Wave-induced motion of rigid bodies: beads, boats and buildings

The determination of rigid-body motion caused by incident waves is a familiar problem in mechanics. Good examples are the motion of a ship in the presence of water waves and the motion of a rigid structure in the presence of seismic waves. The basic goals are to determine the motion of the rigid body and the effects of the motion on the wave field, assuming linear theory is adequate. Although the underlying mathematical problems are similar, several solution methods have evolved, depending on the physical problems of interest. For ship motions, the standard approach is to decompose the problem into seven subproblems, one for each of the six rigid-body modes and one to take account of the incident wave. This approach is reviewed and then adapted to problems in acoustics and to problems in elastodynamics, such as those that arise in simple examples of soil–structure interaction. It is argued that the resulting approach for elastodynamic problems has clear advantages over those currently in use.

We are mainly concerned with time-harmonic problems: the time dependence is taken as e −iωt , where ω is the frequency.A typical problem would be: given an incident wave, determine the scattered waves and the motion of the body.This motion is driven by time-harmonic forces and moments applied to the surface of the rigid body (ignoring body forces, for simplicity).These forces and moments come from the unbounded medium outside the body.The body's motion also generates radiated waves: therefore, the problem is genuinely coupled.
The exterior medium can be fluid or solid.In the former case, suppose first that the fluid is inviscid and compressible: this is the realm of acoustics, where the fluid motion is governed by the Helmholtz equation.Application areas include suspensions or aerosols, with solid particles in a liquid or gas [5,6], sonar [7] and fish bioacoustics: it is thought that the otolith organs in the ears of fish are sensitive to motion [8,9].General formalisms for such acoustic problems were given by Haskind (Khaskind) [10] and Olsson [11].However, almost all calculations have been made for a spherical rigid body and an incident plane wave: the method of separation of variables can be used, and the equations of motion of the body reduce to a single differential equation (see §5a).This basic method was described by Lamb ([12], §298).Subsequent papers include [7,[13][14][15].There are also papers on related time-domain problems [16], where a plane sound pulse interacts with a rigid sphere [17][18][19].(For a good survey of relevant Russian work, see [20], ch.6.) Olsson [11] has given numerical results (using null-field/T-matrix methods) for timeharmonic scattering by rigid bodies of various shapes including spheroids and thick spheroidal shells.
Another class of problems concerns inviscid incompressible fluids (water) with a free surface: gravity waves can propagate along the surface and interact with a floating body.Obviously, such problems arise in ship hydrodynamics, where they have been studied extensively [21,22], but there are more modern applications such as to floating wind turbines [23][24][25].The textbook procedure is to decompose the full problem into seven problems, one for each of the six DOF of the rigid-body motion and one for scattering the incident wave by the body in its fixed equilibrium position (the diffraction problem).(Of course, symmetries can reduce the number of independent problems to be solved.)This decomposition (which dates to the 1940s) is a simple consequence of linearity but, surprisingly, it does not seem to have been exploited for acoustic problems (for which it was introduced by Haskind [10]), or for elastodynamic problems (for which it was introduced by Thau [26]).(Elastodynamic problems will be described below.)It turns out that there is no need to solve the diffraction problem completely: the corresponding forces and moments can be computed in terms of the solutions of the six basic radiation problems; see §4a.This result [27], which is a consequence of Green's theorem, is associated with the name of M. D. Haskind (1913Haskind ( -1963)), an engineer from Odessa, Ukraine.
Suppose next that the medium outside the rigid body is an elastic solid, extending to infinity in all directions (full-space problem).Waves in the solid are incident on the rigid body, causing it to oscillate, scattering waves away from the body.As before, the goal is to calculate the scattered waves and the motion of the rigid body.
In 1900, Lamb ([28], §6) considered the scattering of a shear wave by a movable rigid sphere in an incompressible solid.The general problem of elastodynamic scattering by a movable rigid sphere has been solved [13,29,30], ( [31], p. 622).For spheroids, see [32][33][34][35].For the limiting case of a thin circular disc, see [36,37].(For scattering of flexural waves by movable rigid inclusions in a thin plate, see [38].) Another important class of problems arises when the external medium is a half-space (z > 0, say) with a traction-free flat boundary (at z = 0).The rigid body may be completely buried, it may be partially buried (analogous to the geometry for floating-body problems) or it may be attached to the flat surface.Such problems are basic in the context of soil-structure interaction (SSI); for reviews, see [39][40][41][42][43], ( [44], §5) and ( [45], §7.4).For good descriptions of early work, see [41,42] and ( [46], ch. 7).Much of this is concerned with the forced motion of a massless structure.The literature on circular discs oscillating on the flat boundary, usually with simplifying non-welded continuity conditions between the disc and the half-space, is extensive [41]; for a rigid disc welded to the half-space, see [47].
SSI problems can be tackled directly, as one large coupled system, and this may be appropriate if the structure is complicated or if nonlinear phenomena are important; see [48] for a review.However, SSI problems are often decomposed into subproblems in a variety of ways; see ( [41], §7), ( [42], §2), ( [45], §7.4.3) and §6b.Most of these are distinct from that developed for floatingbody problems.In fact, taking that route was suggested by Thau [26] in 1967; in general, his decomposition leads to seven radiation problems, each with prescribed displacements on the surface of the rigid body (see §6a below).Then, it is perhaps not surprising that there is an analogue of Haskind's relation, implying that the diffraction problem (incident wave but fixed rigid body) does not have to be solved completely; for a statement, see (6.10).
The paper begins in §2 with a simple one-dimensional example, in which a wave along a stretched string interacts with a point mass (a bead) attached to the string.This problem can be solved directly but it can also be solved using the kind of decomposition into subproblems described above.
Then, as we are interested in rigid-body dynamics, we give a summary of the relevant theory in §3.We include this section because we could not find a clear presentation in the literature, keeping in mind that in order to couple to the exterior medium, we need to know the displacement of the rigid body's boundary (without introducing the angular velocity vector).Note that for problems involving fluids, coupling is via the normal velocity but this is proportional to the normal component of the surface displacement for time-harmonic problems.
Section 4 reviews methods for treating problems involving water waves and floating bodies.The velocity potential in the water is decomposed into the sum of seven potentials.Each one corresponds to a certain forced motion of the body, with normal velocity specified on the body's surface and outgoing surfaces waves far away.The potentials are combined in such a way that the body's equations of motion are satisfied.Haskind's relation is derived, so that six potentials have to be computed (rather than seven).These can be computed using standard software for water-wave radiation problems.
The basic ideas coming from resolving floating-body problems in §4 are adapted to other physical situations, starting with acoustic problems in §5.In fact, for those problems, all the formulae from §4 apply without any substantial changes: Laplace's equation is replaced by the Helmholtz equation.One example is worked out in detail (plane wave scattering by a rigid sphere): the calculations are straightforward and serve to show how the general formalism gives the expected results.In the general case, again, the six potentials needed can be computed by solving a standard boundary integral equation.
Elastodynamic problems are the subject of §6.We start with the full-space problem: a rigid body is embedded in an elastic solid.Half-space SSI problems are discussed later ( §6c).The displacement in the exterior region is decomposed into the sum of seven displacement fields: this is Thau's decomposition [26].Each one corresponds to a certain forced motion of the body, with displacement specified on the body's surface and outgoing waves far away.These elastodynamic boundary value problems have to be solved for the associated traction vectors on the body.Again, there is a form of Haskind's relation, which eliminates the need to solve one problem.The remaining six problems can be solved by standard numerical methods [44,49].
Alternative decompositions are discussed in §6b.It is argued that these are inferior to Thau's decomposition, even for half-space SSI problems.There are some concluding remarks in §7.
. A simple one-dimensional example Consider a string stretched along the x-axis, with uniform tension T, density per unit length ̺ and wavespeed c = T/̺.The lateral displacement of the string from its equilibrium position is u(x, t).Time-harmonic waves of the form u(x, t) = Re{U e ±ikx e −iωt } can propagate along the string; here U is a constant and k = ω/c.
where the right-hand side is the restoring force provided by the tension in the string.In addition, the displacement must be continuous at the bead, giving There is a wave incident on the bead from the left.This causes the bead to move laterally, with a reflected wave and a transmitted wave.See figure 1. Suppressing the time dependence, we have where R and T are the (unknown) reflection and transmission coefficients.From (2.2), we have Solving this pair of equations gives (2.4) For an alternative approach, one that will generalize to the rest of the paper, we introduce two wavefields.The first (later called the diffraction field) is the solution of the problem when the mass is held fixed: it is This says the incident wave is perfectly reflected, with no transmitted waves; note u d (0±) = 0.The second wavefield is the solution of the radiation problem when the bead oscillates with unit amplitude, in the absence of the incident wave.It is given by waves are radiated symmetrically; note u 1 (0±) = 1.Finally, to determine the solution of the original problem, we take a linear combination, and then determine the constant ξ 1 using the bead's equation of motion.Note first that, by design, 2) is satisfied.Then (2.1) gives −mω 2 ξ 1 = Tik{ξ 1 − (2U − ξ 1 )}.Solving for ξ 1 gives ξ 1 = UT, with T given by (2.4), as before.This example is very simple because (i) the general solution of the governing wave equation is available, and (ii) the bead can move with just one degree of freedom.These factors make a direct treatment feasible (as seen when we started from (2.3)), whereas the alternative approach has more general applicability.To recap, we introduced two solutions, u d and u 1 , and then we determined the correct linear combination of these two using the bead's equation of motion.In

. Rigid bodies (a) Basic formulation
Consider a rigid three-dimensional body B with boundary S and mass density ̺.Introduce a fixed system of Cartesian coordinates Ox 1 x 2 x 3 with associated unit vectors e 1 , e 2 and e 3 .Letx = x 1 e 1 + x 2 e 2 + x 3 e 3 = x i e i , with the usual summation convention.The body has mass m = B ̺(x)dx.I t s centre of mass is at the point G with position vector X G = m −1 B x̺(x)dx with respect to O.The moment of inertia tensor relative to G, J G ,isdefinedby(see [50, eqns (2.39) and (2.40)]) where see, for example, ( [3], eqn (10.140)), ( [45], eqn (1.29)) and ( [22], p. 156, eqn (144)), together with (3.11).We are interested in motions of B. Suppose points X i in B have position vectors X i with respect to O.During the motion, X i moves to a point with position vector x i (t).As B is a rigid body, where Q is a (dimensionless) rotation tensor.This means that Q is a proper-orthogonal secondorder tensor: it satisfies det Q = 1andQQ T = Q T Q = I.Such tensors can be parameterized using three independent parameters.When x 1 = x and X 1 = X, we can write (3.3) as In particular, where x G (t) locates the centre of mass at time t.Subtracting this equation from (3.4), or using (3.3), gives This expresses a well-known fact: the motion of a rigid body can be decomposed into a translation of the centre of mass together with a rotation about the centre of mass.

(b) Equations of motion
Consider a rigid body subjected to a force and a moment applied to its surface S. (Here, we ignore body forces.)Denoting the applied force by F and time derivatives by overdots, the first equation of motion is Thus the translational equation of motion of the centre of mass involves the vectors q and s.
The second equation of motion governs the rotational motion about the centre of mass G.Casey gives it in a convenient form, namely ( [50], eqn (2.35) 2 ), where M G is the applied moment about G, ε is the alternator, M ∧ is a tensor defined by Casey ( [50], eqn (2.44)), to first order, and E is the Euler tensor in the reference configuration, (3.2).The tensor E is related to the inertia tensor relative to G in the reference configuration, J G , by (3.1).In detail, we have and Returning to (3.10), noting that E is symmetric and S is skew-symmetric, it follows that M ∧ is also skew-symmetric.Then, using components in (3.9), Hence, from (3.10) and (3.7), Downloaded from https://royalsocietypublishing.org/ on 13 December 2023 Collecting the two equations of motion, we have The second of these can be solved for s given M G , and then the first equation gives q.
For some purposes, it may be convenient to express both the applied moment and the inertia tensor with respect to O instead of G; denote these quantities by M • and J • , respectively.We have (see [50], eqn (2.23)), where we have used (3.12) 1 .For the inertia tensors, we have the parallel axis theorem (see [3], eqn (10.147) and [4], eqn (7.25)), When combined with (3.12) 1 ,wehave where G is a 3 × 3 matrix, Later, we denote the 6 × 6 matrix on the right-hand side of (3.14) by L; see (3.18).One feature of the calculations above is that we did not introduce the angular velocity vector (often denoted by ω(t)).One reason is we want to extract the displacement at each point in the body, not the velocity of those points (although we shall require the normal velocity of S in §4).

(c) Time-harmonic motions
We are interested in small time-harmonic oscillations of the body B about its equilibrium position; the time dependence is e −iωt .Put q(t) = Re{ q e −iωt } and s(t) = Re{s e −iωt },w h e r e q and s are constant vectors.Henceforth, we suppress the time dependence.
Rigid-body oscillations of B couple to the exterior via the displacement of the boundary S.This displacement is obtained by substituting Q = I + S in (3.4) 1 , giving x − X = q + SX = q + s × X, after use of (3.8).The quantity x − X on the left is the small displacement of the point at X with position vector X in the reference configuration.In the time-harmonic setting, we denote this displacement by u(X) = q + s × X; (3.16) its value at all points X ∈ S will be needed.As expected, there are six DOF in (3.16), the three components of the two constant vectors, q and s.

(d) Summary
At this stage, we have obtained the time-harmonic equations of motion of the rigid body B in terms of the forces and moments applied to its surface S, F and M. The body can move with six DOF.Its displacement is given by (3.16), which we write as where .

Ship motions
A ship may be considered to be a rigid body floating in an ocean.(If deformations of the ship are considered to be important, the theory described below can be extended by including vibrational modes of the ship; this extension leads to what is known as 'hydroelasticity theory' [53].)Gravity plays an essential role: a floating body is (usually) statically stable, so there will be a restoring force when the body moves from its equilibrium position [54,55].We are interested in deviations from the static state caused by interactions with an incident surface gravity wave: the wave will be scattered and the body will move.This is a classic problem in ship hydrodynamics.The governing equations were derived by John ([56], §2) using a systematic perturbation procedure.For a detailed exposition, see ( [57], §7.2); see also [21], ( [58], §19β)and( [52], ch.3).For bodies that are also translating with a constant mean velocity, see [53,59].
The theory starts by assuming that the ocean is an inviscid incompressible fluid, and that the motion is irrotational.There will be surface gravity waves, and these will interact with the body.The fluid motion can be determined by computing a (scalar) potential Φ(x, t), with ∇ 2 Φ = 0i n the fluid and fluid velocity equal to grad Φ.The dynamic pressure is P(x, t) =−̺ w ∂Φ/∂t,where ̺ w is the fluid (water) density.The kinematic condition on the wetted surface of the body, S,i s that the normal velocity of the water ∂Φ/∂n should match the normal velocity of the body.If we choose coordinates so that the mean free surface is at z = 0, with the z-axis pointing upwards, the linearized free-surface boundary condition is ∂ 2 Φ/∂t 2 + g ∂Φ/∂z = 0a tz = 0, where g is the acceleration due to gravity.Finally, for simplicity, we consider deep water, so that there is no motion as z →−∞.
where p = iω̺ w φ.The displacement of B is given by (3.19), so the kinematic boundary condition becomes where the unit normal vector n on S points into the water and the vectors b I are defined by (3.20).

(a) Decomposition into seven problems
There are seven unknowns, the amplitudes ξ I in (3.19) and the velocity potential φ.To proceed, we decompose φ into the sum of seven potentials, where φ d = φ inc + φ sc is known as the diffraction potential, φ inc is the velocity potential of the (given) incident wave, φ sc represents scattered waves, ∂φ d /∂n = 0onS, and φ I and φ sc must satisfy the radiation condition (ensuring outgoing waves at infinity).Thus the decomposition in (4.3) ensures that the kinematic condition, (4.2), is satisfied.For two-dimensional problems, we decompose φ into the sum of four potentials, as shown in figure 2. The other six potentials φ I correspond to forced motion of B in each of the six modes characterized by the vectors b I but in the absence of the incident wave.This kind of decomposition will be used again later for other physical problems.
Each of the seven potentials, φ I and φ sc , solves a boundary value problem with prescribed normal derivative on S (Neumann problem); see (4.4) and (4.5).They can be computed by solving the same boundary integral equation over S apart from different right-hand side functions ([60], §6.6.3),[61].If a 'direct formulation' ( [60], §6.6.2) is used, the output will be the boundary values of the potentials, and these are what will be needed in the next step: computing the forces and moments exerted on the body so that we can solve the equations of motion for B to find the amplitudes ξ I in (4.3).
It is worth comparing with the simpler problem of potential flow generated by a rigid body moving in an unbounded perfect fluid.For that problem, the corresponding potential is decomposed into the sum of six potentials (as in (4.3) but without φ d ), a decomposition that goes back to Kirchhoff in 1869; see ( [12], §118).These define a 6 × 6m a t r i xF with complex frequency-dependent entries F IJ (ω).The real and imaginary parts of F define added-mass and damping coefficients ( [22], §6.17), and these can be used to assess the contributions of each rigid-body mode to the response of the structure and to the waves radiated by the structure.Note that F does not depend on the incident wave.The matrix F is symmetric, F IJ = F JI .To see this, we use the reciprocity relation for two outgoing potentials, φ 1 and φ 2 ,  F IJ ξ J , I = 1, 2, ..., 6.

(b) Equations of motion
(4.9) These drive the motion of B, and so they appear on the left-hand side of (3.17).Hence 6 which is a 6 × 6 system for ξ I .T h e6× 6 symmetric matrix L with components L IJ is given explicitly by (3.18).

(c) Haskind's relation
The quantities D I on the right-hand side of (4.10) are components of F d and M d , and these are defined in terms of certain integrals of the diffraction potential φ d = φ inc + φ sc .It turns out that these integrals can be evaluated without computing φ d itself.
Let us start with the force F d , (4.6) 1 .Its components are Similarly, ) . So, the six-vector D has components This formula shows that the forces and moments induced by the diffraction potential φ d can be calculated without calculating φ d itself.Using (4.4) again, we can write (4.14) as Then, if φ inc satisfies ∇ 2 φ inc = 0 everywhere in the water, the integration over the wetted surface S on the right-hand side of (4.15) can be replaced by an integration over a control surface S ∞ in the far field where far-field approximations to φ I can be used.The formulae in (4.14) and (4.15), with integration over S or S ∞ , are known as Haskind's relation [27]; the derivation given here follows one given by Newman [62].See also ([21], eqn 31) and ([57], §7.6.3).Chertock ([63], eqn (12)) found a similar formula in a slightly different context; see also [64].The decomposition in (4.

. Acoustic scattering by a movable rigid body
The discussion in §4 extends readily to problems where a sound wave in an inviscid compressible fluid interacts with a movable rigid body; in fact, this extension was also made by Haskind [10].For these acoustic problems, the velocity potential Φ(x, t) satisfies the wave equation so that the corresponding potential φ(x) (see (4.1)) satisfies the Helmholtz equation (∇ 2 + k 2 )φ = 0 in the fluid, where k = ω/c and c is the speed of sound.The corresponding dynamic pressure is given by p = iω̺ w φ where ̺ w is the density of the fluid.
All the remaining calculations in §4 can be repeated without changes.The relevant potentials φ I satisfy the Helmholtz equation outside S, the Sommerfeld radiation condition, and a Neumann boundary condition on S, (4.4).Such problems are standard in acoustics, and they can be solved using boundary integral equations ( [60], ch.5); software is readily available [65].

(a) An example: plane-wave scattering by a sphere
Let us consider the scattering of a plane wave by a movable rigid homogeneous sphere.The purpose is to show that the general formalism presented in §4 leads to known results for this very simple problem.
The sphere B has radius a and is centred at the origin.The plane wave propagates in the zdirection, so that where z = x 3 = r cos θ, j n is a spherical Bessel function, P n is a Legendre polynomial, P is a constant (with the same dimensions as φ inc )a n dr and θ are spherical polar coordinates ([60], §4.6).For n is a spherical Hankel function and c n is a constant; this representation ensures that the Sommerfeld radiation condition is satisfied, recalling the time dependence of e −iωt .The boundary condition, ∂φ d /∂r = 0atr = a, gives c n , and then, evaluating on the sphere, having used a Wronskian relation from ( [66], 10.50.1).The only non-trivial component of the diffraction force is , using orthogonality of Legendre polynomials.We also want φ 3 , a radiating solution of the Helmholtz equation satisfying ∂φ 3 /∂r =−iω cos θ on r = a; see (4.4).Evidently, φ 3 (r, θ) = Ah 1 (kr)cosθ with A determined by the boundary condition.
Hence, the only non-trivial entry in the matrix F is where m w = (4/3)π a 3 ̺ w is the fluid mass displaced by B.
It is worth noting that the sphere problem is not typical, in the following sense: the sphere can rotate in any manner without affecting the fluid motion.This non-uniqueness was avoided above by restricting to one-dimensional motion from the outset.Similarly, for a rigid spheroid (or any other axisymmetric body), rotations about its axis do not affect the fluid motion.

. Soil-structure interaction
In this section, we consider scattering of an incident time-harmonic elastic wave Re{u inc e −iωt } by a movable rigid body B. The problem is to compute the scattered waves and the motion of B.As in §4, this scattering problem will be decomposed into seven (or more) simpler problems.This can be done in several different ways, as we shall see.
We start by assuming that B is embedded in an unbounded solid.This implies that the scattered waves must satisfy an elastodynamic form of the Sommerfeld radiation condition.
In the SSI context, it is usual to embed B in a half-space, with a flat traction-free boundary.(For two-dimensional problems, see figure 2, where the incident waves shown could be body waves or surface waves.)We discuss this situation later, in §6c.

(a) Thau's decomposition
In our opinion, the best decomposition mimics what is done for ship motions ( §4).However, it is not the one used in most of the SSI literature; for an exception, see [67].
We start by defining the diffraction field u d = u inc + u sc ,w h e r eu d = 0 on S.T h u su sc satisfies a radiation condition and the boundary condition u sc =−u inc on S. Denote the corresponding traction vector on S by t d .
The other six problems are associated with the six DOF in the motion of the rigid body; its displacement is given by (3.19), involving the six unknowns ξ I , I = 1, 2, ..., 6.Thus, solve six exterior problems for u I  The seven problems for u sc and u I are standard exterior Dirichlet problems in elastodynamics.They can be solved using the same boundary integral equation over S with different right-hand sides.If a direct formulation is used, the output will be the traction vectors, and (most of) these will be needed later.For some details, see ( [60], §6.5) and [49].
Then, we write the displacement as This ensures continuity of u across S; see (3.19), recalling that u d = 0 and u I = b I on S.
The particular decomposition of u into the solution of seven uncoupled problems as (6.1) was introduced by Thau [26] in 1967.Apparently, the advantages of this decomposition were not noticed at the time.

(i) Equations of motion
In order to determine ξ I in (6.1), we calculate the (time-harmonic) forces and moments on the body.From the diffraction field u d ,wehave We put the components of these two vectors into a six-vector D, with From u I , the corresponding forces and moments are Their components are These define a 6 × 6matrixF with complex frequency-dependent entries F IJ (ω).As noted below (4.7), the real and imaginary parts of F can be used to assess the contributions of each rigid-body mode to the response of the structure and to the waves radiated by the structure.Note again that F does not depend on the incident wave.The matrix F is symmetric, F IJ = F JI .To see this, we use the reciprocal theorem, relating two outgoing elastodynamic states, with the usual summation convention.Start by taking I = j and J = k: Next, take I = j and J = k + 3: on S,wehaveu I i t Finally, take I = j + 3andJ = k + 3: on S,wehave Combining (6.2) and ( 6.3), we find the forces and moments acting on B have components given by (4.9), leading to the 6 × 6 system for ξ I , (4.10).

(ii) Haskind-type relations
At this stage, the problem has been solved in terms of seven elastodynamic vectors u d and u I , I = 1, 2, ..., 6, and these can be computed by solving appropriate boundary integral equations.It is not surprising, perhaps, that we can avoid computing u d , much as we avoided computing φ d in §4 because of Haskind's relation.
Let t inc denote the tractions on S corresponding to u inc with components t inc i and u inc i , respectively.Define t sc , t sc i and u sc i similarly.Then, as t d = t inc + t sc , (6.2) 1 gives Using the reciprocity relation, (6.5), with I = j and J = sc, together with u j i = δ ij and u sc i =−u inc i on S, we can rewrite the second integral in (6.6), giving For the moment, (6.2) 2 gives Using the reciprocity relation with dS.
The integrand on the left-hand side is (e j × x) • t sc = e j • (x × t sc ), as in the second integral in (6.8).Hence Putting the components of F d and M d into a six-vector D, we obtain These are used on the right-hand side of the equations of motion, (4.10).We can regard (6.10) as the elastodynamic versions of Haskind's relation, as derived in §4c for ship motions.We note that, in (6.10b), the integration surface can be moved to infinity if u inc is a valid elastodynamic field everywhere outside S.
Let us summarize.As a consequence of (6.10), we have to solve six exterior problems with u I = b I on S and then compute the corresponding tractions t I on S, I = 1, 2, ..., 6.The motion of B is then determined by solving a 6 × 6 system of linear algebraic equations.We shall compare this workload with alternative formulations below.
(b) Alternative decompositions We write the displacement inside B using (3.19).Outside B, we write (cf.(6.1)) ξ I u I (x), (6.11)where u r is a radiating field that satisfies u r =−u * d on S.This ensures continuity of u across S.W e can then proceed as before.Note that we have to solve an additional radiation problem (to find u r ), and we have to calculate the forces and moments associated with u r .Also, the construction of u * d requires solving a problem with traction (Neumann) data (t * sc =−t inc on S); all other problems have displacement (Dirichlet) data.Thus, we conclude that this approach is not attractive.

(ii) Luco's decomposition
Another option is to introduce an exterior problem that has zero forces and moments on S,butnot zero tractions.Here, we use notation similar to that used by Luco [68].He splits the displacement inside B, writing 68], eqn (3c)); each of these can be written as in (3.19), using coefficients ξ R I and ξ * I , respectively.A radiating exterior field u R is constructed, with u R = u R o on S ( [68], eqn (4a)); evidently, where the coefficients ξ R I are to be found.A second radiating exterior field u s is constructed; it takes account of the incident field u inc , and satisfies u s = u * o − u inc on S,( [68], eqn (5a)).Thus where u sc was defined above: u sc = u d − u inc with u d = 0 on S. In addition, the forces and moments induced by u s are required to cancel those induced by the incident field ( [68], eqn (5c)).Enforcing this condition, using the notation introduced earlier (see (6.2) and (6.Solving this 6 × 6 system gives ξ * I and hence u * o .Next, write the exterior field as in ( [68], eqn (3a)), This matches with the interior field on S. The associated forces and moments come from u R alone (because of the condition imposed on u s ).These drive the motion of B. Then, proceeding as before, we can obtain equations for the remaining coefficients ξ R I .This approach seems much more cumbersome than Thau's approach because one needs to compute nine exterior fields and two sets of coefficients, ξ * I and ξ R I .
(iii) The 'substructure theorem' A standard approach for solving elastodynamic inclusion (transmission) problems proceeds as follows.Let u S denote the (unknown) total displacement of S.
an equation for u S that holds on S; it is the continuous analogue of the 'substructure theorem' in Kausel's book ( [45], eqn (7.150)),where the focus is on discrete formulations.Let us simplify the right-hand side of (6.12), noting first that t inc = Z u inc ; for the latter, we want the outgoing field u out inc , say, satisfying u out inc = u inc on S, and then Z u inc gives the tractions on S corresponding to u out inc .However, comparison with §6a shows that u out inc =−u sc whence Z u inc − t inc =−t d .
Suppose now that the elastic inclusion is replaced by a rigid body B: how does this affect (6.12) and the interpretation of Z int ?The 'welded' condition, t ext = t int on S is replaced by conditions involving integrals of t ext over S (giving the forces and moments acting on B) and the equations of motion of B. In addition, we know that we can write u S as (3.19), whence (6.12) becomes where the six traction vectors t J are defined by solving six exterior problems, as in §6a.equation (6.13) must be projected onto a six-dimensional space in order to determine the six coefficients ξ I ; all the other formulations in §6 do not require projections because they deal directly with forces and moments.Denote the projection operator by P; given a traction vector t(x)definedfor x ∈ S,defineP t to be the six-vector containing the forces and moments acting on B.Inparticular , (P t d ) I = D I and (P t J ) I = F IJ ; see (6.2) and (6.3).Hence, applying P to (6.13) gives Comparison with (4.10) gives (PZ int b J ) I =−ω 2 L IJ , which may be used to interpret Z int ,ifsuchan interpretation is desired.We conclude that the apparent simplicity of (6.12) is not easy to exploit when applied to waveinduced motions of rigid bodies.Nevertheless, discrete formulations based on (6.12) are popular, perhaps because of the ubiquity of finite-element methods.

(c) Half-space problems
Consider a homogeneous elastic half-space, z > 0, with a traction-free boundary at z = 0. We suppose that there is a plane wave propagating towards the flat boundary; this is the incident field u inc .There will be reflected plane waves, both P-waves and S-waves in general.These are readily calculated in the absence of the body ( [69], ch.6), and then the sum of the incident and reflected plane waves is known as the free-field displacement u ff ; in the context of SSI, u ff gives 'the motion that the ground would have experienced if neither the soil had been excavated nor the structure erected' ( [45], p. 514).
For the analogue of Thau's decomposition ( §6a), we replace u inc by u ff , defining the diffraction field by u d = u ff + u sc with u d = 0 on S.T h efi e l du sc is required to be outgoing and to satisfy the zero-tractions condition on z = 0 (outside B if the body intersects z = 0); these conditions are also imposed on the fields u I .Similar alterations are required in the alternative decompositions outlined in §6b.
Construction of the fields u I is more complicated than for full-space problems because they have to satisfy the zero-traction condition on the flat boundary.Appropriate boundary integral equations can be derived, using a 'half-space Green function' (one that satisfies the zero-traction Downloaded from https://royalsocietypublishing.org/ on 13 December 2023

. Conclusion
We reviewed several physical situations where a three-dimensional rigid body interacts with a time-harmonic incident wave.After reviewing the classical theory of rigid-body motions, we discussed problems involving floating bodies (ships) and other offshore structures: a surface gravity wave is incident on the structure, causing it to move and affecting how the incident wave is scattered.These problems have a rich literature, dating back to the 1940s.They require the determination of a scalar potential φ, and the method of choice is to decompose φ into seven potentials, one for each of the six rigid-body modes (three translations and three rotations), and one to take account of the incident wave.The six potentials corresponding to the rigid-body modes are independent; they depend on the geometry and the frequency, but they do not involve the incident wave.The contribution from the seventh potential can be computed in terms of the other six, using Haskind's relation.Exactly the same approach can be used for acoustic problems.
The same basic approach can be used for elastodynamic problems, where an incident elastic wave interacts with a rigid body.For full-space problems, Thau's decomposition [26] into seven subproblems is straightforward: of course, they are vector problems but they can be solved numerically using standard methods; and a form of Haskind's relation in available.The decomposition into subproblems is advantageous because it breaks the original problem into seven problems of the same basic type (exterior problems with different specified displacements of the body's surface), allowing the contributions from the incident waves and the rigid-body modes to be separated: there is also a close analogy with the standard introduction of added-mass and damping matrices in hydrodynamics.
The extension to half-space problems (as usually encountered in problems of SSI) is also straightforward, but now it is not so easy to solve the six subproblems because the traction-free condition on the flat boundary of the half-space has to be imposed.Nevertheless, this approach does provide a rigorous way to analyse such problems, and we believe that further investigations are warranted.
To conclude, we should not lose sight of the underlying assumptions behind the analysis.First, we used linear theory: the motions are assumed to be small.This is standard in many situations involving wave-induced motions and, moreover, the resulting predictions are often better than one might expect.(It is difficult to quantify how small the motions should be.)Large-amplitude motions are certainly of interest (especially those of floating bodies), but nonlinear theories are expensive to deploy because they inevitably require solving partial differential equations numerically in an unknown moving domain coupled to the motion of the rigid body.We assumed time-harmonic motions; motions that are truly time dependent can be studied, in principle, by Fourier analysis, or by other methods [16].We also assumed that we are dealing with rigid bodies.This could be relaxed (e.g.buildings can deform), but then the six rigid-body modes would have to be augmented with the vibrational modes of the body [53].
Data accessibility.This article has no additional data.Authors' contributions.P.A.M.: conceptualization, formal analysis, investigation, methodology, validation, writing-original draft, writing-review and editing.
Con ict of interest declaration.I declare I have no competing interests.Funding.No funding has been received for this article.Acknowledgements.The author thanks Harry Dankowicz, Chris Garrett and three anonymous referees for constructive comments on earlier drafts of the paper.

Figure 1 .
Figure 1.A snapshot of the bead on the displaced string at time t.

Figure 2 .
Figure 2. The solution of the two-dimensional problem is decomposed into four basic subproblems.
Denote the pressure corresponding to φ d by p d = iω̺ w φ d .The corresponding (time-harmonic) force and moment on the rigid body are F d =− S p d n dS and M d =− S p d x × n dS.(4.6)We put the components of these two vectors into a six-vector D, with D i = F d i and D i+3 = M d i .Next, consider the contributions from the potentials φ I , I = 1, 2, ..., 6. Denote the pressure corresponding to φ I by p I = iω̺ w φ I .The corresponding forces and moments on B are F I =− S p I n dS and M I =− S p I x × n dS.Their components are F Ii =− S p I n i dS and M Ii = F I,i+3 =− S p I (x × n) i dS.(4.7)