}^{0}_{ka}+\varphi_k\nabla\tau_a;\qquad \mathbf{u}^{0}_{ka}\perp\nabla\tau_a;\qquad \mathbf{u}^{0}_{ka}= \frac{-\mathbf{M}(\mathbf{u}_{k-1,a})+\mathbf{L}(\mathbf{u}_{k-2,a})}{\lambda+\mu}\,a^2; \tag{3} \]
\[ (\lambda+2\mu)\bigl(2\nabla\varphi_k\nabla\tau_a+\varphi_k\Delta\tau_a\bigr) +\bigl(\mathbf{M}(\mathbf{u}_{ka})-\mathbf{L}(\mathbf{u}_{k-1,a})\bigr)\nabla\tau_a=0, \tag{4} \]
\[ \mathbf{u}_{-1}=\mathbf{u}_{-2}=0, \]
where \(\lambda\) and \(\mu\) are the Lamé parameters,
\[ \mathbf{L}(\mathbf{u})=(\lambda+\mu)\nabla(\nabla\mathbf{u})+\mu\Delta\mathbf{u}. \]
\(\mathbf{M}\) is the operator defined in \((^{2})\).
A transverse ray solution is the series
\[ \mathbf{u}_b=\sum_{k=0}^{+\infty}\mathbf{u}_{kb}(x,y,z)\,f_k\bigl(t-\tau_b(x,y,z)\bigr), \tag{5} \]
where
\[ (\nabla\tau_b)^2=\frac{1}{b^2},\qquad b^2=\frac{\mu}{\rho}; \tag{6} \]
* In the present paper only the case of a homogeneous medium is considered. By the same method one could also consider the case of an inhomogeneous elastic medium.
\[ \mathbf u_{kb}=\mathbf u^0_{kb}+\mathbf u^1_{kb};\qquad \mathbf u^0_{kb}\parallel \nabla \tau_b;\qquad \mathbf u^1_{kb}\perp \nabla \tau_b;\qquad \mathbf u^0_{kb}=\frac{\mathbf M(\mathbf u_{k-1,b})-\mathbf L(\mathbf u_{k-2,b})}{\lambda+\mu}\,b^2, \tag{7} \]
the role of condition (4) in this case is played by the requirement that the component of the vector
\[ \mathbf M(\mathbf u^0_{kb}+\mathbf u^1_{kb})-\mathbf L(\mathbf u_{k-1,b}), \tag{8} \]
perpendicular to \(\nabla \tau_b\), vanish. If the series (2) and (5) admit termwise differentiation twice, then the ray solutions satisfy the equations of elasticity theory.
\(3^\circ\). Let the elastic body be bounded by an analytic surface \(S\). Take for \(\tau_a\) and \(\tau_b\) solutions of equations (2) and (6) satisfying the conditions
\[ \tau_a\big|_S=\tau_b\big|_S=\tau_c;\qquad \operatorname{Im}\frac{\partial \tau_a}{\partial \nu}\bigg|_S<0;\qquad \operatorname{Im}\frac{\partial \tau_b}{\partial \nu}\bigg|_S<0; \tag{9} \]
\[ \operatorname{Im}\tau_c=0,\qquad \nabla(\tau_c,\tau_c)=\frac{1}{c^2}. \tag{10} \]
Here \(\nu\) is the inward normal, \(\nabla(\tau_c,\tau_c)\) is the first differential parameter of the function \(\tau_c\) on the surface \(S\), reducing in the plane case to the square of the derivative with respect to arc length; \(c\) is the velocity of Rayleigh waves.
Obviously,
\[ \frac{\partial \tau_a}{\partial \nu}=(-i)\sqrt{\frac{1}{c^2}-\frac{1}{a^2}},\qquad \frac{\partial \tau_b}{\partial \nu}=(-i)\sqrt{\frac{1}{c^2}-\frac{1}{b^2}}. \tag{11} \]
Introduce on the surface a “semigeodesic” coordinate system \((^4)\): through each point of a fixed analytic curve \(\mathbf r=\mathbf r(\alpha)\) on the surface, draw the geodesic line perpendicular to it. Any point \(M\) on the surface near the curve \(\mathbf r=\mathbf r(\alpha)\) is characterized by two parameters: \(\tau=s/c\), where \(s\) is the arc length of the geodesic from its intersection with the curve \(\mathbf r=\mathbf r(\alpha)\) to \(M\), and \(\alpha\), characterizing this point of intersection. Points \(N\) outside the surface \(S\) are conveniently characterized by the quantity \(\nu\)—the distance along the normal from \(N\) to \(S\)—and by the parameters \(\alpha,\tau\), characterizing the point \(M\) of intersection of the normal passing through \(N\) with the surface \(S\).
\(4^\circ\). It is convenient to seek the vector \(\mathbf u^1_{kb}\) in the form
\[ \mathbf u^1_{kb}=\psi_k\vec{\xi}+\psi_{k\eta}\vec{\eta}, \tag{12} \]
where
\[ \vec{\xi}=-\frac{1}{b\sqrt{1-b^2(\partial \tau_b/\partial \nu)^2}}\left(\vec{\nu}^{\,0}-b^2\nabla\tau_b(\nabla\tau_b\vec{\nu}^{\,0})\right), \tag{13} \]
\[ \vec{\eta}=\vec{\alpha}^{\,0}-b^2\nabla\tau_b(\nabla\tau_b\vec{\alpha}^{\,0}), \tag{14} \]
where \(\vec{\nu}^{\,0}\) (respectively \(\vec{\alpha}^{\,0}\)) is the vector which has, in the curvilinear coordinate system \(\tau,\alpha,\nu\), contravariant components \((0,0,1)\) (respectively \((0,1,0)\)).
Obviously, \(\vec{\xi}\) and \(\vec{\eta}\) are orthogonal to \(\nabla\tau_b\), and the length of \(\vec{\xi}\) is \(1/b\). The condition that vector (8) be parallel to \(\nabla\tau_b\) at the points of the surface \(S\) is equivalent to the equalities
\[ 2\nabla\psi_k\nabla\tau_b+\psi_k\Delta\tau_b-\frac{b_{\alpha\tau}}{c}\psi_{k\eta} +\frac{1}{\rho}\bigl(\mathbf M(\mathbf u^0_{kb})-\mathbf L(\mathbf u_{k-1,b})\bigr)\vec{\xi}=0, \tag{15} \]
\[ 2\nabla\psi_{k\eta}\nabla\tau_b+\psi_{k\eta}\Delta\tau_b+\psi_k\frac{\sqrt{g_{\alpha\alpha}}}{c^2} \left(\frac{\partial\ln\sqrt{g_{\alpha\alpha}}}{\partial\tau}-b_{\alpha\tau}\frac{\partial\tau_b}{\partial\nu}\right) + \]
\[ +\frac{c}{b^2}\frac{b_{\alpha\tau}\psi_k}{\sqrt{g_{\alpha\alpha}}c^2} +\frac{1}{\rho}\bigl(\mathbf M(\mathbf u^0_{kb})-\mathbf L(\mathbf u_{k-1,b})\bigr)\vec{\alpha}^{\,0}=0, \tag{16} \]
where \(\rho\) is the density. Here and below by \(g_{\alpha\alpha}, g_{\alpha\tau}, g_{\tau\tau}\) (respectively \(b_{\alpha\alpha}, b_{\alpha\tau}, b_{\tau\tau}\)) we denote the coefficients of the first (respectively second) quadratic form of Gauss.
5°. Require that the vector \(\mathbf u=\mathbf u_a+\mathbf u_b\) satisfy, on the surface \(S\), the condition of absence of stresses:
\[ -\lambda(\mathbf u_{k+1,a}\nabla\tau_a+\mathbf u_{k+1,b}\nabla\tau_b) -2\mu\left(u^\nu_{k+1,a}\frac{\partial\tau_a}{\partial\nu} +u^\nu_{k+1,b}\frac{\partial\tau_b}{\partial\nu}\right) +\lambda(\nabla\mathbf u_{ka}+\nabla\mathbf u_{kb}) +2\mu\left(\frac{\partial u^\nu_{ka}}{\partial\nu} +\frac{\partial u^k_{kb}}{\partial\nu} +\Gamma^\nu_{\nu\delta}u^\delta_{ka} +\Gamma^\nu_{\nu\delta}u^\delta_{kb}\right)=0, \tag{17} \]
\[ -u^\tau_{k+1,a}\frac{\partial\tau_a}{\partial\nu} -u^\tau_{k+1,b}\frac{\partial\tau_b}{\partial\nu} -u^\nu_{k+1,a}\frac{\partial\tau_a}{\partial\tau} -u^\nu_{k+1,b}\frac{\partial\tau_b}{\partial\nu} +\frac{\partial u^\tau_{ka}}{\partial\nu} +\Gamma^\tau_{\delta\nu}u^\delta_{ka} +\frac{\partial u^\tau_{kb}}{\partial\nu} +\Gamma^\tau_{\delta\nu}u^\delta_{kb} +\frac{1}{c^2}\left(\frac{\partial u^\nu_{ka}}{\partial\tau} +\Gamma^\nu_{\tau\delta}u^\delta_{ka} +\frac{\partial u^\nu_{ka}}{\partial\tau} +\Gamma^\nu_{\tau\delta}u^\delta_{kb}\right)=0 \]
(the third equation is not written out for lack of space; \(\Gamma^\nu_{\tau\delta}\) are the Christoffel symbols of the coordinate system \(\tau,\alpha,\nu\); summation over repeated indices is understood; \(u^\tau,u^\nu\) are the contravariant components of the displacement vector). Substituting in (17), instead of \(\mathbf u_{k+1}\) and \(\mathbf u_k\), their expressions from formulas (3), (7), (12), we obtain, for finding \(\varphi_{k+1},\psi_{k+1}\), a system of equations with determinant equal to zero (\(\psi_{k+1,n}\) is determined uniquely through \(u^0_{k+1}\), \(u_k\) from the third equation, not written out here, and the determinant is equal to zero by virtue of Rayleigh’s equation). From the analogous equation for \(\varphi_k,\psi_k\) it follows that \(\varphi_k\) and \(\psi_k\) are determined up to terms:
\[ \overline{\varphi}_k=e\chi_k,\qquad \overline{\psi}_k=e_2\chi_k,\qquad e_1=\frac{1}{b^2}-\frac{2}{c^2},\qquad e_2=\frac{2i}{c}\sqrt{\frac{1}{c^2}-\frac{1}{a^2}}. \]
Substitute into relation (15)
\[ \varphi_k=\varphi^0_k+e_1\chi_k,\qquad \psi_k=\psi^0_k+e_2\chi_k \]
(\(\varphi^0_k,\psi^0_k\) are any values of \(\varphi_k\) and \(\psi_k\) found from equations analogous to (17), where \(k\) has been replaced by \(k-1\)). Multiplying the first equation (17) by \(l_1\), the second by \(l_2\), where
\[ l_1=\frac{1}{\mu}\left(\frac{1}{b^2}-\frac{2}{c^2}\right),\qquad l_2=-2i\sqrt{\frac{1}{c^2}-\frac{1}{b^2}}, \]
and adding, we obtain the necessary and sufficient conditions for solvability. They have the form
\[ A_1\chi_k+A_2\frac{d\chi_k}{ds} +A_3\frac{\partial\varphi_k}{\partial\nu} +A_4\frac{\partial\psi_k}{\partial\nu}+\ldots=0 \]
(the coefficients \(A_i\) are not written out for lack of space).
By dots are denoted terms depending on \(\varphi^0_k,\psi^0_k,u^a_{ka},u^0_{kb},\mathbf u_{k-1,a},\mathbf u_{k-1,b}\). Substituting here the values of \(\partial\varphi_k/\partial\nu\) and \(\partial\psi_k/\partial\nu\) from formulas (4) and (13), we obtain an ordinary differential equation for \(\chi_k\).
Let us write out this equation in the most important case \(k=0\):
\[ A\frac{d\chi_0}{ds} +\left( B-\frac{d\ln g_{\alpha\alpha}}{ds} +C\frac{b_{\tau\tau}}{c^2} +D\frac{b_{\alpha\alpha}}{g_{\alpha\alpha}} \right)\chi_0=0. \tag{18} \]
Here
\[ A=-\frac{8}{c}\left(\frac{1}{b^2}-\frac{2}{c^2}\right)^2 +\frac{4}{c}\left(\frac{1}{b^2}-\frac{2}{c^2}\right) \left(\frac{1}{a^2}-\frac{2}{c^2}\right) \frac{\sqrt{1/c^2-1/b^2}}{\sqrt{1/c^2-1/a^2}} +\frac{4}{c}\left(\frac{1}{b^2}-\frac{2}{c^2}\right)^2 \frac{\sqrt{1/c^2-1/a^2}}{\sqrt{1/c^2-1/a^2}}, \qquad B=\frac{1}{4}A; \tag{19} \]
\[ C=\left(\frac{1}{b^2}-\frac{2}{c^2}\right) \frac{C_1\partial\tau_a/\partial\nu+C_2\partial\tau_b/\partial\nu} {(1/c^2-1/a^2)(1/c^2-1/b^2)}, \tag{20} \]
where
\[ C_1=2\left(\frac{1}{c^2}-\frac{1}{a^2}\right) \left(\frac{1}{c^2b^2}-\frac{1}{b^4}-\frac{1}{c^4}\right), \]
\[ C_2=\left(\frac{1}{b^2}-\frac{2}{c^2}\right) \left(\frac{1}{c^2}-\frac{1}{a^2}\right) \left(\frac{1}{c^2}-\frac{3}{2b^2}\right) +\frac{2}{c^2}\left(\frac{1}{c^2}-\frac{1}{b^2}\right) \left(\frac{2}{c^2}-\frac{3}{a^2}\right), \tag{21} \]
\[ D=\left(\frac{1}{b^2}-\frac{2}{c^2}\right)\cdot2\cdot \left(\frac{1}{b^2}-\frac{b^2}{c^4}\right)\frac{\partial\tau_a}{\partial\nu} +\left(\frac{1}{b^2}-\frac{2}{c^2}\right)\frac{b^2}{2} \left(\frac{1}{b^4}+\frac{4}{c^4}\right)\frac{\partial\tau_b}{\partial\nu}. \]
It is not difficult to show that \(A>0\). The coefficients \(C\) and \(D\) are purely imaginary, and therefore
\[ |\chi_0(M)|=|\chi_0(M_0)|\sqrt{\frac{J(M_0)}{J(M)}}, \tag{22} \]
where \(J(M)=\sqrt{g_{\alpha\alpha}}\) are the coefficients of divergence of the surface rays, i.e., of the geodesic lines. It is natural to call \(|\chi_0(M)|\) the intensity of the Rayleigh waves. Formula (22) is analogous to the usual formulas of the ray method \((^{2,3})\).
In the plane case the terms with \(B\) and \(D\) drop out, \(b_{\tau\tau}/c^2\) should be replaced by \(1/R\), where \(R\) is the radius of curvature of the boundary. It follows from formula (22) that in the plane case the intensity of the Rayleigh wave does not change in the course of its propagation, as was naturally to be expected.
Leningrad State University
named after A. A. Zhdanov
Received
2 XI 1960
CITED LITERATURE
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- P. K. Rashevskii, Course of Differential Geometry, ch. VII, Moscow, 1956.
- I. G. Petrovskii, DAN, 47, No. 4 (1945).
- J. B. Keller, F. C. Karal, Excitation and Propagation of Surface Waves, Inst. of Math. Sci. Div. EM Res. Research Report NEM-128, Febr. 1959.