## Abstract

An exact expression for the fluid-coupled structural waves that propagate in a three-dimensional, rectangular waveguide with elastic walls is presented in terms of the non-separable eigenfunctions ψ_{n}(*y*,*z*). It is proved that these eigenfunctions are linearly dependent and that an eigenfunction expansion representation of a suitably smooth function *f*(*y*,*z*) converges point-wise to that function. Orthogonality results for the derivatives ψ_{ny}(*a*,*z*) are derived which, together with a partial orthogonality relation for ψ_{n}(*y*,*z*), enable the solution of a wide range of acoustic scattering problems. Two prototype problems, of the type typically encountered in two-part scattering problems, are solved, and numerical results showing the displacement of the elastic walls are presented.

## Introduction

The propagation of acoustic waves along ducts or pipes has long been of interest to scientists and engineers. This is partly because ducting systems, such as those used for heating, ventilation and air conditioning (HVAC), provide ideal channels for the transmission noise through structures such as buildings and aircraft. Acoustic scattering is a feature of ducting systems that becomes relevant whenever there is an abrupt change in geometry or material property. At such a discontinuity, the incident sound field undergoes both reflection and transmission, usually initiating all propagating higher-order modes in the process. Capitalizing on this phenomenon, much research into the effective design of reactive silencers for ducting systems has been carried out. In recent years, there has been increasing interest in the effect that wave-bearing boundaries have on the propagating noise and, in this context, much of the analytic work concerns two-dimensional ducts or those circular cylindrical geometries. A wide range of analytic techniques, including the Wiener–Hopf technique (Rawlins 2007), Fourier methods (Huang 2002) and mode-matching (Lawrie & Guled 2006), have been employed to study this class of problem.

The Wiener–Hopf technique has long been a convenient tool to tackle a wide range of two-part problems, whereas analytic mode-matching has only recently been established as a viable tool for two-dimensional structures in which the boundary conditions contain higher-order derivatives. The reason for this is that the eigenfunctions for such systems do not satisfy standard orthogonality conditions, such as that for the set of functions cos(*nπy*/*a*), *n*=0,1,2, …. Orthogonality relations (ORs) do, however, exist albeit of non-standard form, and Lawrie & Abrahams (1999) were among the first authors to demonstrate this. Since then, this class of OR has arisen in a number of different physical situations (e.g. Evans & Porter 2003; Kaplunov *et al.* 2004; Manam *et al.* 2006).

Problems involving the propagation of guided waves in three-dimensional media are often more difficult to analyse. Recent studies include those of Zakharov (2008), who derived ORs for three-dimensional waves in visco-elastic laminates and Zernov & Kaplunov (2008), who studied three-dimensional edge waves on plates. Three-dimensional guided waves are of particular relevance to the HVAC industry because many HVAC ducts have a rectangular cross section. For this reason, there is a vast body of literature concerning the propagation of acoustic waves in such ducts. Most of the literature, however, deals with ducts that have rigid walls, and the primary reason for this is the inherent difficulties involved in analysing those with flexible walls. Articles that do address the effects of flexible duct walls include those of Cabelli (1985), Lawrie & Abrahams (2002), Huang & Choy (2005) and Martin *et al.* (2004), the latter of which is primarily an experimental investigation of the wall displacement.

This article is concerned with the propagation of sound in a three-dimensional duct, of rectangular cross section, formed by three rigid walls and closed by a thin elastic plate (figure 1). The analysis herein builds upon the work of Lawrie & Abrahams (2002) who first presented a non-separable ansatz for the duct modes together with a partial OR. That article offered a pioneering insight into the three-dimensional case; however, it was clear that the partial OR is insufficient to enable the efficient solution of some typical scattering problems. The partial OR is based on that for the underlying two-dimensional system (Lawrie & Abrahams 1999) and thus, while incorporating the appropriate orthogonality properties in the *y*-direction, it neglects any such properties for the *z*-direction. Recent developments, however, in the theory underpinning such two-dimensional systems, have paved the way for a better understanding of their three-dimensional counterparts (Lawrie 2007). Thus, the aim of this article is twofold. Firstly, it is intended to establish the analytic properties of the eigenfunctions for the three-dimensional waveguide shown in figure 1. Secondly, it is intended to investigate the potential use of these eigenfunctions in mode-matching problems for three-dimensional geometries.

In §2, following Lawrie & Abrahams (2002), the eigenfunctions and dispersion relation are presented. Some significant new results are derived in §3: it is proved that the eigenfunctions are linearly dependent and that an eigenfunction expansion of a suitable function *f*(*y*,*z*) converges point-wise to that function. In §4, the derivatives ψ_{ny}(*a*,*z*), *n*=0,1,2,…, are considered. It is demonstrated that these functions are also linearly dependent, and some new orthogonality results are presented. Two prototype problems are studied in §5. These two involve a semi-infinite duct of the type shown in figure 1. In the first, an acoustic field is generated by a prescribed pressure at the end of the duct, and the edge of the plate is assumed to be simply supported (pin-jointed). The second is forced by a prescribed velocity field, and spring-like edge conditions are applied. Some concluding remarks are presented in §6.

## Travelling wave solutions

In this section, the non-separable ansatz for the duct modes, posed by Lawrie & Abrahams (2002), is presented. The duct occupies the region −∞ < *x* < ∞, 0≤*y*≤*a* and −*b*≤*z*≤*b*, where *x*,*y* and *z* are Cartesian coordinates that have been non-dimensionalized with respect to *k*^{−1} with *k* being the fluid wavenumber. An elastic plate of infinite length and width 2*b* bounds the duct at *y*=*a*, −*b*≤*z*≤*b*, whereas the remaining three sides are rigid (figure 1). A compressible fluid of density ρ and sound speed *c* occupies the interior region of the duct, but the exterior region is *in vacuo*. The travelling waves are assumed to have harmonic time dependence, i.e. exp(−i*t*, where *t* has been non-dimensionalized with respect to ω^{−1}, with ω=*c**k*. Without loss of generality, it can be assumed that the duct modes propagate in the positive *x*-direction, so that the non-dimensional, time-independent velocity potential ϕ(*x*,*y*,*z*) assumes the form
2.1
where *B*_{n} is the amplitude of the *n*th travelling wave, *s*_{n} the axial wavenumber (as yet unknown, but assumed to be either positive real or have positive imaginary part) and the non-separable reduced potential ψ is to be determined. It is convenient, in the first instance, to treat the wavenumber as a continuous variable *s* rather than a discrete set of values *s*_{n}. Thus, when written in terms of ψ(*s*,*y*,*z*), Helmholtz’ equation reduces to
2.2
The normal component of fluid velocity vanishes at the three rigid walls which, on assuming even eigenmodes, ϕ(*x*,*y*,−*z*)=ϕ(*x*,*y*,*z*), implies
2.3
2.4
The boundary condition that describes the deflections of the thin elastic plate bounding the top of the duct is
2.5
where μ is the *in vacuo* plate wavenumber and α a fluid loading parameter. Thus, μ^{4}=*c*^{2}*h*ρ_{p}/*B**k*^{2} and α=*c*^{2}ρ/*B**k*^{3}, where *B*=*E**h*^{3}/12(1−ν^{2}) is the bending stiffness of the plate in which *E* is Young’s modulus, ρ_{p} is the density of the plate and ν is Poisson’s ratio. Details pertaining to the derivation of equation (2.5) are given by Grant & Lawrie (2000) and the references therein.

The vertical displacement of the plate is proportional to ϕ_{y}(*x*,*a*,*z*), where the subscript *y* here and henceforth denotes differentiation. Further, the elastic plate is assumed to be clamped to the adjacent rigid sides of the duct; hence, the displacement and gradient along these edges are zero and it follows that
2.6
2.7
These conditions will henceforth be referred to as the *corner conditions*. Note that alternative conditions, such as free plate edges, could equally well have been chosen.

Lawrie & Abrahams (2002) propose the following ansatz for ψ(*s*,*y*,*z*):
2.8
This respects the fact that ψ is chosen to be even in *z* (equation (2.4)). The functions *Y*_{m}(*y*), *m*=0,1,2,…, from equations (2.2) and (2.3), satisfy the following simple ordinary differential equation and boundary condition:
2.9
which yields *Y*_{m}(*y*) = cosh(γ_{m}*y*), where γ_{m} is related to τ_{m}(*s*) through . A further condition for *Y*_{m}(*y*) can be obtained by ensuring that ψ satisfies equation (2.5). On substituting equation (2.8) into equation (2.5), it is found that the quantities γ_{m}, *m*=0,1,2,…, are defined by
2.10
where the prime symbol denotes differentiation with respect to the argument (in this case *y*). Note that this notation is used to denote the derivative for functions of only one variable, otherwise the full partial derivative or the subscript notation is used.

Although *non-Sturm–Liouville*, the eigen system specified by equations (2.9) and (2.10) is well studied. It is a straightforward procedure to show that the appropriate OR for this system is
2.11
where δ_{jm} is the usual Kronecker delta and
2.12
Furthermore, the eigenfunctions *Y*_{j}(*y*), *j*=0,1,2,…, have the following properties:
2.13
2.14
2.15
where δ(⋅) is the usual Dirac delta function. These results are established by Lawrie (2007)1 and, in view of their significance to the analysis that follows, it is useful to relate the notation of that article to that used herein. The above eigen system is retrieved from the general system considered by Lawrie on setting *P*_{a}(*s*)=*s*^{4}−μ^{4}, *Q*_{a}(*s*)=−α, *P*_{0}(*s*)=1, *Q*_{0}(*s*)=0. It should also be noted that the definition of *C*_{m} above differs from that in Lawrie (2007) by a multiplicative α.

Expression (2.11) can now be employed to impose the condition of no normal velocity on the rigid face at *z*=*b* (equation (2.4)). On multiplying by and summing over *j*, it is found that
2.16
where
2.17
The first and third terms in equation (2.16) are zero (due to boundary condition equation (2.4) and corner constraint equation (2.7)) and it follows that
2.18
The ansatz for ψ(*s*,*y*,*z*) (equation (2.8)), can be multiplied by any arbitrary constant (or function of *s*) and so, without loss of generality, it is chosen that *P*(*s*)=1. It follows that the duct modes have the form
2.19
Note that equations (2.17) and (2.18) confirm equation (2.14). Equation (2.19) contains the quantity *s* that corresponds to the axial wavenumber of the mode and is, as yet, unspecified. Condition (2.6), however, has not been imposed and it is this that gives rise to the dispersion relation, i.e.
2.20
Despite its complicated form, it is a straightforward, if tedious, procedure to numerically solve equation (2.20) and the roots *s*_{n}, *n*=0,1,2,…, are the admissible axial wavenumbers for the duct modes. Further, it is straightforward to show that

for every root

*s*_{n}there is another root −*s*_{n},there is a finite number of real roots,

there is an infinite number of roots located on ℜ(

*s*)=0, ℑ(*s*)>0, andthere is an infinite number of roots with non-zero real and imaginary parts.

In order that equation (2.1) represents only waves that travel in the positive *x*-direction and/or decay exponentially as *x*→∞, the convention is adopted that the +*s*_{n} roots have either ℜ(*s*_{n})>0, ℑ(*s*_{n})=0 or ℑ(*s*_{n})>0. They are ordered sequentially, real roots first, starting with the largest real root and then by increasing imaginary part. Thus, *s*_{0} is always the largest real root. It is important to note that for any complex root, say *s*_{c}, lying in the first quadrant of the complex *s*-plane, then −*s**_{c}, where the asterisk indicates the complex conjugate, also lies in the upper half-plane. Such pairs are positioned in the sequence of roots according to the magnitude of their imaginary part and, furthermore, in the order *s*_{c} followed by −*s**_{c}. It is worthwhile commenting that these roots lie on curves reminiscent of the parabolic arcs observed in studies of two-dimensional elastic waveguides (e.g. Besserer & Malischewsky 2004). Finally, it is assumed that *s*_{n}≠0 and that no root is repeated. Where appropriate, the notation ψ_{n}(*y*,*z*) will henceforth be used to indicate that *s* has been replaced by *s*_{n} in the function ψ(*s*,*y*,*z*). Likewise, τ_{m}(*s*_{n}) becomes τ_{mn}, etc.

In §§3 and 4, several properties of the eigenfunctions ψ_{n}(*y*,*z*), *n*=0,1,2,…, and their derivatives ψ_{ny}(*a*,*z*), *n*=0,1,2,…, are presented. In §5, these results are used to solve two prototype scattering problems.

## Analytic properties of the eigenfunctions

## Lemma 3.1

The set of eigenfunctions ψ_{n}(*y*,*z*), *n*=0,1,2,…, satisfy the following orthogonality relation:
3.1
where *D*_{n}=−*L*′(*s*_{n})/2*s*_{n} and δ_{ℓn} is the Kronecker delta function.

## Proof.

This result (first presented by Lawrie & Abrahams 2002) is verified by substituting the series expressions for ψ_{n}(*y*,*z*) and ψ_{m}(*y*,*z*) (i.e. equation (2.19) with *s* replaced by *s*_{n} and *s*_{m}, respectively) into the left-hand side of equation (3.1). On interchanging the order of summation and integration and using equation (2.11), expression (3.1) follows. ▪

## Theorem 3.2

The eigenfunctions ψ_{n}(*y*,*z*) described above are linearly dependent for 0≤*y*≤*a* and 0≤*z*≤*b*.

## Proof.

In order to prove this result, it is necessary to determine only one infinite sum of the eigenfunctions which is zero. Such a sum can be constructed by analysing the families of poles in the integrand of a suitably chosen integral. The method is the same as that used by Lawrie (2007). The appropriate integral is
3.2
where 0≤*y*≤*a*, 0≤*w* and *z*≤*b* and the path of integration is indented above (below) any poles on the negative (positive) real axis. Note that ψ_{y}(*s*,*a*,*w*) is obtained directly on differentiating equation (2.19) with respect to *y*, *L*(*s*) is given by equation (2.20) and
3.3
which, on differentiating with respect to *z*, yields ψ(*s*,*y*,*z*).

The integrand has three families of poles:

Those corresponding to the admissible wavenumbers

*s*_{n},*n*=0,1,2…, and defined by*L*(*s*)=0.Those corresponding to .

A doubly infinite family of poles defined by . That is, occurring when ,

*m*=0,1,2,…,*n*=1,2,3….

The path of integration can be deformed onto a semi-circular arc of radius *R*≫1 in the upper half-plane. On evaluating the residue contributions for the poles crossed and noting that there is no contribution from the arc as , it is found that
3.4
It is straightforward to show that
3.5
and thus expression (3.4) becomes
3.6
where ε_{0}=2 and ε_{n}=1, *n*>0. It is interesting to observe that this now takes the form of a separable function: an infinite sum of the eigenfunctions *Y*_{m}(*y*) times an infinite sum of the functions . The latter can be evaluated as
3.7
where *H*(⋅) is the usual Heaviside function, defined such that *H*(0)=1/2. On differentiating this with respect to *z*, it is found that
3.8
Thus, the sum of interest has been expressed as a product of a sum of the eigenfunctions *Y*_{m}(*y*) and some delta functions. Due to the linear dependence of the functions *Y*_{m}(*y*) (equation (2.13)), the sum in the right-hand side of equation (3.8) is in fact zero. It follows that
3.9
▪

This result demonstrates that the eigenfunctions ψ_{n}(*y*,*z*) are linearly dependent. In fact, equation (3.9) is not the only such sum. On replacing ψ_{y}(*s*,*a*,*w*) with ψ_{yyy}(*s*,*a*,*w*) in the integrand of equation (3.2) and repeating the above calculation, it is found that
3.10
which on using equation (2.13) reduces to
3.11

## Lemma 3.3

The eigenfunctions ψ_{n}(*y*,*z*) described by equations (2.19) and (2.20) have the property
3.12

## Proof.

As for theorem 3.2, this result can be proved by analysing the families of poles in the integrand of a suitably chosen integral. The appropriate integral is
3.13
where 0≤*y*≤*a* and 0≤*w*,*z*≤*b* and the path of integration is indented above (below) any poles on the negative (positive) real axis and
3.14
Note that on differentiating equation (3.14) with respect to both *y* and *z* and putting *s*=*s*_{n}, ψ_{n}(*y*,*z*) is retrieved. The pole structure of this integrand is the same as that of *I*_{1}(*y*,*z*,*w*) and, on deforming the path of integration onto a semi-circular arc of radius *R*≫1 in the upper half-plane and evaluating the residue contributions for the poles crossed, it is found that, as ,
3.15
Again the variables have been separated and thus, the sum of interest has been expressed in the form ‘a sum dependent on *y* times a sum that is dependent on *z*’. It follows that
3.16
Then, on differentiating with respect to both *y* and *z*, and using equations (2.9) and (2.15), it is found that
3.17
▪

## Theorem 3.4

Given that the coefficients
exist, where *f*(*y*,*z*) is any function that is three times differentiable on the domain 0≤*y*≤*a* and 0≤*z*≤*b*, and the eigenfunctions ψ_{n}(*y*,*z*) are defined by equations (2.19) and (2.20), then the series
3.18
converges point-wise to *f*(*y*,*z*) for 0≤*y*≤*a* and 0≤*z*≤*b*.

## Proof.

Assume that a suitably smooth function *f*(*y*,*z*), where 0≤*y*≤*a* and 0≤*z*≤*b*, can be expressed as an eigenfunction expansion in terms of ψ_{n}(*y*,*z*), *n*=0,1,2,…. Let *F*_{N}(*y*,*z*) be the sum of the first *N* terms of this eigenfunction expansion, thus
3.19
On substituting the coefficients *B*_{n} into equation (3.19), it is found that
3.20
which, after interchanging the order of differentiation and integration and then letting , may be written as
3.21
where
3.22
Now, by using equations (3.9) and (3.11) and lemma 3.3, it is found that
3.23
where 0≤*y*≤*a* and 0≤*z*≤*b*. ▪

## The derivatives of the eigenfunctions

It has been proved that a function *f*(*y*,*z*) can be represented as an eigenfunction expansion in terms of the eigenfunctions ψ_{n}(*y*,*z*). The partial OR (equation (3.1)) uses, however, only the orthogonality properties of the functions *Y*_{n}(*y*). That is, it neglects any orthogonality properties in the *z* dependence of ψ_{n}(*y*,*z*). Yet, by differentiating equations (3.8) and (3.10) with respect to *y*, setting *y*=*a* and using equations (2.13) and (2.14), the following two results are obtained:
4.1
and
4.2
where 0≤*w*,*z*≤*b*.

Expressions (4.1) and (4.2) provide useful information about the set of functions ψ_{ny}(*a*,*z*). The first of these expressions indicates that these functions are linearly dependent, whereas the other hints at the existence of an OR for this set of functions. Further, by differentiating equation (3.21) with respect to *y*, setting *y*=*a* and using equations (3.9) and (4.1), it is found that
4.3
where 0≤*z*≤*b* and
4.4
It may be deduced from equation (4.3) that an arbitrary, suitably differentiable function *q*(*z*) defined for 0≤*z*≤*b* can be expressed as an eigenfunction expansion in terms of the functions ψ_{ny}(*a*,*z*), i.e.
4.5
where
4.6

At this point, it is expedient to remark that, unlike equations (4.1) and (4.2), the sum
4.7
is neither convergent nor can it be expressed in terms of delta functions. This is an important implication for the class of function that can be represented as an eigenfunction expansion of the form of equation (3.18). If equation (3.21) is differentiated three times with respect to *y*, and *y* is then set to *a*, the divergent sum (equation (4.7)) arises. Thus, in order for the eigenfunction expansion (equation (3.18)) to be three times differentiable, it is necessary that *f*_{y}(*a*,*z*)=0,0≤*z*≤*b*. Expression (4.7) also impinges upon the results that follow immediately below.

In view of equations (3.9) and (4.1), it is clear that the expansion (equation (4.5)) is not unique. The coefficient , *n*=0,1,2,…, can be generalised as
4.8
where *p*(*w*) and λ(*w*) are suitable but arbitrary functions. Then *q*(*z*) is still given by equation (4.5) but with replaced by *A*_{n}. A natural question to ask is: given the coefficients *A*_{n}, can *p*(*z*) and λ(*z*) be constructed as an eigenfunction expansion similar in form to equation (4.5)? In fact, it is relatively straightforward to construct such a representation for the function *p*(*z*). On multiplying equation (4.8) by αψ_{n}(*y*,*z*) and summing over *n*, then integrating with respect to *y*, 0≤*y*≤*a*, and using equation (3.17), it is found that
4.9
where
4.10
It is a little more complicated, however, to construct the function λ(*z*). As the sum (equation (4.7)) is not defined, it is not possible to multiply through by ψ_{nyyy}(*a*,*z*) and then sum. Instead it is necessary first to construct the coefficient
4.11
where equations (4.5) and (4.9) have been used implying that
4.12
and
4.13
Then
4.14
and it follows that
4.15

The following example is included to demonstrate the effectiveness of the three eigenfunction expansions (4.5), (4.9) and (4.15). It should be noted that *q*(*z*) and λ(*z*) must be smooth and satisfy and ; otherwise they are arbitrary. The function *p*(*z*), on the other hand, must satisfy . These restrictions are consistent with the corner conditions satisfied by ψ_{ny}(*a*,*z*), *n*=0,1,2,…. Thus, the functions *p*(*z*), *q*(*z*) and λ(*z*) are chosen to be ; *q*(*z*)=(*z*^{2}−*b*^{2}/16)(*z*^{2}/*b*^{2}−1)^{4}; and λ(*z*)=(4*z*^{2}/*b*^{2}−1)(36*z*^{2}/*b*^{2}−1)(*z*^{2}/*b*^{2}−1)^{6}. The eigenfunctions used to generate the series representations of these functions are those corresponding to a duct with physical dimensions *a**k*^{−1}=0.15 m, *b**k*^{−1}=0.31 m and bounded by an aluminium plate of thickness 0.002 m. The frequency is 85 Hz, for which the dispersion relation has two real roots. Figure 2*a,b* shows the functions *p*(*z*) and *q*(*z*) together with their eigenfunction representations (evaluated using 30 and 45 terms, respectively). Figure 3 shows λ(*z*) and its eigenfunction representation evaluated using 60 terms. In each case, the eigenfunction representations overlie the actual functions. Note that although it is required only that *q*(*z*) and λ(*z*) have a zero of order 2 at *z*=*b*, the eigenfunction expansions generally converge faster the higher the order of this zero.

## Two prototype scattering problems

In this section, the use of equations (3.1) and (4.2) for solving two typical scattering problems is demonstrated. The problems are drawn from a class often encountered when using mode-matching to solve a two-part problem involving acoustic propagation in a duct though normally for the two-dimensional case (e.g. Lawrie & Kirby 2006). For both prototype problems the duct is semi-infinite, lying in the region *x*>0, but in all other respects is identical to that described at the beginning of §2. In the first case, the scattered field is generated by a prescribed pressure distribution applied to the surface *x*=0, 0≤*y*≤*a*, 0≤*z*≤*b* and the plate is simply supported along *x*=0, 0≤*z*≤*b*, whereas for the second, the velocity is prescribed and spring-like edge conditions are applied. In both cases, numerical results are generated for a duct with physical dimensions *a**k*^{−1}=0.15 m, *b**k*^{−1}=0.31 m and bounded by an aluminium plate of thickness 0.002 m.

### Prescribed pressure

The fluid velocity potential can be expressed in terms of the eigenmodes ψ_{ℓ}(*y*,*z*) as
5.1
where the amplitude *B*_{ℓ} is determined from the applied pressure and the ‘pin-jointed’ edge conditions:
5.2
5.3
As the edge conditions are applied on *x*=0 and *y*=*a* for 0≤*z*≤*b*, it follows that ϕ_{yz}(0,*a*,*z*)=ϕ_{yzz}(0,*a*,*z*)=0, where 0≤*z*≤*b*. Thus, due to Helmholtz’s equation, ϕ_{yxx}(0,*a*,*z*)+ϕ_{yyy}(0,*a*,*z*)=0 which enables the edge conditions to be expressed as ϕ_{y}(0,*a*,*z*)=ϕ_{yyy}(0,*a*,*z*)=0, where 0≤*z*≤*b*. On multiplying equation (3.1) by *B*_{ℓ} and summing over ℓ and applying the edge conditions, it is found that
5.4
in which *f*(*y*,*z*) is the (known) non-dimensional pressure applied to the end face *x*=0, 0≤*y*≤*a* and −*b*≤*z*≤*b*.

To demonstrate the above solution, numerical results are presented for the case in which the pressure distribution applied at *x*=0 is
and the frequency is 155 Hz (for which the dispersion relation has three real roots). Figure 4*a,b* shows the ‘plate displacement’ ϕ_{y}(*x*,*a*,*z*) obtained using equation (5.1) with the coefficients given by equation (5.4). A section of the displacement (taken in the *x**y*-plane at *z*=*b*/3) is shown in figure 4*c,d*. This confirms that, as expected for a simply supported plate, the plate displacement is zero, whereas the gradient is non-zero at *x*=0. Note that, for this example, all the eigenfunction expansions have been evaluated using 50 terms. Note also that the applied pressure is separable, ‘even’ in *z*, and has been chosen such that the *z* dependence satisfies the constraints discussed in §4, i.e. *f*(*y*,0)≠0 and *f*(*y*,*b*)=*f*_{z}(*y*,*b*)=0. In addition to these constraints, it is necessary that *f*_{y}(0,*z*)=*f*_{y}(*a*,*z*)=0. The first of the last two is a reflection of the fact that ψ_{ny}(0,*z*)=0, *n*=0,1,2…, and the second, as mentioned above, is required in order that *f*_{yyy}(*a*,*z*) can be expressed as an eigenfunction expansion in terms of ψ_{nyyy}(*a*,*z*), *n*=0,1,2,….

### Prescribed velocity

Similar to the above case, the fluid velocity potential can be expressed in terms of the eigenmodes ψ_{n}(*y*,*z*) as
5.5
where λ(*w*) is unrelated to that of §4. The integral in equation (5.5) is an arbitrary eigensolution and, in view of equation (3.9), it is clear that the *x* derivative of this term vanishes when *x*=0. Thus, the boundary condition at *x*=0, 0≤*y*≤*a* and −*b*≤*z*≤*b* is expressed as
5.6
where *f*(*x*,*y*) is the (known) velocity distribution applied to the end of the duct. The edge conditions to be applied are a spring-like condition, i.e. ϕ_{yxxx}+βϕ_{y}(0,*a*,*z*)=0, 0≤*z*≤*b*, where β is proportional to the spring stiffness, together with the zero gradient condition ϕ_{yx}(0,*a*,*z*)=0, 0≤*z*≤*b*. Note that as , this pair of conditions tend to those for a clamped edge. Note also that the clamped edge conditions cannot be applied directly as they give rise to a Fredholm integral equation of the first kind, for which it is unclear if a unique solution exists.

Thus, on multiplying equation (3.1) by *B*_{ℓ}*i**s*_{ℓ}, summing over ℓ and applying the zero gradient edge condition, we obtain
5.7
Analogous to the previous problem, ϕ_{yxxx}(0,*a*,*z*)+ϕ_{xyyy}(0,*a*,*z*)=0 and hence the remaining edge condition can be expressed as ϕ_{xyyy}(0,*a*,*z*)−βϕ_{y}(0,*a*,*z*)=0, 0≤*z*≤*b*, which on using equation (5.5) can be expressed as an integral equation. It is found that
5.8
where equation (4.2) has been used in order to isolate the unknown function λ(*z*) on the left-hand side and *B*_{n} is given by equation (5.7). This is a Fredholm integral of the second kind and is straightforward to solve numerically.

To demonstrate the above solution, numerical results are presented for the case in which the velocity distribution applied at *x*=0 is *f*(*y*,*z*)=(*z*^{2}/*b*^{2}−1)^{6}(*y*^{2}/*a*^{2}−1)^{2}. The frequency is taken to be 50 Hz (for which the dispersion relation has only one real root). Figure 5 shows the function λ(*z*), 0≤*z*≤*b*, obtained by solving equation (5.8) numerically. It is worth noting that λ(*z*) satisfies the conditions and , and that, in order to ensure this, it was necessary to choose the applied velocity distribution to satisfy *f*_{yyy}(*a*,*b*)=0. Figure 6*a,b* shows the plate displacement ϕ_{y}(*x*,*a*,*z*) obtained using equation (5.5) having determined λ(*z*) with β=100 000. It is clear that ϕ_{y}(0,*a*,*z*)≈0,−*b*≤ *z* ≤ *b*. A section of the plate displacement (taken in the *x**y*-plane at *z*=*b*/3) is shown in figure 6*c,d*. This confirms that the plate displacement is close to zero and the gradient is zero at *x*=0. Note that, for this example, the eigenfunction expansions have been evaluated using a maximum of 45 terms. Finally, figure 7 shows the absolute value of the non-dimensional plate displacement at the edge *x*=0 plotted against β for three fixed values of *z*, i.e. *z*=0, *b*/3, 3*b*/4. When β=0 the spring-like condition reduces to the zero force condition; as β increases it is clear that, after an modest initial augmentation, the edge displacement tends to zero.

## Discussion

The initial aim of this article was to establish the analytic properties of the eigenfunctions ψ_{n}(*y*,*z*), *n*=0,1,2…. This has been accomplished in §§3 and 4 where several pertinent results are derived. In §3, theorem 3.2 proved that the eigenfunctions ψ_{n}(*y*,*z*), *n*=0,1,2…, are linearly dependent. It is interesting to note that, although non-separable themselves, certain sums involving products of the eigenfunctions ψ_{n}(*y*,*z*) (and/or their derivatives) can be reduced to a separable form which enables them to be evaluated using known properties of the underlying two-dimensional system. Sums (equation (3.9) and (3.11)) are of this form and, furthermore, are valid for 0≤*w*≤*b* which enables arbitrary eigensolutions, such as that used in equation (5.5), to be assembled. These eigensolutions satisfy equations (2.2)–(2.7) and are useful for constructing solutions that satisfy specified edge conditions, such as the spring-like condition used in §5. Theorem 3.4 proved that, with appropriate choice of the amplitude coefficients, an eigenfunction expansion of a suitable function *f*(*x*,*y*) converges point-wise to that function. This result is particularly significant in that it justifies the use of such expansions as a means of representing a wide class of functions.

In §4, the derivatives of the eigenfunctions were investigated and new orthogonality properties were presented. These results are not expressed in the form of an OR (indeed it is not clear whether an OR, as such, exists). Instead it was demonstrated that three suitably smooth functions *p*(*z*), *q*(*z*) and λ(*z*) can be expressed as an eigenfunction expansion in terms of the functions χ_{n}(*z*), ψ_{ny}(*a*,*z*) and ψ_{nyyy}(*a*,*z*), respectively, but with the same (or closely related) amplitude coefficient in each case.

The second aim of this article was to investigate the potential use of the eigenfunctions ψ_{n}(*y*,*z*) in mode-matching problems. This issue is addressed in §5 where two prototype problems have been solved. These prototypes have been selected to be representative of the class of problem that arises in a typical mode-matching problem in which the pressure and velocity must usually be enforced at an interface between two adjacent duct sections. Here the interface is taken to lie at *x*=0. The first prototype deals with a prescribed pressure and pin-jointed edge conditions and this is solved exactly using the partial OR (equation (3.1)). The second prototype deals with a prescribed velocity and spring-like edge conditions which tend to the usual clamped conditions as the spring constant increases. The solution is constructed as an eigenfunction expansion containing an additive eigensolution (equation (5.5)), which is expressed in terms of an unknown function λ(*z*). On applying the edge conditions, a Fredholm integral equation is obtained which is then solved numerically to determine λ(*z*) and hence the solution to the problem.

The purpose of this paper is not to present a parametric investigation of the prototype problems, but instead to assess whether the eigenfunction expansions considered herein provide a viable solution method to three-dimensional mode-matching problems. Nevertheless, numerical results showing the plate displacement are presented for both prototype problems, thereby demonstrating that this approach is indeed a potentially powerful tool for tackling three-dimensional mode-matching problems. There are, however, some issues that need to be addressed. For example, sufficient roots of the dispersion relation must be found in order for the method to be accurate. In fact, it is generally not difficult to find the complex roots but obtaining the imaginary roots is a tedious procedure. The reason for this is the structure of *L*(*s*). When *s* is imaginary, *L*(*s*) has an infinite number of asymptotes and the zeros generally lie very close these and are consequently difficult to locate numerically because of the rapidly changing gradient. Furthermore, owing to the poles of *L*(*s*), it is difficult to apply the argument principle as a means of testing whether all roots in a given region of the complex plane have been found. On account of equations (4.1) and (4.2), however, the quantity *R*_{nm} (equation (4.12)), has some interesting properties, one of which is
6.1
The sum on the right-hand side is highly sensitive to missing roots and thus, this expression provides a useful check for whether sufficient roots of the dispersion relation have been found. With these points in mind, however, it would be useful to have some means of predicting the number and location of roots. A Weyl series (e.g. Howls & Trasler 1998) would be helpful in this respect. Alternatively, following Lawrie & Kirby (2006), it may be possible to construct a ‘root-free’ approach for certain duct geometries.

In summary, this article presents an analytic investigation into a class of eigenfunctions that is potentially of use in mode-matching problems involving the propagation of sound in three-dimensional rectangular ducts. The thrust of the article is in establishing the underlying properties of these eigenfunctions. It remains to be seen whether the proposed methodology is numerically efficient when applied to problems of real engineering significance, particularly when compared with modern numerical approaches such as the semi-analytic finite element method (Moreau *et al.* 2006; Castaings & Lowe 2008). Nevertheless, based on such comparisons for the two-dimensional case (Kirby & Lawrie 2005), it is anticipated that this three-dimensional mode-matching method should prove a useful qualitative tool while simultaneously providing a benchmark against which to test a variety of numerical strategies.

## Footnotes

↵1 The reader is advised that there are errors of sign in equations (3.24) and (3.26) of Lawrie (2007). These do not, however, impinge upon the results presented in that article.

- Received February 6, 2009.
- Accepted April 7, 2009.

- © 2009 The Royal Society