Institut für Numerische und Angewandte...

Institut für Numerische und Angewandte Mathematik

Trapped modes and Fano resonances in two-dimensionalacoustical duct-cavity systems

S. Hein, W. Koch, L. Nannen

Nr. 2011-13

Preprint-Serie des

Instituts für Numerische und Angewandte Mathematik

Lotzestr. 16-18

D - 37083 Göttingen

Under consideration for publication in J. Fluid Mech. 1

Trapped modes and Fano resonancesin two-dimensional

acoustical duct-cavity systems

STEFAN HEIN1, WERNER KOCH1

AND LOTHAR NANNEN2,1Institut fur Aerodynamik und Stromungstechnik, DLR Gottingen, 37073 Gottingen, Germany

2Institut fur Numerische und Angewandte Mathematik, Universitat Gottingen, 37083Gottingen, Germany

(Received 30 June 2011)

Revisiting the classical acoustics problem of rectangular side-branch cavities in a two-dimensional duct of infinite length we use the finite element method to numericallycompute the acoustic resonances as well as the sound transmission and reflection foran incoming fundamental duct mode. To satisfy the requirement of outgoing waves inthe far field we use two different forms of absorbing boundary conditions, namely thecomplex scaling method and the Hardy space method. In general the resonances aredamped due to radiation losses, but there also exist various types of localized trappedmodes with nominally zero radiation loss. The most common type of trapped mode isantisymmetric about the duct axis and becomes quasi-trapped with very low dampingif the symmetry about the duct axis is broken. In this case a Fano resonance resultswith resonance and antiresonance features and drastic changes in the sound transmissionand reflection coefficient. Two other types of trapped modes, termed embedded trappedmodes, result from the interaction of neighbouring modes or Fabry–Perot interferencein multi-cavity systems. These embedded trapped modes occur only for very particulargeometry parameters and frequencies and become highly localized quasi-trapped modesas soon as the geometry is perturbed. We show that all three types of trapped modesare possible in duct-cavity systems. If several cavities interact the single-cavity trappedmode splits into several trapped supermodes which might be useful for the design oflow-frequency acoustic filters.

1. IntroductionIn our recent publication Hein et al. (2010) we showed that the scattering of acoustic

waves by an object in a duct can change drastically if the symmetry about the duct axisis broken. The reason for this is that by breaking the symmetry, a trapped mode whichis antisymmetric about the duct axis and therefore cannot be excited by a plane wave,becomes a highly localized quasi-trapped mode which allows the coupling of modes witheven and odd symmetry. Opening up an alternative transmission path a Fano interfer-ence results with typical resonance-antiresonance features of the transmission coefficient.First used to explain autoionisation in atoms (see Fano (1935) or Fano (1961)) Fano res-onances are a generic phenomenon in many different fields of physics (see the excellentreview by Miroshnichenko et al. (2010)) and can be considered a generalisation of cou-pled mechanical oscillators, originally employed by Ormondroyd & Den Hartog (1928)

2 S. Hein, W. Koch and L.Nannen

for vibration damping. Detailed discussions of this vibration damping mechanism canbe found in textbooks of mechanical vibrations such as Den Hartog (1947) (see also therecent article by Joe et al. (2006)).

A prerequisite for the occurrence of Fano interference is a localized, high quality factorresonance which can interfere with the incoming continuous spectrum propagating wave.The quality factor Q of a resonator is proportional to the resonant frequency and inverselyproportional to the resonance damping. It is here where the trapped modes with infinitelylarge Q become of importance. In acoustics, as well as other wave propagation problems,trapped modes with nominally zero radiation loss are possible for laterally boundedproblems such as waveguides or cascades, e.g. Evans & Linton (1991) or Porter & Evans(1999) (a survey of the mathematical physics behind resonances and trapped modes inquantum wires as well as in acoustics can be found in the monography of Hurt (2000)).

In quantum mechanics trapped modes are known as bound states in the continuum(BSCs or BICs), see for example Sadreev et al. (2006) or Moiseyev (2009) and referencestherein. Gonzalez et al. (2010) listed several mechanisms producing bound states in anopen quantum dot and similar mechanisms lead to trapped modes in acoustical duct-cavity systems. The most common mechanism is based on the symmetry of the system:the localized trapped mode and the continuous spectrum mode have different symmetryabout the duct axis. By separating the symmetric and antisymmetric problem the con-tinuous spectrum of the latter begins at the cutoff frequency of the first duct crossmodeand the corresponding trapped modes are the discrete real resonances below the continu-ous spectrum of the propagating mode. Most publications about acoustic trapped modesincluding the seminal papers by Parker (1966) and Evans & Linton (1991) are in this cat-egory. These acoustic trapped modes have a frequency below the cutoff frequency of thefirst duct crossmode and, being antisymmetric about the duct axis, cannot be excitedby an incoming plane wave. Ladron de Guevara et al. (2003) call such trapped-moderesonances with vanishing damping ’ghost Fano resonances’. By breaking the symme-try the trapped modes become highly localized quasi-trapped modes, as demonstratedby Aslanyan et al. (2000), which can couple with an incoming plane wave.

A second mechanism is due to the destructive interference between two resonantmodes via a common continuum. The embedded trapped modes reviewed by Linton &McIver (2007) are examples of this mechanism. For coupled-channel problems in atomicphysics Friedrich & Wintgen (1985) proved that bound states in the continuum can occurdue to the interference of two resonances belonging to two different channels (modes).As a consequence of this interference avoided crossings of resonances are observed asa continuous parameter off the system is varied. Such embedded trapped modes havebeen found also for quantum dots where Dirichlet boundary conditions are prescribed,see Sadreev et al. (2006) or the review by Oko�lowicz et al. (2003) who introduced theterm resonance trapping for a bifurcation of resonance widths into long-lived and short-lived resonances. The third mechanism is associated with Fabry–Perot interference byadjusting the distance between two ’mirrors’, see for example Ordonez et al. (2006). Thefrequency of an embedded trapped mode can be above or below the cutoff frequency ofthe first duct crossmode and the corresponding mode can be symmetric or antisymmetricabout the duct axis. But this mechanism is operative only at very particular geometryparameters and frequencies. We shall encounter examples of all three mechanisms in ourinvestigation of duct-cavity systems.

Already Evans & Linton (1991) showed that trapped modes are possible not onlyfor two-dimensional objects in a duct but also for two-dimensional cavities which aresymmetric about the duct axis. As long as the cavities are located symmetrically aboutthe duct axis the shape of the cavity does not matter. For example Sugimoto & Imahori

Trapped modes and Fano resonances in ducts with cavities 3

(2006) showed analytically the existence of trapped modes for Helmholtz resonators. Fora circular cylindrical acoustic waveguide containing an axisymmetric obstacle Linton &McIver (1998) proved the existence of an infinite sequence of trapped modes. The sameis true for axisymmetric cavities as demonstrated by the numerical computations of Hein& Koch (2008). Hein & Koch (2008) also showed that trapped modes are possible incylindrical ducts with symmetrical side branches, so-called symmetric cross junctions,where they can be excited by shear layers leading to severe vibration and noise problemsin gas and steam pipe lines, see for example Jungowski et al. (1989); Ziada & Buhlmann(1992); Kriesels et al. (1995); Ziada & Shine (1999) or the recent survey by Tonon et al.(2011). Whereas trapped and nearly trapped acoustic modes are important for shear-layer excitation we consider in the present study the no-flow case and are interested inthe influence trapped modes have on the transmission and reflection of sound waves.

Very similar resonant-cavity configurations have been the subject of intensive researchin various other wave-propagation systems and led to important applications. For ex-ample the transmission of electrons in quantum wave guides with stubs has applicationsin advanced computer technology, see for example Sols et al. (1989), Akis et al. (1997),Debray et al. (2000) or Cattapan & Lotti (2007). In such systems the asymmetry canbe tuned by applying a magnetic field, see for example the graphene quantum-dot-likestructures investigated by Gonzalez et al. (2010). Electron waveguides consisting of arectangular cavity and straight leads are even used to model the interaction betweendiscrete-system electrons and continuous-spectrum photons in an atom as demonstratedby Petrosky & Subbiah (2003) or Ordonez & Kim (2004). Rotter et al. (2004) employedthe equivalence between microwave transport and single-electron motion to demonstratethe controlled variation of Fano resonances in a microwave cavity via variable diaphragmsat the duct-cavity junction.

Using light instead of electrons, photonic crystal waveguides with side-coupled cavitiesbecame of increasing importance for optical filters or switches in communication tech-nology, see amongst others Xu et al. (2000) or Fan et al. (2006). Here photonic crystalsare periodically structured optical materials in which resonant cavities are generated byremoving or changing one or several structural elements, e.g. by introducing defects. Theanalogous propagation of acoustic and elastic waves in phononic crystals with waveguide-cavity structures can be used for acoustic or elastic filters and has been investigatedamongst others by Fellay et al. (1997), El Boudouti et al. (2008) and Hladky-Hennionet al. (2008).

In classical acoustics rectangular cavities, so-called quarter-wavelength resonators, areused as effective narrow-band silencers at low frequencies, see Pierce (1981) or Munjal(1987). However, in the literature large discrepancies have been reported between mea-sured and calculated resonance frequencies in certain parameter ranges. Various authorstried to remedy this by introducing improved end corrections or form factors, see forexample Alster (1972); Tang & Sirignano (1973); Chanaud (1994) or Ji (2005). Further-more, for cavity-duct configurations the end correction is also a function of the ratio ofcavity width/length to duct width. But, as noted by Singh (2006), the end-correctionfactor for duct-cavity configurations has not been well documented. Therefore, if two-and three-dimensional effects predominate, one has to resort to numerical computation.

In the present paper we shall limit ourselves to single and multiple, rectangular, hard-walled acoustic resonators and use the finite-element method to numerically compute thecomplex resonances and their relation to the physical observables such as sound trans-mission and reflection. Contrary to some of the other wave-propagation problems theparameters of the acoustical problem can be controlled easily by changing the geome-try. We shall show that as soon as the symmetry about the duct axis is broken Fano


resonances occur with an asymmetrical resonance-antiresonance line shape in additionto simple resonances with a symmetrical, so-called Lorentzian line shape. Furthermore,embedded trapped modes which are embedded in the continuous spectrum are possiblefor resonators with particular geometry parameters and frequencies which can be aboveor below the frequency of the first duct crossmode. In the present paper we treat thetwo-dimensional problem, the results of which are not only of interest by themselvesby providing examples of various types of trapped modes and their relation to physicalobservables, but also serve as guide for the more complicated case of three-dimensionalcavities to be treated in a subsequent paper.

The paper is organized as follows: after formulating the problem and giving a briefoutline of the solution method in §2 we treat the single-cavity problem in §3 first assum-ing symmetry about the duct axis before introducing asymmetry. The effects of cavityinteraction are treated in §4 and the concluding §5 gives a brief summary of the mainresults.

2. Formulation of the problem and outline of the solutionAcoustic disturbances in an infinitely long duct of uniform height h∗ with zero mean

flow and several cavities, as shown in figure 1 for N = 2 cavities, are governed by thewave equation, the propagation velocity being the ambient speed of sound c0

∗. Here,and in the following, the star superscript marks dimensional quantities. We formulateour problems in two-dimensional Cartesian coordinates (x, y) nondimensionalized with acharacteristic reference length l∗ref which in our waveguide examples is chosen to be theduct height h∗ such that h = h∗/l∗ref = 1. All velocities are nondimensionalized with c0

∗,densities with the ambient density ρ0

∗ and pressures with ρ0∗c∗0

2. Assuming harmonictime dependence exp(−iω∗t∗), where ω∗ is the circular frequency, the wave equation intwo dimensions reduces to the nondimensional Helmholtz equation

(∂2/∂x2 + ∂2/∂y2 + K2)φ = 0 (2.1)

for the velocity potential φ(x, y). The dimensionless frequency is defined as K = ω∗l∗ref/c0∗,

with K/2π being the Helmholtz number. The time-independent dimensionless distur-bance velocity and pressure are then given by v(x, y) = ∇φ and p(x, y) = iKφ respec-tively.

On the sound-hard walls of the duct as well as the cavities we prescribe Neumannboundary conditions

∂φ

∂n= 0. (2.2)

In a uniform waveguide the solution of (2.1) is in the form of duct modes

φ(x, y) =∞∑

n=0

{AneiKnx + Bne−iKnx} cos{nπ(y + h/2)/h} (2.3)

where An, Bn are the (complex) modal amplitudes of waves travelling in +x directionand −x direction respectively. The modal wavenumber Kn = {K2 − (nπ/h)2}1/2 withpositive real part denotes a propagating wave for K > (nπ/h), otherwise the wave isevanescent (cutoff).

For the (inhomogeneous) scattering problem the frequency K is prescribed and anincoming wave φ(i) is given in modal form

φ(i)(x, y) = EreiKrx cos{rπ(y + h/2)/h} (2.4)


with Er being the prescribed amplitude of the rth incoming propagating duct mode. Theformulation of the scattering problem is completed by imposing a radiation condition forthe scattered wave φ(s) = φ−φ(i). Allowing only waves φ(s) with positive outward energyflux guarantees that the cavities are the sources of a scattered wave travelling to infinity.

The energy flux of φ(s) through a cross section Γ of the waveguide is given by (see forexample Pierce (1981), p.40)

12

Re{∫

Γ

p(s) v(s) · dΓ}

= ±K

2Im

{∫ h/2

−h/2

φ(s)(x, y)∂φ(s)(x, y)

∂xdy

}. (2.5)

The sign ± depends on the orientation of Γ in +x or −x direction. The propagatingmodes (Kn real and positive) in (2.3) with eiKnx are outgoing in +x direction, while themodes with e−iKnx are outgoing in −x direction. The cutoff modes do not contribute tothe energy flux but the same classification can be used for the cutoff modes because forIm(Kn) > 0 the modes with e±iKnx are exponentially decreasing for x → ±∞.

For the resonance problem we obtain the frequency K as solution of an eigenvalueproblem (see Zworski (1999)): we solve for the eigenvalues K2 with the correspondingradiating eigenfunctions φ which satisfy the Helmholtz equation (2.1) together with van-ishing Neumann boundary conditions on the duct and cavity walls. In other words, wecompute the (complex) resonances K with positive real part for which the homogeneousproblem (no incoming wave) has nontrivial solutions. We call Re(K) the resonant fre-quency, φ the resonant function and | Im(K)| the damping. From this the quality factor

Q = Re(K)/(2| Im(K)|) (2.6)

of a resonance can be computed which is proportional to the ratio of stored energy overenergy loss per cycle.

Historically resonances were often defined as large response of an oscillating system toa given source. A resonance defined this way depends on the source and the objectivefunction. For example, using (2.5) we can compute the energy reflection and transmissionfactors τr and ρr which are defined as the ratio between the transmitted (or reflected) en-ergy and the energy of a prescribed rth incoming wave. The frequency of the transmissionresonance is usually different from the frequency of the reflection resonance and cannotbe computed directly from the resonance. The eigenvalue approach gives a unique defini-tion for the resonant frequencies. Numerically, we truncate the unbounded waveguide byartificial boundaries creating a bounded domain which can be discretized by finite ele-ment methods. In this paper we apply the high-order code NGSolve of Joachim Schoberltogether with his grid generation code NETGEN, cf. Schoberl (1997). The resulting ma-trix eigenvalue problem is solved using a shifted Arnoldi algorithm. The accuracy of thefinite-element solution is controlled by the maximal mesh size Δ of the grid and the orderp of the finite element polynomial on an individual triangle. In NGSolve the mesh sizeΔ can be varied locally.

The implementation of the radiation condition needs special attention because at theartificial boundaries unphysical reflections occur unless special boundary conditions areapplied. Basically there are two methods to overcome this problem: the first uses so-callednon-reflecting boundary conditions on the surface bounding the computational domain.They are approximations to the so-called Dirichlet-to-Neumann (DtN) operator, map-ping the Dirichlet values of the unknown solution on the artificial boundary onto theirNeumann values. Due to (2.3) this mapping is given by mapping the modal amplitudesA

(s)n of φ(s) in +x direction and B

(s)n in −x direction to the amplitudes iKnA

(s)n and

−iKnB(s)n respectively. Since the cutoff waves decay exponentially in n, a simple approx-


imation to the exact Dirichlet-to-Neumann operator uses only the first N modes (seefor example Harari et al. (1998)). Such a boundary condition depends nonlinearly on Kbecause Kn depends nonlinearly on K. In the case of the resonance problem this leadsto a nonlinear eigenvalue problem and is therefore not advisable.

The second class of methods consists of a special treatment of the unbounded domainoutside the bounded computational domain. For our numerical analysis we use two dif-ferent methods of this kind: the scattering problem is solved using the so-called Hardyspace method of Nannen (2008); Hohage & Nannen (2009); Nannen & Schadle (2011).The method has been implemented by Lothar Nannen in the high-order finite elementcode NGSolve of Joachim Schoberl (cf. Schoberl (1997)) and relies on the fact that theLaplace transform of an outgoing solution to the Helmholtz equation belongs to a certainHardy space, while an incoming solution does not. Hence, using special finite element ba-sis functions out of this Hardy space guarantees that the solution is outgoing. Althoughthe theoretical justification and the implementation of the method is complicated, it isvery easy to use and converges super-algebraically. Only the number of degrees of free-dom in the unbounded domain and a parameter depending on K have to be chosen. TheHardy space method depends linearly on K2 and therefore leads to a linear eigenvalueproblem for the resonance problem.

In case of a scattering with an incoming wave φ(i), we have to consider that only thescattered wave φ(s) is outgoing. Hence, in the bounded interior domain we solve for thetotal wave φ and in the unbounded exterior domain for the scattered wave φ(s). Thisleads to jump terms on the artificial boundaries (see Nannen & Schadle (2011) for thedetails). Moreover, if Kn ≈ 0, i.e. if an evanescent wave becomes a propagating wave,the convergence of the method will be very slow. Therefore, a variant of the Hardy-spacemethod is used in this case: roughly speaking the critical mode is incorporated into theGalerkin basis of the method. Since this variant depends nonlinearly on K, it cannot beused for resonance problems.

For the solution of the resonance problem we use the complex scaling method becauseit also leads to a linear eigenvalue problem for K2 and has the advantage that it is well es-tablished in the atomic and molecular physics literature. It adds a non-physical layer (theshaded domains in figure 1) to the computational domain in which outgoing waves areabsorbed (therefore, some authors use the term absorbing boundary conditions). A per-fectly matched layer (PML) is a refined absorbing layer introduced by Berenger (1994)with no spurious reflections at the PML interface due to perfect impedance matching(however, weak reflections are possible due to numerical discretization). Instead of thefrequency-dependent PML formulation of Berenger (1994) in the form of the complexcoordinate stretching formulation of Chew & Weedon (1994), Hein et al. (2004) usedthe much older and essentially identical complex scaling method of atomic and molecu-lar physics with frequency-independent PML coefficients. The complex scaling method,also termed Aguilar-Baslev-Combes-Simon theory, was introduced by Aguilar & Combes(1971), Baslev & Combes (1971) and Simon (1973), see the monograph by Hislop & Sigal(1996) or the recent review by Moiseyev (1998). In our past publications we described theimplementation of complex scaling in more detail so that we refer to these publications(see for example Hein et al. (2004) or Hein et al. (2010)) as well as the original papersfor more details.

3. Two-dimensional duct with a single cavityFirst we consider a single cavity (N = 1) of depth b = h + 2d and length l located

in an infinitely long two-dimensional duct of height h, see figure 1 with d = d1, d2 = 0,

Trapped modes and Fano resonances in ducts with cavities 7y

x

d1

d1

d2

d2

h b1 = h + 2d1

b2 = h + 2d2

l1 l2

c1

s1 s2

PML PML

xPMLl

xPMLr

dPML dPML Er

Rs;r Ts;r

Figure 1. Sketch of acoustic scattering by N = 2 cavities in a two-dimensional duct withPMLs.

s = s1 and l = l1. If the cavity is symmetric about the duct axis (s = 0) it protrudes oneach side of the duct by the same amount d. One possibility of breaking the symmetryis to keep the cavity depth b = h + 2d constant and move the cavity in y-direction bythe amount s, indicated by the dashed cavity outline in figure 1 (similar structures havebeen considered in electronic and photonic stub tuners, see for example Akis et al. (1995)or Akis & Vasilopoulos (1996)).

For such a duct-cavity configuration the incoming multi-modal sound is prescribed interms of the modal amplitudes Er of the propagating duct modes. However, in this paperwe limit ourselves to a single incoming mode Er as given by (2.4). Then the amplitudeof the sth reflected and transmitted mode due to the rth incident mode is denoted byRs;r and Ts;r respectively and the modal energy reflection and transmission coefficientsare defined by

ρs;r =(2 − δr,0)Ks

(2 − δs,0)Kr

∣∣∣∣Rs;r

Er

∣∣∣∣2

, τs;r =(2 − δr,0)Ks

(2 − δs,0)Kr

∣∣∣∣Ts;r

Er

∣∣∣∣2

. (3.1)

Here δm,n denotes Kronecker’s delta function. The total energy reflection and trans-mission coefficients ρr and τr due to a single incoming mode r are then obtained bysumming the above defined modal reflection and transmission coefficients over all prop-agating modes s. Conservation of energy requires

ρr + τr = 1, (3.2)

which provides an excellent check of the numerical accuracy. In our computations wedo not use the modal coefficients (3.1) but compute the total energy reflection andtransmission coefficients directly by using (2.5). In noise control usually the so-calledtransmission loss TL (in decibels) is used which is defined by

TL[dB] = 10log10(1/τr). (3.3)

Before we investigate the scattering problem we compute the resonances which are asimportant in open systems as in closed systems because they are responsible for maximain the response. Whereas in open systems most resonances are radiation damped (leakymodes) there exist also trapped modes in duct-cavity systems with nominally zero radi-ation loss. Acoustic resonances are closely related to the size and shape of the resonator.For a rectangular cavity the cavity depth d and cavity length l are the dominating ge-


ometric parameters and we investigate in the following the dependence of the acousticresonances on these two parameters for a symmetric duct-cavity system with s ≡ 0.

3.1. Symmetrical duct-cavity system (s ≡ 0)

In electronic waveguides such symmetrical structures are termed double stub tuners, seefor example Akis et al. (1997). For a single rectangular cavity of depth d and length l inan infinite wall radiating into the open half space the acoustic resonances can be classifiedby two integers (m, n), where m = 0, 1, 2, . . . , is the number of cavity pressure nodes inthe x direction (longitudinal mode number) and n = 0, 1, 2, . . . , is the number of cavitypressure nodes in the y direction (depth mode number), see Tam (1976) or Koch (2005).Similarly, the acoustic resonances of a single cavity-duct system can also be classified by(m, n), but now m and n are the longitudinal and depth mode numbers of the extendedcavity of length l and depth b = h + 2d radiating to infinity through the duct (comparethe analogous modal classification in a photonic T-stub by Danglot et al. (1998)). Ifthe cavity openings into the duct are closed by solid walls the resonances of this closed,hard-walled cavity are real and are given by

Re(K/2π) = 0.5{

(m/l)2 + (n/b)2}1/2

, m, n = 0, 1, 2, . . . . (3.4)

However, there is a fundamental difference between the scattering by a cavity in ahalf space and the scattering by a cavity located symmetrically in a duct: the resonantmodes of the former have symmetry properties only about the mid-cavity axis whereas themodes of the latter are also symmetric or antisymmetric about the duct axis. The possibleantisymmetry about the duct axis is the reason for the existence of localized trappedmodes as shown by Evans & Linton (1991). Strictly speaking our modal classification isnot exact because the nodal lines of leaky modes are not fixed in time and the modalnumbers can even change near avoided crossings where two neighbouring resonancesinteract with each other. Nevertheless, this modal classification allows us to keep track ofindividual resonances and in the following we define resonant modes which are symmetricin both x and y by SSmn, modes which are antisymmetric in x and symmetric in y byASmn, modes which are symmetric in x and antisymmetric in y by SAmn, and finally,modes which are antisymmetric in both x and y by AAmn. In the following plots wemark x-symmetric resonances by solid curves and x-antisymmetric resonances by dashedcurves. Due to symmetry properties of the modes all computations can be performed inthe quarter domain x ≤ l/2, y ≥ 0.

3.1.1. Influence of cavity depth d/h

From the simple quarter-wavelength resonator theory, see for example Munjal (1987),we know that the cavity depth d (enlarged by the corresponding end correction lend)determines the resonant frequency. Therefore, we fix first the cavity length l/h = 0.5and vary the cavity depth d/h. The resonant frequencies Re(K/2π) as well as the corre-sponding damping Im(K/2π) of a few modes are depicted in figures 2 and 3 as functionof cavity depth d/h. Here figure 2 shows the y-symmetric resonances and figure 3 they-antisymmetric resonances. Also shown are the corresponding real resonances of theclosed cavity (3.4) by the dotted curves.

For d/h = 0 the resonant frequencies of the cavity depth modes (0, r) start at thecorresponding y-symmetric or y-antisymmetric cutoff frequencies of the duct modes,marked on the right-hand side of figures 2 and 3 respectively, and decrease rapidly withincreasing d/h. Of particular importance are the y-antisymmetric modes because belowthe cutoff frequency of the first duct crossmode, marked by the shaded domain in figure


0.0 0.4 0.8 1.2 1.6 2.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Re(

K/2

π)

(a)

r = 0

r = 2

r = 4

SS02

SS04SS06

AS12AS14 AS16

AS16 AS18

AS18

0.0 0.4 0.8 1.2 1.6 2.0-0.20-0.16-0.12-0.08-0.04 0.00

Im(K

/2π)

(b)

0.0 0.4 0.8 1.2 1.6 2.0

-0.4-0.3-0.2-0.1 0.0

Im(K

/2π)

d/h

(c)

Figure 2. Symmetric duct-cavity system with N = 1: variation of y-symmetric resonances withvariable cavity depth d/h keeping the cavity length l/h = 0.5 fixed. (a) resonant frequencies,(b) damping of x-symmetric modes, (c) damping of x-antisymmetric modes. Solid curves markx-symmetric modes, dashed curves mark x-antisymmetric modes. The dotted curves show thereal resonances (3.4) of the closed system.

3, the antisymmetric modes cannot radiate to infinity through the duct and therefore aretrapped near the cavity as shown by Evans & Linton (1991).

For d/h → ∞ the resonant frequencies approach Re(K/2π) = μ/2l, μ = 0, 1, 2, . . . ,which are the cutoff frequencies of the propagating duct modes in the finite length sideduct of width l/h = 0.5 formed by the cavity. An exception seems to be the mode markedby diamond symbols in figure 2: this mode is highly damped and seems to constantlyinterchange its modal character with the crossing AS modes. The physical reason behindthe limiting frequency value of this damped mode is not clear (it is not a resonance ofinfinitely long crossing ducts).

The values at d/h = 1 in our figures 2 and 3 should agree with the results of Duan et al.


0.0 0.4 0.8 1.2 1.6 2.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Re(

K/2

π)

(a)

r = 1

r = 3

SA01

SA03SA05

AA11AA13

AA15AA17

0.0 0.4 0.8 1.2 1.6 2.0-0.20-0.16-0.12-0.08-0.04 0.00

Im(K

/2π)

(b)

0.0 0.4 0.8 1.2 1.6 2.0-0.40

-0.30

-0.20

-0.10

0.00

Im(K

/2π)

d/h

(c)

Figure 3. Symmetric duct-cavity system with N = 1: variation of y-antisymmetric resonanceswith variable cavity depth d/h keeping the cavity length l/h = 0.5 fixed. (a) resonant frequencies,(b) damping of x-symmetric modes, (c) damping of x-antisymmetric modes. Solid curves markx-symmetric modes, dashed curves mark x-antisymmetric modes. The dotted curves show thereal resonances (3.4) of the closed system.

(2007) at l/h = 0.5 (their figures 10 and 12) which have been computed by the complexscaling method but using the multi-domain Chebyshev collocation method instead of thepresent finite element method. As expected s1 and s2 in figure 12 of Duan et al. (2007)agree with our modes SS02 and SS04 in figure 2. However, in figure 10 of Duan et al.(2007) our mode SA03 at Re(K/2π) = 0.5 is missing. This mode is special because ford/h = 1 the resonant cavity mode SA03 coincides with the first crossmode in the mainduct and it was not clear in Duan et al. (2007) if it is a true resonance or merely amember of the continuous spectrum. But from our figure 3 it can be seen very clearlythat as soon as d/h = 1 this mode is a true resonant mode which is trapped for d/h > 1and damped for d/h < 1.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Re(

K/2

π)

(a)

μ = 0

μ = 2

μ = 4

μ = 6

ν = 1

ν = 2

SS02

SS02

SS04

SS06

AS10

SS20

SS20

AS12

AS14 SS24

SS42

AS32SS22

AS10

AS120.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.20-0.16-0.12-0.08-0.04 0.00

Im(K

/2π)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.30-0.25-0.20-0.15-0.10-0.05 0.00

Im(K

/2π)

l/h

(c)

Figure 4. Symmetric duct-cavity system with N = 1: variation of y-symmetric resonances withvariable cavity length l/h keeping the cavity depth d/h = 0.5 fixed. (a) resonant frequencies,(b) damping of x-symmetric modes, (c) damping of x-antisymmetric modes. Solid curves markx-symmetric modes, dashed curves mark x-antisymmetric modes. The dotted curves show thereal resonances (3.4) of the closed system. The encircled cross marks an embedded trappedmode.

3.1.2. Influence of cavity length l/h

The second important geometric parameter is the cavity length l/h. In the followingwe keep d/h = 0.5 constant and vary l/h. Figures 4 and 5 show the y-symmetric andy-antisymmetric resonances respectively. For l/h = 0 the resonant frequencies Re(K/2π)of the (0, n) resonances approach the one-dimensional organ pipe resonances (see East(1966))

Re(K/2π) = (2ν − 1)/4d, ν = 1, 2, . . . , (3.5)


0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Re(

K/2

π)

(a)

μ = 1

μ = 3

μ = 5

ν = 1

ν = 2

SA01AA11

SA21AA31

SA03 SA03

AA13

SA21

AA31SA41

AA13

SA05

AA15

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.20-0.15-0.10-0.05 0.00

Im(K

/2π)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.10-0.08-0.06-0.04-0.02 0.00

Im(K

/2π)

l/h

(c)

Figure 5. Symmetric duct-cavity system with N = 1: variation of y-antisymmetric resonanceswith variable cavity length l/h keeping the cavity depth d/h = 0.5 fixed. (a) resonant frequencies,(b) damping of x-symmetric modes, (c) damping of x-antisymmetric modes. Solid curves markx-symmetric modes, dashed curves mark x-antisymmetric modes. The dotted curves show thereal resonances (3.4) of the closed system. Encircled crosses mark embedded trapped modes.

marked by arrows on the left-hand side of figures 4 and 5. Again all y-antisymmetricresonances with frequencies below the cutoff frequency Re(K/2π) = 0.5 of the first ductcrossmode (shaded domain in figure 5) are trapped. Figure 6 shows the eigenfunctionscorresponding to the first x-symmetric and first x-antisymmetric trapped mode at l/h =1.5 (see also Koch (2005)).

As shown by Duan et al. (2007), aside from these y-antisymmetric trapped modeswith frequencies below the cutoff frequency of the first duct crossmode also discrete em-bedded trapped modes occur with frequencies above the cutoff frequency of the firstduct crossmode (see also the recent survey by Linton & McIver (2007)). These embed-


(a) (b)

Figure 6. Eigenfunctions φ(x, y) corresponding to (a) the first x-symmetric trapped modeSA01 and (b) the first x-antisymmetric trapped mode AA11 at l/h = 1.5 in figure 5a.

(a) (b) (c)

Figure 7. Eigenfunctions φ(x, y) corresponding to three of the four embedded trappedmodes marked in figures 4 and 5: (a) y-symmetric, x-symmetric embedded trapped mode atl/h = 1.97, Re(K/2π) = 0.5078, (b) y-antisymmetric, x-symmetric embedded trapped modeat l/h = 1.408, Re(K/2π) = 0.7541, (c) y-antisymmetric, x-antisymmetric embedded trappedmode at l/h = 1.986, Re(K/2π) = 0.7964.

ded trapped modes exist only for particular combinations of the geometry parameters(d/h, l/h) and may be y-symmetric or y-antisymmetric. Duan et al. (2007) computedthese embedded trapped modes directly for a rectangular cavity. Here we show only a fewexamples, marked by encircled crosses in figures 4 and 5 which were computed numeri-cally by searching for locations where the damping of the resonance is less than 10−7. Toillustrate the localized character of embedded trapped modes we plotted in figure 7 threeeigenfunctions corresponding to three out of the four embedded trapped modes markedin figures 4 and 5. Unlike the eigenfunctions of the trapped modes with frequencies be-low the cutoff frequency of the first duct crossmode the eigenfunctions of the embeddedtrapped modes can no longer be classified by our two modal numbers (m, n). This seemsto be due to the fact that these embedded trapped modes are generated by the interactionof two resonant modes. It is also interesting to note that the embedded trapped modesoccur very close to the point of modal degeneracy of the closed system, e.g. where thedotted curves of modes with the same symmetry cross (compare also the analogous re-sults of Sadreev et al. (2006) for a rectangular quantum billiard with Dirichlet boundaryconditions, or Duan et al. (2007)).

For l/h → ∞ the resonant frequencies approach Re(K/2π) = μ/[2(h + 2d)], μ =0, 1, 2, . . . , which are the cutoff frequencies of the duct modes in a duct of width b = h+2d.The corresponding cutoff frequencies are marked by arrows on the right-hand side offigures 4 and 5.


3.2. Asymmetrical duct-cavity system (s = 0)For the symmetrical duct-cavity system (s ≡ 0) treated in the previous section thereexist y-symmetric and y-antisymmetric resonances, but an incoming y-symmetric ductmode cannot excite an y-antisymmetric trapped mode. This changes drastically if thesymmetry about the duct axis is broken. In this section we investigate the behaviour ofthe resonances as the symmetry in y is broken and compute the corresponding soundtransmission and reflection coefficients τ0 and ρ0 respectively. For this purpose we vary theasymmetry parameter s/h keeping the cavity geometry, i.e. cavity depth b/h = 1 + 2d/hand cavity length l/h fixed and select three representative values d/h = 0.25, 0.5 and 1.0in figures 2 and 3 to start with at s/h = 0. The corresponding initially y-symmetric andy-antisymmetric resonances at s/h = 0 are marked by arrows on the left hand side ofthe figures 8, 10 and 12. From figures 3 and 2 we observe that for all three chosen valuesof d/h the frequency of the trapped mode SA01 is always below the cutoff frequency ofthe first duct crossmode, whereas the frequency of the first y-symmetric resonance SS02is above the cutoff frequency of the first duct crossmode for d/h = 0.25, but below ford/h = 0.5 and approaches the resonant frequency of the SA01 mode for d/h = 1.

3.2.1. Asymmetrical duct-cavity system with d/h = 0.25For the asymmetrical duct-cavity system s/h = 0 there is no more symmetry or anti-

symmetry about the duct axis and correspondingly we use the half domain x ≤ l/2 in ourcomputation. Figure 8 shows the variation of the resonances for 0 ≤ s/h ≤ 0.25 keepingd/h = 0.25 and l/h = 0.5 constant, where the limiting value s/h = 0.25 corresponds toa one-sided cavity. Due to the changing location of the cavity opening the resonanceschange slightly with s/h, whereas the real resonances of a closed cavity (indicated bythe dotted lines in figure 8 for comparison) remain constant. All resonant frequencies de-cay slowly with increasing s/h and the trapped mode SA01 becomes a slightly dampedquasi-trapped mode. The damping of the mode originating from SS02 increases withincreasing s/h whereas the damping of the mode originating from AS12 decreases withincreasing s/h and therefore this resonance should show up in the response (note thatAS10 and AS12 interchange their modal character near the avoided crossing at l/h ≈ 1.2in figure 4a).

To see the influence of the resonances on the response to an incoming fundamentalduct wave we computed the acoustic transmission and reflection coefficients τ0 and ρ0

for several values of s/h in figure 9 as function of frequency Re(K/2π) keeping d/h =0.25 and l/h = 0.5 fixed. In contrast to the scattering by objects in a waveguide, asdiscussed by Hein et al. (2010), a duct-cavity system is basically transparent. Therefore,instead of transmission resonances we obtain reflection resonances. We see that for they-symmetric duct-cavity system with s/h = 0 the reflection coefficient shows maximaonly at the resonances of the y-symmetric modes SS02 and AS12. As soon as s/h =0 very distinct Fano resonances with typical resonance-antiresonance features appearin the shaded frequency domain of figure 9 near the frequency of the quasi-trappedmode. Corresponding to the decreasing resonant frequency of the quasi-trapped modethe frequency of the maximum of the reflection coefficient ρ0 also decreases slightly inthe shaded domain.

3.2.2. Asymmetrical duct-cavity system with d/h = 0.5Next we consider a cavity with d/h = 0.5 and l/h = 0.5 fixed. Figure 10 shows

the variation of the corresponding resonances with 0 ≤ s/h ≤ 0.5 where the limitingvalue s/h = 0.5 denotes again a one-sided cavity. The resonant frequency of the modeemerging from SA01 decreases with increasing s/h and the trapped mode becomes a


0.00 0.05 0.10 0.15 0.20 0.25 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4R

e(K

/2π)

(a)

r = 0

r = 1

r = 2

r = 3

SA01

SS02

SA03

AS12SS04

0.00 0.05 0.10 0.15 0.20 0.25-0.20-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.02 0.00

Im(K

/2π)

s/h

(b)

Figure 8. Asymmetric duct-cavity system with N = 1: variation of resonances with antisym-metry parameter s/h keeping cavity depth d/h = 0.25 and cavity length l/h = 0.5 fixed. (a)resonant frequencies, (b) damping. The dotted lines show the real resonances (3.4) of the corre-sponding closed cavity.

slightly damped quasi-trapped mode. The frequency of the mode originating from SS02approaches the cutoff frequency of the first duct crossmode with decreasing damping.The one-sided cavity at s/h = 0.5 is a special case because the first cutoff mode of theduct is also a solution of the duct-cavity system and therefore has zero damping. Thefrequency of the mode originating from SA03 increases slightly with increasing s/h andthe corresponding damping decreases till it reaches a minimum around s/h = 0.31 beforeincreasing again.

Figure 11 shows the corresponding transmission and reflection coefficients τ0 and ρ0

respectively, as well as the transmission loss TL for d/h = 0.5, l/h = 0.5 and various s/has function of frequency Re(K/2π). Again Fano resonances appear in the shaded domainof figure 11 as soon as s/h = 0. The frequency of the maximum of the reflection coeffi-cient ρ0 in the shaded domain also decreases slightly as anticipated from the decreasingresonance frequency of the quasi-trapped mode in figure 10. Clearly visible is also thegrowing influence of the mode originating from SA03 which reaches a maximum of ρ0 fors/h = 0.3 near Re(K/2π) = 0.8 as a consequence of the decreasing damping of this modewith increasing s/h. But most striking is that the maximum of the mode originating from


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0

0.2

0.4

0.6

0.8

1.0

τ 0

r = 0 r = 1 r = 2

s/h = 0

0.05

0.1

0.15

0.2

0.25(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0

0.2

0.4

0.6

0.8

1.0

ρ 0

(b)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 10 20 30 40 50 60

TL

[dB

]

Re(K/2π)

(c)

Figure 9. Asymmetric duct-cavity system with N = 1: variation of antisymmetry parameter s/hkeeping cavity depth d/h = 0.25 and cavity length l/h = 0.5 fixed. (a) transmission coefficientτ0, (b) reflection coefficient ρ0 and (c) transmission loss TL.

SS02 remains large, shifting towards the cutoff frequency of the first duct crossmode,such that we obtain now two narrow-frequency peaks for the transmission loss. This is aconsequence of this mode becoming a quasi-trapped mode with very low damping.


0.0 0.1 0.2 0.3 0.4 0.5 0.0

0.2

0.4

0.6

0.8

1.0

1.2R

e(K

/2π)

(a)

r = 0

r = 1

r = 2

SA01

SS02

SA03

SS04

AS12

AA11

0.0 0.1 0.2 0.3 0.4 0.5-0.20-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.02 0.00

Im(K

/2π)

s/h

(b)

Figure 10. Asymmetric duct-cavity system with N = 1: variation of resonances with anti-symmetry parameter s/h keeping cavity depth d/h = 0.5 and cavity length l/h = 0.5 fixed.(a) resonant frequencies, (b) damping. The dotted lines show the real resonances (3.4) of thecorresponding closed cavity.

3.2.3. Asymmetrical duct-cavity system with d/h = 1.0Finally, we show in figure 12 the variation of the resonances as function of 0 ≤ s/h ≤ 1.0

keeping d/h = 1.0 and l/h = 0.5 constant. Now we obtain in addition to the quasi-trapped mode originating from the trapped mode SA01 a second quasi-trapped modeoriginating from the second trapped mode SA03 with the resonant frequencies of bothmodes remaining below the threshold frequency of the first duct crossmode. On theother hand the frequency of the mode originating from SS02 increases with increasings/h and reaches a maximum near s/h = 0.667. The corresponding damping decreases andapparently forms an embedded trapped or quasi-trapped mode, marked by the encircledcross in figure 12, where the damping is about 10−7. The possible existence of embeddedtrapped modes for asymmetrical duct-cavity systems has been conjectured by Sadreevet al. (2006) for a related quantum billiard problem with Dirichlet boundary conditions.The frequency of the damped mode originating from SS04 varies slightly as s/h increasesand the corresponding damping reaches a minimum near s/h = 0.75 after increasing first.

The corresponding transmission and reflection coefficients τ0 and ρ0 respectively areshown in figure 13 for d/h = 1 and l/h = 0.5. The spectra mirror the results obtained for


0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

τ 0

r = 0 r = 1 r = 2

s/h = 00.10.20.30.40.5

(a)

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

ρ 0

(b)

0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60

TL

[dB

]

Re(K/2π)

(c)

Figure 11. Asymmetric duct-cavity system with N = 1: variation of antisymmetry parameters/h keeping cavity depth d/h = 0.5 and cavity length l/h = 0.5 fixed. (a) transmission coefficientτ0, (b) reflection coefficient ρ0 and (c) transmission loss TL.

the resonances in figure 12. Already for s/h = 0 we observe two peaks of ρ0 correspond-ing to the first and second y-symmetric resonances SS02 and SS04 respectively. Thefrequency of the first peak increases steadily with increasing s/h up to the frequency ofthe embedded (or quasi-trapped) mode at s/h = 0.3749, marked by the arrow with the


0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0R

e(K

/2π)

(a)

r = 0

r = 1

r = 2

SA01

SS02

SA03

SS04

SA05

0.0 0.2 0.4 0.6 0.8 1.0-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Im(K

/2π)

s/h

(b)

Figure 12. Asymmetric duct-cavity system with N = 1: variation of resonances with anti-symmetry parameter s/h keeping cavity depth d/h = 1.0 and cavity length l/h = 0.5 fixed.(a) resonant frequencies, (b) damping. The dotted lines show the real resonances (3.4) of thecorresponding closed cavity. The encircled cross marks a possible embedded trapped mode.

encircled cross on top of figure 13a, where we observe a very narrow resonance. After thatthe frequency of the maximum of the reflection coefficient decreases again with increasings/h and the resonance width becomes wider again. Due to the other two quasi-trappedmodes emerging from SA01 and SA03 we observe Fano resonances in the two shadeddomains such that we have three sharp peaks in the transmission loss TL for s/h = 0.The Fano resonance near Re(K/2π) = 0.5 becomes increasingly narrower as the dampingof the corresponding quasi-trapped mode approaches zero near s/h = 1.

4. Two-dimensional duct with multiple cavitiesIn this section we investigate the coupling between cavities and how this effects the

acoustic resonances as well as the transmission and reflection of sound. It is well knownthat if several structures are arranged in series a band structure with transmission stopbands emerges. Often very few structures are necessary to achieve this effect. For aone-dimensional acoustic waveguide with slender side branches, i.e. the cross section ofthe waveguide and side branches is much smaller than the wavelength, this has been


0.0 0.2 0.4 0.6 0.8 0 10 20 30 40 50

TL

[dB

]

Re(K/2π)

(c)0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

ρ 0

(b)0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

τ 0

r = 0 r = 1

s/h = 00.20.40.60.81.0

(a)

Figure 13. Asymmetric duct-cavity system with N = 1: variation of antisymmetry parameters/h keeping cavity depth d/h = 1.0 and cavity length l/h = 0.5 fixed. (a) transmission coefficientτ0, (b) reflection coefficient ρ0 and (c) transmission loss TL.

demonstrated for example by Wang et al. (2001) or El Boudouti et al. (2008). The trans-mission gaps become more pronounced the more side branches are added. In addition,El Boudouti et al. (2008) found sharp Fano-like resonances within these transmissiongaps without incorporating any defects in the structure. Essentially the side branches


act as Bragg reflectors and therefore these resonances depend on the distance betweenthe side branches. Similar results have been obtained for the related electronics problemof two or several quantum dots in series, see for example Ordonez et al. (2006) or Cat-tapan & Lotti (2008). Our present numerical computation extends the one-dimensionaltreatment of El Boudouti et al. (2008) to the two-dimensional case where the wavelengthis of the order of the channel and cavity width.

4.1. Symmetrical multi-cavity duct systemWith the multitude of geometric parameters we are forced to limiting ourselves to specialcavity parameters. Starting with a multiple cavity-duct system which is symmetricalabout the duct axis we choose N identical cavities with d1/h = d2/h = . . . = dN/h =d/h = 0.5 and l1/h = l2/h = . . . = lN/h = l/h = 0.5.

4.1.1. Influence of inter-cavity distance c/h

First we consider only N = 2 identical cavities and vary the inter-cavity distance c/hwhich can be taken as measure of coupling strength, where small c/h implies strongcoupling. Making use of symmetries we limit our computation to the quarter plane x ≥l + c/2, y ≤ 0. The results for the first few y-symmetric and y-antisymmetric resonancesare shown in figures 14 and 15 respectively as function of c/h. Again, solid curves denotex-symmetric resonances and dashed curves mark x-antisymmetric resonances. In figure14a we notice that with increasing c/h the resonant frequencies decrease. For particularinter-cavity distances resonances with vanishingly small damping occur below the cutofffrequency of the first duct crossmode. We marked these resonances by encircled crossesand consider them to be embedded trapped modes if Im(K/2π) < 10−7. These embeddedtrapped modes appear at nearly periodic inter-cavity distances c/h which is typical forFabry–Perot interference.

Figure 15 shows the y-antisymmetric resonances. All resonances below the cutoff fre-quency of the first duct crossmode (shaded domain) are trapped. In addition, someembedded trapped modes above the cutoff frequency of the first duct crossmode aremarked by encircled crosses. To get a feeling for these Fabry–Perot-type trapped modesthe eigenfunctions of three embedded trapped modes marked in figures 14 and 15 areshown in figure 16. Furthermore, an important effect of cavity coupling becomes evidentin figure 15: for small inter-cavity distance c/h, corresponding to strong cavity coupling,the trapped mode SA01 of the single cavity in figure 5 splits into an x-symmetric andx-antisymmetric ’supermode’. Such interaction-dependent mode splitting has been ob-served also for the electronic states in diatomic molecules with bonding and antibondingorbitals as well as for photonic molecules, see for example Bayer et al. (1998) and refer-ences cited therein. For large c/h (and identical cavities) the two trapped ’supermodes’degenerate into the single trapped mode SA01 of the single cavity, as marked on theright-hand side of figure 15. If instead of two cavities N cavities are present then asplitting into N ’supermodes’ occurs for small c/h.

Next we compute the sound transmission and reflection coefficient τ0 and ρ0 respec-tively for N = 2 identical cavities with d/h = l/h = 0.5 separated by variable c/h. For anincoming fundamental duct mode only y-symmetric resonances can be excited. Figure 17shows the results for several values of c/h as a function of frequency 0 ≤ Re(K/2π) ≤ 0.9.We chose the values of c/h such that the computed range includes the first x-symmetricembedded trapped mode near c/h = 0.71 in figure 14. For c/h = 0.1 a transmission gapemerges which is centred around the transmission minimum of the corresponding singlecavity (compare the result for s/h = 0 in figure 11). With increasing c/h the transmis-sion gap widens and the first x-symmetric resonance, marked by solid circles in figure


0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8R

e(K

/2π)

(a)

r = 2

r = 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.20-0.16-0.12-0.08-0.04 0.00

Im(K

/2π)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.20-0.16-0.12-0.08-0.04 0.00

Im(K

/2π)

c/h

(c)

Figure 14. Symmetric duct-cavity system with N = 2 interacting identical cavities withd/h = 0.5, l/h = 0.5: variation of inter-cavity distance c/h for y-symmetric resonances. (a) res-onant frequencies, (b) damping of x-symmetric modes, (c) damping of x-antisymmetric modes.Solid curves mark x-symmetric modes, dashed curves mark x-antisymmetric modes. The encir-cled crosses denote embedded trapped modes.

14, becomes a quasi-trapped, strongly localized mode similar to a defect mode. The fre-quency of this quasi-trapped mode decreases with increasing c/h, a typical consequenceof Fabry–Perot interference, and therefore traverses the transmission gap from the upperto the lower edge of the transmission gap. At the first x-symmetric embedded trappedmode near c/h = 0.71, marked by the encircled cross on top of figure 17, this resonancebecomes a Fano-like resonance with zero damping. For larger values of c/h it becomesagain a quasi-trapped mode with small damping. A closer inspection near the Fano-likeresonance shows that the transmission loss has a double peak (see the enlarged insert infigure 17c where the computed points for c/h = 0.75 are shown by small filled circles).


0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4R

e(K

/2π)

(a)

SA01

r = 1

r = 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.25-0.20-0.15-0.10-0.05 0.00

Im(K

/2π)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.20

-0.15

-0.10

-0.05

0.00

Im(K

/2π)

c/h

(c)

Figure 15. Symmetric duct-cavity system with N = 2 interacting identical cavities withd/h = 0.5, l/h = 0.5: variation of inter-cavity distance c/h for y-antisymmetric resonances.(a) resonant frequencies, (b) damping of x-symmetric modes, (c) damping of x-antisymmetricmodes. Solid curves mark x-symmetric modes, dashed curves mark x-antisymmetric modes.The shaded domain marks the region of trapped modes and encircled crosses denote embeddedtrapped modes.

First we thought this to be due to numerical inaccuracy, but increasing the accuracy ofIm(K/2π) to less than 10−9 identical results were obtained. The physical meaning of thisdouble peak is still unclear. When c/h approaches the value of the first x-antisymmetricembedded trapped mode the second x-antisymmetric resonance, marked by open trian-gles in figure 14, traverses the transmission gap, and so on.

4.1.2. Influence of number of interacting cavities N

As mentioned before, the transmission gaps become more pronounced the more cavitiesare added. This can be seen in figure 18 for d/h = 0.5, l/h = 0.5, c/h = 0.1 and N = 1


(a) (b) (c)

Figure 16. Eigenfunctions φ(x, y) corresponding to three of the Fabry–Perot-type trappedmodes marked in figures 14 and 15: (a) first y-symmetric, x-symmetric embedded trapped modeat c/h = 0.71, Re(K/2π) = 0.4366, (b) first y-symmetric, x-antisymmetric embedded trappedmode at c/h = 1.856, Re(K/2π) = 0.4367, (c) y-antisymmetric, x-symmetric embedded trappedmode at c/h = 2.2, Re(K/2π) = 0.5364.

up to N = 4. The mid-gap frequency is determined by the single cavity transmissionminimum and already as few as N = 4 cavities suffice to obtain a very flat transmissiondip which can be used for the design of an acoustic band-filter. Very similar observationshave been made for optical resonators coupled to a waveguide, see for example figure 11in Xu et al. (2000).

For N cavities we observe N − 1 quasi-trapped modes (x-symmetric and x-antisym-metric) forming a transmission band traversing the transmission gap. At the c/h valuecorresponding to an embedded trapped mode these N − 1 quasi-trapped modes seem tocoalesce forming a narrow transmission resonance in the transmission gap, see figure 19for the example N = 4. A closer inspection for N = 4 near the embedded trapped modereveals that whereas the imaginary parts of all three quasi-trapped mode resonances atc/h = 0.71 are less than 10−7, the frequencies differ in the fourth digit. Cattapan &Lotti (2008) found a similar miniband at slightly different energies in their investigationof serial quantum dots, see their figure 4, and attributed it to the presence of evanescentmodes in the connecting bridges and leads. The transmission band with N − 1 peakswithin the transmission gap can be seen more clearly in figure 20 depicting the transmis-sion coefficient τ0, reflection coefficient ρ0 and transmission loss for N = 4. For claritywe included only three values for c/h. Enlarging the figure around the frequency of theembedded trapped mode (see the insert in figure 20c) we find that the transmission coef-ficient τ0 has 3 peaks but the transmission loss TL has 4 peaks which is in agreement withthe findings in the insert in figure 17c for N = 2. A similar appearance of multiple peaksinstead of a transmission band plateau, depending on the number of optical resonators,has been shown by Xu et al. (2000) in their figure 13.

4.2. Asymmetrical multi-cavity duct system

As long as the multi-cavity duct system is symmetric about the duct axis the trappedmodes with frequencies below the cutoff frequency of the first duct crossmode have noinfluence on the transmission and reflection of the fundamental duct mode. Limitingourselves to N = 2 equal cavities with d/h = 0.5 and l/h = 0.5 we study the effectof breaking the symmetry about the duct axis in this section. Breaking the symmetrycan be done in several ways, with two limiting cases namely either both cavities movingnormal to the duct axis by the same amount s in the same direction, i.e s1 = s2 = s, orin the opposite direction with s1 = −s2 (see figure 1). In order to clearly demonstrate theeffect of modal splitting we choose a small inter-cavity distance c/h = 0.1. If both cavitiesmove in the same direction by the same amount s/h the effect of trapped mode splittingin figure 15 can be observed only for small inter-cavity distances c/h; with increasing


0.0 0.2 0.4 0.6 0.8 0 20 40 60 80 100

TL

[dB

]

Re(K/2π)

(c)

0.42 0.44 0.46 0

40

80

0.0 0.2 0.4 0.6 0.8 0.0

0.2

0.4

0.6

0.8

1.0

ρ 0

(b)0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0τ 0

r = 0 r = 1

c/h = 1

0.75

0.5

0.25

0.1

(a)

Figure 17. Symmetric duct-cavity system with N = 2 interacting identical cavities withd/h = 0.5 and l/h = 0.5 for various inter-cavity distances c/h. (a) transmission coefficientτ0, (b) reflection coefficient ρ0 and (c) transmission loss TL.

c/h the double reflection peaks disappear and, surprisingly, the system becomes almosttransparent with only small sound reflection.

On the other hand, if the two cavities move normal to the duct axis in oppositedirections by the same amount s/h the effect of modal splitting can be observed for all


0.0 0.2 0.4 0.6 0.8 0.0

0.2

0.4

0.6

0.8

1.0τ 0

Re(K/2π)

N = 4 32

1

Figure 18. Symmetric duct-cavity system with N interacting identical cavities and d/h = 0.5,l/h = 0.5, c/h = 0.1: influence of number of cavities N on transmission coefficient τ0.

s/h. In figure 21 we plotted the corresponding resonances as function of 0 ≤ s/h ≤ 0.5.At s/h = 0 the two y-antisymmetric split modes of figure 15 are marked by arrows onthe left-hand side of figure 21. For s/h > 0 both trapped modes become quasi-trappedwith the x-symmetric mode being marked by solid circles and the x-antisymmetric modeat slightly higher frequencies being marked by open circles. With increasing s/h the x-antisymmetric mode becomes slightly more damped. The first y-symmetric resonance offigure 14, which governs the transmission dip in the y-symmetric configuration, is markedby open triangles in figure 21 and approaches the cutoff frequency of the first crossmodeas s/h increases.

The influence of these resonances can be seen clearly in the transmission and reflectioncoefficient shown together with the transmission loss in figure 22. For s/h = 0, depicted infigure 22 by the thick solid curve, we recover the curve for N = 2 in figure 18. The doublepeak in the transmission loss in the shaded domain of figure 22c is due to the splittingof the trapped mode. The frequencies of the resulting quasi-trapped modes decreasewith increasing s/h as predicted by the resonances (compare this with the single peakfor a single cavity in figure 11c). The reflection resonance due to the first y-symmetricresonance becomes quite narrow as this resonance, depicted by open triangles in figure21, approaches the first cutoff frequency. Through the double peak the low-frequencyattenuation becomes more broad-band. In analogy to the problem of vibration dampingby coupled oscillators, see Den Hartog (1947), we hypothesize that one might achievean even wider broad-band sound absorption by covering the cavities with a sheet ofabsorptive facing material as is customary in the design of jet engine liners (see also Jinget al. (2007)).

5. ConclusionIn traditional acoustics resonators in a duct are used as low-frequency filters. In the

present paper we reconsidered the two-dimensional problem of single and multiple rect-angular cavities in an infinitely long duct with zero mean flow. We solved the multi-modalproblem numerically by means of the finite element method and computed the (complex)resonances as well as the sound transmission and reflection coefficients using absorbing


0.0 0.4 0.8 1.2 1.6 2.0 0.0

0.2

0.4

0.6

0.8

1.0R

e(K

/2π)

(a)

0.0 0.4 0.8 1.2 1.6 2.0-0.12-0.10-0.08-0.06-0.04-0.02 0.00

Im(K

/2π)

(b)

0.0 0.4 0.8 1.2 1.6 2.0

-0.12

-0.08

-0.04

0.00

Im(K

/2π)

c/h

(c)

Figure 19. Symmetric duct-cavity system with N = 4 interacting identical cavities withd/h = 0.5, l/h = 0.5: variation of inter-cavity distance c/h for y-symmetric resonances. (a) res-onant frequencies, (b) damping of x-symmetric modes, (c) damping of x-antisymmetric modes.Solid curves mark x-symmetric modes, dashed curves mark x-antisymmetric modes. The encir-cled crosses denote embedded trapped modes.

boundary conditions in order to approximate the radiation condition. As is well knownfrom classical acoustics the resonant frequencies decrease with increasing size of theresonator, i.e. with increasing cavity depth d or increasing cavity length l. In additionto radiation-damped resonances we found three different types of trapped modes withnominally zero radiation loss: the first type is the well known trapped mode which isantisymmetric about the duct axis with a resonant frequency below the cutoff frequencyof the first duct crossmode. Such a trapped mode cannot be excited by the fundamen-tal duct mode. However, as soon as the symmetry about the duct axis is broken thesetrapped modes become highly localized quasi-trapped modes causing the appearance of


0.0 0.2 0.4 0.6 0.8 0

40

80

120

160

TL

[dB

]

Re(K/2π)

(c)

0.42 0.44 0.46 0 40 80 120 160

0.0 0.2 0.4 0.6 0.8 0.0

0.2

0.4

0.6

0.8

1.0

ρ 0

(b)0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0τ 0

r = 0 r = 1

c/h = 1

0.75

0.5

(a)

Figure 20. Symmetric duct-cavity system with N = 4 interacting identical cavities withd/h = 0.5 and l/h = 0.5 for three inter-cavity distances c/h: (a) transmission coefficient τ0,(b) reflection coefficient ρ0 and (c) transmission loss TL.

Fano resonances with typical resonance-antiresonance features and drastic changes in thetransmission and reflection coefficient.

The second type of trapped mode is due to the destructive interference of two reso-nant modes and is connected with the phenomenon of avoided crossings of resonances.


0.0 0.1 0.2 0.3 0.4 0.5 0.0

0.2

0.4

0.6

0.8

1.0R

e(K

/2π)

(a)

r = 0

r = 1

r = 2

0.0 0.1 0.2 0.3 0.4 0.5-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Im(K

/2π)

s/h

(b)

Figure 21. Asymmetric duct-cavity system with N = 2 interacting identical cavities withd/h = 0.5, l/h = 0.5, c/h = 0.1 moved by the same amount s/h normal to the duct axis inopposite direction: (a) resonant frequencies, (b) damping. Solid curves mark x-symmetric modes,dashed curves mark x-antisymmetric modes.

Usually termed embedded trapped modes, because these resonances are embedded in thecontinuous spectrum, they cause a bifurcation of resonance widths into long-lived andshort-lived resonances. Near these embedded trapped mode frequencies the two inter-acting modes interchange their modal identity. The third type of trapped mode occursin multi-cavity systems and is associated with Fabry–Perot interference between cavitiesby adjusting the distance between cavities. The latter two types of trapped modes areoperative only at very special geometry parameters and frequencies. As soon as thesespecial conditions are disturbed Fano resonances appear with symmetry-antisymmetryfeatures. In the present paper we presented examples of all three types of trapped modesand showed the effect of various geometry variations. Of particular interest for the designof low-frequency acoustic filters is the trapped-mode splitting into distinct ’supermodes’if several cavities interact. These trapped ’supermodes’ become quasi-trapped modes ifthe symmetry about the duct axis is broken and show up as multiple peaks in the trans-mission loss resulting in damping over a larger frequency domain. Similar effects can beobserved in other physical systems such as quantum waveguides, optical or electronicwaveguides where they led to important applications.


0.0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 60

TL

[dB

]

Re(K/2π)

(c)0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

ρ 0

(b)0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0τ 0

r = 0 r = 1

s/h = 00.10.20.30.40.5

(a)

Figure 22. Asymmetric duct-cavity system with N = 2 interacting identical cavities withd/h = 0.5, l/h = 0.5, c/h = 0.1 moved normal to the duct axis by the same amount s/h in op-posite direction: (a) transmission coefficient τ0, (b) reflection coefficient ρ0 and (c) transmissionloss TL for various asymmetry parameters s/h.


REFERENCES

Aguilar, J. & Combes, J. 1971 A class of analytic perturbations for one-body SchrodingerHamiltonians. Commun.Math.Phys. 22, 269–279.

Akis, R. & Vasilopoulos, P. 1996 Large photonic band gaps and transmittance antiresonancesin periodically modulated quasi-one-dimensional waveguides. Physical Review E 53, 5369–5372.

Akis, R., Vasilopoulos, P. & Debray, P. 1995 Ballistic transport in electron stub tuners:Shape and temperature dependence, tuning of the conductance output, and resonant tun-neling. Physical Review B 52, 2805–2813.

Akis, R., Vasilopoulos, P. & Debray, P. 1997 Bound states and transmission resonances inparabolically confined cross structures: Influence of weak magnetic fields. Physical ReviewB 56, 9594–9602.

Alster, M. 1972 Improved calculation of resonant frequencies of Helmholtz resonators. J.SoundVibration 24 (1), 63–85.

Aslanyan, A., Parnovski, L. & Vassiliev, D. 2000 Complex resonances in acoustic wave-guides. Q.Jl Mech.Appl.Math. 53, 429–447.

Baslev, E. & Combes, J. 1971 Spectral properties of many body Schrodinger operators withdilation analytic interactions. Commun.Math.Phys. 22, 280–294.

Bayer, M., Gutbrod, T., Reithmaier, J., Forchel, A., Reinecke, T., Knipp, P., Dremin,

A. & Kulakovskii, V. 1998 Optical modes in photonic molecules. Phys.Rev.Lett. 81,2582–2585.

Berenger, J. 1994 A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys. 114, 185–200.

Cattapan, G. & Lotti, P. 2007 Fano resonances in stubbed quantum waveguides with impu-rities. Eur.Phys.J.B 60, 51–60.

Cattapan, G. & Lotti, P. 2008 Bound states in the continuum in two-dimensional serialstructures. Eur.Phys.J.B 66, 517–523.

Chanaud, R. 1994 Effects of geometry on the resonance frequency of Helmholtz resonators.J.Sound Vibration 178 (3), 337–348.

Chew, W. & Weedon, W. 1994 A 3-D perfectly matched medium from modified Maxwell’sequation with stretched coordinates. Microwave Optical Technol.Lett. 7 (13), 599–604.

Danglot, J., Carbonell, J., Fernandez, M., Vanbesien, O. & Lippens, D. 1998 Modalanalysis of guiding structures patterned in a metallic phononic crystal. Appl.Phys.Lett. 73,2712–2714.

Debray, P., Raichev, O., Vasilopoulos, P., Rahman, M., Perrin, R. & Mitchell, W.

2000 Ballistic electron transport in stubbed quantum waveguides: Experiment and theory.Phys.Rev.B 61, 10950–10958.

Den Hartog, J. 1947 Mechanical Vibrations. New York: McGraw Hill.Duan, Y., Koch, W., Linton, C. & McIver, M. 2007 Complex resonances and trapped

modes in ducted domains. J.Fluid Mech. 571, 119–147.East, L. 1966 Aerodynamically induced resonance in rectangular cavities. J.Sound Vibration

3 (3), 277–287.El Boudouti, E., Mrabti, T., Al-Wahsh, H., Djafari-Rouhani, B., Akjouj, A. & Do-

brzynski, L. 2008 Transmission gaps and Fano resonances in an acoustic waveguide: an-alytical model. J.Phys.: Condens.Matter 20, 255212–1–255212–10.

Evans, D. & Linton, C. 1991 Trapped modes in open channels. J.Fluid Mech. 225, 153–175.Fan, S., Yanik, M., Wang, Z., Sandhu, S. & Povinelli, L. 2006 Advances in theory of

photonic crystals. J.Lightw.Technol. 24, 4493–4501.Fano, U. 1935 Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco.

Nuovo Cimento 12, 154–161.Fano, U. 1961 Effects of configuration interaction on intensities and phase shifts. Phys.Rev.

124 (6), 1866–1878.Fellay, A., Gagel, F., Maschke, K., Virlouvet, A. & Khater, A. 1997 Scattering of

vibrational waves in perturbed quasi-one-dimensional multichannel waveguides. PhysicalReview B 55, 1707–1717.

Friedrich, H. & Wintgen, D. 1985 Interfering resonances and bound states in the continuum.Phys.Rev.A 32, 3231–3242.


Gonzalez, J., Pacheco, M., Rosales, L. & Orellana, P. 2010 Bound states in the contin-uum in graphene quantum dot structures. ArXiv preprint cond-mat.mes-hall/1009.2711v1 .

Ladron de Guevara, M., Claro, F. & Orellana, P. 2003 Ghost Fano resonance in a doublequantum dot molecule attached to leads. Phys.Rev.B 67, 195335–1–195335–6.

Harari, I., Patlashenko, I. & Givoli, D. 1998 Dirichlet-to-Neumann maps for unboundedwave guides. J.Comp.Phys. 143, 200–223.

Hein, S., Hohage, T. & Koch, W. 2004 On resonances in open systems. J.Fluid Mech. 506,255–284.

Hein, S. & Koch, W. 2008 Acoustic resonances and trapped modes in pipes and tunnels.J.Fluid Mech. 605, 401–428.

Hein, S., Koch, W. & Nannen, L. 2010 Fano resonances in acoustics. J.Fluid Mech. 664,238–264.

Hislop, P. & Sigal, I. 1996 Introduction to Spectral Theory . Springer.

Hladky-Hennion, A.-C., Vasseur, J., Djafari-Rouhani, B. & de Billy, M. 2008 Sonicband gaps in one-dimensional phononic crystals with a symmetric stub. Physical Review B77, 104304–1–104304–7.

Hohage, T. & Nannen, L. 2009 Hardy space infinite elements for scattering and resonanceproblems. SIAM J.Num.Anal. 47 (2), 972–996.

Hurt, N. 2000 Mathematical Physics of Quantum Wires and Devices. Kluwer.

Ji, Z. 2005 Acoustic length correction of closed cylindrical side-branched tube. J.Sound Vibration283, 1180–1186.

Jing, X., Wang, X. & Sun, X. 2007 Broadband acoustic liner based on the mechanism ofmultiple cavity resonance. AIAA–Paper 2007–3537. 13th AIAA/CEAS Aeroacoustics Con-ference, Rome, Italy.

Joe, Y., Satanin, M. & Kim, C. 2006 Classical analogy of Fano resonances. Physica Scripta74, 259–266.

Jungowski, W., Botros, K. & Studzinski, W. 1989 Cylindrical side-branch as tone genera-tor. J.Sound Vibration 131 (2), 265–285.

Koch, W. 2005 Acoustic resonances in rectangular open cavities. AIAA Journal 43 (11), 2342–2349.

Kriesels, P., Peters, M., Hirschberg, A., Wijnands, A., Iafrati, A., Riccardi, G.,

Piva, R. & Bruggeman, J. 1995 High amplitude vortex-induced pulsations in a gastransport system. J.Sound Vibration 184 (2), 343–368.

Linton, C. & McIver, M. 1998 Trapped modes in cylindrical waveguides. Q.JlMech.Appl.Math 51, 389–412.

Linton, C. & McIver, P. 2007 Embedded trapped modes in water waves and acoustics. WaveMotion 45, 16–29.

Miroshnichenko, A., Flach, S. & Kivshar, Y. 2010 Fano resonances in nanoscale structures.Rev.Modern Physics 82, 2257–2298.

Moiseyev, N. 1998 Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Physics Reports 302, 211–293.

Moiseyev, N. 2009 Suppression of Feshbach resonance widths in two-dimensional waveguidesand quantum dots: a lower bound for the number of bound states in the continuum.Phys.Rev.Lett. 102, 167404–1–167404–4.

Munjal, M. 1987 Acoustics of Ducts and Mufflers. New York: John Wiley & Sons.

Nannen, L. 2008 Hardy-Raum Methoden zur numerischen Losung von Streu- und Res-onanzproblemen auf unbeschrankten Gebieten. PhD thesis, Georg-August UniversitatGottingen, Der Andere Verlag, Tonning.

Nannen, L. & Schadle, A. 2011 Hardy space infinite elements for Helmholtz-type problemswith unbounded inhomogeneities. Wave Motion pp. 116–129.

Oko�lowicz, J., P�loszajczak, M. & Rotter, I. 2003 Dynamics of quantum systems embed-ded in a continuum. Phys.Rep. 374, 271–383.

Ordonez, G. & Kim, S. 2004 Complex collective states in a one-dimensional two-atom system.Physical Review A 70, 032701–1–032701–13.

Ordonez, G., Na, K. & Kim, S. 2006 Bound states in the continuum in quantum-dot pairs.Physical Review A 73, 022113–1–022113–4.


Ormondroyd, J. & Den Hartog, J. 1928 Theory of the dynamic vibration absorber.Trans.ASME 50, 9–25.

Parker, R. 1966 Resonance effects in wake shedding from parallel plates: Some experimentalobservations. J.Sound Vibration 4 (1), 62–72.

Petrosky, T. & Subbiah, S. 2003 Electron waveguide as a model of a giant atom with adressing field. Physica E 19, 230–235.

Pierce, A. 1981 Acoustics. McGraw Hill.Porter, R. & Evans, D. 1999 Rayleigh–Bloch surface waves along periodic gratings and their

connection with trapped modes in waveguides. J.Fluid Mech. 386, 233–258.Rotter, S., Libisch, F., Burgdorfer, J., Kuhl, U. & Stockmann, H.-J. 2004 Tunable

Fano resonances in transport through microwave billiards. Phys.Rev.E 69, 046208–1–046208–4.

Sadreev, A., Bulgakov, E. & Rotter, I. 2006 Bound states in the continuum in openquantum billiards with a variable shape. Phys.Rev.B 73, 235342–1–235342–5.

Schoberl, J. 1997 NETGEN An advancing front 2D/3D-mesh generator based on abstractrules. Computing and Visualization in Science 1, 41–52.

Simon, B. 1973 The theory of resonances for dilation analytic potentials and the foundationsof time dependent perturbation theory. Ann.Math. 97, 247–274.

Singh, S. 2006 Tonal noise attenuation in ducts by optimising adaptive Helmholtz resonators.Master’s thesis, The University of Adelaide.

Sols, F., Macucci, M., Ravaioli, U. & Hess, K. 1989 Theory for a quantum modulatedtransistor. J.Appl.Phys. 66, 3892–3906.

Sugimoto, N. & Imahori, H. 2006 Localized mode of sound in a waveguide with Helmholtzresonantors. J.Fluid Mech. 546, 89–111.

Tam, C. 1976 The acoustic modes of a two-dimensional rectangular cavity. J.Sound Vibration49 (3), 353–364.

Tang, P. & Sirignano, W. 1973 Theory of a generalized Helmholtz resonator. J.Sound Vibra-tion 26 (2), 247–262.

Tonon, D., Hirschberg, A., Golliard, J. & Ziada, S. 2011 Aeroacoustics of pipe systemswith closed branches. Aeroacoustics 10, 201–276.

Wang, X., Kushwaha, M. & Vasilopoulos, P. 2001 Tunability of acoustic spectral gaps andtransmission in periodically stubbed waveguides. Physical Review B 65, 035107–1–035107–10.

Xu, Y., Li, Y., Lee, R. & Yariv, A. 2000 Scattering-theory analysis of waveguide-resonatorcoupling. Phys.Rev.E 62, 7389–7404.

Ziada, S. & Buhlmann, E. 1992 Self-excited resonances of two side-branches in close proximity.J.Fluids Struct. 6, 583–601.

Ziada, S. & Shine, S. 1999 Strouhal numbers of flow-excited acoustic resonance of closed sidebranches. J.Fluids Struct. 13 (1), 127–142.

Zworski, M. 1999 Resonances in physics and geometry. Notices of the AMS 46 (3), 319–328.

Institut für Numerische und Angewandte MathematikUniversität GöttingenLotzestr. 16-18D - 37083 Göttingen

Telefon: 0551/394512Telefax: 0551/393944

Email: [email protected] URL: http://www.num.math.uni-goettingen.de

Verzeichnis der erschienenen Preprints 2011:

2011-1 M. Braack, G. Lube, L. Röhe Divergence preserving interpolation on anisotro-pic quadrilateral meshes

2011-2 M.-C. Körner, H. Martini, A.Schöbel

Minsum hyperspheres in normed spaces

2011-3 R. Bauer, A. Schöbel Rules of Thumb � Practical Online-Strategies forDelay Management

2011-4 S. Cicerone, G. Di Stefano, M.Schachtebeck, A. Schöbel

Multi-Stage Recovery Robustness for Optimiza-tion Problems: a new Concept for Planning un-der Disturbances

2011-5 E. Carrizosa, M. Goerigk, M.Körner, A. Schöbel

Recovery to feasibility in robust optimization

2011-6 M. Goerigk, M. Knoth, M.Müller-Hannemann, A. Schö-bel, M. Schmidt

The Price of Robustness in TimetableInformation

2011-7 L. Nannen, T. Hohage, A.Schädle, J. Schöberl

High order Curl-conforming Hardy space in�niteelements for exterior Maxwell problems

2011-8 D. Mirzaei, R. Schaback, M.Dehghan

On Generalized Moving Least Squares and Dif-fuse Derivatives

2011-9 M. Pazouki, R. Schaback Bases of Kernel-Based Spaces

2011-10 D. Mirzaei, R. Schaback Direct Meshless Local Petrov-Galerkin(DMLPG) Method: A Generalized MLSApproximation

2011-11 T. Hohage, F. Werner Iteratively regularized Newton methods with ge-neral data mis�t functionals and applications toPoisson data

2011-12 D. Rosca, G. Plonka Uniform spherical grids via equal area projectionfrom the cube to the sphere

2011-13 S. Hein, W. Koch, L. Nannen Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems

Date post:	19-Oct-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Institut für Numerische und Angewandte...

Documents