The completeness of the set of modes for various waveguides and its significance for the near-field interaction with vibrating structures

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ABSTRACT: Modal decomposition is often used in geophysics and acoustics for the solution of problems related to wave propagation in elastic or acousto-elastic waveguides. One of the key elements of this method is the solution of an eigenvalue problem for obtaining the roots of the characteristic equation, which may signify either frequencies or wavenumbers. The nature of the roots for the majority of the elastic systems allows for a line search along the real or the imaginary axis in the complex plane. Nonetheless, there exist cases in which eigenvalues become complex-valued requiring thus the use of more advanced algorithms. Most up-to-date algorithms for the solution of complex eigenvalue problems are based on the principle of the argument method for a first gross estimation of the location and the number of the roots within a predefined region, followed by a bisection or steepest descent method for a refinement of the final root position. These techniques are shown to be efficient when the roots are located distinctly apart. In the case of elastic or acousto-elastic waveguides, complex roots do exist and often lie close to each other, thereby not allowing for an efficient application of such algorithms. In this study, an approach is presented in which the real and the imaginary parts of the characteristic equation of several types of waveguides are treated separately in order to define the locus of points in the complex plane where roots may lie. It is shown that next to the real-valued roots, which correspond to propagating modes in the medium, infinitely many imaginary and complex-valued ones do exist which complete the eigenvalue spectrum. The contribution of the latter is rather significant in the vicinity of a load and is very essential for the source-waveguide interaction.

KEY WORDS: Characteristic equation; Complex eigenvalues; Acousto-elastic waveguides; Principle of the argument method 1 INTRODUCTION

Modal methods are often used for solving problems involving wave propagation in acoustic or acousto-elastic waveguides [1]. They are preferred over wavenumber integration techniques mainly because of the fact that they are robust and computationally efficient. Waveguides, in contrast to un-bounded domains, are characterized by two (closely) spaced surfaces in one of the principal directions. The energy, once released in such systems, is guided through multiple reflections at the upper and lower boundary parallel to the two surfaces [2].

In the linear regime, a solution in the frequency domain normally suffices since any transient response can be expressed as a superposition of the various harmonics via an inverse Fourier transformation. A solution to the system of equations describing propagation of mechanical disturbances in waveguides requires the simultaneous satisfaction of a system of equations of motion together with a number of boundary and interface conditions for the domain of interest. Under such restrictive conditions, a non-trivial solution of the system of equations does not exist for arbitrary values of both the excitation frequency ω and the wavenumber k which describes the propagation parallel to the waveguide boundaries. It can exist, however, for discrete values of k as a function of ω, i.e. k(ω). The discrete values k(ω) can be found by solving a classical eigenvalue problem [3].

Whereas the solution of such problems is rather easy for layered acoustic waveguides, this is hardly the case for elastic or poro-elastic media. The principal difference between the

two cases is that in the latter one, complex eigenvalues do exist even for purely elastic media [4-5]. In this paper an attempt is made to understand the origin, as well as the physical significance of the complex-valued eigenvalues on the basis of a discussion involving complex contour integration over the wavenumbers. To clarify this, the change in the location of the complex eigenvalues for an elastic layer is studied for gradually increasing depths and for several boundary conditions.

In the literature, several methods have been developed for the solution of complex eigenvalue problems [6-10]. These are mainly restricted to analytical polynomial functions in which the number as well as the multiplicity of the roots is known a priori. In cases involving wave propagation in layered elastic media, there is first the challenge that infinitely many roots exist and second that these roots are closely spaced in the complex plane. Thus, the majority of the numerical algorithms that are robust in several other cases, are failure-prone in the cases involving elastic or acousto-elastic waveguides.

In this paper, a method is presented for the solution of the eigenvalue problem in elastic and acousto-elastic layered media. In contrast to other techniques [6-10], which are purely mathematical in nature, the proposed method is based on a physical understanding of the location, type and number of roots to be expected in each examined case. At first, the real and the imaginary parts of the characteristic equation are treated separately in order to define the locus of points in the wavenumber plane where roots may lie. Next, the roots are

The completeness of the set of modes for various waveguides and its significance for

the near-field interaction with vibrating structures

A. Tsouvalas1, H. Hendrikse1, A.V. Metrikine1

1

Department of Civil Eng., Faculty of Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands

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interpreted as the intersection points of the two contours. The principle of the argument method is applied around each root in order to verify its existence within the specified domain. Further refinement of the roots is also possible at this stage.

In section 2, the basic theoretical background of the method is discussed with the help of a simple example involving plane waves in an elastic layer with stress-free boundaries. In section 3, the case of a purely elastic waveguide is treated and the orthogonality condition of the obtained eigenmodes is derived based on the reciprocity theorem of elastodynamics. In section 4, the method is expanded to the case of an acousto-elastic waveguide. The existence of a fluid layer results in an infinite number of purely imaginary eigenvalues which are not present in the elastic case. The orthogonality relation is also expanded to include the presence of an inviscid fluid layer. Finally, in section 5, the forced response of a beam in contact with a fluid medium is analysed in order to show that the evanescent spectrum, which corresponds to purely imaginary wavenumbers in this case, needs to be considered for obtaining realistic results in the vicinity of the vibrating structure. In fact, it is shown that the displacement compatibility at the beam-fluid interface can be satisfied only when the evanescent spectrum is accounted for in the solution.

2 THEORETICAL BACKGROUND

The method proposed in this paper for the evaluation of the roots of the characteristic equation for several types of waveguides is based on the following steps:

I. Examination of the type and number of roots for the problem under investigation based on a physical interpretation of the roots for each domain of interest. In this aspect, the problems analysed here are divided into those involving (a) only acoustic media; (b) only elastic layers; (c) both elastic and acoustic layers. II. Graphical interpretation of the locus of points where

complex roots may lie based on the results of step I above.

III. Determination of the real and imaginary roots of the characteristic equation using a classical bisection algorithm.

IV. Determination of the complex roots, based on the results of step II above, using a numerical tool developed in Fortran for each particular case. V. Evaluation of the number of roots located within a

predefined region in the complex plane based on the principle of the argument.

Step III, will not be discussed in detail since it is a well-known procedure and presents no particular difficulty [1]. Steps I, II and IV-V are discussed in some detail through a simple example involving plane waves in a solid layer in the remaining part of this section.

2.1 Plane harmonic waves in a solid layer with stress-free boundaries

The chosen example is based on the case of plane harmonic waves propagating in the global x-direction in a waveguide bounded by two stress-free surfaces located at a distance of 2h apart along z-coordinate (Figure 1). This example is quite generic in the sense that other cases, as for example cylindrical waves spreading outwards away from a source,

share the same vertical dependence with plane waves in a 3-D Cartesian coordinate frame [11].

Following the notation used by Achenbach [4], the Rayleigh-Lamb frequency spectrum, involving motion in the

x-z plane of the layer, can be divided into two families of

modes, namely the symmetric and antisymmetric ones. The frequency equation for the symmetric modes is given by:

 

_

4

_

0 tan tan 2 2 2 2    p q pq k ph qh ₍₁₎

and for the antisymmetric ones as:

 





0 4 tan tan 2 2 2 2    pq k p q ph qh , (2) in which _p2 2_/_c2 _k2 L  and 2 2 2 2 /c k q  T are vertical wavenumbers, k is the horizontal wavenumber and ω is the circular frequency. The constants cL and cT correspond to the phase speeds of compressional and shear waves, respectively. The task is to obtain the horizontal wavenumbers k(ω) for a given real and positive frequency ω (ω≥0).

Figure 1 Plane waves in an elastic layer

At first, the existence of complex-valued roots (wavenumbers) needs to be verified. It is to be expected that in the vicinity of ω=0, all roots will be either complex-valued or imaginary, since no wave propagation can exist in the layer. In addition, infinitely many roots should exist in accordance with the notion of a continuum. An expansion of the equations above using Taylor series around ω=0 and truncation with the accuracy O(ω4), yields the following two expressions:

sinh

 

kh cosh

 

kh kh0 (3) sinh

 

kh cosh

 

kh kh0, (4) for the symmetric and the antisymmetric modes of the layer, respectively, which are functions of the thickness of the layer

h and of the wavenumber k. By setting the real and the

imaginary parts of the above equations individually equal to zero, a family of curves can be drawn in the complex plane, which is referred to as zero contour plots. In Figure 2, the zero contour plots of Eq.(3) for h=1m are shown for the region 0 < Re(k) < 4 and 0 < Im(k) < 20. The roots can be interpreted graphically as the intersection points of the two family of curves. As can be seen, there exist six roots in the predefined region and none of them is imaginary. The roots are located symmetrically with respect to the real and the imaginary axes in the other quadrants and lie on top of two parabola which are formed symmetrically with respect to the horizontal axis. The position of the roots for ω~0 actually dictates the locus of points where the various branches of the frequency equation originate for ω=0. A branch is defined in this context as the graphical representation of the relationship between the

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frequency ω and the wavenumber k(ω) for a particular mode. The collection of branches constitutes the so-called frequency

spectrum.

As already mentioned, Eq.(3) and (4) depend solely on the thickness of the layer. Therefore, it is of interest to examine the location of the complex-valued roots of Eq.(3) for increasing depths of the layer. In Figure 3, the obtained roots are shown for three different layer depths ranging from 2m to 20m. As can be seen, increasing depths displace the origin of the various branches towards the imaginary axis. In addition, the spatial distance between the roots in the complex plane decreases when the layer thickness increases, i.e. the roots are closer to each other.

Figure 2 Zero contour plots of the real part (red curves) and of the imaginary part (blue curves) of the characteristic equation

for the symmetric modes at the first quadrant

Thus, for z→∞ one would reasonably expect that Re(k)→0. If one defines the branch cuts for the case of a half space as shown in Figure 4, then it can be concluded that the four branches, in which the roots lie for the case of a waveguide, move onto the branch cuts on the imaginary axis for z→∞. Regarding the antisymmetric modes as given by Eq. (4), similar results are obtained and the discussion is omitted here for the sake of brevity.

Figure 3 Position of the roots for varying depth of the solid layer for the symmetric modes

On the basis of the discussion above, one can conclude that complex-valued roots will also exist for other values of ω>0 for two primary reasons. At first, since all branches of the frequency spectrum originate somewhere in the complex

wavenumber plane for ω=0, they will necessarily follow a route via the complex plane before turning into propagating modes (real-valued wavenumbers) for gradually increasing frequency. Second, the case of an elastic layer has shown that no imaginary roots exist for ω=0. But it is well-known that a continuum contains infinitely many roots. Since no purely imaginary roots exist in this case, there should be an infinite number of complex-valued ones to complete the spectrum. Please note that the non-existence of imaginary roots at ω=0 does not imply that their existence is prohibited for other values of ω>0. In fact, as will be shown in the sequel, imaginary roots may exist for ω>0, since some of the branches may follow a short route along the imaginary axis before turning into propagating modes.

Figure 4 Definition of branch cuts for the case of a half space Having verified the existence of complex-valued roots around ω~0, we can now proceed with the solution for non-zero frequencies (ω>0). It is to be expected that next to the complex-valued roots, a finite number of real roots will exist provided that the frequency ω is chosen larger than the cut-off frequency of the waveguide (ω>ω0). In Figure 5, the zero

contour plots of the real and the imaginary parts of Eq.(1) are shown for an excitation frequency of ω=100 rads-1 (f~16Hz) and for a layer with the following properties: h=10m, cL=297ms-1 and cT=121ms-1.

Figure 5 Zero contour plots for the real (red curves) and the imaginary part (blue curves) of Eq.(1) at ω=100 rad s-1 As can be seen, Eq. (1) yields a finite number of real-valued wavenumbers which correspond to the propagating modes in the layer. Next to the real-valued roots, an infinite number of complex-valued ones exists, which corresponds to evanescent waves in x-coordinate. Thus, an efficient way of locating the complex-valued roots can be based on a search confined only at a strip positioned at the vicinity of the imaginary axis in one of the four quadrants of the complex wavenumber plane since

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the complex-valued roots are characterized by an imaginary part much larger than the real part, i.e. │Im(k)│>>│Re(k)│.

2.2 Eigenvalues for other types of waveguides

For the case of a waveguide consisting of fluid layers, the roots of the characteristic equation can either be real or imaginary. Thus, classical algorithms based on the bisection method can be applied [1]. One needs to perform a line search along the real axis to locate the eigenvalues corresponding to propagating modes and another search along the imaginary axis for the evanescent ones. The aforementioned procedure suffices in locating all eigenvalues within a predefined region in the complex wavenumber plane.

On the contrary, for the case of an acousto-elastic waveguide both imaginary and complex-valued roots exist. The exposure of infinitely many imaginary roots in this case is attributed to the presence of the fluid layer. One can locate all roots of an acousto-elastic waveguide in the following way: At first, a line search along the real axis of the complex plane allows to locate one by one all the propagating modes. The upper limit for which real modes do exist is dictated by the slowest wave in the medium. Second, a line search along the imaginary axis in the wavenumber plane allows to locate the imaginary eigenvalues corresponding to purely evanescent waves. For the case of an elastic layer, the number of imaginary values is either finite or zero, whereas for the acousto-elastic case, the number of imaginary eigenvalues is infinite (Figure 6). At last, a search in the complex plane is required for the complex-valued roots. Results have shown that a search in a narrow strip close to the imaginary axis at one of the quadrants suffices in locating all the roots within a predefined region. A refinement of the roots can be based on the minimization of the modulus method as discussed in the sequel.

Figure 6 Indicative position of the roots of the characteristic equation for various types of waveguides

3 ELASTIC WAVEGUIDES

In this section, the case of an elastic waveguide bounded by a stress free surface at z=0 and a rigid bottom at z=H is examined. The case of a single solid layer is first introduced and subsequently the case of a multi-layered soil is analysed.

3.1 Case of a single solid layer

It is convenient to analyse the response of the layer in terms of

P-SV modes (also known as Rayleigh modes) and SH modes

(also known as Love modes). The split of the motion of the layer into the aforementioned types of modes is originally introduced for plane waves in a 3-D Cartesian frame but it has been shown that a similar distinction holds for other types of coordinate systems [11]. For the P-SV modes the following problem is addressed in the cylindrical coordinate system. The motion of the soil medium is described by the following set of linear equations: 2 2 2 ) ( t            u   u  u  , (5)

in which u(r,z,t) is the displacement vector in the solid, λ and

μ are the Lame coefficients and ρ is the mass density. The

constitutive and geometrical relations for the soil medium read: ij   kk   ij 2  ij (6)



The determinant of the coefficient matrix, formed by introducing the displacement and stresses as functions of the potentials into the boundary conditions, should be set equal to zero, i.e.:

 

0

det D  (17) We expand once more the resulting expression, i.e. Eq.(17), around ω~0 using Taylor series and we keep only the lowest

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order terms in the final expression. The zero contour plot in this case is shown in Figure 7 for a layer with the following properties: H=20m, cL=297ms-1 and cT=121ms-1. The complex roots in the region are depicted in Figure 8.

Figure 7 Zero contour plots for the real (red curves) and the imaginary part (blue curves) of Eq.(17) at ω~0 rad s-1

Figure 8 Roots of the characteristic equation at ω~0 rad s-1 As can be seen, all roots are located close to the imaginary axis. Based on these results, a robust algorithm in the case of an elastic layer can be based on the following steps:

I. A line search along the real and the imaginary k-axis for locating real and imaginary roots based on a classical bisection algorithm. The upper limit for which roots are found on the real axis is dictated by the Rayleigh pole. Since the Rayleigh wave speed is approximately 0.9 c_T, the search can be terminated at a wavenumber equal to Re(k)/0.85c_T. II. A search in the complex plane for the values of k for

which Eq.(17) is equal to zero. The search can be confined at a strip in one of the four quadrants which makes the algorithm computationally efficient. The search in the complex plane is based on the principle of the argument method for a first estimation of the total number of roots within a predefined region in the complex wavenumber plane. Subsequently, the total region is divided into a number of sub-regions each containing a single root. The subdivision into single-root sections is again based on the principle of

the argument method. Finally, a refinement of the location of each root is based on the minimization of the modulus of the complex determinant.

To illustrate the robustness of the proposed method, the located roots are shown for two excitation frequencies f=1Hz and f=33Hz in Figure 9 for the waveguide described previously. At f=1Hz, no propagating modes exist in the layer. On the contrary at f=33Hz, 15 modes with real wavenumbers exist. The largest real-valued root corresponds to the Rayleigh wave at the free surface (z=0) of the elastic layer. An estimation of the wavenumber corresponding to this mode is

ω/cR~1.83 rad m-1. This corresponds exactly to the largest

real-valued wavenumber as shown in Figure 9.

Figure 9 Roots of the Eq.(17) at f=1Hz (red dots) and at

f=33Hz (blue dots)

For the SH modes of the layer the problem is very similar to the one of a single fluid layer. Since only real- and imaginary-valued roots exist in this case, classical numerical algorithms based on the bisection method can be used.

3.2 Orthogonality of Rayleigh eigenfunctions for an elastic layer

Orthogonality relations in the case of elastic waveguides can be derived on the basis of the reciprocity theorem of elastodynamics [13]. It can been shown that the following orthogonality condition holds for the Rayleigh modes:

i ij H i i zr j j zz i i i i j j dz k v k u k u u k        _        0 , , (18) in which ζ=ρ·(cL 4 -(cL 2 -2·cT 2 ))/cL 2 and η=(cL 2 -2·cT 2 )/cL 2 . The indices i and j correspond to two different Rayleigh modes, ui

is the radial displacement of the soil, vj is the vertical

displacement and σzz,i, σzr,i are the normal and shear stresses of

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3.3 Case of a waveguide consisting of two solid layers

The case of a multi-layered solid waveguide can be treated in a similar way. The only difference is in the number of branches in which complex-valued roots lie in each of the quadrants of the complex wavenumber plane. The number of branches is always equal to the number of layers. In Figure 10, the roots of the characteristic equation are shown for

f=10Hz and for a waveguide consisting of two layers with the

properties given in Table 1.

Table 1. Properties of the layers for an elastic waveguide Layer number Thickness (m) ρ (kgm-3) cL (ms-1) cT (ms-1) Layer 1 5 1700 297 121 Layer 2 15 1900 443 195

Figure 10 The roots of the characteristic equation for a waveguide consisting of two layers

As can be seen, the complex-valued roots are clustered in two branches, each associated with a solid layer. The vertical eigenfunctions belonging to wavenumbers of a particular branch have the largest amplitude in the correspondent layer. In this case, the modes corresponding to the outer branch have the largest amplitude in the upper layer, whereas the ones of the inner branch show a larger amplitude in the lower layer. The orthogonality condition for a multi-layered solid is given by Eq.(18) in which the integration is taken now over the total solid depth.

4 ACOUSTO-ELASTIC WAVEGUIDES

In this section, the case of an acousto-elastic waveguide is analysed and the main differences with the previous cases are highlighted. The example corresponds to the case of a water column resting on top of a soil sediment. This is a typical example in the field of underwater acoustics for oceanic environments.

4.1 Fluid layer on top of an elastic soil layer

The case analysed here is shown in Figure 11. A fluid layer of depth D overlies a solid layer of thickness H. The material properties are summarized in Table 2.

Figure 11 Geometry of the acousto-elastic waveguide An analytical derivation of an equation similar to Eq.(17) which accounts for the presence of the fluid layer is discussed by the authors in [3].

Table 2. Properties for the acousto-elastic waveguide Layer number Thickness (m) ρ (kgm-3) cL (ms-1) cT (ms-1) Fluid layer 10 1000 1500 - Solid layer 20 1700 297 121

An analysis of the position of the roots for ω~0 using Taylor series, following the steps described previously, yields the contour plots shown in Figure 12. The existence of infinitely many imaginary roots is verified in this case. Due to symmetry, the search for the complex-valued roots can be limited to just one of the four quadrants as discussed previously. The location of the roots is shown for an excitation frequency of f=20Hz in Figure 13 using the algorithm developed in Fortran for this case.

Figure 12 Zero contour plots for the real and the imaginary parts for an acousto-elastic waveguide at ω~0 rad s-1 In contrast to the elastic layer, an infinite number of purely imaginary roots exists. In addition, the slowest wave in the medium is the so called Scholte wave which is an interface

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wave travelling along the solid-fluid interface with an amplitude that decreases exponentially away from the interface in both media. The wave speed of the Scholte mode is about 0.87 times the speed of the shear waves in the solid layer. This corresponds to a wavenumber of ω/cR~1.20 rad m-1

as shown in Figure 13. Note that the speed of the Scholte wave is slightly lower than that of the Rayleigh wave for the same solid medium due to the presence of the fluid pressure on the surface of the solid. This has been observed in other studies [14] and is well-known in the scientific community.

Figure 13 Roots of the characteristic equation of an acousto-elastic waveguide at f=20 Hz

4.2 Orthogonality of eigenfunctions

The orthogonality relation can be generalized to include the presence of the fluid layer on top of the elastic layer. Following a similar approach as in [13], it can be shown that the orthogonality relation in this case can be expressed as:

ij i H i i zr j j zz i i i i j j s D i j i dz k v k u k u u k dz k p           _           0 , , 0i , (19) in which υi is the radial velocity of the fluid and pj is the

pressure in the fluid region. Eq.(19), is a generalization of the orthogonality relation derived in [13].

5 SIGNIFICANCE OF THE EVANESCENT FIELD Even though the evanescent spectrum, which consists of purely imaginary and/or complex-valued modes, decays rapidly for increasing horizontal distances from the source, its contribution is rather significant in the vicinity of a load and is very essential for the source-waveguide interaction, should this be considered. To illustrate this, the simple case of a vibrating beam in contact with a 2-D fluid layer is examined here. The situation is schematically shown in Figure 14 and the parameters are summarized in Table 3. The beam is of

finite length and lies within 0<x<L. The fluid occupies the region y>0 and 0<x<L. A load is applied on the beam at

x=L/3. The load has an amplitude of 1kN and is purely

sinusoidal with a frequency of 200Hz.

Figure 14 Geometry of the beam-fluid model Table 3. Parameters of the beam model

Parameter Value Unit

L 10 m

EI 7.2 x 106 N m2

ρA 300 kgm-1

cf 1500 ms-1

ρf 1000 kgm-3

The vibrations of the beam are described in terms of the in

vacuo beam modes satisfying the simple-supported boundary

conditions. The fluid response is expressed in terms of modes (both propagating and evanescent) as explained previously. The problem is then solved in a similar way to that described in [15] using the orthogonality of the beam modes and that of the fluid eigenfunctions.

Figure 15 Displacement mismatch in the y-direction at the beam-fluid interface with propagating modes only In Figure 15, the modulus of the transversal displacement of the beam and of the fluid is shown together with the mismatch (difference between the two) at the beam-fluid interface. In this case only the propagating modes of the fluid are accounted for in the modal summation. As can be seen, the displacement mismatch is quite large. In Figure 16, the same

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results are shown but now a large number of evanescent modes is included in the modal sum. The mismatch error at the interface is less than 0.1% for all points along the length of the beam. Thus, it can be concluded that the evanescent spectrum needs to be accounted for, if the exact solution is needed at the interface. Similar results are obtained for more complicated structures vibrating in elastic or acousto-elastic media as discussed by the authors in [3].

Figure 16 Displacement mismatch in the y-direction at the beam-fluid interface including the evanescent spectrum 6 CONCLUSIONS

In this paper, a method is presented for the determination of the eigenvalues of various waveguides within a predefined region in the complex wavenumber plane. In contrast to other methods, the present approach is based on a physical interpretation of the number and type of roots to be expected in various cases. Two cases are thoroughly examined namely, an elastic waveguide and an acousto-elastic waveguide.

It has been shown that for the case of a single elastic layer bounded by a stress release surface and a rigid bottom real-, imaginary- and complex-valued roots may exist. For frequencies larger than the cut-off frequency of the layer, a finite set of real-valued roots exists, which corresponds to the propagating modes in the layer. Next to the real-valued roots, a finite number of imaginary ones may exist which together with an infinite number of complex-valued ones forms the so-called evanescent spectrum. An efficient algorithm for locating the roots for each excitation frequency, can be based on a line search along the real and the imaginary axis of the complex wavenumber plane, followed by a search along a narrow strip in one of the quadrants in the complex wavenumber plane for locating the complex-valued roots. For a waveguide consisting of two elastic layers, the situation is similar to the above with the only difference that an additional branch of roots in the complex plane exists. For a multi-layered soil, the number of branches where complex-roots lie is equal to the number of layers of the waveguide.

In addition, the case of an acousto-elastic waveguide is examined. In contrast to purely elastic layers, an infinite number of imaginary roots exists due to the presence of the acoustic domain. An example is analysed and the presence of real-, imaginary- and complex-valued modes is shown. An extension of the classical bi-orthogonal relation to account for the presence of the acoustic layer is also included.

Finally, the case of a vibrating beam in a 2-D fluid is discussed in order to illustrate the significant contribution of the evanescent spectrum in the modal response. It has been shown that if accurate results are needed at the interface, a large number of evanescent modes needs to be accounted for.

Although the focus in this work was on a particular example involving wave propagation in layered media, it is believed that a similar approach for locating complex eigenvalues can be followed for a class of problems in dynamics. In essence, once the type and the number of roots to be expected in each problem can be determined, efficient algorithms can be developed for locating the roots. In contrast to methods which are purely mathematical in nature, a physical understanding of the locus of points where roots may lie, allows one to construct efficient and robust numerical algorithms for various cases.

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