rough wave guides, modal couplings

In collaboration with

Laboratoire d'Acoustique Ultrasonore et d'Electronique (LAUE), UMR CNRS 6068, Le Havre, France

and the Institut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), UMR CNRS 8520 , Lille, France

in the frame of the GDR 2501 "Study of the ultrasonic propagation in non homogeneous media for non destructive testing" ("Etude de la propagation ultrasonore en milieux non-homogènes en vue du contrôle non destructif" in french).

Motivating of the study

Rough solid guide

Mechanisms of the decreasing of a Lamb mode

Propagation in a fluid waveguide with small corrugation

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Motivating of the study

A Non Destructive Evaluation and Testing problem
- of plates having an irregular surface condition,
- of corrosion,
- of quality of bonding in a structure,

has led the Research Group (GDR in french) 2501 of CNRS "Study of the ultrasonic propagation in non-homogeneous media for non destructive testing" to study the propagation in anisotropic solid [A16], then fluid, waveguides, with non regular shape profile boundaries (roughness or corrugation).

According to the size and to the frequency, this last study [A15, A17] can be applied to other fields, including urban acoustics for instance (propagation in U streets, tunnels, ...).

Rough solid guide

The materials used in industry do not always have a smooth surface quality: they include imperfections, even a roughness on one or the other face. The roughness can also come from a corrosion, or can be required to improve the quality of a welding between two materials. Industrial testing is done currently mainly in normal incidence (echography), the structure to be controlled being inspected point by point. The use of plate waves, called Lamb waves, which take into account all the thickness of material and which propagate in a direction parallel to the interfaces, permits to test a whole material strip, in only one shoot, which thus considerably saves time.

The characterization of the roughness of solid plates is thus an important problem in term of ultrasonic non destructive testing, which requires the understanding of the acoustic wave propagation of Lamb waes in solid guides with non regular shape boundaries (roughness); the analysis of the decrease of the amplitude of the modes of Lamb is likely to provide information on the properties of this roughness.

Analytical studies (LAUM) on anisotropic plates of which one of the interfaces comprises a 2D roughness allowed, by writing the boundary conditions in any point of the profile and then by use of a small perturbation method, to obtain the dispersion curves for Lamb waves in such a medium (figure 1), and to evaluate the decrease of the amplitude of these modes by calculation of the imaginary part of the wave number [B13-B15, C11, A16, A17]. These studies are partially corroborated by experimental results (LAUE), but are complex enough so that a more advanced model is developed in a easier case, that of a fluid plate.

Figure 1 : Dispersion curves for Lamb modes in a shot blasted glass plate. On the left) frequency as a function of the real part of the wave number, on the right) frequency as a function of the imaginary part of the wave number. This imaginary part is zero when there is no roughness.

Mechanisms of decreasing of a Lamb mode (basic physical phenomena)

Fig. 2

When the interfaces are smooth, for a given pulsation omega

and a given wave number k0, there is a reciprocal conversion between the six plane waves and the Lamb mode.

When the interfaces are rough, there is a scattering of all the waves by the roughness. Let us take one of the six waves, with the wave number k0 corresponding to the Lamb mode without roughness (figure 2). When impacting on the roughness, the scattering of this wave creates other Lamb modes by coupling, with different wave numbers (ka, kb, kc, kd, ...). Due to the roughness, while all the waves are propagating, the energy of the main Lamb mode k₀is transfered between all the possible modes (and inversely), which can be expressed in terms of energy loss of the main Lamb mode k₀, and therefore by a decreasing of its amplitude.

Figure 3 : Observation at point ksi

₁ of the scattering phenomenon of an acoustic mode on the roughness (total depth of the roughness denoted
H( ksi

)). The first square bracket represents the field which propagates in the direction of increasing ksi

(summation of the primary wave and of "upstream" coupled waves); the second square bracket represents the backscattered waves from ksi

₁ to + infinity.
Click on the image for a zoom

Qualitatively, when the guide is a fluid waveguide, and without taking into account the intermodal coupling (only one mode is considered here), the 1st-order amplitude of the acoustic pressure (denoted

) makes appear the primary wave exp(-ik1m ksi

₁) which propagates when the interfaces are smooth, and the coupling due to the scattering of this primary wave on the roughness, which leads to a reallocation of the acoustical energy having the same modal behaviour.

The primary wave and the coupling waves arrive at the observation point ksi

1. The coupling waves are those generated upstream ksi

1 and which propagate in the same direction as the primary wave, and those generated downstream ksi

1 which are backscattered (in red on figure 3). The waves indicated in green on figure 3 do not contribute to the pressure field at the observation point ksi

1. The details of the calculus can be found in Ref. [A15]. See also figures 9 to 11.

Propagation in a fluid waveguide with geometrically perturbed rigid walls (small corrugation)

A numerical study carried out at IEMN made it possible to visualize the field formed by the effect of roughness on the incident wave, for a periodic rough profile. A first analytical study highlights the coupling of modes by the roughness. This roughness is modeled in the form of acoustic reaction of each of the small fluid volumes which are delimited by the rough profile, and is represented by an impedance-like operator, which does not depend on the shape of these small fluid cavities [B16, C14, C16].

A more quantitative second study permits the slopes of the rough profile to be taken into account [A15].

Figure 4 : Geometry of the problem and associated equations Click on the image for a zoom	The fluid waveguide presented on figure 4 (density ₀ and speed of sound c₀, is bounded by two dimensional shape perturbations (depth H₀ from x₃=0 and depth H_d from x₃=). The presence of this roughness (total depth H=H₀+H_d) leads to define two characteristic thicknesses: that of the "interior guide" d and that of the "outer guide" . The associated eigenvalue problem will be subsequently considered is the Neumann problem associated to this outer guide . The well-posed problem is the following. - propagation equation with bulk source strength, - boundary conditions expressed by a relationship between the normal derivative of the acoustic pressure and the acoustic pressure itself, which involves a complex operator which describes any type of roughness (derivative operator with respect to x₁ and x₂, and function of the derivative of the profile, thus of the slope), - Sommerfeld radiation condition following the directions of propagation. Subsequently, the profile variations (ratio depth H to thickness and also slopes) are assumed to be small.
Figure 5 Click on the image for a zoom	Expansion of the acoustic pressure on the transverse eigenmodes of the guide. The method is based on the "projection" of the propagation equation on the transverse eigenmodes of the guide. The one dimensional eigenmodes (following x₃ coordinate) are those of the 3D dimensional guide bounded by the smooth outer plane surfaces (thickness ). It should be noted that a "multimodal method" can be found in the literature. This method utilizes a modal basis which is different at each point x1; this modal basis is thus a local basis associated to the real domain. The present method, called "inter-modal method", utilizes a unique modal basis, using the distance of the outer waveguide which encloses the corrugated considered waveguide (eigenfunctions _m and associated wave numbers k_m), the "projection" (the integration) being done on the real domain.
Figure 6 : Use of the Green theorem. Click on the image for a zoom	Modal formulation Equation given on figure 6 (particular case of a monochromatic source) is obtained by the projection of the propagation equation onto the eigenfunctions _m, then by the use of the 1D Green theorem which involves the boundary conditions (figure 4). This equation makes appear the modal coefficients of mode m (in green) and also those of the other modes µ (created by coupling), the source strength term (in blue), and two coupling terms (in red): one is related to a boundary operator which includes the slope of the profile and a spatial derivative operator, and the other is related to a "bulk" operator, since the integration takes place over the perturbed lateral dimensions of the guide.
Figure 7 : Solution using integral formulation (1/2). Click on the image for a zoom	Solution using integral formulation The solution is obtained by carrying out successive approximations, using at each stage the integral formulation with an appropriate Green function. In the case of a semi-infinite plate, and assuming a monochromatic source flush-mounted at x₁=0 and x₂=0, the chosen 1D Green function is given on figure 7.
Figure 8 : Solution using integral formulation (2/2). Click on the image for a zoom	Intermodal couplings - At zero-order, the solution is just the incident field which is the only field which exists when the interfaces are smooth. - At 1st-order, other terms appear, which represent the intermodal coupling effects, due to the roughness, all over the length of the corrugation. The integrals from -infinite to +infinite show that the coupling source (here the roughness) acts as a secondary extended source. Moreover, the expression of the Green function (figure 7) highlights the fact that the effect of the corrugation at the observation point x₁ of the coupling upstream and downstream this point is not the same. It should be noted that the system of equations (µ,m) is not a linear system because of the presence of the integrals which represent the non local effect of the roughness; this is the reason why the solution makes use of the perturbation theory. - And so on until the nth-order.
Figure 9 : Direct mono-mode solution (1/3). Click on the image for a zoom	Mono-mode direct solution (out of sources) In the particular case of a single mode, that created by the source and numbered m (intermodal couplings µ are neglected), the equation on figure 6 takes the form of that on figure 9 (the source strength term cancel because it is assumed to be out of the observation domain). This particular case is just a qualitative model is order to provide (in particular) a very simple analytic solution which helps to understand the physical phenomena : the mode m is coupled to itself, through the factors indicated in red in the equation at the bottom of figure 9.
Figure 10 : Direct mono-mode solution (2/3). Click on the image for a zoom	Mono-mode direct solution (continuation) A first-order expansion as a function of the total height of the roughness H=H₀+H_d, then a change of the variable x₁ --> ₁ (translation to the lower order) permit to obtain the framed equation on figure 10. This equation is truncated only to the 1st order expansion with respect to the small quantity H/. The approximate solution, denoted , can be written as , where the function is the zero-order expansion solution (Born approximation). Finaly, the 1st-order perturbation term is solution of the ordinary differential equation , leading to an integral over the roughness.
Figure 11 : Direct mono-mode solution (3/3). Scattering phenomena of a single mode on the corrugation: a sensor, located at point ₁ receives the primary waves, and the coupled waves which are created upstream and downstream from this point by scattering on the roughness. Click on the image for a zoom	Mono-mode direct solution (continuation) Finaly, the 1st-order acoustic pressure amplitude makes appear the primary wave exp(-ik1m ₁) which propagates when the interfaces are smooth, and the coupling due to the scattering of this primary wave on the roughness, which leads to a reallocation of the acoustical energy having the same modal behaviour (since this is here a mono-mode approach). The primary wave and the coupling waves arrive at the observation point 1. The coupling waves are those generated upstream 1 and which propagate in the same direction as the primary wave, and those generated downstream 1 which are backscattered (in red on figure 3). The waves indicated in green on figure 3 do not contribute to the pressure field at the observation point 1. This little qualitative model, which does not take into account the intermodal coupling, just permits to highlight the basic physical phenomena. See also figures 2 and 3.
Figure 12 : Intermodal couplings in the particular case of a sawtooth periodic profile. The modes created by coupling due to the roughness are denoted µ and the mode generated by the source is denoted m. Click on the image for a zoom	Intermodal couplings - Particular case of a periodic sawtooth profile (spatial period ) The periodic profile (figure 12) consists of a regularly distribution of N small cavities (teeth) with a spatial period denoted . The acoustic sources are assumed to generate only the mode m=1, the characteristics of the roughness and the frequency being given on figure 12 for air and glass (isotropic solid in which here only pressure waves propagate). The modes created by coupling due to the roughness are denoted µ. Four modes are taken into account here, the mode µ=3 being evanescent for the chosen frequency. The order n=3 is generally enough in order to make the model converge. The results are presented on figures 13 to 17. The phonon relationships (see figure 18 and associated text) permit to predict either a strong or a weak coupling between the modes, and also the oscillation periods.
Figure 13 : Modulus of the acoustic pressure amplitude of the mode µ=0 created by coupling. The period of the periodic oscillations involves the spatial period of the profile, and the wave numbers of each mode (see phonon relationship). Click on the image for a zoom	Figure 14 : Modulus of the acoustic pressure amplitude of the mode m=1 created by the source. The coupling due to the roughness creates other modes than the mode generated by the source, which leads to the decreasing of its acoustic pressure amplitude as a function of x₁. Click on the image for a zoom
Figure 15 : Modulus of the acoustic pressure amplitude of the mode µ=2 created by coupling. The amplitude of this mode is much less important than those of modes µ=0 and m=1, as it is predicted by the phonon relationship. Click on the image for a zoom	Figure 16 : Modulus of the acoustic pressure amplitude of the (evanescent) mode µ=3 created by coupling. Click on the image for a zoom
Figure 17 : Modulus of the total acoustic pressure amplitude. Click on the image for a zoom
Figure 18 : Dispersion curves (red thick lines) of the smooth fluid waveguide and curves (blue thin lines) corresponding to the phonon relationship. Click on the image for a zoom	Phonon relationship The oscillation periods of the acoustic pressure amplitudes of the different modes (figures 13-17), appear without any difficulty in the case of a sinusoidal rough profile: phase terms and denominators of the form which permit to give an interpretaion of these results. This relation can be interpreted as a law of conservation of the phonon momentum in the dual space, leading to the term "phonon relationship". The red thick lines of figure 18 are the classical dispersion curves for guided waves (with cut-off frequencies) for the four first modes. The blue thin lines are the curves coming from the phonon relationship, for modes generated by the source. The intersection between the two curves allow to predict when a strong self-coupling of the mode m=1 generated by the source with itself could occur (this is the case for example at fd/c₀=1.26), or when a strong coupling of this mode m=1 with the mode µ=0 created by coupling (this occurs at fd/c₀=1.31). As these two frequencies are very closed to each other, it can easily be seen that, in the former example (figures 12-17), there are both a strong self-coupling of the mode m=1 generated by the source with itself, and a strong coupling of this mode m=1 with the mode µ=0 created by coupling.
Figure 19 : Intermodal couplings in the particular case of a symmetric sawtooth periodic profile. The modes created by coupling due to the roughness are denoted µ and the mode generated by the source is denoted m. Click on the image for a zoom	Intermodal couplings - Particular case of two symmetric (in phase) periodic sawtooth profiles (spatial period ) en dents de scie The periodic profile (figure 19) consists of a regularly distribution of N small cavities (teeth) with a spatial period denoted , on both sides of the guide (symmetric profile) . The acoustic sources are assumed to generate only the mode m=0, the characteristics of the roughness and the frequency being given on figure 19 for air and glass (isotropic solid in which here only pressure waves propagate). The modes created by coupling due to the roughness are denoted µ and the mode generated by the source is denoted m. The modes such as (m+µ) is an odd number have a null amplitude (figures 21 and 23). Four modes are taken into account here, the mode µ=3 being evanescent for the chosen frequency. The order n=3 is generally enough in order to make the model converge. The phonon relationships permit to predict either a strong or a weak coupling between the modes, and also the oscillation periods.
Figure 20 : Modulus of the acoustic pressure amplitude of the mode m=0 created by the source. The coupling due to the roughness creates other modes than the mode generated by the source, which leads to the decreasing of its acoustic pressure amplitude as a function of x₁. Click on the image for a zoom	Figure 21 : Modulus of the acoustic pressure amplitude of the mode µ=1 created by coupling. The amplitude of this mode is zero because m+µ=1 (odd). Click on the image for a zoom
Figure 22 : Modulus of the acoustic pressure amplitude of the mode µ=2 created by coupling. The amplitude of this mode is much less important than that of the mode m=0, as it can be predicted by the phonon relationship. Click on the image for a zoom	Figure 23 : Modulus of the acoustic pressure amplitude of the mode µ=3 created by coupling. The amplitude of this mode is zero because m+µ=3 (odd). Click on the image for a zoom
Figure 24 : Modulus of the total acoustic pressure amplitude. Click on the image for a zoom

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