We construct microlocal solutions of Rayleigh and Stoneley waves in isotropic linear elasticity with the density and the Lamé parameters smooth up to a curved boundary or interface. We compute the direction of the microlocal polarization and show a retrograde elliptical motion of these two type of waves.
Rayleigh waves in linear elasticity are a type of surface waves. They are first studied by Lord Rayleigh in [16] and can be the most destructive waves in an earthquake. They propagate along a traction-free boundary and decay rapidly into the media. By the geophysical literatures, Rayleigh waves have a retrograde elliptical particle motion for shallow depth in the case of flat boundary and homogeneous media, see [14,17]. Stoneley waves are a type of interface waves that propagate along the interface between two different solids. They are first predicted in [24]. Roughly speaking, Rayleigh waves can be regarded as a special (limit) case of Stoneley waves. Both geophysical and mathematical works have been done for these two kinds of waves, see [1,2,2,4,7–10,13,14,16,18,19,22,23,26,28,31] and their references. Most geophysical works on them are considering specific situations, for example, the case of flat boundaries, plane waves, or homogeneous media. The propagation phenomenon of Rayleigh waves in an isotropic elastic system is first studied by Michael Taylor in [26] from a microlocal analysis point of view. Kazuhiro Yamamoto in [31] shows the existence of Stoneley waves as the propagation of singularities in two isotopic media with smooth arbitrary interfaces. Sönke Hansen in [8] derives the Rayleigh quasimodes by the spectral factorization methods for inhomogeneous anisotropic media with curved boundary and then in [10] shows the existence of Rayleigh waves by giving ray series asymptotic expansions in the same setting. In particular, the author derives the transport equation satisfied by the leading amplitude which represents the term of highest frequency. In [3], the authors develop a semiclassical analysis of elastic surface wave generated by interior (point) source and in [4] the inverse spectral problem of Rayleigh waves is studied. Most recently in [21], the authors describe the microlocal behavior of solutions to the transmission problems in isotropic elasticity with curved interfaces. Surface waves are briefly mentioned there as possible solutions of evanescent type which propagate on the boundary, see [21, Section 8.2].
In this work, we describe the microlocal behaviors of Rayleigh waves and Stoneley waves for an isotropic elastic linear system with variable coefficients and a curved boundary or interface. We construct the microlocal solutions of these two waves and compute the direction of their polarization explicitly. The magnitude of the polarization depends on an amplitude constructed by the geometric optics, which satisfies a transport equation. We explain how to compute the zero order term involved in the transport equation in Appendix A. In particular, we show a retrograde elliptical particle motion in both waves as an analog to the flat case. Essentially, the existence of Rayleigh waves comes from the nonempty kernel of the principal symbol of the Dirichlet-to-Neumann map (DN map) Λ in the elliptic region. In Section 2, we briefly state some relevant results in [21] that we are going to use later. In Section 3, based on the analysis in [21], one can see the Rayleigh waves are corresponding to the solution to on the boundary, where l is a source microlocally supported in the elliptic region. Next, inspired by the diagonalization of the Neumann operator for the case of constant coefficients in [19], we diagonalize Λ microlocally up to smoothing operators by a symbol construction in [27]. Note that the DN map Λ is a pseudodifferential operator with a matrix-valued principal symbol , see (3). One can see this diagonalization is global and it gives us a system of one hyperbolic equation and two elliptic equations on the boundary with some metric. The solution to this system applied by a ΨDO of order zero serves as the Dirichlet boundary condition on the timelike boundary of the elastic system. Then we construct Rayleigh waves in the same way as we construct parametrices for elliptic evolution equations, as the boundary data is microlocally supported in the elliptic region. The wave front set and the direction of the microlocal polarization of the Rayleigh waves on the boundary can be derived during the procedure and they explain the propagation of Rayleigh waves and show a retrograde elliptical particle motion. All these results are based on the diagonalization of the DN map. More specifically, in Section 3.3, we derive microlocal Rayleigh waves on the boundary if we have the Cauchy data at . The polarization is given in Theorem 3.1 and the leading term shows a retrograde elliptical motion of the particles, same as that of the case of homogeneous media in [14,17], as is explained after Theorem 3.1. In Section 3.4, the inhomogeneous problem, i.e., when there is a source on the boundary, is studied and the microlocal solution and the polarization on the boundary are presented in Theorem 3.2. In the second part of this work, Stoneley waves are analyzed in a similar way with a more complicated system on the interface. We construct the microlocal solutions without justifying the parametrix. The parametrices are verified in [6] and therefore the exact solution has the same microlocal behavior as the solution we construct. The main results of Stoneley waves for Cauchy problems is in Theorem 4.1 and for inhomogeneous problems is in Theorem 4.2. The microlocal Stoneley waves derived there have similar patterns as that of Rayleigh waves and one can see the leading term shows a retrograde elliptical motion of the particles as well.
Preliminaries
Suppose is a bounded domain with a smooth boundary . Let the density ρ and the Lamé parameters be smooth functions up to , depending on the space variable x, and even the time t.
In this section, we recall some notations and results in [21]. For fixed , one can choose the semigeodesic coordinates such that is locally given by . For this reason, we view u as a one form and write the elastic system in the following invariant way in presence of a Riemannian metric g. Let ∇ be the covariant differential in Riemannian geometry. We define the symmetric differential and the divergence δ as
where u is a covector field and v is a symmetric covariant tensor field of order two. The stress tensor is given by
Then the operator E and the normal stress are
where is the outer unit normal vector on the boundary. The elastic wave equation can be written as
and near some fixed one can decouple this system up to smoothing operators as
where U is a ΨDO of order zero, is a matrix ΨDO of order one, is a scalar ΨDO of order one, the S wave and P wave speed are
The principal symbol of U locally is
when . There is an invariant way to choose , for more details see [21, Section 4].
Then let . The elastic system decouples into two wave equations with the S wave and P wave speed respectively. This decoupling indicates that the solution u has a natural decomposition into the S wave and P wave modes, see [21, Proposition 4.1].
The boundary value problem in the elliptic region
When we solve the boundary value problems for the elastic system, the construction of the microlocal solutions depends on in which region the wave front set of the boundary data belongs to. Note that we always consider the outgoing solutions, i.e., the solutions whose singularities propagates to the future only. The Rayleigh waves happen when there is a free boundary and the singularities of the boundary data are in the elliptic region.
In this case, given the boundary data , first we get by inverting , the restriction of U to the boundary, which maps to f. It is shown that is an elliptic ΨDO on and therefore it is microlocally invertible, see [21, Section 5.1]. Then we seek the outgoing microlocal solution w to the two wave equations with the boundary data . Since the wave front set of is in the elliptic region, the Eikonal equations have no real valued solutions. Instead, the microlocal solution is constructed by a complex valued phase function, see [21, Section 5.3] and Section 3.5 for more details. After we construct w, we have as the microlocal solution to the elastic system.
Rayleigh wave
The main goal of this section is to construct the microlocal solution of Rayleigh waves and to analyze their microlocal polarization. We follow Denker’s notation to denote the vector-valued distributions on a smooth manifold X with values in by . Similarly is the set of distributions with compact support in X with values in .
Suppose there is a source on the timelike boundary , only for a limited time . Let u be an outgoing solution to the boundary value problem with the inhomogeneous Neumann boundary condition, i.e.
where is microlocally supported in the elliptic region. Let Λ be the Dirichlet-to-Neumann map (DN map) which maps the Dirichlet boundary data to the Neumann boundary data . Then solving (1) microlocally is equivalent to solving the Dirichlet boundary value problem , with f being the outgoing solution on Γ to
In other words, as long as we solve f from (2), we can construct the microlocal outgoing solution u to (1) as an evanescent mode with a complex valued phase function. The construction of the microlocal solution to the boundary value problem with wave front set in the elliptic region is well studied in [21, Section 5.3] and also see Section 3.5, as is mentioned before. The main task of this section is to solve (2) microlocally and study the microlocal polarization of its solution.
In the elliptic region, the DN map Λ is a matrix ΨDO with principal symbolwhere
By [21], the operator that maps to is an elliptic ΨDO with principal symbol and the parametrix
The operator that maps to the Neumann boundary data is an ΨDO with principal symbol
We emphasize that the notation we use differs from that in [21, Section 6.1] by a scalar , see (6.1) there. Then the principal symbol of the DN map
can be computed and is given in (3). □
Diagonalization of the DN map
In the following, we first diagonalize in the sense of matrix diagonalization and then we microlocally decouple the system up to smoothing operators. By [19], first we have
where
Let
Notice that we always have the following equalities
Then we conclude that the principal symbol can be diagonalized as
by a matrix
where
with
More specifically, we compute that
which is an unitary matrix. Here are the eigenvalues of smoothly depending on . The eigenvalues of are given by
for . Notice that are always positive. It follows that only could be zero and this happens if and only if the determinant of the blocks in (5) equals zero, i.e.
Notice the elliptic region has two disconnected components . We consider the analysis for and the other case is similar. Define and let
Then (9) is equivalent to
It is well-known that at fixed point , there exists a unique simple zero satisfying for . This zero corresponds to a wave called the Rayleigh wave and it is called the Rayleigh speed. Rayleigh waves are first studied in [16]. Since is simple, i.e., , by the implicit function theorem we have the root of can be written as a smooth function near a small neighborhood of the fixed point. Then we can write as a product of and an elliptic factor, i.e.
where is nonzero and homogeneous in of order zero
The last equality is from (6), (9), and (11). There is a characteristic variety
corresponding to . In particular, by (6) along we have
In order to fully decouple the system up to smoothing operators, we would like to show the three eigenvalues are distinct. Notice this is not necessary in our situation, since with one can always decouple the system into a hyperbolic one and an elliptic system near .
Near, the eigenvaluesare distinct.
We already have . Additionally, one can show that is always true by the following calculation
where are defined in (10). The values of and might coincide but near they are separate, since is close to zero while has a positive lower bound. Therefore, near we have three distinct eigenvalues. □
Let be an elliptic ΨDO of order zero as constructed in [27] with the principal symbol equal to . Let the operators with symbol and with symbols , for . Near some fixed , the DN map Λ can be fully decoupled as
where are the lower order terms. If we define
in what follows, then the first entry in the first row can be written as .
Each entry of the matrix is homogeneous in of order 1 and that of is homogeneous of order 0. The operator has a homogeneous symbol, which implies its parametrix will have a classical one. After the diagonalization of the system, the operator have a classical symbol, and so does .
If the density ρ and the Lamé parameters are time-dependent, then depend on . Otherwise, the eigenvalues only depends on , and therefore we have and instead of the functions and operators above.
Now let
where is the component of any vector-valued distribution u for . Solving is microlocally equivalent to solving the following system
In the last two equations, the operators are elliptic so we have , for . The first equation is a first-order hyperbolic equation with lower order term.
Inhomogeneous hyperbolic equation of first order
For convenience, in this subsection we abuse the notation x instead of using . Here we suppose and remember in our case one has .
Let be an elliptic operator with a classical symbol smoothly depending on a parameter t and the lower term with a classical symbol. Moreover, assume the principal symbol of is real.
In this subsection we are solving the inhomogeneous hyperbolic equation
where with microsupport in the elliptic region.
Generally, the operator is not a ΨDO unless the principal symbol of λ is smooth in ξ at . However, since we only consider the elliptic region, we can always multiply it by a cutoff ΨDO whose microsupport is away from and this gives us a ΨDO. Therefore, following the theorem of propagation of singularities by Hörmander, we have if w is the solution to (17), where is given by the flow of , for a more explicit form see (30).
Homogeneous equations
We claim that the homogeneous first-order hyperbolic equation with an initial condition
has a microlocal solution
by the geometric optics construction, where we require and is a phase function that is smooth, real valued, homogeneous of order one in ξ with on the conic support of a. These assumptions guarantee the oscillatory integral (19) is a well-defined Lagrangian distribution.
More explicitly, the procedure presented below is based on the construction in [27, VIII.3]. If we suppose
then we have
where
have the asymptotic expansions according to the Fundamental Lemma. In the following we use the version given in [30].
[
30
, Chapter VI, Theorem 3.1] Supposeis a smooth, real-valued function forand the gradient. Supposeis a pseudodifferential operator of order m. Write, where. Then we have the asymptotic expansionwhere we use the notation
To compute for , we can write
which implies equals to the last term above. One can show is a polynomial w.r.t. ρ with degree . In particular, we have
From Lemma 1, we have the following asymptotic expansions of by writing , with . We use the notation and to have
and
Indeed for each fixed α, the order of each term in the asymptotic expansion of b is no more than , that of d is no more than .
To construct the microlocal solution, we are finding proper in form of , where is homogeneous in ξ of degree . We also write as asymptotic expansion such that
where we separate the term of order 1 since it gives us the eikonal equation
for the phase function. Then equating the zero order terms in ξ we have
where we set
be the vector field and
Then for lower order terms, i.e. , we have
where is expressible in terms of . This finishes the construction of microlocal solutions in (19).
This construction is valid in a small neighborhood of , since the Eikonal equation is locally solvable. However, we can find some such that the solution v is defined and use the value at as the Cauchy data to construct a new solution for , for the same arguments see [21, Section 3.1]
Inhomogeneous equations when
Now we are going to solve the inhomogeneous equation with zero initial condition. A simpler case would be when the lower order term vanishes, i.e.
In this way the microlocal solution can be obtained by the Duhamel’s principle. Indeed, let the phase function , the amplitude to be constructed for solutions to the homogeneous first order hyperbolic equation with an initial condition as in (18). More specifically, suppose the phase solves the eikonal equation (22) and the amplitude solves the transport equation (23) and (26) with . Let H be the Heaviside function, i.e., for and for . Then the solution to (27) up to a smooth error is given by
where we define as the solution operator to (27) with the phase φ and the amplitude a. Here, the kernel of
can be formally regarded as the product
of a conormal distribution and a Lagrangian distribution that we describe in the following claim. This description is a basic fact about FIOs.
The kernelofdefined by (
29
) is a Lagrangian distribution associated with the canonical relationIt follows thatis an FIO of order.
In Euclidean case, we have . Then
Further, we show that is a distribution kernel such that the microlocal solution (28) is well-defined for any supported in the elliptic region.
The kernelis a well-defined distribution, with the twisted wave front set satisfying, whereandis defined in (
30
). Ifmicrolocally supported in the elliptic region, thenis a distribution with.
We use [12, Theorem 8.2.10], with the assumption that the principal symbol of is homogeneous in ξ of order one. By [11, Theorem 2.5.14], for , since has no zero sections, we have
In particular, the first term in the right side is not ignorable in general. However, if we assume there is no , which is true if g is microlocally supported in the elliptic region, then for w satisfying equation (27), the wave front set . Especially for , we have . □
Inhomogeneous equations with nonzero
When the lower order term is nonzero, the Duhamel’s principle does not work any more. Instead, we can use the same iterative procedure as in [30, Section 5] to construct an operator such that
Here each has a classical symbol so does their sum. In particular, the principal symbol of is . The similar trick is performed for λ-pseudodifferential operators in [19].
In this way, the microlocal solution to the inhomogeneous hyperbolic equation can be written as
where is the solution operator for the inhomogeneous first order hyperbolic equation with zero initial condition. Since is an elliptic ΨDO with principal symbol equal to 1, we have the same conclusion for the wave front set of w as the simpler case.
Assumemicrolocally supported in the elliptic region. Then the inhomogeneous first-order hyperbolic equation (
17
) admits a unique microlocal solution given by (
33
), where the phase functionand the amplitudeare constructed for the operatorin (
31
), as in Section
3.2.1
. More specifically, the phasesolves the eikonal equation (
22
); the amplitudesolves the transport equation (
23
) and (
26
) with.
To justify the parametrix, we still need to show that if w is the solution to (17) with and , then as well. By (31) it suffices to show this is true when the lower order term can be reduced to the form or vanishes. One can verify that the operator is symmetric hyperbolic as is defined in [27]. Then following the same arguments there, by a standard hyperbolic estimates, one can show w is smooth. □
The Cauchy problem and the polarization
In this subsection, before assuming the source and solving the inhomogeneous equation (2) with zero initial condition, we first assume that the source exists for a limited time for and we have the Cauchy data at , i.e.
Recall the diagonalization of Λ in (15), (16). The homogeneous equation implies
where solves the homogeneous first-order hyperbolic equation in (16). Notice in this case the hyperbolic operator is with r given by (14). If we have the initial condition , then by the construction in Section 3.2.1 we have
where the phase function φ solves the eikonal equation (22); the amplitude solves the transport equations (23) and (26) with γ defined in (25) and .
To find out how the initial condition of f is related to that of h, we plug (36) into (35), use the Fundamental Lemma in Lemma 1, and set . Since , after these steps we get three ΨDOs of order zero, of which the symbols can be computed from the Fundamental Lemma, such that
where the principal symbols are
This indicates not any arbitrary initial conditions can be imposed for (34). Instead, to have a compatible system, we require that there exists some distribution such that can be written in form of (37).
Supposesatisfies (
37
) with some. Then microlocally the homogeneous problem with Cauchy data (
34
) admits a unique microlocal solutionwhereis the Rayleigh speed,are defined in (
10
), andis the amplitude from the geometric optics construction with the highest order term.
With f, one can construct the actual displacement u as an evanescent mode. Notice this theorem gives us a local representation of f in the sense of Remark 4.
To understand the polarization of microlocal solution in the theorem above, if we write the real and imaginary part of the term separately, then we have
where we assume . It follows that the solution in (38) can be regarded as a superposition of and . We are going to show that each of them has a retrograde elliptical motion in the following sense. Here by retrograde motion, we mean the rotational motion of a particle is in the direction opposite to the direction of its primary motion. We say it is elliptical, if the particle moves in an elliptic trajectories. For more details, see [17, Chapter 5.3].
Take the real part as an example, by assuming is real. Then the real part of the displacement on the boundary is the superposition of . On the one hand, we have the following equation satisfied by the components
which describes an ellipsoid.
On the other hand, the rotational motion of the particle has the direction opposite to its orbital motion, i.e. the direction of the propagation of singularities. Indeed, if we consider the Euclidean case and additionally assume for convenience. we have and . It follows that . Locally the singularities propagate along the path in the direction of on the boundary , where is the initial point. W.O.L.G. assume in what follows. In this case, we have
Notice is the angle between the vector , as the direction of strongest singularities of the solution, and the plane , see Fig. 1. At each fixed point, the polarized vector rotates in the direction that the angle increases as t increases. Consider the plane that is perpendicular to and contains with normal vector . The rotational motion is clockwise in this plane, while the direction of the propagation of singularities can be regarded as counterclockwise, as is the earth. Therefore, we have a retrograde motion.
Propagation of the singularities and the rotation of the microlocal polarization.
In the general case, notice by (23) the leading amplitude satisfies
where X and γ are defined in (24) and (25) with . In the following assume the Rayleigh speed only depends on the space variable and we write . Let
be the decomposition of the real and imaginary part. We emphasize that the imaginary part comes from that of , the lower order term in the decoupled system, and it is a classical symbol of order zero.
The integral curve of X is the unit speed geodesic . Here is the unit speed geodesic in Ω with the initial point and the initial direction . For simplification we omit and to write it as for the moment. Then the solution of the transport equation is
which implies and υ is a classical symbol of order zero. To find out at each fixed point how the polarization rotates as the time changes, we compute the time derivative
which is positive for large enough since is a symbol of order zero. This indicates for large the angle increases as t increases and the rotation of the polarization vector is still clockwise, if we assume at the fixed point points in the direction that increase.
In this case, the singularities propagates along the null characteristics of . Particularly, the singularities propagate along the geodesics , where is the initial point. By Remark 6, along the geodesics we have , which is exactly the opposite direction where the singularities propagate. Since we assume points in the direction that increase, the singularities propagates in the counterclockwise direction. Therefore, we have a retrograde elliptical motion as before.
Notice in (39), the phase function φ is always the dominant term for large . By (22), the phase function φ satisfies the Eikonal equation
By [20] or [29, Chapter XII Proposition 3.1], we solve it locally using the method of characteristics. Let be the Hamiltonian. First we find the Hamiltonian curves by solving the system
where s is the parameter and we set . This system corresponds to the unit speed geodesic flow
where .
The inhomogeneous problem
In this subsection, we solve the inhomogeneous problem (2). We apply Proposition 3 to the first equation in (16) with zero initial condition. Recall is defined in (15) and in (12). Then the first equation with zero initial condition has a unique microlocal solution
where the phase function and the amplitude are given by Proposition 3 with the hyperbolic operator . We can also write the solution as by (32). Therefore we have the following theorem.
Supposemicrolocally supported in the elliptic region. The inhomogeneous system (
2
) with zero initial condition atadmits a unique microlocal solution
Note that the projection of the microsupport of is supported in . Since for is a matrix-valued ΨDO, so is that of . Similarly are microlocally supported in as well.
In the following, we are going to find the polarization of the microlocal solution (41) when it is away from the microsupport of . In other words, we only consider the solution when the source vanishes (mod ). For convenience, we can assume . With this assumption, the Heaviside function in is negligible, i.e.
where is the FIO defined in (3.2.2) associated with the canonical relation . Additionally, we have when . Then for , the microlocal solution (41) is
where . To find out the leading term of the amplitude of the solution, we need to find the leading term of the amplitude of , the solution to the following transport equation
Here the zero order term of the ΨDO is involved. The procedure of computing is in the Appendix A. Then by Lemma 1, the leading term is given by
Recall given in (8) and in (13). The leading term equals to
with
where we use the notation defined in (10) and combine (6), (9), (13). Further, taking the phase function into consideration, we have the real and the imaginary part of the integrand equal to
where we set and write . This implies a retrograde elliptical motion of f as we stated before.
Moreover, if we expand the term , then the microlocal solution in (42) can be written as
Similarly, by the Fundamental Lemma we compute the leading term in the amplitude
where the last equality comes from (6), (9), (13), (7). Therefore, we have the following theorem.
Assume everything in Theorem
3.2
. For, the displacement on the boundary equals to
This theorem describes the microlocal polarization up to lower order terms of the displacement of Rayleigh waves on the boundary, when there is a source microlocally supported in the elliptic region with compact support. Up to lower order terms, the displacement can also be regarded as a supposition of the real part and imaginary part of the leading term and each of them has a elliptical retrograde motion as we discussed before. Indeed, the leading term has a similar pattern of that in Theorem 3.1, except different scalar functions in each component. Additionally, one can see the compatibility condition (37) satisfied by the Cauchy data there actually means the Cauchy data can be regarded as produced by certain source, according to (42). We mention that one can compute the polarization set defined in [5] and it corresponds to the direction of the polarization that we have here.
The microlocal solution u
With the boundary displacement f given in Section 3.3 or 3.4, in this subsection we are going to construct the microlocal solution u to the elastic system (1) in the elliptic region. This can be done by solving the boundary value problem, as is mentioned in Section 2.1.
Recall in [21] with we have the decoupled system
where is a matrix-valued ΨDO and is a scalar one. The boundary value on is given by . For more details of U and see [21]. We construct the microlocal solution w to (45) by the same procedure of constructing parametrices to the elliptic boundary value problems, and then we get . One can follow the construction in [21, Section 3.2.3] by solving the Eikonal equations with a complex phase function or use the following steps based on the construction in [25, Section 7.12].
First, in the semi-geodesic coordinates , we have for , which enables us to regroup the above equations as
where refers to g restricted to and we abuse the notations to denote the new lower order terms. In the elliptic region, the operators and are self-adjoint operators with real positive symbols. Therefore they have a square root respectively with
Further, we apply the elliptic operators and respectively to the equations above, to have
where are the new lower order terms of order zero. Since is elliptic itself, one can write . The same works for . This implies
where Q is a matrix-valued ΨDO with
By [25, Section 7.12], we look for
where is constructed inductively. More specifically, the leading term is given by
For , we solve from
where is the remainder term related to . It is shown in [25] that for there exists such that
where C is a positive constant satisfying . Such C exists since there exists a conically compact neighborhood K containing in the elliptic region.
When f is given in (41), the parametrix we construct above should satisfy
In particular, it has the same microlocal behavior as the true solution to (1) by the justification of parametrices established in [26].
Flat case with constant coefficients
In this subsection, suppose the boundary Γ is flat given by with Euclidean metric. Suppose the parameters are constants.
In this case, by using the partition of unity, the elastic wave equation can be fully decoupled as
and the solution is , where
with as the solution to and , where
To solve , we perform the same procedure as before. By verifying , we have
where is given in (13) and we use the notations in (10) as before. Combining the microlocal solution (19) to the homogeneous hyperbolic equation and that of the inhomogeneous one with zero initial condition in (40), we have the solution to the inhomogeneous one with arbitrary initial condition
has the following solution
and one can show
by directly taking the Fourier transform. Since all ΨDOs involved here have symbols free of , then they are Fourier multipliers and we have
where given in (8) is unitary and therefore .
The last two equations after we diagonalize the DN map have the following solutions
Thus, the displacement on the boundary is given by .
In the following assume we have a time-periodic source
where p is a positive number and are constants. This gives us a line source periodic in t on the boundary. Furthermore, we assume . In this case . Since the amplitude , by (44), (47) the displacement away from the support of the source up to lower order terms equals to
If we choose the Cauchy data as the inverse Fourier transform of the distribution
then we have
which coincides with (77) and (78) in [14].
In this example assume we have , where are constants. This gives us a source supported at on the boundary. In this case, since and , by (44) the displacement for equals to
where
Stoneley waves
In this section, we assume Γ is an interface between two domains . Locally, Γ can be flatten as and suppose is the positive part. For the density and Lamé parameters, we have in and in , which are functions smooth up to Γ. Let be u restricted to .
Suppose there are no incoming waves but boundary sources microlocally supported in the elliptic region, i.e. we are finding the outgoing microlocal solution for the elastic equation with transmission conditions
where denote the jump of v from the positive side to the negative side across the surface Γ. By (9.2) in [21], with no incoming waves the transmission conditions can be written in the form of
if we set
This implies that if we can solve from (49), then the solution to (48) can be solved by constructing microlocal outgoing solutions to the boundary value problems with Dirichlet b.c. in respectively. Since has positive sign in and negative sign in , to have evanescent modes in both domain, we choose with opposite signs
where
Then the principal symbols are
where
To solve (49), first we multiply the equation by an invertible matrix to have
In the following, first we solve from
microlocally and then we have . This gives the microlocal outgoing solution to (49).
Diagonalization of
Recall the calculation before, the principal symbol of the DN map can be partially diagonalized by the matrix . By the same trick, we have
where
Let as before. We follow the similar argument as in [2] to show the matrix M has two distinct eigenvalues and only one of them could be nonpositive for . Define the following functions of s
which are related to . Set
and it follows the matrix M can be represented by
Note M is a Hermitian matrix and it has two eigenvalues . The eigenvalues vanish if and only if
is satisfied for some , where
Notice the factor is always positive and therefore it is equivalent to . If such exists, it corresponds to the propagation speed of the so called Stoneley waves, first proposed in [24], see also [31, Lemma 3.1]. We call the Stoneley speed and it is a simple zero by Proposition 4 in the following. The following proposition of uniqueness of Stoneley waves is proved by the definiteness of and first appears in [7] and then is used in [2]. Here we present it in our notations.
For, the eigenvaluesdecrease as s increases. Only one of them can be zero and we denote this eigenvalue by. This happens when there is somesuch that. In particular, if suchexists, it is unique and is a simple zero of.
We claim the matrix and their transposes satisfy
the limit exist and are positive definite,
for , the derivative is negative definite,
for , the trace is always positive.
Then M satisfy these conditions as well, which indicates its eigenvalues decrease as s increases but their sum is always positive, i.e. at most one of them could be zero. The monotonic decreasing of eigenvalues implies the zero should be a simple one.
Now we prove the claim. For (a), we compute
Assuming , we have . Then and .
To prove (b), for convenience we change the variable and with . Then it is sufficient to show is negative definite. Indeed, we have
where we set
Then
and
Therefore, the determinant and the trace of are
which indicates is negative definite. For (c), obviously we have
□
If for all , then is an elliptic ΨDO and the microlocal solution to (50) is . The singularities do not propagate. Otherwise, if there exists such that , then (49) has a nontrivial microlocal solution that propagates singularities, analogously to the case of Rayleigh waves.
In the following, we always assume there exists a such that . By Proposition 4, this zero is simple so by the implicit function theorem it is a smooth function in a small neighborhood of a fixed point. This time the eigenvalues of the system is simply for . Then we can write as a product similar to what we have before
where
To decouple the system as what we did in Section 3, we would like to show the following claim. Notice even without this claim, the analysis still holds since with only vanishing the last two eigenvalues always give us an elliptic 2 by 2 system.
The three eigenvaluesof the matrixare distinct near.
Obviously near we have and . The values of may coincide but near they are separate by the following calculation
where the last inequality holds, since by (51) we have and at . □
To diagonalize , this time we define
where is the entry of M in (52) and
for . We define
Let the operators with symbol in (53) and with symbols , for . By [27], there exists an elliptic ΨDO of order zero with the principal symbol equal to , such that near some fixed , the operator can be fully decoupled as
where are the lower order term. If we define
in what follows, then the first entry in the first row can be written as
The Cauchy problem and the polarization
In this subsection, similar to Section 3.3, we first assume that the source exists for a limited time for and we have the Cauchy data at , i.e.
Recall the diagonalization of before. Let
The homogeneous equation implies
where solves the homogeneous first-order hyperbolic equation in (58). Notice in this case the hyperbolic operator is with r given by (57). If we have the initial condition , then by the construction in Section 3.2.1 we have
where the phase function φ solves the eikonal equation (22), and the amplitude solves the transport equation (23) and (26) with γ defined in (25) with .
By the same analysis before, the initial condition of is related to that of by
where the principal symbols are
We have the following theorem as an analog to Theorem 3.1.
Supposesatisfies (
62
) with some. Then microlocally the homogeneous problem with Cauchy data (
59
) admits a unique microlocal solutionwhere we defineas smooth functions w.r.t., with the Stoneley speedanddefined in (
51
),defined in (
55
).
This theorem describes the microlocal polarization of the displacement of a Stoneley wave on the intersurface. Up to lower order terms, the displacement can also be regarded as a supposition of and , each of which has a elliptical retrograde motion as we discussed before. Indeed, the leading term of f has a similar pattern of that of Rayleigh waves in Theorem (3.1), except different scalar functions in each component.
The inhomogeneous problem
In this subsection, first we are going to microlocally solve the inhomogeneous problem
and then the microlocal solution to (49) and (48) can be derived as we stated before. We perform the same analysis as that for the Rayleigh waves in the previous section.
Let
where is the component of any vector valued distribution u for . Solving is microlocally equivalent to solving the following system
In the last two equations, the operators are elliptic so we have for . The first equation is a first-order inhomogeneous hyperbolic equation with lower order term, which can be solved by Duhamel’s principle. We apply Proposition 3 to have
where the phase function and the amplitude are given by Proposition 3 with the hyperbolic operator being ; and is defined in (15) for the new hyperbolic operator. We also write the solution as by (32). This proves the following theorem, as an analog to Theorem 3.2, 3.3.
Supposemicrolocally supported in the elliptic region. The inhomogeneous system (
64
) with zero initial condition atadmits a unique microlocal solutionwherehas the principal symbol in (
56
),defined in (
53
), andis constructed as in (
31
). Then the microlocal solution to the transmission problem (
48
) can be constructed as evanescent modes from the boundary valueand. In particular, for, the displacement on the boundary has the leading termwhere we defineas smooth functions w.r.t.withas the Stoneley speed,as the eignvalue of M in Proposition
4
,defined in (
55
).
With given above and , we can construct the solution satisfying (48), as is discussed in Section 3.5.
Footnotes
The principal symbol of r
In this section, we explain how to compute the principal symbol of r, the lower order term in the hyperbolic equation in Section 3. It determines the transport equation satisfied by the amplitude that we constructed there.
By (14) we have , where is the lower order term in the fully decoupled system. Recall the decoupling procedure in [27] and the diagonalization of Λ in Section 3. First we have
where is the lower order term given by the asymptotic expansion of the product of ΨDOs, see (69). We denote the entries of R by , for and will compute them in the following. One can see only the entry is involved in the principal symbol of r. Then we decouple the system further by finding a in form of
such that
and the off diagonal terms of vanish. There exists a unique solution for since all eigenvalues are distinct. After this step, the diagonal terms in R remains and they form the diagonal of the zero order terms of the decoupled system. We introduce the notations and for a matrix-valued symbol in the following. Then the principal symbol of R are the second highest order term of the full symbol
where the product of symbols are multiplication of matrices. Observe that any terms that are multiples of vanish along . It follows that
where we use to denote the symbol of second highest order. The third term depends on , which can be computed from Appendix B.
Additionally, since we write in the principal symbol level, there is an extra term in besides . Let . Then we have
and therefore
The second highest order term of σ ( Λ )
In this section, we explain how to compute , the second highest order term of the symbol . We show in the following that when the boundary is flat and the coefficients are constant (the flat case), we have . More generally, the full symbol of Λ in a compact Riemannian manifold with smooth boundary and constant Lamé parameters can be found in [15].
First, recall is constructed from the its boundary value by the geometric optics construction with a complex phase function and an matrix-valued amplitude in [21]. One can regard as the result of applying the solution operator, an FIO with a complex phase, to . In the flat case, especially we have the exact solution with the amplitude , see (46).
Then recall in [21] we have
where the Neumann operator N, U are defined and computed there. It follows that can be computed by
where can be computed by the asymptotic expansion of the composition of a ΨDO with an FIO. Indeed, by the Fundamental Lemma for the complex phase in [30], one can see this composition is actually a ΨDO if restricted on the boundary . More specifically, one can derive
where can be derived from the geometric optics construction. Similarly we have
We note that and are involved in above computation. The latter can be found in [21] and can be computed from the procedure of fully decoupling the operator E. Additionally, the second term in multiplied by on the left will vanish along in the first row, which implies it has no contribution to .
Notice in the flat case, we have are homogeneous in of order 1 and . Therefore in this case .
Acknowledgements
The author would like to thank Prof. Plamen Stefanov for suggesting this problem and for lots of helpful discussions with him throughout this project, and to thank Prof. Vitaly Katsnelson, Prof. Mark Williams for helpful suggestions on the transmission problems part.
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