On the T-matrix for thermo-visco-elastic scattering

Summary

1. Introduction

The literature on the scattering of time-harmonic elastic waves by a spherical elastic inclusion (of radius a) is extensive; for reviews, see (1, section 4.10) and (2, section 2.3). For such problems, there are displacements |$ {\boldsymbol{u}} $| and stresses |$ {\boldsymbol{\tau}} $| outside the sphere with corresponding quantities, |$ {\boldsymbol{u}}^{\circ} $| and |$ {\boldsymbol{\tau}}^{\circ} $|⁠, inside the sphere. Each field is expanded using appropriate wavefunctions, involving spherical harmonics |$ Y_{n}^{m}(\theta , \phi) $| (see section 5 for details), where |$ r $|⁠, |$ \theta $| and |$ \phi $| are spherical polar coordinates. Then, for each pair |$ \{n, m\} $|⁠, there are six continuity (transmission) conditions to be imposed at |$ r\,=\,a $|⁠: these are continuity of three displacement components and continuity of three traction components. For most problems of this type, the dependence on the azimuthal coordinate |$ \phi $| can be eliminated, spherical harmonics |$ Y_{n}^{m} $| can be replaced by Legendre polynomials |$ P_{n} $|⁠, and the number of continuity conditions reduces to four.

More generally, suppose that the elastic media are replaced by thermo-visco-elastic (TVE) media, implying the introduction of two more (scalar) variables, the (reduced) temperatures outside (⁠|$ \Theta $|⁠) and inside (⁠|$ \Theta^{\circ} $|⁠) the inclusion. This leads to eight continuity conditions at |$ r\,=\,a $|⁠, the additional two coming from enforcing continuity of temperature and heat flux across the interface. This reduces to six after eliminating the dependence on |$ \phi $| (3 to 6).

The literature on dynamic thermo-elastic and visco-elastic problems is also extensive (7, 8). For a useful review, see (9). A referee noted work by Danilovskaya from the early 1950s; her papers are in Russian, but there is a short description in (10). For additional work on thermo-elastic problems involving spherical cavities, see (7, section 2.16) and (11 to 14). For visco-elastic problems involving spherical cavities, see, for example (8, section 2.10) and (15 to 17). For TVE problems with a spherical cavity and spherical symmetry, see (18,  19).

For non-spherical scatterers, we could use integral equation methods, as developed for thermo-elastic problems in (7, section 2.17) and (20, chapter X); see, for example (21,  22). Here, we focus on the use of T-matrix methods, as developed in the 1970s for elastodynamic problems (23 to 25). These methods are especially attractive for multiple scattering problems (1, Chapter 7). We are only aware of one previous paper on T-matrix methods for thermo-elastic problems (26); we discuss this paper briefly at the end of section 9.

After reviewing the governing equations for TVE media in section 2, we derive a reciprocal theorem in section 3, connecting any two TVE solutions. Solving the TVE equations is reduced to solving (scalar and vector) Helmholtz equations in section 4, and this is effected using vector spherical wavefunctions (section 5). All this can be seen as preliminary material, closely related to analogous results for elastodynamic media and for thermo-elastic media. The main difference is that the various constants involved (such as stiffnesses) are complex.

A scattering problem for a penetrable scatterer is formulated in section 6, and the T-matrix is introduced. It relates the coefficients in the expansion of the incident wave in terms of regular vector spherical wavefunctions to the coefficients in the expansion of the scattered wave in terms of outgoing vector spherical wavefunctions.

Given two different incident waves, the corresponding total fields (incident plus scattered) satisfy a certain constraint stemming from the reciprocal theorem. This constraint amounts to the vanishing of a certain integral over any sphere enclosing the scatterer, an integral (denoted by |$ I_{r} $|⁠) that can be evaluated exactly (section 7): this is a rather tedious calculation, with most details relegated to an appendix. One consequence of this result is a proof that the T-matrix is (almost) symmetric (section 8). The proof of this symmetry for elastodynamic problems relies on time-reversal (23,  24), but that argument is not available for TVE media.

The reciprocal theorem can also be applied to the solution of a scattering problem and a vector spherical wavefunction. Doing this gives a relation between |$ I_{r} $| and a similar integral over the boundary of the scatterer. Combining relations of this kind leads to an algorithm for computing the T-matrix (section 9). This strategy, which does not require the introduction of a fundamental solution (Green function), is due to Pao (25). Alternative algorithms may be developed, perhaps using boundary integral equations, but that remains to be done.

2. Governing equations

We consider TVE media and take the governing equations from (9); see also (6). For time-harmonic motions, the displacement is |$ {\rm Re}\{{\boldsymbol{u}}({\boldsymbol{r}})\,{\rm e}^{-{\rm i}\omega t}\} $| and the dimensionless temperature is |$ {\rm Re}\{\Theta({\boldsymbol{r}})\,{\rm e}^{-{\rm i}\omega t}\} $|⁠, with similar definitions for the time-harmonic stress (⁠|$ {{\boldsymbol{\tau}}} $|⁠) and strain (⁠|$ {{\boldsymbol{\varepsilon}}} $|⁠). From (9, eq. (2.16)), we have

where |$ \hat{\lambda}=\lambda-{\rm i}\omega\eta_{\lambda},\hat{\mu}=\mu-{\rm i}\omega\eta_{\mu},\lambda $| and |$ \mu $| are the usual (isothermal) Lamé moduli, |$ \eta_{\lambda}=\eta_{K}-\frac{2}{3}\eta_{\mu},\eta_{K} $| is the bulk viscosity, |$ \eta_{\mu} $| is the shear viscosity, |$ \rho_{0} $| is the ambient mass density, |$ \alpha $| is the (volume) coefficient of thermal expansion, |$ K=\lambda+\frac{2}{3}\mu $| is the elastic bulk modulus and |$ T_{0} $| is the ambient temperature. The time-harmonic form of the energy equation (9, eq. (2.10)) is

(after noting that |$ {\rm tr} \ {{\boldsymbol{\varepsilon}}}={\rm div} \ {\boldsymbol{u}} $|⁠), where |$ k_{{\rm th}} $| is the thermal conductivity and |$ c_{v} $| is the specific heat at constant volume.

To simplify notation a little, put

Then, after defining a dimensionless potential |$ \Phi $| by |$ \Phi={\rm div} \ {\boldsymbol{u}} $|⁠, the pair (2.1) becomes

For the time-harmonic stress, we have |$ {\boldsymbol{\tau}}=(\hat{\lambda} \ \Phi-{\mathbb{B}}\Theta){\boldsymbol{I}}+\hat{\mu}((\nabla{\boldsymbol{u}}+(\nabla{\boldsymbol{u}})^{{\rm T}}) $|⁠; see (3.1) and (9, eq. (2.14)).

Apart from the complexified Lamé moduli, |$ \hat{\lambda} $| and |$ \hat{\mu} $|⁠, all the equations above occur in the classical time-harmonic theory of thermo-elasticity; see, for example, (20, p. 53), (7,  27,  28).

3. A reciprocal theorem

Let us use Cartesian coordinates. In terms of (complex) stiffnesses |$ c_{ijkl} $|⁠, we have

At this point, we could permit anisotropic elasticity (29), but we are mainly interested in isotropy, so that

whence

The governing equations for a TVE field |$ ({\boldsymbol{u}},{\boldsymbol{\tau}},\Theta) $| are

Introduce a second TVE field |$ (\tilde{{\boldsymbol{u}}},\tilde{{\boldsymbol{\tau}}},\tilde{\Theta}) $|⁠, governed by

Multiply (3.2) by |$ \mu^{-1}\tilde{u}_{i} $|⁠, (3.4) by |$ \mu^{-1}u_{i} $|⁠, and subtract; multiply (3.3) by |$ \kappa_{2}^{-2}\tilde{\Theta} $|⁠, (3.5) by |$ \kappa_{2}^{-2}\Theta $|⁠, and subtract; add the final results to obtain the (dimensionless) equation

Use of (3.3) and (3.5) shows that the last two terms cancel, leaving

and then

using (3.3) and (3.5) again. The first term vanishes because of the symmetry |$ c_{ijkl}=c_{klij} $|⁠, whereas for the second term, we have

Therefore, we obtain the identity

This holds everywhere in the TVE medium.

Consider a bounded scatterer with smooth boundary |$ S $|⁠. Enclose |$ S $| by a sphere |$ S_{r} $| of radius |$ r $|⁠, centred at |$ O $|⁠. Integrate (3.6) over the volume bounded by |$ S $| and |$ S_{r} $|⁠. Then, application of the divergence theorem gives a reciprocal theorem,

where

|$ ({\boldsymbol{t}})_{i}=\tau_{ij}\hat{r}_{j} $|⁠, |$ (\tilde{{\boldsymbol{t}}})_{i}=\tilde{\tau}_{ij}\hat{r}_{j} $|⁠, |$ \hat{{\boldsymbol{r}}} $| is a unit vector in the outward radial direction, |$ ({\boldsymbol{t}}_{+})_{i}=\tau_{ij}n_{j} $|⁠, |$ (\tilde{{\boldsymbol{t}}}_{+})_{i}=\tilde{\tau}_{ij}n_{j} $| and |$ {\boldsymbol{n}} $| is the outward unit normal vector on |$ S $|⁠. We have also introduced the operators |$ {\mathcal{D}}_{r} $| and |$ {\mathcal{D}}_{n} $|⁠, defined by

as |$ {\mathbb{B}}\kappa_{2}^{-2}=({\rm i}\omega)^{-1}T_{0}k_{{\rm th}} $|⁠, |$ {\mathcal{D}}_{n}\Theta $| is proportional to |$ T_{0}k_{{\rm th}} \ {\partial\Theta}/{\partial n} $|⁠, which is the heat flux through |$ S $|⁠.

Evidently, the reciprocal theorem (3.7) gives a relation between two TVE fields, |$ ({\boldsymbol{u}},{\boldsymbol{\tau}},\Theta) $| and |$ (\tilde{{\boldsymbol{u}}},\tilde{{\boldsymbol{\tau}}},\tilde{\Theta}) $|⁠. Several applications will be made later.

The thermo-elastic version of (3.7) is (7, p. 56) and (20, p. 531).

4. Solving the governing equations

Computing the divergence of (2.2a) gives

where |$ \kappa_{3}^{2}={\rho_{0}\omega^{2}}/{{\mathbb{A}}} $| and |$ {\mathbb{C}}={{\mathbb{B}}}/{{\mathbb{A}}} $|⁠. We note that |$ \kappa_{1} $|⁠, |$ \kappa_{2} $| and |$ \kappa_{3} $| are inverse lengths in magnitude, whereas |$ {\mathbb{C}} $| is dimensionless.

Eliminating |$ \Theta $| from (2.2 b) and (4.1) gives

with |$ 2a=\kappa_{1}^{2}+\kappa_{3}^{2}+{\mathbb{C}}\kappa_{2}^{2} $|⁠; |$ \Theta $| satisfies the same equation. Factorizing,

where

Then, in general, we can write

Furthermore, using (2.2b) or (4.1), we obtain

where

Next, return to (2.2a), which we rearrange as

where |$ {\boldsymbol{w}}={\rm curl} \ {\boldsymbol{u}} $|⁠. Using |$ {\mathbb{B}}={\mathbb{A}}{\mathbb{C}} $|⁠, |$ \Phi=\phi_{{\rm c}}+\phi_{{\rm t}} $|⁠, |$ {\mathbb{A}}\kappa_{3}^{2}=\rho_{0}\omega^{2} $| and (4.5), the last term simplifies:

Taking the curl of (4.7) gives

where |$ k_{{\rm s}}^{2}=\rho_{0}\omega^{2}/\hat{\mu} $|⁠. Equation (4.8) is a vector Helmholtz equation for |$ {\boldsymbol{w}} $|⁠.

Returning to (4.7), we find that

This representation for |$ {\boldsymbol{u}} $| will be correct dimensionally if |$ {\boldsymbol{w}} $|⁠, |$ \phi_{{\rm c}} $| and |$ \phi_{{\rm t}} $| are dimensionless. For the dimensionless temperature |$ \Theta $|⁠, use (4.5).

Let us summarize. Suppose that |$ \phi_{{\rm c}} $| and |$ \phi_{{\rm t}} $| solve the scalar Helmholtz equations (4.4), and that |$ {\boldsymbol{w}} $| solves (4.8). Then (4.9) defines a valid displacement field, |$ {\boldsymbol{u}}_{1} $|⁠, say. But then it is easy to check that |$ {\rm curl} \ {\boldsymbol{u}}_{1}={\boldsymbol{w}} $| is also a valid displacement field, one with |$ \Theta=\Phi = 0 $|⁠.

5. Spherical wavefunctions

Choose an origin |$ O $|⁠. Let |$ {\boldsymbol{r}} $| be the position vector of a typical point |$ P $| with respect to |$ O $|⁠. The spherical polar coordinates of |$ P $| are |$ (r, \theta, \phi) $| with |$ r=|{\boldsymbol{r}}| $|⁠. Introduce spherical harmonics |$ Y_{n}^{m}(\hat{{\boldsymbol{r}}})=Y_{n}^{m}(\theta, \phi) $|⁠, where |$ \hat{{\boldsymbol{r}}}={\boldsymbol{r}}/r $|⁠, |$ m = 0, \pm 1, \pm 2, \ldots, \pm n $| and |$ n = 0,1,2, \ldots $|⁠. These functions form a complete orthonormal system in |$ L^{2}(\Omega) $|⁠,

where the overline denotes complex conjugation and |$ \Omega $| is the unit sphere.

Next, define scalar spherical wavefunctions

where |$ j_{n} $| is a spherical Bessel function, |$ h_{n}\equiv h_{n}^{(1)} $| is a spherical Hankel function and |$ k $| is a constant. Each wavefunction satisfies |$ (\nabla^{2}+k^{2})u = 0 $|⁠.

We will also require (dimensionless) vector spherical wavefunctions (1, Definition 3.37),

where |$ z_{n} $| is |$ j_{n} $| or |$ h_{n} $|⁠. We note that |$ {\bf M}_{0}^{0}={\bf N}_{0}^{0}={\bf 0} $|⁠,

The functions |$ {\bf L}_{n}^{m} $| are irrotational; they can be used to expand |$ {\rm grad}\,\phi_{{\rm c}} $| in (4.9) when |$ k = k_{{\rm c}} $| and they can be used to expand |$ {\rm grad}\,\phi_{{\rm t}} $| when |$ k = k_{{\rm t}} $|⁠. The functions |$ {\bf M}_{n}^{m} $| and |$ {\bf N}_{n}^{m} $| are solenoidal and they satisfy the vector Helmholtz equation,

They can be used to expand |$ {\rm curl}\,{\boldsymbol{w}} $| in (4.9) when |$ k = k_{{\rm s}} $|⁠.

For future use, define

Thus, |$ {\bf J}_{nm}^{[p]} $| is a regular vector spherical wavefunction. We shall refer to |$ {\bf H}_{nm}^{[p]} $| as an outgoing vector spherical wavefunction.

5.1. Vector spherical harmonics

These are defined as follows. For |$ m = 0, \pm 1, \pm 2, \ldots, \pm n $| and |$ n = 1,2, \ldots $|⁠,

where |$ B_{n}^{m}=\eta^{n}\,{\partial Y_{n}^{m}}/{\partial\theta} $|⁠, |$ C_{n}^{m}=\eta_{n}(\sin\theta)^{-1}\,{\partial Y_{n}^{m}}/{\partial\phi} $|⁠, |$ \eta_{n}=[n(n\,+\,1)]^{-1/2} $| and |$ \hat{{\boldsymbol{r}}} $|⁠, |$ \hat{{\boldsymbol{\theta}}} $| and |$ \hat{{\boldsymbol{\phi}}} $| are spherical polar unit vectors. When |$ n = 0 $|⁠, we have |$ {\bf C}_{0}^{0}={\bf B}_{0}^{0}={\bf 0} $| and |$ {\bf P}_{0}^{0}(\hat{{\boldsymbol{r}}})=(4\pi)^{-1/2}\hat{{\boldsymbol{r}}} $|⁠.

Vector spherical harmonics are orthonormal in |$ L^{2}(\Omega) $|⁠,

We also have |$ {\bf P}_{n}^{m}\cdot{\bf B}_{n}^{m}={\bf B}_{n}^{m}\cdot{\bf C}_{n}^{m}={\bf C}_{n}^{m}\cdot{\bf P}_{n}^{m}=0 $|⁠, |$ \overline{{\bf P}_{n}^{m}}=(-1)^{m}{\bf P}_{n}^{-m} $|⁠, |$ \overline{{\bf C}_{n}^{m}}=(-1)^{m}{\bf C}_{n}^{-m} $| and |$ \overline{{\bf B}_{n}^{m}}=(-1)^{m}{\bf B}_{n}^{-m} $|⁠.

In terms of vector spherical harmonics, we have (1, p. 107)

5.2. Traction and temperature

We are interested in the tractions on a spherical surface. We have

Define |$ {\boldsymbol{t}}=\tau_{rr}\hat{{\boldsymbol{r}}}+\tau_{r\theta}\hat{{\boldsymbol{\theta}}}+\tau_{r\phi}\hat{{\boldsymbol{\phi}}} $|⁠. We evaluate |$ {\boldsymbol{t}} $| for |$ {\bf L}_{n}^{m} $|⁠, |$ {\bf M}_{n}^{m} $| and |$ {\bf N}_{n}^{m} $|⁠, defined by (5.8).

• •
  Suppose |$ {\boldsymbol{u}}=k_{{\rm s}}^{-1}{\bf M}_{n}^{m}({\boldsymbol{r}}; k_{{\rm s}}) $|⁠. Then |$ {\rm div} \ {\boldsymbol{u}}=0 $|⁠, |$ u_{r}=0 $| and |$ \Theta=0 $|⁠. Hence (5.9) gives

  $$\begin{align*} {\boldsymbol{t}}=\hat{\mu}\eta_{n}^{-1}\,{\mathcal{Z}}_{n}(k_{{\rm s}}r){\bf C}_{n}^{m}\quad\mbox{with}\quad{\mathcal{Z}}_{n}(\xi)=z_{n}^{\prime}(\xi)-\xi^{-1}z_{n} (\xi).\end{align*}$$
• •
  Suppose |$ {\boldsymbol{u}}=k_{{\rm s}}^{-1}{\bf N}_{n}^{m}({\boldsymbol{r}}; k_{{\rm s}}) $|⁠. Then |$ {\rm div} \ {\boldsymbol{u}}=0 $| and |$ \Theta=0 $|⁠. Hence (5.9a) gives

  $$\begin{align*}\tau_{rr}=\frac{2\hat{\mu}}{\eta_{n}^{2}k_{{\rm s}}^{2}}\,\frac{{\rm d}}{{\rm d}r}\left(r^{-1}z_{n}(k_{{\rm s}}r)\right)Y_{n}^{m}=\frac{2\hat{\mu}}{\eta_{n}^ {2}k_{{\rm s}}r}{\mathcal{Z}}_{n}(k_{{\rm s}}r)Y_{n}^{m}.\end{align*}$$

Continuing, (5.9b) gives

where

and we have used the following results: |$ \eta_{n}^{-2}=n(n\,+\,1) $|⁠,

Similarly, |$ \tau_{r\phi}=\hat{\mu}\eta_{n}^{-1}{\mathcal{Z}}_{n}^{(2)}(k_{{\rm s}}r)C_{n}^ {m} $|⁠, and so

• •
  Suppose |$ {\boldsymbol{u}}=k^{-1}{\bf L}_{n}^{m}({\boldsymbol{r}}; k) $|⁠, where |$ k=k_{{\rm c}} $| or |$ k=k_{{\rm t}} $|⁠. From (4.4) and (5.3), we can take |$ \phi={\rm div} \ {\boldsymbol{u}}=-z_{n}(kr)Y_{n}^{m} $| and then (4.5) gives

  $$\begin{align}\Theta=\frac{\kappa_{2}^{2}\,\phi}{k^{2}-\kappa_{1}^{2}}=\frac{k^{2}-\kappa_{3}^{2}}{{\mathbb{C}}\, k^{2}}\phi\quad\mbox{with}\quad\phi=-z_{n}(kr)Y_{n}^{m}(\hat{{\boldsymbol{r}}}).\end{align}$$

  (5.10)

Hence, as |$ {\mathbb{B}}\Theta={\mathbb{A}}(\kappa_{3}^{2}-k^{2})k^{-2}z_{n}(kr)Y_{n}^{m} $| and |$ {\partial u_{r}}/{\partial r}=z_{n}^{\prime\prime}(kr)Y_{n}^{m} $|⁠, (5.9a) gives

where we have used |$ {\mathbb{A}}(k^{2}-\kappa_{3}^{2})-k^{2}\hat{\lambda}=\hat{\mu}(2k^{2}-k_{{\rm s}}^{2}) $| and

From (5.9b), we obtain

with a similar expression for |$ \tau_{r\phi} $|⁠. Hence

All these formulas agree with those in (25, eq. (18)) and (30, section 6.2) when |$ {\mathbb{B}}=0 $|⁠.

6. Scattering by an obstacle

Consider a bounded scatterer, centred at |$ O $|⁠, with arbitrary smooth boundary |$ S $|⁠. For a general incident wave |$ {\boldsymbol{u}}_{{\rm inc}} $|⁠, we expand around |$ O $| using regular vector spherical wavefunctions

where |$ n_{1}=n_{2}=0 $|⁠, |$ n_{3}=n_{4}=1 $| and the coefficients |$ d_{nm}^{[p]} $| are known. The expansion (6.1) is assumed to converge inside the sphere |$ S_{r} $|⁠.

In order to limit a profusion of subscripts and superscripts, we introduce a compressed notation, starting with a multi-index |$ L=\{n, m, p\} $| and write (6.1) compactly as

Later, we will also use |$ \Lambda=\{\nu, \mu, \varrho\} $|⁠.

The scattered field |$ {\boldsymbol{u}}_{{\rm sc}} $| can be expanded in a similar manner. Thus

where |$ {\bf H}_{L}={\bf H}_{nm}^{[p]} $| is an outgoing vector spherical wavefunction (see (5.5)) and the coefficients |$ c_{L}=c_{nm}^{[p]} $| are unknown. The expansion (6.3) converges at points outside |$ S_{+} $|⁠, the smallest sphere centred at |$ O $| that encloses |$ S $|⁠.

The tractions and temperatures corresponding to |$ {\bf J}_{L} $| and |$ {\bf H}_{L} $| are given below by (A.2), (A.8), (A.12) and (A.15).

The coefficients |$ c_{L} $| and |$ d_{L} $| are related by a T-matrix,

The entries in |$ T_{L\Lambda}=T_{nm, \nu\mu}^{\, p\,\varrho} $| can be calculated explicitly when |$ S $| is a sphere (⁠|$ S = S_{+} $|⁠). In general, numerical methods are required when |$ S $| is not spherical. We describe one algorithm for computing the T-matrix for TVE problems in section 9.

6.1. Scattering by a sphere

Suppose in this section (only) that |$ S $| is a sphere of radius a. The total field outside the sphere is |$ {\boldsymbol{u}}={\boldsymbol{u}}_{{\rm inc}}+{\boldsymbol{u}}_{{\rm sc}} $|⁠; summing (6.2) and (6.3) gives

For each pair |$ \{n, m\} $| in |$ L=\{n, m, p\} $|⁠, there are four unknown coefficients, |$ c_{nm}^{[p]} $|⁠, |$ p = 1,2,3,4 $|⁠. Therefore, if there are no fields inside the sphere, we need four boundary conditions at |$ r = a $|⁠. For example, if the sphere is a thermally insulated cavity, the traction vector and the heat flux both vanish at |$ r = a $|⁠; the analogous thermo-elastic problem is solved in (7, section 2.16).

More generally, suppose there are also fields inside the sphere, with displacement |$ {\boldsymbol{u}}^{\circ} $|⁠, stress |$ {\boldsymbol{\tau}}^{\circ} $|⁠, temperature |$ \Theta^{\circ} $|⁠, and wavenumbers |$ k_{{\rm c}}^{\circ} $|⁠, |$ k_{{\rm t}}^{\circ} $|⁠, and |$ k_{{\rm s}}^{\circ} $|⁠. Then, we write

where the regular vector spherical wavefunctions |$ {\bf J}_{L}^{\circ} $| are defined by (5.5) with appropriate choices for wavenumbers and |$ d_{L}^{\circ} $| are unknown coefficients. These coefficients (together with |$ c_{L} $| in (6.5)) are to be determined using eight continuity (transmission) conditions across the interface at |$ r = a $|⁠.

6.2. Transmission conditions

Let us return to scattering by a penetrable non-spherical obstacle with boundary |$ S $|⁠. The total displacement field in the exterior is |$ {\boldsymbol{u}} $| with temperature |$ \Theta $|⁠; the traction on |$ S $| is |$ {\boldsymbol{t}}_{+} $| and the heat flux on |$ S $| is (proportional to) |$ {\mathcal{D}}_{n}\Theta $| (see (3.10)). Similarly, the corresponding quantities inside |$ S $| are |$ {\boldsymbol{u}}^{\circ} $|⁠, |$ \Theta^{\circ} $|⁠, |$ {\boldsymbol{t}}_{-}^{\circ} $| and |$ {\mathcal{D}}_{n}^{\circ}\Theta^{\circ} $|⁠. Across |$ S $|⁠, we have the transmission conditions

Suppose we solve the scattering problem with two different incident fields, |$ {\boldsymbol{u}}_{{\rm inc}} $| and |$ \tilde{{\boldsymbol{u}}}_{{\rm inc}} $|⁠. Denote the corresponding interior fields by |$ {\boldsymbol{u}}^{\circ} $| and |$ \tilde{{\boldsymbol{u}}}^{\circ} $|⁠, with similar notations for the other fields. As all fields are regular inside |$ S $|⁠, an application of the reciprocal theorem (integrate (3.6) over the interior of |$ S $|⁠) gives

Then, as the solutions of both scattering problems satisfy the same transmission conditions, (6.6), we deduce that

Furthermore, from (3.7), we can move the surface of integration from |$ S $| to a sphere |$ S_{r} $| enclosing the scatterer, giving

where |$ I_{r} $| is defined by (3.8). The integral defining |$ I_{r} $| can be evaluated explicitly in terms of the coefficients |$ d_{L} $| in (6.2) and |$ c_{L} $| in (6.3); we do this next. The calculations are lengthy (and can be skipped), but the final result is quite simple: see (7.11) below.

7. Exact evaluation of |$ I_{r} $|

Introduce the notations

(see (3.10) for |$ {\mathcal{D}}_{r} $|⁠) so that (3.8) becomes

We have |$ \left[{\boldsymbol{u}};\tilde{{\boldsymbol{u}}}\right]_{r}=-\left[\tilde{{\boldsymbol{u}}};{\boldsymbol{u}}\right]_{r} $| and |$ [\Theta;\tilde{\Theta}]_{r}=-[\tilde{\Theta};\Theta]_{r} $|⁠.

Take two total displacement fields |$ {\boldsymbol{u}}={\boldsymbol{u}}_{{\rm inc}}+{\boldsymbol{u}}_{{\rm sc}} $| and |$ \tilde{{\boldsymbol{u}}}=\tilde{{\boldsymbol{u}}}_{{\rm inc}}+\tilde{{\boldsymbol{u}}}_{{\rm sc}} $|⁠. Then

We can make a similar decomposition for the temperatures,

Hence

7.1. Two incident fields

Choose two regular incident fields,

The corresponding tractions on |$ S_{r} $| are denoted by |$ {\boldsymbol{t}}_{{\rm inc}} $| and |$ \tilde{{\boldsymbol{t}}}_{{\rm inc}} $|⁠.

The reciprocal theorem implies that

where |$ \Theta_{{\rm inc}} $| is computable from |$ {\rm div} \ {\boldsymbol{u}}_{{\rm inc}} $|⁠; see (A.9). In  Appendix A.1, we check this result by direct integration over |$ S_{r} $|⁠. This is a prelude to introducing the scattered fields.

7.2. Two scattered fields

The scattered fields corresponding to (7.6) can be expanded as (6.3):

The corresponding tractions on |$ S_{r} $| are denoted by |$ {\boldsymbol{t}}_{{\rm sc}} $| and |$ \tilde{{\boldsymbol{t}}}_{{\rm sc}} $|⁠.

Direct integration over |$ S_{r} $| (see  Appendix A.3) shows that

This result cannot be obtained from the reciprocal theorem (because |$ {\boldsymbol{u}}_{{\rm sc}} $| and |$ \tilde{{\boldsymbol{u}}}_{{\rm sc}} $| are singular at the origin) but it could be obtained by letting |$ r\to\infty $|⁠.

7.3. Cross terms

From (7.3) and (7.4), we need cross terms, such as |$ \left[{\boldsymbol{u}}_{{\rm inc}};\tilde{{\boldsymbol{u}}}_{{\rm sc}}\right]_{r} $| and |$ [\Theta_{{\rm inc}};\tilde{\Theta}_{{\rm sc}}]_{r} $|⁠. Direct integration (see  Appendix A.4) shows that

where

7.4. Synthesis

Assembling the results above, namely, (7.5), (7.7) and (7.10), we obtain

This quantity is also equal to |$ I_{S} $|⁠, an integral over |$ S $| defined by (3.9); see (3.7).

The reciprocal formula, |$ I_{r}=I_{S} $|⁠, will be used in section 8 to extract information on the T-matrix. It will also be used in section 9 to derive a formula for computing the T-matrix.

8. The T-matrix is (almost) symmetric

When |$ {\boldsymbol{u}} $| and |$ \tilde{{\boldsymbol{u}}} $| are solutions of the same scattering problem but with different incident fields, |$ {\boldsymbol{u}}_{{\rm inc}} $| and |$ \tilde{{\boldsymbol{u}}}_{{\rm inc}} $|⁠, we know that |$ I_{S}({\boldsymbol{u}},\Theta;\,\tilde{{\boldsymbol{u}}},\tilde{\Theta})=0 $|⁠, implying that |$ I_{r}({\boldsymbol{u}},\Theta;\,\tilde{{\boldsymbol{u}}},\tilde{\Theta})=0 $|⁠; see (6.8) and (6.9). Then, from (7.11), we obtain

Introducing the T-matrix, (6.4), so that |$ \tilde{c}_{L}=\sum_{\Lambda}T_{L\Lambda}\tilde{d}_{\Lambda} $|⁠, we obtain

Similarly, as |$ c_{L}=\sum_{\Lambda}T_{L\Lambda}d_{\Lambda} $|⁠,

Interchanging |$ L $| and |$ \Lambda $|⁠, this sum is

Substituting in (8.1) gives

As this must hold for all possible pairs of incident fields, we obtain

with |$ \beta_{n}^{[p]} $| defined by (7.10c).

The result (8.2) is known for pure elastodynamic problems (23 to 25). In those papers, the authors eliminate the factors |$ \beta_{n}^{[p]} $| by adjusting the normalizations of |$ {\bf J}_{L} $| and |$ {\bf H}_{L} $|⁠; see, for example, (25, eq. (25)). The same could be done here, if desired.

9. Computing the T-matrix

If we choose |$ \tilde{{\boldsymbol{u}}}={\bf J}_{L} $| in (7.11), and combine with (3.7), we obtain

where |$ \hat{\Theta}_{L} $| is the temperature corresponding to |$ {\bf J}_{L} $|⁠. Similarly, the choice |$ \tilde{{\boldsymbol{u}}}={\bf H}_{L} $| gives

Applying the reciprocal theorem inside |$ S $| to |$ {\boldsymbol{u}}^{\circ} $| and |$ {\bf J}_{L}^{\circ} $|⁠, (6.7), we obtain

and then the transmission conditions (6.6) give

Next, we solve the pair, (9.2) and (9.3), for the unknown boundary quantities. Define two dimensionless 4-vectors,

both defined on |$ S $|⁠. (Note that if we knew |$ {\bf U} $| away from |$ S $|⁠, we could compute |$ {\bf Z} $| from |$ {\bf U} $|⁠; but we do not.) Then (9.2) and (9.3) become

where

these are known dimensionless 4-vectors. Similarly, (9.1) becomes

where |$ \hat{{\bf Z}}_{L}=\hat{\mu}^{-1}(k_{{\rm s}}^{-1}\hat{{\bf T}}_{L},{\mathcal{D}}_{n}\hat{\Theta}_{L}) $| and |$ \hat{{\bf U}}_{L}=({\bf J}_{L},k_{{\rm s}}^{-1}\hat{\Theta}_{L}) $|⁠. Note that we have used |$ k_{{\rm s}} $| and |$ \hat{\mu} $| so as to achieve dimensionless quantities; other choices could be made.

The next step is to choose a set of basis functions, 4-vectors |$ {\boldsymbol{\Phi}}_{\nu} $|⁠, suitable for representing 4-vectors defined on |$ S $|⁠. (A combination of vector spherical harmonics and scalar spherical harmonics could be used.) Thus, we can write

(We could use different basis functions for |$ {\bf U} $| and |$ {\bf Z} $|⁠.) Substituting in (9.4), we obtain

where

and |$ d_{L}^{-}=k_{{\rm s}}d^{[p]}_{n,-m} $|⁠. In an obvious matrix-vector notation, write (9.7) as

with formal solution

where |$ G=\hat{Q}^{\circ}Q^{-1}-\hat{M}^{\circ}M^{-1} $|⁠. Then |$ c_{n,-m}^{[p]} $| can be expressed in terms of |$ {\bf a} $| and |$ {\bf b} $| from (9.5) leading to a (complicated) formula for the T-matrix, via (6.4).

The point here is to show that the T-matrix could be computed, in principle, although it is not clear that the specific formulas derived are useful—but similar formulas have been used effectively for related scattering problems (31). Other methods for computing the T-matrix could be developed, perhaps using boundary integral equations (1, section 7.8.5); such methods could be checked against the symmetry relation (8.2).

Kathreptas (26) has constructed the T-matrix for a thermally insulated cavity (so that |$ {\bf Z}=\mu^{-1}({\boldsymbol{t}}_{+},\, k_{{\rm s}}{\mathcal{D}}_{n}\Theta)={\bf 0} $| on |$ S $|⁠) in the context of thermo-elasticity. He starts with the bilinear expansion for the appropriate fundamental solution, and uses regular spherical wavefunctions for |$ {\boldsymbol{\Phi}} $| in (9.6), thus generalizing the approach used by Waterman (24).

10. Discussion

In this paper, we have investigated the T-matrix and its properties for TVE media. Such media are of increasing importance for technological applications, where multiple scattering is often of interest: the properties of each scatterer can be encoded in its T-matrix. Most studies assume that the scatterers are identical spheres (32,  33), for which the T-matrix is diagonal, but non-spherical scatterers are of interest. Multiple scattering problems often require the use of addition theorems for vector spherical wavefunctions; see (34) for a recent review.

Acknowledgements

This work was provoked by discussions with Valerie Pinfield and Art Gower at the Isaac Newton Institute for Mathematical Sciences, Cambridge, during the programme on ‘Mathematical theory and applications of multiple wave scattering’. The author thanks the Institute for its support and hospitality. This work was supported by EPSRC grant no EP/R014604/1.

References

APPENDIX A

Evaluating integrals over the sphere|$ \boldsymbol{S_{r}} $|

A.1. Direct evaluation of |$ I_{r}({\boldsymbol{u}}_{{\rm inc}},\Theta_{{\rm inc}};\,\tilde{{\boldsymbol{u}}}_{{\rm inc}},\tilde{\Theta}_{{\rm inc}}) $|

Expand the incident fields (7.6) in terms of vector spherical harmonics. Thus

where |$ \hat{{\mathcal{P}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{C}}}_{n}^{[p]} $| and |$ \hat{{\mathcal{B}}}_{n}^{[p]} $| can be found using (5.5) and (5.8). Similarly, denote the traction on |$ S_{r} $| corresponding to |$ {\bf J}_{L} $| by

where |$ \hat{{\mathcal{Q}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{R}}}_{n}^{[p]} $| and |$ \hat{{\mathcal{S}}}_{n}^{[p]} $| can be found using (5.9). Then

Integrating over the sphere |$ S_{r} $| (noting that |$ {\rm d}S = r^{2}\,{\rm d}\Omega $|⁠), using (5.7), gives

and so

Hence

where

The quantities |$ \hat{{\mathcal{P}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{C}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{B}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{Q}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{R}}}_{n}^{[p]} $| and |$ \hat{{\mathcal{S}}}_{n}^{[p]} $| are given in terms of spherical Bessel functions (which explains the use of the caret notation; see (5.2)). Specifically,

where

and we used |$ j_{n}^{\prime\prime}(\xi)=-(2/\xi)j_{n}^{\prime}(\xi)+\{(\eta_{n}\xi)^{-2}-1\} j_{n}(\xi) $|⁠, the differential equation satisfied by |$ j_{n}(\xi) $|⁠. Later we will use |$ {\mathcal{H}}_{n} $|⁠, |$ {\mathcal{H}}_{n}^{+} $|⁠, |$ {\mathcal{H}}_{n}^{(2)} $| and |$ {\mathcal{H}}_{n}^{(3)} $|⁠, defined as in (A.5) but with |$ j_{n} $| replaced by |$ h_{n} $|⁠.

By inspection, the only non-trivial terms in (A.3) are |$ \hat{\Omega}^{{\rm P}}_{n}(1; 2) $|⁠, |$ \hat{\Omega}^{{\rm P}}_{n}(1; 4) $|⁠, |$ \hat{\Omega}^{{\rm P}}_{n}(2; 4) $|⁠, |$ \hat{\Omega}^{{\rm B}}_{n}(1; 2) $|⁠, |$ \hat{\Omega}^{{\rm B}}_{n}(1; 4) $| and |$ \hat{\Omega}^{{\rm B}}_{n}(2; 4) $|⁠. Moreover, direct calculation (see  Appendix A.2) shows that

which leaves

From (A.10), we have

In order to evaluate |$ I_{r}({\boldsymbol{u}}_{{\rm inc}},\Theta_{{\rm inc}};\,\tilde{{\boldsymbol{u}}}_{{\rm inc}},\tilde{\Theta}_{{\rm inc}}) $|⁠, (7.5), we require |$ [\Theta_{{\rm inc}};\tilde{\Theta}_{{\rm inc}}]_{r} $|⁠. Let |$ \hat{\Theta}_{L}=\hat{\Theta}_{nm}^{[p]} $| denote the temperature corresponding to |$ {\bf J}_{L} $|⁠. From (4.5), (4.6) and (5.10), we obtain

Hence, (7.6) gives

and then

where |$ E={\mathbb{B}}\kappa_{2}^{-2}\alpha^{[1]}\alpha^{[2]}k_{{\rm c}}^{2}k_{{\rm t}}^{2}=-\hat{\mu}k_{{\rm s}}^{2} $| and |$ {\mathbb{J}}_{n} $| is defined by (A.7). When combined with (A.6) and (A.7), we confirm (7.7).

A.2. Evaluation of |$ \hat{\Omega}^{{\rm P}}_{n}+\hat{\Omega}^{{\rm B}}_{n} $|

With |$ p = 1 $| and |$ \varrho = 4 $|⁠, (A.3) gives

where |$ {\mathcal{A}} $|⁠, |$ {\mathcal{B}} $|⁠, |$ {\mathcal{C}} $| and |$ {\mathcal{D}} $| are to be calculated. We find that |$ {\mathcal{A}}={\mathcal{B}}={\mathcal{C}}={\mathcal{D}}=0 $|⁠, whence |$ \hat{\Omega}^{{\rm P}}_{n}(1; 4)+\hat{\Omega}^{{\rm B}}_{n}(1; 4)=0 $|⁠.

Similarly, when |$ p = 1 $| and |$ \varrho = 2 $|⁠, we have

After some calculation, we find |$ {\mathcal{A}}=0 $|⁠, |$ {\mathcal{B}}=-k_{{\rm s}}^{2}/k_{{\rm t}} $|⁠, |$ {\mathcal{C}}=k_{{\rm s}}^{2}/k_{{\rm c}} $| and |$ {\mathcal{D}}=0 $|⁠. Hence

A.3. Evaluation of |$ I_{r}({\boldsymbol{u}}_{{\rm sc}},\Theta_{{\rm sc}};\,\tilde{{\boldsymbol{u}}}_{{\rm sc}},\tilde{\Theta}_{{\rm sc}}) $|

Expanding |$ {\bf H}_{L} $| in terms of vector spherical harmonics gives

where |$ {\mathcal{P}}_{n}^{[p]} $|⁠, |$ {\mathcal{C}}_{n}^{[p]} $| and |$ {\mathcal{B}}_{n}^{[p]} $| are the same as |$ \hat{{\mathcal{P}}}_{n}^{[p]} $|⁠, |$ \hat{{\mathcal{C}}}_{n}^{[p]} $| and |$ \hat{{\mathcal{B}}}_{n}^{[p]} $|⁠, respectively, as given by (A.4), but with the spherical Bessel function |$ j_{n} $| replaced by the spherical Hankel function |$ h_{n} $|⁠. Similarly, denote the traction corresponding to |$ {\bf H}_{L} $| by

where, again, |$ {\mathcal{Q}}_{n}^{[p]} $|⁠, |$ {\mathcal{R}}_{n}^{[p]} $| and |$ {\mathcal{S}}_{n}^{[p]} $| are given by (A.4), but with |$ j_{n} $| replaced by |$ h_{n} $|⁠. Then

Integrating over the sphere |$ S_{r} $|⁠, using (5.7), gives

and so

with |$ \Omega^{{\rm P}}_{n} $|⁠, |$ \Omega^{{\rm C}}_{n} $| and |$ \Omega^{{\rm B}}_{n} $| defined by (A.3) but with |$ j_{n} $| replaced by |$ h_{n} $|⁠. Then, as in  Appendix A.1, the only non-trivial terms in (A.3) are |$ \Omega^{{\rm P}}_{n}(1; 2) $|⁠, |$ \Omega^{{\rm P}}_{n}(1; 4) $|⁠, |$ \Omega^{{\rm P}}_{n}(2; 4) $|⁠, |$ \Omega^{{\rm B}}_{n}(1; 2) $|⁠, |$ \Omega^{{\rm B}}_{n}(1; 4) $| and |$ \Omega^{{\rm B}}_{n}(2; 4) $|⁠. Moreover, direct calculation shows that

which leaves

From (A.10), we have

To evaluate |$ I_{r}({\boldsymbol{u}}_{{\rm sc}},\Theta_{{\rm sc}};\,\tilde{{\boldsymbol{u}}}_{{\rm sc}},\tilde{\Theta}_{{\rm sc}}) $|⁠, (7.5), we require |$ [\Theta_{{\rm sc}};\tilde{\Theta}_{{\rm sc}}]_{r} $|⁠. Similar to (A.8), let |$ \Theta_{L}=\Theta_{nm}^{[p]} $| denote the temperature corresponding to |$ {\bf H}_{L} $|⁠. Thus

where |$ \alpha^{[1]} $| and |$ \alpha^{[2]} $| are defined by (4.6). Hence, (7.8) gives

followed by

where |$ E=-\hat{\mu}k_{{\rm s}}^{2} $| and |$ {\mathbb{H}}_{n} $| is defined by (A.14). When combined with (A.13) and (A.14), we confirm (7.9).

A.4. Evaluation of |$ I_{r}({\boldsymbol{u}}_{{\rm inc}},\Theta_{{\rm inc}};\,\tilde{{\boldsymbol{u}}}_{{\rm sc}},\tilde{\Theta}_{{\rm sc}}) $|

Start with |$ \left[{\boldsymbol{u}}_{{\rm inc}};\tilde{{\boldsymbol{u}}}_{{\rm sc}}\right]_{r} $|⁠. From (7.6), (7.8), (A.1), (A.2)  (A.11) and (A.12),

Integrating over the sphere |$ S_{r} $|⁠,

and so

where

Let us start with |$ X^{{\rm C}}_{n}(p;\varrho) $|⁠. As |$ {\mathcal{C}}_{n}^{[p]}=\hat{{\mathcal{C}}}_{n}^{[p]}={\mathcal{R}}_{n}^{[p]}=\hat{{\mathcal{R}}}_{n}^{[p]}=0 $| for |$ p = 1 $|⁠, 2 and 4 (see (A.4)) the only non-trivial value is |$ X^{{\rm C}}_{n}(3; 3) $|⁠:

where |$ {\mathcal{H}}_{n} $| is defined below (A.5) and we have used the Wronskian |$ j_{n}(\xi)h_{n}^{\prime}(\xi)-j_{n}^{\prime}(\xi)h_{n}(\xi)={\rm i}\,\xi^{-2} $| (35, 10.50.1).

We have |$ X^{{\rm P}}_{n}(p;\varrho)=X^{{\rm B}}(p;\varrho)=0 $| when |$ p = 3 $| or |$ \varrho = 3 $|⁠. Otherwise, we evaluate directly (see  Appendix A.5 for some details). We find as follows:

where

In summary, the only non-trivial terms in (A.17) come from the diagonal terms with |$ p=\varrho $|⁠, and those with |$ p = 1 $|⁠, |$ \varrho\,=\,2 $| and |$ p\,=\,2 $|⁠, |$ \varrho\,=\,1 $|⁠. Thus

The factors of |$ r^{-2} $| in (A.20) are to be expected because of the prefactor |$ r^{2} $| in (A.17), but the presence of |$ {\mathbb{W}}_{n} $| suggests cancellation with corresponding terms coming from |$ [\Theta_{{\rm inc}};\tilde{\Theta}_{{\rm sc}}]_{r} $| (defined in (7.1)). We confirm this next.

Recall that, from (7.2),

where the first term on the right-hand side has been evaluated as (A.17) with (A.20). For the second term, starting from the expansions (A.9) and (A.16), we obtain

where

|$ \alpha^{[1]} $| and |$ \alpha^{[2]} $| are defined by (4.6), and |$ {\mathbb{W}}_{n}(k; K) $| is defined by (A.19). As |$ {\mathbb{B}}\kappa_{2}^{-2}\alpha^{[1]}\alpha^{[2]}k_{{\rm c}}^{2}k_{{\rm t}}^ {2}=-\hat{\mu}k_{{\rm s}}^{2} $| (see above (7.7)), we see that the |$ {\mathbb{W}}_{n} $| terms cancel when substituted in (A.21). Therefore, we arrive at (7.10a), in which

and |$ \beta_{n}^{[3]}=\beta_{n}^{[4]}=\eta_{n}^{-2}=n(n\,+\,1) $|⁠. These four constants are dimensionless.

A.5. Evaluation of |$ X^{{\rm P}}_{n} $| and |$ X^{{\rm B}}_{n} $|

Using (A.18) and (A.4),

where |$ {\mathbb{W}}_{n} $| is defined by (A.19). Continuing,

where |$ {\mathbb{X}}_{n}(k)=2j_{n}^{\prime}(kr)\{h_{n}^{\prime}(\xi_{{\rm s}})+\xi_{{\rm s}}^{-1}h_{n}(\xi_{{\rm s}})\}+(kk_{{\rm s}})^{-1}\{k_{{\rm s}}^{2}-2(\eta _{n}r)^{-2}\}j_{n}(kr)h_{n}(\xi_{{\rm s}}) $|⁠. Similarly,

where |$ {\mathbb{Y}}_{n}(k)=2j_{n}^{\prime}(\xi_{{\rm s}})h_{n}^{\prime}(kr)+2\xi_{{\rm s}}^{-1}j_{n}(\xi_{{\rm s}})h_{n}^{\prime}(kr)+(kk_{{\rm s}})^{-1}\{k_{{\rm s}}^{2}-2(\eta_{n}r)^{-2}\}j_{n}(\xi_{{\rm s}})h_{n}(kr) $|⁠.