## Abstract

The problem of a ribbed membrane or plate submerged in a fluid with mean flow is studied. We first derive a method which can be used to reduce this, and similar problems to a band matrix inversion. We then find the pass and stop band structure found in the case of static fluid persists when a mean flow is introduced, and we give an explanation in terms of the eigenvalues of the transfer matrix of the system. We then study disordered structures and observe the phenomenon of Anderson localization. In some parameter régimes the addition of disorder causes significant delocalization.

## 1. Introduction

Consider an elastic membrane or plate supported by an array of ribs and immersed in incompressible inviscid fluid with mean flow speed *U*. When one of the ribs is excited by a time-harmonic force, we wish to ask whether this excitation spreads along the body of the array, or remains localized near the driven rib. This represents a simple model of the propagation of structural vibration on a ship hull or an aeroplane wing, and extends the large body of work on structure-borne noise in a static fluid (Crighton 1984; Spivack 1991; Spivack & Barbone 1994; Cooper & Crighton 1998) to include the effects of non-zero mean flow.

When the rib array is equispaced, the zero mean flow ribbed membrane has been shown to exhibit pass and stop band behaviour; there are bands of frequencies in which an excitation can pass along the whole system, and bands of frequencies in which an excitation is trapped near its source. This occurs in both infinite (Crighton 1984) and finite (Spivack 1991; Cooper & Crighton 1998) systems. When the rib array becomes non-equispaced the pass bands are destroyed and a weak localization effect is observed (Spivack & Barbone 1994; Cooper & Crighton 1999). In all cases, the far-field decay of the system response is algebraic if the algebraically decaying branch line component of the Green function of the system is included in the analysis, but is exponential if the branch line component is neglected. However, near-field decay behaviour can still be exponential, and there can be significant exponential decay before the crossover into the far field.

The extra complication in this work comes from the mean flow, because the membrane (or plate) Green function with non-zero mean flow—the response to point forcing of the unribbed system—is more complex. Not only does it have more wave modes than the zero mean flow problem, but it also exhibits some particularly unusual behaviour (Crighton & Oswell 1991; Kelbert & Sazonov 1996). For mean flow speeds greater than a critical speed *U*_{c}, the Green function is exponentially growing in time, independently of the frequency of the forcing. When *U*<*U*_{c}, the Green function is temporally harmonic, with the nature of the Green function dependent on the frequency of the forcing, *ω*, and three distinct types of behaviour are found: at low frequencies the system is convectively unstable; at moderate frequencies the system is neutrally stable, but with an anomalous response in which a neutral mode possesses a group velocity directed towards the driver; and at high frequencies the mean flow becomes irrelevant and the system is found to be stable. Since the zero mean flow ribbed membrane has been extensively studied, our primary interest is in the region of low to moderate frequencies.

The instability and unusual energy flux behaviour observed in the Green function are not the major difficulties in extending zero mean flow results to the non-zero mean flow problem. Analytically, the difficulty is that the Green function consists of more than one surface wave mode. This breaks the simple zero mean flow analysis of Spivack & Barbone (1994), but with a little effort a generalization of their method can be made to work, and this extension will be described here.

In §2 we start by defining the problem to be solved and its mathematical formulation. In §3 we show how the problem can be simplified by a simple transformation that has the effect of localizing the surface wave coupling. In §4 we apply this simplification to the periodically ribbed membrane, showing how the pass and stop band structures found in the zero mean flow problem persist as the mean flow speed is increased. In §5 we apply our simplification to the more complex problem of the periodically ribbed plate, when both forces and torques must be solved for. In §6 we move back to the ribbed membrane, and discuss the effects of disorder on the response of the system.

## 2. Statement of problem

We mainly consider the case of a fluid-loaded membrane. The response of the fluid-loaded plate will be briefly discussed in §5. The parameters of the system are the fluid density *ρ*, fluid mean flow speed *U*, membrane mass per unit area *m* and membrane tension *T*. We will non-dimensionalize with the fluid loading length-scale *m*/*ρ* and fluid loading time-scale (*m*/*ρ*)(*m*/*T*)^{1/2}. This has the effect of setting *m*=*ρ*=*T*=1, leaving us with the parameter *U*′=*U*(*m*/*T*)^{1/2}, which is the ratio of the mean flow speed to the wavespeed on an unloaded membrane. Henceforth, we drop the primes on dimensionless quantities.

We will not consider the absolutely unstable parameter régime, so we must impose the restriction , which will be applied implicitly throughout the following. This critical flow speed can be found by noting that at the onset of absolute instability, in this problem, three of the roots of its dispersion relation are coincident (Crighton & Oswell 1991; Kelbert & Sazonov 1996). For forcing at angular frequency *ω* we are therefore able to factor out a time dependence of e^{−iωt}; this has been done in everything that follows.

### (a) Governing equations

Suppose that we have a membrane supported by ribs at points *x*=*x*_{1},*x*_{2},…,*x*_{N}, where *x* is the coordinate along the membrane, and driven on the *D*th rib by a transverse force *F*_{D}. We may safely assume that *x*_{1}<*x*_{2}<⋯<*x*_{N}. For simplicity, we will suppose that the non-driven ribs have infinite mechanical impedance and so must remain fixed, and we aim to find the forces *F*_{1},*F*_{2},…,*F*_{N} on the rib array. Note that finite mechanical impedance can be easily included, as is done by Cook (1998).

Now, let *G*(*x*) be the Green function of the membrane, i.e. the response to a single line force. By linearity, the ribbed membrane displacement *η* is found to be(2.1)which gives us the set of *N*−1 linear equations(2.2)when we observe that the non-driven ribs do not move.

We can view either the force applied to the *D*th rib, or the displacement of the *D*th rib, as an unknown. If we set the displacement of the *D*th rib to be 1, we get the additional equation(2.3)which just sets the scale of the forces in our linear system. Equations (2.2) and (2.3) give us the matrix equation **G**.**F**=** η**, where

**G**

_{ij}=

*G*(

*x*

_{i}−

*x*

_{j}), which is a set of

*N*linear equations in

*N*unknowns to be solved for the forces

*F*

_{i}.

It is in theory possible to numerically invert the matrix **G** by methods such as LU or QR factorization, but unfortunately the matrix **G** is ill-conditioned—the neutral surface waves which propagate on the membrane maintain a constant amplitude. Thus, the matrix **G** is far from diagonally dominant, and one cannot make *N* particularly large before catastrophic loss of precision occurs in standard factorization routines. The problem is even worse in the convectively unstable régime. We will see that it is possible to recombine these equations to give a numerically well-conditioned matrix. This rearrangement of the equations will also give us a means of understanding the qualitative form of the response of the system.

### (b) The Green function

Since we are not in the absolutely unstable régime the Green function for a fluid-loaded membrane is(2.4)where(2.5)is the dispersion relation for free waves on the fluid-loaded membrane with mean flow (Kelbert & Sazonov 1996), and are the wavenumbers of the two surface waves found in *x*>0, and and are the wavenumbers of the two surface waves found in *x*<0. The branch cut for the square root always has a positive real part; this corresponds to a cut along the imaginary axis in the *k*-plane. The branch line integrals ∫_{ubc} and ∫_{lbc} thus correspond to integrals around the positive and negative imaginary axes, respectively.

We see that we can write the Green function matrix **G** as(2.6)where **G**_{sw} contains the surface wave parts of **G**, and **G**_{bc} almost all of the branch line contributions. As is usual (Crighton 1984), we include the line admittances *G*(0) in **G**_{sw}, so that the diagonal elements of **G**_{bc} are zero.

We also observe that it is trivial to evaluate the Green function numerically. Previous studies of this problem have tended to use asymptotic results, in order to produce equations that can be manipulated analytically. In this problem there is little hope of doing any analytical manipulation and we will use numerically computed values for the Green function in all of the calculations in this paper.

## 3. Reduction of Green function matrix

The only true long range coupling in the Green function is that provided by the branch line component. This component of the Green function is weak, and is sufficiently strongly decaying to be trivially invertible. The surface wave components, although apparently a long range coupling, can be recast as a purely local term.

We consider the matrix , where **S** is a pentadiagonal matrix with **S**_{i(i−2)}=*a*_{i}, **S**_{i(i−1)}=*b*_{i}, **S**_{ii}=1, **S**_{i(i+1)}=*c*_{i} and **S**_{i(i+2)}=*d*_{i}, and observe that we can make the bulk of pentadiagonal if *a*_{i}, *b*_{i}, *c*_{i} and *d*_{i} satisfy the equations(3.1)for *i*=3…*N*−2, where(3.2)

The left end of the array gives the equations(3.3)and the right end of the array gives(3.4)

Note that *b*_{2}, *c*_{2} and *d*_{2}, and *a*_{N−1}, *b*_{N−1} and *d*_{N−1} are indeterminate. Although it might be useful to set *d*_{2}=*a*_{N−1}=0, we observe that when the ribs are periodically spaced this choice will set *c*_{1}=*b*_{2}, *d*_{1}=*c*_{2}, *b*_{N−1}=*a*_{N} and *c*_{N−1}=*b*_{N}. Therefore, the matrix **S** has zero determinant and we are not able to invert the matrix . It proves convenient instead to set *d*_{2}=*a*_{N−1}=1.

If we temporarily ignore , we see that this transformation produces the pentadiagonal system of equations , which is(3.5)for *i*=3…*N*−2, for some values *α*_{i,j} which it is not useful to write explicitly. There are also four other equations from the endpoints of the rib array.

It is clear that obvious variants of this reduction scheme are applicable to other similar wavebearing systems and will allow the reduction of a general wave-like (or exponential) coupling to a purely local effect. The matrix is pentadiagonal because the Green function has four wave modes, which should be contrasted with the corresponding result for a static fluid, when the Green function has two wave modes and the equivalent of is tridiagonal (Spivack & Barbone 1994).

### (a) Transfer matrix form

Note that away from the driven rib the recurrence (3.5) can be cast in a transfer matrix form by writing(3.6)where the *i*th transfer matrix **T**_{i} governs the propagation from the *i*th rib to the (*i*+1)st rib and is simply given as(3.7)

This formulation will prove to be useful later.

### (b) Reconstruction of solution, energy flux and other useful tools

In this section we briefly derive some useful results that will be helpful later. Spivack & Barbone (1994) were able to produce some strong results for a fluid-loaded membrane in zero mean flow, and some of these will be seen to carry over to this problem. However, their direct methods are much less useful here, and we must proceed from more general principles.

Firstly, we note that we can reconstruct the displacement of the membrane between two ribs, given the forces on the ribs and neglecting the branch line component of the Green function. Suppose that *x*_{i−1}<*x*<*x*_{i}, and that neither the (*i*−2)th, (*i*−1)th, *i*th or (*i*+1)th ribs are driven. Then, by applying the ideas that led to equations (3.1) and (3.2), we find that *η*(*x*) can be written in the form(3.8)for some vector **B**(*x*) for which an expression can be found.

Next we observe that, between two ribs, the pressure on the membrane is a linear functional of *η*(*x*),(3.9)and similarly we note that we can find the fluid velocity,(3.10)

Lastly, under the assumption that the branch line component of the solution is not present, we observe that we can write *ϕ* as a linear functional of *η*, *η*_{x}, *η*_{xx} and *η*_{xxx}, and we see that *η*, *ϕ*_{y}, *ϕ* and *p* are all linear in the state vector (*F*_{i+1}*F*_{i}*F*_{i−1}*F*_{i−2})^{T}.

To use all of these observations, we next note that the system allows us to define an energy flux (Crighton & Oswell 1991),(3.11)and since we work with time-harmonic solutions, we define an average energy flux,(3.12)in which (.)^{*} denotes the complex conjugate. At the midpoint of a bay we can evaluate all of the functions in the mean energy flux (3.12) as linear functionals of (*F*_{i+1}*F*_{i}*F*_{i−1}*F*_{i−2})^{T}, and so we observe that conservation of energy implies the existence of a set of unitary matrices **Σ**^{(i)}, such that(3.13)is independent of *i* away from the driven rib, which is equivalent to the statement(3.14)(Note that (.)^{†} denotes the conjugate transpose.)

When the ribs are equispaced, the matrices **Σ**^{(i)} are constant and so the transfer matrix **T** preserves the Hermitian form (** x**,Σ.

**). It is useful to investigate this special case. The main point to note is that if**

*y***e**is an eigenvector of

**T**with eigenvalue

*λ*, then

**Σ**.

**is an eigenvector of**

*e***T**

^{*}with eigenvalue 1/

*λ*. Thus, if

*λ*is an eigenvalue of

**T**, so is 1/

*λ*

^{*}, and if |

*λ*|≠1, this construction provides a pair of eigenvalues of

**T**. This unitary structure will be exploited later.

### (c) Branch line contribution

Observing that left multiplication by **S** just consists of row operations, we see that has the same decay rate away from the central diagonal as **G**_{bc}. Thus numerical inversion of is accurate, if not cheap.

We now have two choices. If the number of ribs is not too large we can directly invert the matrix using LU factorization. This direct inversion is well-conditioned because we have localized the surface wave component; the surface wave inversion acts as a preconditioner for the whole system. For arrays of up to a few hundred ribs there is no need to resort to iterative methods, and so direct LU factorization of the matrix is the best way to proceed. For longer arrays an iterative scheme may be necessary, but an excessively complicated scheme should not be needed and the iteration(3.15)should be sufficient, since we expect to be a small correction. This iteration also allows us to interpret results found by LU factorization; we will see later that the dominant coupling is actually contained in ; the most important part of the branch line contribution is contained in the line admittance *G*(0).

This idea is equivalent to the manipulation of Cooper & Crighton (1998), who took the iteration (3.15) as far as **F**^{(2)}. Numerically, of course, there is no reason to do this and we might as well iterate (3.15) to convergence. This iterative scheme was found to be unnecessary for the sizes of problems studied here.

## 4. Periodically ribbed membranes

As derived above, our simplifying transformation is applicable to both periodic and aperiodic rib arrays, but for the next few sections we only consider periodic rib arrays. Suppose that the inter-rib spacing *x*_{i+1}−*x*_{i} is a constant, say *h*. In this case the coefficients *α*_{i,j} in equation (3.5) and the transfer matrices **T**_{i} are found to be independent of *i* (and the *i* suffixes on both these quantities will be omitted throughout the rest of §4). Similarly, the coefficients *a*_{i}, *b*_{i}, *c*_{i} and *d*_{i} are constant for *i*=3…*N*−2 and will be written *a*, *b*, *c* and *d*.

We can invert the matrix using standard routines and the results are shown in figure 1. We observe a very distinctive pass and stop band structure. In the pass bands the response of the system is of the same order on the whole array. In the stop bands we observe exponential decay of the response away from the driven rib. Near the ends of the array the decay does reduce to algebraic decay, but by the ends of the array the response has decayed so much that this crossover is somewhat irrelevant (Flach 1998).

We note that in the stop bands the rate of doing work at the driven rib ( modulo constants) is zero to numerical precision.

Note also the significant upstream–downstream asymmetry. This asymmetry, although due to the mean flow, is not simply controlled by the mean flow. It is easy to find parameter values at which the array response upstream of the driver is large and the array response downstream of the driver is small.

This pass–stop structure can be explained by the system of equations ; the branch line parts of the Green function matrix just produce a change in the quantitative response; the form of the response is controlled by the recurrence (3.5) and the endpoints of the rib array. This can be seen clearly in figure 6.

We find that of the four eigenvalues *λ*_{1…4} of the transfer matrix **T**, |*λ*_{3}|≫1 and |*λ*_{1}|≈1, with |*λ*_{3}*λ*_{4}|=|*λ*_{1}*λ*_{2}|=1. (Note that in §3*b* we predicted that eigenvalues off the unit circle come in pairs (*λ*,*μ*) such that *λμ**=1; this relation was satisfied with an error of 10^{−10} at most.) In a pass band the eigenvalues *λ*_{1} and *λ*_{2} lie on the unit circle and move around it as the frequency is varied. When the system leaves a pass band, the two eigenvalues collide and move off the unit circle. When the system re-enters a pass band, they collide again, attach themselves to the unit circle and begin to move around it. This collision of eigenvalues on the unit circle is just a Krein crunch (Lalonde & McDuff 1997), but in the unitary group rather than the symplectic group. The eigenvalue collision can be observed in figure 2, which is a plot of |*λ*_{1}| and |*λ*_{2}| as functions of frequency. Note how closely figures 1 and 2 coincide; almost everywhere we see a localized response where the eigenvalues are off the unit circle and a delocalized response when they are on the unit circle.

It is not difficult to produce an argument (Borland 1963) which suggests that eigenmodes whose eigenvalues are not on the unit circle are localized near the driven rib and endpoints of the array. Thus, only eigenmodes whose eigenvalues lie on the unit circle can pass along the whole of the array. This does not necessarily mean that they will do so with a large amplitude—we observe ‘pass bands’ in which the amplitude in the bulk of the array is less than 1% of the maximum amplitude on the array. Compare, for instance, the low‐frequency regions of figures 1 and 2, where figure 2 shows that there are two eigenvalues on the unit circle, but figure 1 shows a localized response. This low‐amplitude response is simply because the boundary conditions and driver conditions can be satisfied by the localized modes without resort to the extended modes.

One other point to note is that the low‐frequency régime of convective instability can lie within a stop band, so we get little response in the bulk of the system even though the Green function has exponential growth in space.

## 5. Periodically ribbed plates

Since the Green function of the elastic plate under mean flow is qualitatively similar to that of the elastic membrane, and the plate problem is more physically plausible, we would also like to apply these ideas to the problem of a ribbed fluid-loaded plate. The only complication is that we must solve for both a force and a torque on each rib, but the same ideas carry through to this more complicated problem.

Assuming that the plate is clamped at each rib (Guo 1993), so that *η* and *η*_{x} are both zero on each non-driven rib, we find that(5.1)where *G*_{p} is the plate Green function, a prime denotes ∂/∂*x*, *F*_{j} is the force on the *j*th rib and *T*_{j} is the torque on the *j*th rib.

This gives us the set of equations:(5.2)(5.3)(5.4)where equations (5.2) and (5.3) come from the plate displacement on the rib array and equation (5.4) comes from the clamped rib condition *η*_{x}(*x*=*x*_{i})=0. We can rewrite equations (5.2)–(5.4) as a set of linear equations of the form:(5.5)

Next, we observe that the surface wave part of the plate Green function *G*_{p}(*x*) is qualitatively similar to the membrane Green function, that all the derivatives of the plate Green function have waves of the same wavelength and that the matrix **S** is only a function of wavelength and rib position. This means that we can apply the simplifying transformation to the plate equations by left-multiplying equation (5.5) by(5.6)to obtain(5.7)where , and are all pentadiagonal. Neglecting the branch line contributions, we can rewrite the simplified plate equation (5.7) as a matrix of bandwidth 11, which can be cheaply and accurately solved using standard library routines.

Results for the force and the torque on a ribbed plate, neglecting the branch line component of the Green function, are shown in figures 3 and 4, respectively. The frequency range shown covers the entire range of convective instability and anomalous propagation of the forced plate. We observe pass and stop band behaviour, as for the ribbed membrane, although the distinction is now much less sharp.

## 6. Aperiodicity on the fluid-loaded membrane

Most structures, of course, are not exactly periodic and have some degree of disorder. When a periodic wavebearing system becomes disordered, we expect to observe the phenomenon of Anderson localization (Anderson 1958). Generally, one expects the effect of localization to confine any response of the system to the region near the driver, and in our case we expect the response of the system to be localized near the driver.

Note that the matrix is not Hermitian. Typical Anderson models arising from quantum mechanics use Hermitian matrices. The idea that non-Hermitian Anderson models are relevant is more recent; such models appear in population biology (Nelson & Shnerb 1998) and the study of superconductors (Hatano & Nelson 1997). We note that the model of Spivack & Barbone (1994) is another non-Hermitian Anderson model.

For definiteness, we introduce disorder by displacing each rib from its mean position by a random amount,(6.1)where the random variables *R*_{i} are independent, each with uniform distribution on [−1,1], and *σ* is a disorder parameter.

### (a) Omitting the branch line

In this subsection we will neglect the branch line part of the Green function. This is not strictly necessary, and we will consider the effect of the branch line part of the Green function later.

It is trivial to numerically solve the linear equations , and some results are shown in figure 5. Figure 6 compares the response of ordered and disordered arrays, at a frequency which is in a pass band of the ordered system.

Interestingly, we observe a very significant delocalization at low frequency in figure 5; although the recurrence (3.5) allows a pass band response in the periodic system, the boundary conditions at the ends of the rib array and the driver conditions prevent any extended modes from appearing. This can be seen clearly in figure 7. A physical explanation is simple: when the system becomes aperiodic the centre of the array cannot feel these end effects, wants to propagate and can now, to some extent, do so. Although we expect the system response to become localized when the disorder is sufficiently large, it appears that there is a window in which disorder obscures the end effects, but is not strong enough to kill off propagation altogether. We also note that this delocalized response vanishes when we average over realizations (figure 8). This, then, is the ‘sensitivity to boundary conditions’ used by Thouless (1974) as the characteristic of extended modes. This particularly dramatic manifestation of this sensitivity does not seem to have been previously observed in studies of similar systems (Anderson 1958; Thouless 1978; Flach & Willis 1998).

It is useful to introduce the Lyapunov exponents of the system (Kottos *et al*. 1999), which give a measure of its localization behaviour. Also, the inverse of the smallest Lyapunov exponent (in absolute value) gives a kind of localization length of the system. If we introduce an ‘end-to-end’ transfer matrix(6.2)the product of the transfer matrices (3.7) of the whole system, with singular values *s*_{1},…,*s*_{4}, we can define the Lyapunov exponents as(6.3)where 〈*f*〉 denotes an average over realizations.

It is generally found that the addition of some kind of disorder to such a system causes the Lyapunov exponents to repel each other. This can be clearly seen by comparing figures 9 and 10, which show the Lyapunov exponents as functions of frequency for the same parameter values as figures 1 and 5, respectively. Nevertheless, even with this repulsive effect, we see that as the disorder is introduced, there are frequency intervals in which the smallest Lyapunov exponents remain very small. This means that the system is still able to propagate signals even when disordered. In these régimes the effect of disorder is simply to obscure the end effects, which, in this system, can allow a signal to propagate, as seen in figure 7.

### (b) Branch line contribution

Numerically, it is simple to include the branch line part of the Green function when solving the disordered system. Averaging over realizations, we obtain a result much the same as figure 8. The effect of the branch line part tends to decrease the amount of localization attained, sometimes quite drastically, but to understand the qualitative response of the system it suffices to consider the Green function with the branch line omitted.

We see the effect of the branch line component on localization in figure 11. While we obtain true exponential localization in the far field without the branch line component, when we include the branch line component, the far-field behaviour becomes algebraic. This behaviour is also found in the zero mean flow problem (Cooper & Crighton 1999).

## 7. Discussion

The pass and stop band structures found when the fluid is taken to be static are shown to persist as the mean flow speed is increased from zero. We see strong asymmetries at low frequency, when the effect of the fluid loading is important.

An interesting result is that the branch line part of the membrane Green function has little qualitative effect on the response of the system at the flow speeds in which we are interested. It becomes more significant when the system is disordered, although to obtain a qualitative understanding it suffices to restrict attention to the surface wave parts of the Green function.

One other point to note is that the Green function grows exponentially downstream at sufficiently low frequencies when *U*>1. Thus, it is not immediately obvious that the infinite rib array problem of Crighton (1984) is well defined, and we must consider the finite array problem. Another point to note is that the infinite array problem can only be thought of as a model of the start-up of forcing, when reflections from the ends of the array have not yet returned to the driver. We only have the Green function in the long-time limit, so we can only consider a fully developed solution and thus, necessarily, a finite array.

Finally, we observe that we have a complete description of the response of a ribbed fluid-loaded elastic plate or membrane in the time-harmonic steady state. In passing, we have derived a means to produce similar results in any one-dimensional wavebearing system, and we see that this work could be trivially extended to consider, say, a ribbed elastic cylinder (Peake 1997; Photiadis & Houston 1999).

The next major part of this problem is to study the transient response to the switch-on of forcing, which would allow us to investigate the absolutely unstable parameter régime.

## Acknowledgments

The author gratefully acknowledges the financial support of the Engineering and Physical Sciences Research Council, BAE Systems, the ONR and the Isaac Newton Trust.

## Footnotes

- Received April 20, 2004.
- Accepted August 16, 2004.

- © 2005 The Royal Society