# Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

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## Abstract

Elastic bodies with cuspidal singularities at the surface are known to support wave processes in a finite volume which lead to absorption of elastic and acoustic oscillations (this effect is recognized in the engineering literature as Vibration Black Holes). With a simple argument we will give the complete description of the phenomenon of wave propagation towards the tip of a three-dimensional cusp and provide a formulation of the Mandelstam (energy) radiation conditions based on the calculation of the Umov-Poynting vector of energy transfer. Outside thresholds, in particular, above the lower bound of the continuous spectrum, these conditions coincide with ones due to the traditional Sommerfeld principle but also work at the threshold frequencies where the latter principle cannot indicate the direction of wave propagation. The energy radiation conditions supply the problem with a Fredholm operator of index zero so that a solution is determined up to a trapped mode with a finite elastic energy and exists provided the external loading is orthogonal to these modes. An example of a symmetric cuspidal body is presented which supports an infinite sequence of eigenfrequencies embedded into the continuous spectrum and the corresponding trapped modes with the exponential decay at the cusp tip. We also determine (unitary and symmetric) scattering matrices of two types and derive a criterion for the existence of a trapped mode with the power-low behavior near the tip.

## Keywords

Vibration Black Holes Cuspidal singularity Elastic waves Energy radiation conditions Trapped modes Fredholm operator Weighted spaces with detached asymptotics## Mathematics Subject Classification

35B40 74J20 35C20 35C07 35P05## 1 Introduction

### 1.1 Formulation of the Problem

^{1}Fig. 2.

### 1.2 The Mandel-Voigt Notation

### 1.3 Energy Space and Korn’s Inequality

### 1.4 Structure of the Paper

Formal asymptotic expansions of solutions to the problem (1.10), (1.11) are derived in Sect. 2 while the dimension reduction procedure leads to the limit system of ordinary differential equations of Euler’s type in the longitudinal coordinate \(z\). These expansions have been justified in the paper [7] but Theorem 1 in Sect. 3 provides a reformulated result which crucially simplifies all further considerations, namely we prove that the formal asymptotic expansion gives rise to certain three-dimensional elastic fields which entirely satisfy the problem (1.10), (1.11) in the \(d^{\prime}\)-neighborhood of the cusp tip \(\mathcal {O}\) with some \(d^{\prime}\in(0,d)\) and have power-law solutions of the limit system as the main asymptotic terms. It is important that any solution of the whole problem with a slow exponential growth can be represented as a linear combination of these solutions and a remainder which decays exponentially as \(x\to{\mathcal{O}}\).

Furthermore, in Sect. 3 we classify the above-mentioned solutions and pay the most attention to those who have the singularities \(z^{\pm i \kappa-5/2}\) with certain \(\kappa\in{\mathbb{R}}\), are called oscillatory waves, and transport elastic energy along the cusp (1.1). Based on the Mandelstam radiation conditions, we moreover classify the waves as incoming from and outgoing to the tip \(\mathcal {O}\).

In Sect. 4 we proceed with computing the Umov-Poynting vector of the elastic energy transfer which implies the symplectic form \(q\), see (4.4), indicates the direction of wave propagation and supports the above-mentioned Mandelstam classification. Moreover, the form \(q\) allows us to normalize the incoming and outgoing waves, cf., Sect. 4.3, and, as a result, to introduce the unitary and symmetric scattering matrix \(S\). In a similar way, we introduce the packets of the energy (\(\kappa>0\)) and non-energy (\(\kappa<0\)) waves with the singularities \(z^{\kappa-5/2}\) which involve a real exponent \(\kappa\) and make the total energy functional finite and infinite, respectively. These packets generate the augmented scattering matrix \(\mathfrak{S}\). It is an artificial object but imitates the classical scattering matrix, inherits its natural unitary and symmetry properties, and plays the most important role in our description of elastic trapped modes in the solid \(\varOmega\).

In Sect. 5.2 we prove that in the rotational symmetric isotropic bodies the problem (1.10), (1.11) have an infinite unbounded sequence of eigenvalues while the corresponding elastic eigenmodes enjoy the exponential decay as \(x\to{\mathcal{O}}\). It is remarkable that an infinite part of this sequence belongs to the continuous spectrum (1.5), i.e., we present an example of an unbounded family of embedded eigenvalues.

In Sect. 5.3 we determine the augmented scattering matrix \(\mathfrak{S}\) and verify its properties. It gives in Theorem 2 a criterion of the existence of trapped modes with the power-law behavior near the tip \(\mathcal {O}\). Moreover, a simple formula (5.41) reproduces the traditional scattering matrix \(S\) which describes the wave processes in the elastic body \(\varOmega\). Finally, in Sect. 5.5 we formulate the (energy) Mandelstam radiation conditions and provide a Fredholm operator of index zero for the problem (1.17), (1.18) in weighted spaces with detached asymptotics (Theorem 3).

It is worth mentioning that the detailed investigation in Sects. 2–4 of asymptotics inside the cusp allows us to reduce the mathematical tools in Sect. 5 to quite simple algebraic operations and does not require a complicated analysis of the original boundary value problem.

## 2 Formal Asymptotic Analysis

### 2.1 Preamble

### 2.2 Asymptotic Ansatz

### 2.3 Splitting Differential Operators

^{2}

### 2.4 Solving the Recurrent Sequence of Problems

As was shown in [1], see also [7, 17, 22] and [15, Sect. 7.3], the sequence of the problems (2.15) and their compatibility conditions lead to one-dimensional limit spectral ordinary differential equations in the variable \(z\). Let us implement this procedure.

### 2.5 The Limit System of Ordinary Differential Equations

### 2.6 Isotropic Homogeneous Elastic Material

## 3 Waves

### 3.1 Reduction of the Resulting System of Ordinary Differential Equations

### 3.2 Power-Logarithmic Solutions

All power-logarithmic solutions to the system (2.31) have been constructed.

### 3.3 Elastic Waves in the Cusp

### 3.4 Propagation of Waves

The above classification of waves (the outgoing wave \(\mathbf{u}^{j-}\) and the incoming wave \(\mathbf{u}^{j+}\)) is mainly based on the classical Sommerfeld radiation principle, see, e.g., [26, 27]. However, it is known (see [28] as well as [29, Ch. 1], [30] and other publications), that in elastic waveguides, even of cylindrical shape, direction of the wave propagation according to Sommerfeld can differ from the direction of the energy transfer because of the opposite sign of the group velocity. In this way, the limiting absorption principle [26, 27] is usually applied although it does not work at the thresholds of the continuous spectrum (see a discussion in [29, Ch. 1] and an example in [30, §5]). Clearly, the Sommerfeld radiation principle also does not serve for our threshold case \(\kappa=0\). Since we want to cover the whole continuous spectrum, in the next section we employ the universal approach which refers to the Umov-Poynting vector [31, 32] and the energy radiation conditions due to L. Mandelstam [33] (see also [29, Ch. 1] and [30]) and adapt them to wave propagation in cuspidal elastic solids. Furthermore, we will demonstrate in Sect. 5 that the corresponding radiation conditions guarantee a correct formulation of the problem in weighted spaces with detached asymptotics.

### 3.5 Notation and Terminology of Further Use

In the one-dimensional model we will call *oscillatory* the complex-valued waves (3.18) with \(\kappa>0\). In the case \(\kappa=0\) these waves become *standing* while waves with the components (3.12) and (3.14) are *resonant* due to presence of the additional growing factor \(\log z\); both the waves can be chosen real. Notice that we systematically ignore the factors \(z^{-5/2}\) in \(w_{\sharp}(z)\) and similar factors in \(w_{\bullet}(z)\) whose role is to provide the linear energy (3.20) on the cross-section \(\varpi(z)\) with the independence property discussed in Sect. 3.3. Without the weighting factors the introduced terminology is quite similar to the traditional terminology in cylindrical waveguides, cf. [26, 27, 29] and others.

This terminology also applies to the displacement fields \(u\) defined according to (2.2)–(2.4), (2.21) in the three-dimensional cusp (1.1). In this way, all above-mentioned waves make the elastic energy infinite in the intact set \(\varPi_{d}\), see the representation (3.22) with \(K\rightarrow+0\). The same property is attributed to waves initiated by the component (3.6) with \(\kappa<0\) but in the case \(\kappa>0\) the elastic energy computed in \(\varPi_{d}\) for the corresponding three-dimensional ansatz (2.2) becomes finite. The latter waves with \(\kappa>0\) and \(\kappa<0\) will be recognized as *energy* and *non-energy* waves.

In the paper [7] a procedure is described to construct the complete asymptotic series for these elastic fields. Moreover, according to Sect. 4.9 in [7], any solution \(w\) of the system (2.31) gives rise to the three-dimensional displacement vector \(u\) which has the sum (3.25) as the main asymptotic term and satisfies the differential equations (1.2) in \(\varPi_{d^{\prime}}\) and the boundary conditions (1.3) on \(\varGamma_{d^{\prime}}\) with some \(d^{\prime}\in(0,d]\) but of course no boundary condition is imposed on \(u\) at the cross-section \(\overline{\omega(d^{\prime})}= \partial\varPi_{d^{\prime}}\setminus\varGamma_{d^{\prime}}\).

### Theorem 1

*There exists a positive constant*\(\delta\)

*with the following property*.

*If a solution*\(u\)

*to the problem*(1.10), (1.11)

*restricted on the cusp*\(\varPi_{d}\)

*satisfies the condition*

*then this solution admits the asymptotic representation in the form*

*where*\(\widehat{u}^{k}\)

*are the vector functions*(3.29), (3.30),

*the remainder enjoys the decay property*

*and the coefficients in*(3.32)

*meet the estimate*

### Proof

Using variables \(\eta\) and \(t=z^{-1}\), we can write the problem (1.10), (1.11) restricted on the cusp \(\varPi_{d}\) as a variational problem with constant coefficients in a half-cylinder perturbed by an differential operator with coefficients vanishing at infinity, see Sect. 10.8 [8] for an abstract treatment of such variational problems. Now we want to apply Theorem 9.3.2 together with Remark 10.8.14 in [8]. In our case the spectrum of the operator pencil corresponding to the Neumann problem for the elasticity operator in the cylinder \(\varpi\times\mathbb{R}\) consists of eigenvalues of finite algebraic multiplicity and there is only one eigenvalue \(\lambda=0\) of multiplicity 12 on the imaginary axis (we note that the notation in [8] uses the additional factor \(i\) in the definition of eigenvalues). Moreover, there exist \(\delta>0\) such that there is no eigenvalues in the strip \(\{\lambda\in{\mathbb{C}}:| \operatorname{Im}\lambda|\leq\delta\}\) with exception of \(\lambda=0\). Using the notation from Theorem 9.3.2 [8], we can set \(k_{+}^{(1)}=k_{-}^{(2)}=0\), \(k_{-}^{(1)}=\delta\), \(k_{+}^{(2)}=- \delta\). Moreover, we observe that the space \({\mathcal{X}}(L)\) coincides with the linear combination of the solutions \(\widehat{u} ^{k}\) constructed above.^{3} Now the reference to Theorem 9.3.2 [8] proves our theorem. □

## 4 Energy Radiation Conditions

### 4.1 The Umov-Poynting Vector

### 4.2 Energy Radiation Conditions

### 4.3 Normalization of Oscillatory Waves

### 4.4 Logarithmic Packets of Standing and Resonance Waves

### 4.5 Packets of Non-energy Waves

The same structure (4.16) applies to the energy (\(\kappa>0\)) and non-energy (\(\kappa<0\)) waves (3.6), (3.12) generated by real roots of the quadratic equation (3.7) in the case of its positive right-hand side. Namely, fixing a power-law solution \(w^{j0}=(w^{j0} _{\sharp},w^{j0}_{\bullet})\) with some root \(\kappa_{j}>0\) we find its “companion” \(w^{j1}=(w^{j1}_{\sharp},w^{j1}_{\bullet})\) with the root \(-\kappa_{j}<0\) such that the relations (4.13) and (4.14) are valid. Now the definition (4.16) gives us the outgoing \(w^{j-}\) and incoming \(w^{j+}\) waves in the one-dimensional model which in turn produce the three-dimensional waves \(u^{j-}\) and \(u^{j+}\) in the cusp (1.1).

It should be mentioned that the packets (4.16) are artificial objects needed for intermediate calculations in the next section.

## 5 Solvability of the Elasticity Problem in Domains with Cusps

### 5.1 The Spectral Parameter is Below the Cutoff Value

### Proposition 1

*Let*\(F\in{\mathcal{W}}(\varOmega)^{\ast}\)

*be an anti*-

*linear continuous functional in the space*\(\mathcal{W}(\varOmega)\)

*such that*

*Then the problem*(5.1)

*with the parameter*\(\varLambda\in(0, \varLambda^{\dagger})\)

*has a solution*\(u\in{\mathcal{W}}(\varOmega)\)

*which is determined up to an addendum in*\(\mathcal{L}_{\varLambda}\),

*an eigenmode*.

*The orthogonality conditions*

*make the solution unique and assure the estimate*

*with a factor*\(c\)

*which depends on the domain*\(\varOmega\),

*the elastic moduli matrix*\(A\),

*and the spectral parameter*\(\varLambda\)

*but is independent of the functional*\(F\).

### 5.2 Embedded Eigenvalues and Trapped Modes

_{∢}and (1.11)

_{∢}the differential equations (1.10) and the boundary conditions (1.11) restricted on the set (5.9) and its lateral surface \(\varGamma_{\sphericalangle}=\{x\in\varGamma:\varphi\in(0,\pi /3)\}\), respectively. The corresponding problem is closed by the following artificial boundary conditions:

_{∢}, (1.11)

_{∢}, (5.10), (5.11) refers to the integral identity

The first distinguishing feature of the artificial boundary conditions (5.10), (5.11) proposed in [35] is the weighted Korn inequality, a simplified version of which is verified in the next assertion.

### Proposition 2

*The anisotropic weighted inequality*

*is valid for any vector function*\(v\in{\mathcal{W}}_{\sphericalangle}( \varOmega_{\sphericalangle})\),

*where*\(r=|x|\)

*is the distance to the cusp tip*\(\mathcal {O}\)

*and the multiplier*\(K_{\sphericalangle}\)

*depends on the sectorial domain*\(\varOmega_{\sphericalangle}\)

*only*.

### Proof

First of all, we mention that, under the Dirichlet conditions from (5.13) on parts of the surfaces \(\varSigma_{0}\) and \(\varSigma_{\pi/3}\) with positive area, any rigid motion, either a translation \(c\in{\mathbb{R}}^{3}\), or rotation \(c\times x\), becomes null.^{4}

### Proposition 3

*Any eigenmode*\(u^{(k)}_{\sphericalangle}\)*in the problem* (5.12) *satisfies the inclusions*\(e^{-\delta/z}\nabla u^{(k)}_{\sphericalangle }\in L^{2}(\varOmega_{\sphericalangle})\)*and*\(z^{-2}e^{-\delta/z}u^{(k)} _{\sphericalangle}\in L^{2}(\varOmega_{\sphericalangle})\), *where*\(\delta\)*is a* (*small*) *positive number*.

### Proof

The second distinguishing feature of the imposed artificial boundary conditions allows us to construct an eigenmode of the original problem (1.10), (1.11) in the intact domain \(\varOmega\) from any eigenmode \(u^{(k)}_{\sphericalangle}\) in the sectorial subdomain (5.9). Indeed, the odd for \((u^{(k)}_{\sphericalangle})_{n}\) and even for \((u^{(k)}_{\sphericalangle})_{s^{j}}\) extensions in the \(n\)-direction through the surface \(\varSigma_{0}\) remain smooth and keep the differential equations (1.10) in \(\{x\in\varOmega: \varphi\in(-\pi/3,0)\}\) and the boundary conditions (1.11) on \(\{x\in\varGamma: \varphi \in(-\pi/3,0)\}\) in view of the mirror symmetry accepted by homogeneous isotropic media. At the same time, the even for \((u^{(k)}_{\sphericalangle})_{n}\) and odd for \((u^{(k)}_{\sphericalangle })_{s^{j}}\) extensions in the \(n\)-direction through the surface \(\varSigma_{\pi/3}\) do the same with the differential equation (1.10) in the next sector \(\{x\in\varOmega: \varphi\in(\pi/3,2 \pi/3)\}\) and the boundary conditions (1.11) on the fragment \(\{x\in\varGamma: \varphi\in(\pi/3,2\pi/3)\}\) of the sector boundary. Repeating this two-fold extension procedure three times turns \(u^{(k)}_{\sphericalangle}\) into a smooth eigenmode in \(\varOmega\) corresponding to the same eigenvalue. In this way, the point spectrum \(\wp_{po}\) contains the whole infinite sequence (5.16). In [1, 35] some other shapes of the cross-section and orthotropic elastic materials are listed which also support the point spectrum of infinite total multiplicity in a cuspidal solid.

### 5.3 The Augmented Scattering Matrix

### Proposition 4

*For any*\(\varLambda>0\),

*there exists the row*\({\mathfrak {Z}}=({\mathfrak{Z}} ^{1}, \dots,{\mathfrak{Z}}^{6})\)

*of solutions to the problem*(1.10), (1.11)

*in the asymptotic form*

*where*\(\chi\)

*is a smooth cut*-

*off function such that*

*the remainder*\(\widetilde{{\mathfrak{Z}}}(x)\)

*decays exponentially as*\(x\to{\mathcal{O}}\),

*and the*\(6\times6\)-

*matrix*\({\mathfrak{S}}\)

*is unitary and symmetric*.

*The solutions*\({\mathfrak{Z}}^{1},\dots,{\mathfrak{Z}} ^{6} \)

*constitute a basis in the subspace*\(\mathcal{Z}_{\varLambda}\).

### Proof

We call \(\mathfrak{S}\) in (5.33) the *augmented scattering matrix* and will use it in the next section.

### 5.4 The Scattering Matrix and a Criterion for the Existence of Power-Law Trapped Modes

### Theorem 2

*The component*\(\mathcal{L}^{pow}_{\varLambda}\)

*of the decomposition*(5.23)

*satisfies the relation*

*where the right*-

*hand side is nothing but the multiplicity of the eigenvalue*1

*of the right below block of the augmented scattering matrix*(5.37).

### Proof

The proof is completed. □

*scattering matrix*in the problem (1.10), (1.11) because algebraic operations on its solutions (5.36) provide the new row of solutions

It is worth to mentioning that the augmented scattering matrix \({\mathfrak{S}}\) in Proposition 4 generates the traditional scattering matrix \(S\) as well as the condition \(\dim\ker({\mathfrak{S}} _{\blacklozenge\blacklozenge}- {\mathbb{I}}_{\aleph(\varLambda)})>0\) in Theorem 2 for the existence of trapped modes with a power-law behavior at the cusp tip \(\mathcal {O}\) rejecting the exponential decay rate as was in Sect. 5.2. In this way, the augmented scattering matrix keeps a lot of information on the solvability of the elasticity problem in the cuspidal solid \(\varOmega\), although it is an artificial object.

### 5.5 The Mandelstam (Energy) Radiation Conditions and the Solvability of the Elasticity Problem Above the Threshold

*artificial Mandelstam radiation*conditions because according to Sect. 4 it transports energy towards the cusp tip. The adjective “artificial” is added due to the appearance in (5.46) non-energy wave packets.

*artificial radiation conditions*.

### Proposition 5

*The operator*\(B^{art}_{\varLambda}(\beta)\)

*is Fredholm of index zero*,

*that is*,

*the problem*(1.17), (1.18)

*with the right*-

*hand sides*\((f,g)\in{\mathcal{R}}_{-\beta}(\varOmega)\)

*enjoying the compatibility conditions*(5.44),

*has a solution*(5.46)

*in the space*\(\mathcal{D}^{art}_{\beta}(\varOmega)\)

*which is defined up to an addendum in*\(\mathcal{L}^{exp}_{\varLambda}\),

*an exponentially decaying trapped mode*.

*Moreover*,

*under the orthogonality conditions*

*the solution becomes unique and meets the estimate*

### Proof

### Proposition 6

### Proof

### Theorem 3

*Let*\((f,g)\in{\mathcal{R}}_{-\beta}(\varOmega)\)

*satisfy the compatibility conditions*(5.2),

*where*\(\mathcal{L}_{\varLambda}\)

*is the whole space*(5.23)

*of trapped modes*.

*Then problem*(1.17), (1.18)

*has a solution in the form*

*where the remainder*\(\widetilde{u}^{out}\)

*falls into the space*\(\mathcal{D}_{-\beta}(\varOmega)\),

*i*.

*e*.,

*decays exponentially*,

*and*\(c^{out}_{\thickapprox}\in{\mathbb{C}}^{6-\aleph(\varLambda)}\), \(c^{out}_{\blacktriangle}\in{\mathbb{C}}^{\aleph(\varLambda)}\).

*The solution*(5.48)

*is defined up to an addendum in*\(\mathcal {L}_{\varLambda }\),

*a trapped mode*,

*but under the orthogonality conditions*(5.3)

*becomes unique and meets the estimate*

### Proof

## Footnotes

- 1.
This picture is published with permission of Professor M.A. Mironov.

- 2.
Although the symbols \(L^{2}(\partial_{z})\) and \(L^{2}(\varOmega)\) of the introduced differential operator and the Lebesgue space look quite similar, no misunderstanding may occur in what follows.

- 3.
- 4.
It is not the case for the perpendicular planes \(\varSigma_{0}\) and \(\varSigma_{\pi/2}\). To support the inference, the angle must be acute, for example, \(\pi/N\) with \(N=3,4,5,\dots\), cf., [35].

## Notes

### Acknowledgements

This work was written within the project 17-11-01003 of Russian Science Foundation. The authors are grateful to the anonymous referee whose comments helped to improve the presentation.

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