Resonance modes in a one-dimensional medium with two purely resistive boundaries: Calculation methods, orthogonality, and completeness

Resonance modes in a one-dimensional medium with two purely

resistive boundaries: Calculation methods, orthogonality,

and completeness

Jean Kergomarda兲 and Vincent Debut

Laboratoire de Mécanique et d’Acoustique, CNRS UPR 7051, 31 Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France

Denis Matignon

Télécom Paris/Département TSI, CNRS UMR 5141 37-39, rue Dareau 75014 Paris, France

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for “resonance modes” or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the one-dimensional propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then, the method of separation of variables共space and time兲 leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or/and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode.

PACS number共s兲: 43.20.Ks, 43.40.Cw, 02.30.Tb, 02.30.Jr 关MO兴

I. INTRODUCTION

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for “resonance modes”共see Ref. 1兲, or free oscillations regimes. These modes can be nonorthogonal for the ordinary scalar product, entailing some difficulties depending on the math-ematical treatment, made either in the time or frequency do-main. Two classical methods exist for such a problem, and can be used either for a scalar, second-order differential equation, or for a system of two equations of the first order. They have been especially used for the problem of a one-dimensional 共1D兲 medium with one resistive boundary, the other boundary condition being of Dirichlet type:

共i兲 In the time domain, the use of time and space as separate variables leads directly to the basis of modes, but they are nonorthogonal for the most common product, and difficulties occur when searching for the coefficients depend-ing, for instance, on initial conditions. Nevertheless, for a particular case, Oliverto and Santini,2 and Guyader3 have solved the problem, and Rideau,4using a system of equations of the first order, found a scalar product making the modes orthogonal共see also Refs. 5–7兲, and gave the proof of com-pleteness.

共ii兲 In the frequency domain, the equations to be solved are ordinary differential equations with boundary conditions

depending on frequency, but the use of orthogonal decompo-sition is possible. This leads to eigenmodes and eigenfre-quencies depending on frequency. It is the case for the clas-sical theory of room acoustics共see, e.g., Morse and Ingard兲,8 using biorthogonality. To return to time domain in order to deduce the resonance modes is a rather delicate task, espe-cially because of the calculation of the derivation of eigen-frequencies with respect to frequency. Biorthogonality has been used also for duct modes 共see, e.g., Ref. 9兲 Another approach has been recently used by Trautmann and Rabenstein,10,11using a system of first-order equations共these authors treat the case of two resistive boundary conditions兲. The present article is devoted to the study of the simple 1D case, when the two boundaries are resistive. One goal is to exhibit how the different methods articulate. We start by using the fact that a straightforward solution exists for the wave equation with source, by applying the residue calculus to the closed form of the Fourier domain solution: As dis-cussed by Levine,12 this closed-form solution, avoiding the sum of a series, is “relatively poorly, if not entirely, unknown to the general acoustics community.” All calculations can be carried out analytically without any approximation, exhibit-ing the properties of the different methods 共however many previous papers restrict their content to small impedance, or admittance, at one extremity, using perturbation methods兲. The case under study corresponds to 1D propagation in a homogeneous medium bounded by two other semi-infinite media with different characteristic impedances, dissipation being therefore due to radiation at infinity. It is especially a兲_{Electronic address: kergomard@lma.cnrs-mrs.fr}

interesting because of its physical significance共it is probably the simplest radiation problem兲, and also because it realizes one of the possible transitions between Neumann and Dirich-let boundary conditions. Notice that in the context of optics and quantum mechanics, the problem has been studied in-cluding the outside media by Leung et al.13,14the resonance modes being called quasinormal modes.

In Sec. II, the equations to be solved are stated, with some possible physical interpretations. As a first step, the classical, closed-form solution of the Green function in the frequency domain is established 共Sec. III兲, with its inverse Fourier transform 共FT兲 corresponding to the successive re-flections共Sec. IV兲. The second step is the residue calculus in order to determine the resonance modes 关Sec. V, the basic result being given by Eq.共26兲兴. Then, results are compared to those of the two aforementioned methods, i.e.:共i兲 The method of separation of variables共Sec. VI兲, which gives the result for given initial conditions关the corresponding results being Eqs.共37兲, 共48兲, and 共49兲兴; in this section, the question of orthogonality and completeness of the modes is investi-gated; and 共ii兲 the method of eigenmodes in the frequency domain 共section VII兲. For the two methods, both second-order scalar equation and first-second-order system of two equations are used successively, with emphasis on the existence of a constant mode.

II. STATEMENT OF THE PROBLEM, PHYSICAL INTERPRETATION

The Green function g共x,t兩x0, t0兲 for the wave equation is

solution of the following equation: 关⳵xx

2

− c−2⳵tt

2_{兴g共x,t兲 = −␦共x − x}

0兲␦共t − t0兲, 共1兲

where x and x0are the spatial coordinates of the receiver and

source, respectively 共or vice versa兲, t and t₀ the times of observation and excitation, respectively, c is the speed of sound. ␦共x兲 is the Dirac function.

For sake of simplicity, x₀ and t₀ are considered to be fixed. Moreover in the whole paper, the choice of t₀= 0 is made. For negative t, the function is zero, as well as its first derivative. The Green function satisfies the following bound-ary conditions:

c␨⳵xg共x,t兲 =⳵tg共x,t兲 at x = 0, 共2兲

c␨_ᐉ⳵xg共x,t兲 = −⳵tg共x,t兲 at x = ᐉ, 共3兲 where ␨= Z /␳c, ␳ is the density of the fluid, and Z is the impedance at x = 0, which is assumed to be a real quantity, independent of the frequency. Similarly,␨_ᐉ= Z_ᐉ/␳c, where Z_ᐉ

is the impedance at x =ᐉ 共ᐉ being positive兲.

An obvious physical interpretation for quantities ␨ and ␨ᐉis the following: Consider for x⬍0 and x⬎ᐉ 共see Fig. 1兲

two media with characteristic impedances␳−c−and␳+c+,

re-spectively. If the media are nondissipative, impedances are real, and can be larger or smaller than the impedance of the bounded medium, ␳c. Moreover, they are positive, because

they correspond to waves outgoing from the bounded me-dium. Therefore, this is the problem of planar pressure waves in a stratified medium, the direction of propagation being

normal to the interfaces. A generalization to more complex stratified media would be possible, at least numerically. In this problem, the Green function corresponds to the acoustic pressure: Of course, it does not have the dimension of a pressure, but the solution for a “concrete” problem with source can be easily solved, as explained in standard text-books, and discussed in a recent paper by Levine.12

Other problems correspond to the previous equations: 共i兲 In an approximate way, ignoring higher-order duct

modes, the problem of planar waves in a rigid walled duct terminating in two semi-infinite ducts with dif-ferent cross sections areas, the quantities␨and␨_ᐉ be-ing the ratios of the areas. The approximation is good at low frequencies.

共ii兲 The problem of a dissipative termination: The termi-nal impedances Z and Z_ᐉ can be the impedances of dissipative media 共at low frequencies, a porous me-dium open to a large space can be an approximation of a pure resistance, due to viscous effects兲.

In all of the previous problems, the quantities ␨and␨_ᐉ are real and positive, the terminations being passive. For active terminations, they can be negative. An example is the beginning of self-sustained oscillations in musical instru-ments: A nonlinear excitator, such as a reed for a clarinet, can be linearized as a pure resistance. When the main control parameter, i.e., the pressure in the mouth of the musician, increases, the resistance becomes negative, the static regime becomes unstable, and an oscillation starts as an increasing exponential共see e.g., Refs. 15 and 16兲.

Obviously, analogous problems for mechanical vibra-tions or other wave fields are numerous.

III. CLOSED-FORM SOLUTION FOR THE FOURIER TRANSFORM

The FT of g共x,t兲 is denoted G共x,␻兲 共throughout the ar-ticle, functions of time are written in small characters, and their FTs are written in capital characters兲. It is equal to

G共x,␻兲 =

冕

−⬁

+⬁

g共x,t兲e−i␻tdt, where 共4兲

g共x,t兲 = 1

2␲

冕

−⬁ +⬁

G共x,␻兲ei␻td␻. 共5兲

The FT of Eqs.共1兲–共3兲 are found to be

共⳵xx2 +␻2/c2兲G共x,␻兲 = −␦共x − x0兲, 共6兲

c␨⳵xG共x,␻兲 = i␻G共x,␻兲 at x = 0; 共7兲

c␨_ᐉ⳵xG共x,␻兲 = − i␻G共x,␻兲 at x = ᐉ. 共8兲 While terminal impedances are independent of fre-quency; boundary conditions are frequency dependent. Nev-ertheless a classical closed-form solution is already known, which has been especially used in Ref. 17. If x⫽x0solutions

of Eq.共6兲 can be written as

G共x,␻兲 = A−cosh关i␻x/c +␩兴, if x ⬍ x0; G共x,␻兲 = A+cosh关i␻共ᐉ − x兲/c +␩_ᐉ兴, if x ⬎ x0.

For the boundary conditions, the following definitions are used:

␨= coth␩; r = e−2␩=共␨− 1兲/共␨+ 1兲,

␨ᐉ= coth␩ᐉ; rᐉ= e−2␩ᐉ=共␨ᐉ− 1兲/共␨ᐉ+ 1兲, 共9兲

where r and r_ᐉare the reflection coefficients. The quantity␩ satisfies: 2␩= −共ln兩r兩+i arg共r兲兲 关2␲兴. Because r is real, we choose the following definition:

␩=␩r+ i␮␲/2; ␮= 0 or 1. 共10兲

Two cases exist: 共i兲 if 兩␨兩⬎1, r⬎0, ␮= 0; 共ii兲 if 兩␨兩⬍1, r ⬍0,␮= 1. A similar remark and definition can be applied to boundary x =ᐉ:

␩ᐉ=␩ᐉr+ i␮ᐉ␲/2; ␮ᐉ= 0 or 1. 共11兲

The case␨= 1共semi-infinite tube or medium兲 corresponds to ␩=⬁: It is discussed in the next sections. Except for the last one, most of the following calculations are valid for all cases. At x = x0, writing the continuity of the function and the

jump of its first derivative, the following result is obtained:

G共x,␻兲 = c i␻

cosh关␩+ i␻x0/c兴cosh关␩ᐉ+ i␻共ᐉ − x兲/c兴

sinh共i␻ᐉ/c +␩+␩_ᐉ兲 , 共12兲 if x艌x0and a similar result if x艋x0, by interchanging x and x0.

IV. SOLUTION IN THE TIME DOMAIN_{„SUCCESSIVE}

REFLECTIONS_…

Equation 共12兲 can be transformed in the time domain, leading to a solution corresponding to the successive reflec-tions of the Green function in infinite space at the two boundaries. It will be the reference solution for the check of the validity of the modal expansion. The sinh function of the denominator can be written as

sinh共i␻ᐉ/c +␩+␩_ᐉ兲 =1 − e

−2␩−2␩_ᐉ−2i␻ᐉ/c

2e−␩−␩ᐉ−i␻ᐉ/c ,

and, if the modulus of the exponential at the denominator is less than unity共this is discussed hereafter兲, as

sinh−1共i␻ᐉ/c +␩+␩_ᐉ兲 = 2e−␩−␩ᐉ−i␻ᐉ/c关1 + F共␻兲 + F2_共␻兲

+ F3共␻兲 + ¯ 兴. 共13兲

F共␻兲=exp共−2␩− 2␩_ᐉ− 2i␻ᐉ/c兲 is the function correspond-ing to a complete round trip of a wave in the tube, of duration 2ᐉ/c. Concerning the numerator of Eq. 共12兲, it can be written: exp共+␩+␩_ᐉ+ i␻ᐉ/c兲Gp共x,␻兲c/4, where:

Gp共x,␻兲 = e−i␻共x−x0兲/c+ re−i␻共x+x0兲/c+ rᐉe−i␻共2ᐉ−x−x0兲/c + rr_ᐉe−i␻共2ᐉ−x+x0兲/c_. 共14兲 Therefore, the Green function is:

G共x,␻兲 = c

2i␻Gp共x,␻兲关1 + F共␻兲 + F

2_{共␻兲 + ... 兴. 共15兲}

The factor Gp共x,␻兲/i␻ corresponds to the four “primary” waves arriving during the first cycle of duration 2ᐉ/c, and this packet is simply reproduced at times 2ᐉ/c, 4ᐉ/c, 6ᐉ/c, etc. 共for a detailed explanation, see e.g., Kergomard兲.15 The inverse FT of the function Gp共x,␻兲/i␻, denoted hp共x,t兲, is obtained by taking into account the zero condition for nega-tive times. The result is found to be, whatever the sign of 共x−x0兲:

hp共x,t兲 = H关t − 兩x − x0兩/c兴 + rH关t − 共x + x0兲/c兴 + rH关t

−共2ᐉ − x − x0兲/c兴 + rrᐉH关t − 共2ᐉ − 兩x − x0兩兲/c兴,

共16兲 where H共t兲 is the step function. Finally

g共x,t兲 =c

2hp共x,t兲 ⴱ 关␦共t兲 + f共t兲 + 共f ⴱ f兲共t兲 + ... 兴; 共17兲

f共t兲 = rr_ᐉ␦共t − 2ᐉ/c兲. 共18兲

The condition of validity of expansion 共13兲 is 兩rr_ᐉ兩⬍1. We notice that if␨is real and positive,兩r兩⬍1, and similarly for ␨_ᐉ. Therefore, the condition is satisfied when the two boundaries are dissipative, or, more precisely, if the combi-nation of the two reflections is dissipative. The case 兩rr_ᐉ兩 ⬎1 will be discussed in Sec. V C. Other comments can be made:

共a兲 The article is limited to purely resistive boundaries, but Eqs.共16兲 and 共18兲 can be generalized to various bound-ary conditions defined by a reflection coefficient, R共␻兲. This is done by replacing the products like rH共t兲 by the convolution product共r*H兲共t兲, where r共t兲 is the inverse FT of R共␻兲.

共b兲 For the case under study, we notice that the convolution product of n times function f共t兲 is 共rr_ᐉ兲n_{␦共t−2nᐉ/c兲.} 共c兲 If␨共respectively␨ᐉ兲 is unity, the reflection coefficient r

共respectively rᐉ兲 vanishes, as well as f共t兲: The first term

of the Green function is the Green function of an infi-nite medium, the first two terms correspond to a semi-infinite medium, etc¼ As it will be seen in the next section, no modes can be found for these cases, because no reflections exist, either ␩or ␩_ᐉtending to infinity. 共d兲 Finally, multiplying both members of Eq. 共15兲 by the

that a closed-form exists in the time domain, which is the basis for the study of the Helmholtz motion of bowed string instruments共see, e.g., Woodhouse兲.18It is a recurrence relationship:

⳵tg共x,t兲 − rrᐉ⳵tg共x,t − 2ᐉ/c兲 = gp共x,t兲c/2.

V. EXPANSION IN RESONANCE MODES USING THE INVERSE FT

Putting expression 共12兲 of the frequency domain in Eq. 共5兲 leads to the modal expansion of the time domain sion. The tool is the residue calculus. If all poles of expres-sion共12兲 are simple and located on or above the real axis, the following equation can be used:

g共x,t兲 = i⌺, if t ⬎ 0 and 0 if t ⬍ 0, 共19兲

where⌺ is the sum of the residues of G共x,␻兲exp共i␻t兲 共see

e.g., Morse and Ingard8 p. 17, changing i to −i兲. A. Calculation of the poles

Zeros of function sinh satisfy:

i␻n=关−␩−␩ᐉ+ in␲兴c/ᐉ, 共20兲

where n is an integer. In order for the poles to be above the real axis, the condition is␩r+␩ᐉr⬎0. It is equivalent to the condition previously obtained for the successive reflections expansion:兩rr_ᐉ兩⬍1. Using definition 共10兲, Eq. 共20兲 is rewrit-ten as

␻n=关n − 共␮+␮_ᐉ兲/2兴␲c/ᐉ + i共␩r+␩_ᐉr兲c/ᐉ. 共21兲 As already remarked by several authors, the imaginary part of the complex frequency is independent of n, and the real part is independent of the dissipation. Depending on the values of␨and␨_ᐉ, different cases must be distinguished: 共i兲 If兩␨兩⬎1 and 兩␨_ᐉ兩⬎1 共real␩and␩_ᐉ兲: The real part of

the frequency corresponds to the values for pure Neu-mann conditions共infinite␨and␨_ᐉ兲.

共ii兲 If兩␨兩⬎1 and 兩␨_ᐉ兩⬍1 共mixed case with either complex ␩ or complex ␩_ᐉ: either ␮ or ␮_ᐉ is unity兲: The real part corresponds to a problem with different condi-tions共Neumann and Dirichlet兲 at x=0 and x=ᐉ. The real part of eigenfrequencies is an odd harmonic of

c / 4ᐉ.

共iii兲 If兩␨兩⬍1 and 兩␨_ᐉ兩⬍1 共complex␩and␩_ᐉ: ␮=␮_ᐉ= 1兲: the real part corresponds to the values for pure Dirich-let conditions共zero␨and␨_ᐉ兲.

Except for case共ii兲, a purely imaginary eigenfrequency exists for n =共␮+␮_ᐉ兲/2.

B. Calculation of the residues

In all cases, the Taylor expansion of the function sinh in Eq. 共12兲 at the first order of the quantity 共␻−␻n兲 can be determined. The result is

sinh关i␻ᐉ/c +␩+␩_ᐉ兴 ⯝ i共− 1兲n_共␻−␻

n兲ᐉ/c. 共22兲

We get for␻ close to the pole␻n:

G共x,␻兲 = − c 2 ␻nᐉ fn共x兲fn共x0兲 共␻−␻n兲 ; 共23兲 fn共x兲 = cosh共␩+ i␻nx/c兲, 共24兲

or fn共x兲=共−1兲ncosh共i␻n共ᐉ−x兲/c+␩ᐉ兲. The residue corre-sponding to the pole ␻= 0, remains to be calculated. For small␻, G共x,␻兲 = c i␻ cosh␩cosh␩_ᐉ sinh共␩+␩_ᐉ兲 = c i␻ 1 ␨−1_+␨ ᐉ −1. 共25兲

Using Eq.共19兲, the inverse FT of G共x,␻兲 is obtained:

g共x,t兲 = H共t兲c 2 ᐉ

兺

n fn共x兲fn共x0兲 i␻n ei␻nt₊ cH共t兲 ␨−1_+␨ ᐉ −1. 共26兲

Some comments can be made.

共a兲 The formula is valid for all aforementioned cases. 共b兲 The mode shapes fn共x兲 are complex-valued functions of the space variable, meaning that the shape is varying with time共for a discussion on complex modes, see e.g; Ref. 19兲. The question of their orthogonality will be discussed in Sec. VI. Notice that functions fn共x兲 do not fulfill the same bound-ary conditions than G共x,␻兲: the boundary conditions are 共7兲 and共8兲, but where␻is replaced by␻n.

共c兲 The last term in Eq. 共26兲 is a constant mode. If one of the impedances␨or␨_ᐉis zero, it disappears, as it is intuitive, in order to satisfy a Dirichlet condition. It is a trivial solution of the wave equation and the boundary conditions, and can be compared to the dc component of a periodic signal. When both ␨ and␨_ᐉ tend to infinity, the boundaries tend to Neu-mann boundaries, and the combination of the nonoscillatory mode of frequency ␻0 and the constant mode results in a

uniform 共i.e., constant in space兲 mode which increases lin-early with time. The calculation is done as follows: If␩and ␩ᐉtend to zero,␻0tends to zero, and the factor exp共i␻0t兲 can

be written as: 1 + i␻0t. The zeroth-order term is equal to the

opposite of the constant mode, and only the linear term re-mains. The result is H共t兲tc2_{/ᐉ, and its FT is −c}2_/ᐉ␻2_{. This}

mode is the classical uniform mode existing for instance in three-dimensional cavities with rigid walls: Curiously, it ex-ists in the standard textbooks 共see, e.g., Ref. 8, page 571兲, but the time domain expression is not given. This mode is similar to the well-known planar guided mode, existing in ducts with rigid walls, whatever the geometrical shape.

共d兲 The imaginary part of the complex frequencies being independent of n, the decay is identical for all nonconstant modes.

共e兲 For the above-considered case 共i兲, we notice that ␻−n= −␻n

*

and f−n共x兲= fn

*_{共x兲, and, more generally,} i␻_␯=共i␻n兲*; f␯共x兲 = 共− 1兲␮fn

*_{共x兲, 共27兲}

where␯= − n +␮+␮_ᐉ is an integer. 共28兲 As a consequence, the solution g共x,t兲 is real. It could be possible to transform the sum by adding the two oscillating terms corresponding to n and␯, when n⫽␯, as it is usually done for nondissipative boundaries. Nevertheless, it appears that the formulas become intricate.

共f兲 Equations 共23兲 and 共25兲 lead directly to another form of the FT of result共26兲, written as a series:

G共x,␻兲 = −c 2 ᐉ

兺

n fn共x兲fn共x0兲 ␻n共␻−␻n兲 + c i␻ 1 ␨−1_+␨ ᐉ −1. 共29兲

An example of comparison of the successive reflections method and modal expansion is shown in Fig. 2共a兲, for con-ditions close to Neumann and Dirichlet. We notice that it is satisfactory. The Gibbs phenomenon appears, because of the truncated series of modes, ensuring the correct accordance between the two methods. Moreover, this accordance con-firms the existence of the constant mode. Decreasing of the maxima is due to the dissipation at the boundaries: For pure Neumann and Dirichlet conditions, the shape would be simi-lar, but perfectly periodical.

C. The case of active boundaries

What happens when the combination of boundaries is active, i.e., when兩rr_ᐉ兩⬎1, or␩r+␩ᐉr⬍0 共at least one of the impedances␨or ␨_ᐉis negative兲? It is possible to prove that Eqs. 共17兲 and 共26兲 remain valid for active boundary

condi-tions, as explained hereafter. The real part of i␻nbeing inde-pendent of n, this suggests to use a new function g˜共x,t兲

= g共x,t兲exp共−␩˜ t兲, where␩˜⬎−␩r−␩ᐉr⬎0, which can be sub-stituted in the initial problem in order for the poles to be located again on or above the real axis. Equation 共1兲 be-comes ⳵xx 2 g ˜共x,t兲 − c−2_关⳵ t+␩˜兴2˜g共x,t兲 = −␦共x − x0兲e−␩˜ t␦共t兲,

and similarly for Eqs.共2兲 and 共3兲. It is equivalent to use an appropriate Laplace transform. Going in the frequency do-main leads to Eqs. 共6兲–共8兲, where G共x,␻兲 is replaced by

G˜ 共x,␻兲 and i␻by共i␻+␩˜兲, and a similar result for Eq. 共12兲.

The analysis of both successive reflections and poles and residues leads to the result g˜共x,t兲=g共x,t兲exp共−␩˜ t兲, where g共x,t兲 is given by Eqs. 共17兲 and 共26兲, respectively, and the

proof is achieved. Here, we do not repeat the complete procedure. We notice that for the case ␩r+␩ᐉr= 0, one boundary is active and the other one is passive: Eigenfre-quencies ␻n are real, while modes are complex. Figure 2共b兲 shows an example of result.

VI. METHOD OF SEPARATION OF VARIABLES A. Second-order homogeneous equation with initial conditions

1. Derivation of the modes

Oliverto and Santini,2and Guyader3 have treated a par-ticular case of the problem 共zero ␨, large ␨_ᐉ兲 using the method of separation of variables. He gets nonorthogonal modes for the common scalar product 兰₀ᐉfn共x兲fm共x兲dx. We will see that the method is valid whatever the values of the two boundary conditions, and how the derivation can be sim-plified.

We are searching for solutions p共x,t兲 of homogeneous Eq. 共1兲 共without second member兲, with boundary conditions 共2兲 and 共3兲, and with given initial conditions. Assuming that, the general solution is a superposition of solutions with sepa-rate variables, the solutions with sepasepa-rate variables are writ-ten in the following form:

p共x,t兲 = f共x兲h共t兲; 共30兲

h共t兲 = B+ei␻t+ B−e−i␻t; 共31兲

f共x兲 = cosh共i␻x/c +␸兲. 共32兲

FIG. 2.共Color online兲 Normalized Green function as a function of time: comparison between the successive reflections method共dotted line兲 and modal expansion 共solid line, 102 modes, i.e., maximum n=50兲. Locations of the source and receiver are x0/ᐉ=0.15 and x/ᐉ

= 0.76, respectively. 共a兲 Passive boundaries ␨= 4; ␨_ᐉ = 0.1 共constant mode=0.097兲. 共b兲 Active boundary ␨ =␨_ᐉ= −4.共constant mode=−2兲.

Decomposition 共30兲 differs from the ordinary FT, be-cause a priori ␻ is a complex quantity, depending on the boundary conditions. Considering first the solution B+ei␻t, this leads to:

␨␻sinh␸=␻cosh␸; 共33兲

␻␨ᐉsinh共i␻ᐉ/c +␸兲 = −␻cosh共i␻ᐉ/c +␸兲. 共34兲

␻= 0 is a solution, corresponding to the constant mode. The other modes are given by Eq.共33兲: sinh共␸−␩兲=0, thus:

f共x兲 = cosh共i␻x/c +␩兲. 共35兲

Actually, there is a sign⫾ in the right-hand side member of Eq. 共35兲, but it is without importance, because it can be included in the coefficient B+of the solution. The eigenval-ues equation is deduced from Eqs.共33兲 and 共34兲, as follows:

sinh共i␻ᐉ/c +␩+␩_ᐉ兲 = 0, 共36兲

the solutions being given by Eq. 共20兲. The solution in time

B−_e−i␻t _{does not lead to new solutions for f共x兲, therefore,}

assuming the solutions form a basis of solutions共this is dis-cussed in Sec. VI B兲, the general solution of a problem with initial conditions can be written as:

p共x,t兲 =

兺

n

Anfn共x兲ei␻nt+ A, 共37兲 where␻nand fn共x兲 are given by Eqs. 共20兲 and 共24兲, respec-tively, and the coefficients An and A depend on the initial conditions, and can be determined using the orthogonality relation of the modes. A is the coefficient of the constant mode.

2. Orthogonality relationship between the modes: First approach

In order to derive an orthogonality relationship between the modes the common product is first calculated:

⌳nm=

冕

0 ᐉ

fn共x兲fm共x兲dx. 共38兲

Because fn共x兲=共−1兲␮f_␯*共x兲, the calculation of the quantities defined in Eq.共38兲 for all values of the index n is equivalent to the calculation of the quantities defined when replacing

fm共x兲 by its conjugate. Writing

冕

0 ᐉ

冋

fn共x兲 d2fm共x兲 dx2 − fm共x兲 d2fn共x兲 dx2

册

dx =

冋

fn共x兲 dfm共x兲 dx − fm共x兲 d2fn共x兲 dx2

册

0 ᐉ ,

and using Eq.共24兲, the following result is obtained: 共␻m 2 −␻n 2_兲 c2 ⌳nm= i共␻m−␻n兲

冋

fn共0兲fm共0兲 ␨ + fn共ᐉ兲fm共ᐉ兲 ␨ᐉ

册

. For ␻m⫽␻n, because␻m+␻n⫽0, the expression of ⌳nm is deduced. For␻m=␻n, the calculation is straightforward. The general formula is found to be

⌳nm= ci ␻m+␻n

冋

fn共0兲fm共0兲 ␨ + fn共ᐉ兲fm共ᐉ兲 ␨ᐉ

册

+1 2ᐉ␦nm, 共39兲 where␦nmis the Kronecker symbol, or:

⌳nm= − c 2 sinh 2␩+共− 1兲n+msinh 2␩_ᐉ i共␻m+␻n兲 +1 2ᐉ␦nm. 共40兲 Modes are found to be nonorthogonal for the product defined by Eq.共38兲, but, as shown by Guyader,2it is possible to solve the problem from the knowledge of initial condi-tions. When dissipation tends to zero 共␩r and ␩ᐉr tend to zero兲, the first term does not vanish, tending to 共−1兲␮ 1₂ᐉ␦_n,␯. This is due to the choice of considering separately the modes ␻nand␻␯.

Otherwise, formula共39兲 remains valid when one of the modes is the constant mode f共x兲=1, and the other one a non constant mode: ⌳n=

冕

0 ᐉ fn共x兲dx = − c i␻n

冋

fn共0兲 ␨ + fn共ᐉ兲 ␨ᐉ

册

= − c i␻n 共sinh␩+共− 1兲nsinh␩_ᐉ兲. 共41兲

Finally, the product of the constant mode by itself isᐉ.

3. Solution with respect to initial conditions

According to Eq.共37兲, the initial conditions are:

p共x,0兲 =

兺

n Ancosh共i␻nx/c +␩兲 + A; 共42兲 ⳵tp共x,0兲 =

兺

n Ani␻ncosh共i␻nx/c +␩兲. 共43兲 Using Eq. 共40兲 for a nonconstant mode m, the following results are obtained:

冕

0 ᐉ p共x,0兲fm共x兲dx =

兺

n An⌳nm+ A⌳m; 共44兲

冕

0 ᐉ ⳵tp共x,0兲fm共x兲dx =

兺

n Ani␻n⌳nm. 共45兲 Multiplying Eq.共44兲 by i␻m, then adding Eq.共45兲, leads to

冕

0 ᐉ 关i␻mp共x,0兲 +⳵tp共x,0兲兴fm共x兲dx = i

兺

n An共␻m+␻n兲⌳nm+ iA␻m⌳m = − c

兺

ˆ n An

冋

fn共0兲fm共0兲 ␨ + fn共ᐉ兲fm共ᐉ兲 ␨ᐉ

册

+ iAmᐉ␻m, 共46兲

=− c

冋

fm共0兲p共0,0兲

␨ +

fm共ᐉ兲p共ᐉ,0兲

␨ᐉ

册

+ iAmᐉ␻m. 共47兲 Notation兺ˆ for the series in Eq.共46兲 indicates that it involves the constant mode. As noticed by Guyader,2 this series is related to the initial conditions at the two ends x = 0 and x =ᐉ. Thus for a nonconstant mode:

Anᐉi␻n=

冕

0 ᐉ

关i␻np共x,0兲 +⳵tp共x,0兲兴fn共x兲dx

+ cp共0,0兲sinh␩+ cp共ᐉ,0兲␧nsinh␩ᐉ. 共48兲 The following property is deduced from Eq. 共27兲: A_␯f_␯共x兲

= An * fn

*_{共x兲, thus p共x,t兲 is real. Calculating 兰} 0 ᐉ_⳵

tp共x,0兲dx, we similarly get coefficient A:

A =c −1_兰 0 ᐉ_⳵ tp共x,0兲dx + p共0,0兲tanh␩+ p共ᐉ,0兲tanh␩ᐉ tanh␩+ tanh␩_ᐉ . 共49兲 What is the condition for which this coefficient vanishes? If for instance at x = 0,␨is zero,␩is infinite, and, according to the boundary condition, p共0,0兲 vanishes, thus A vanishes too. This confirms the remark concerning result共26兲.

Using the initial conditions for the Green function found in Eq.共17兲, it is possible to check result 共26兲, but this will be done hereafter using the equation with source.

B. First-order system of equations, orthogonality, and completeness of the modes

1. Introduction

In this section, we will prove that the modes form a Riesz basis in the space of solutions of a closely related problem, and give the expression of a scalar product making the modes orthogonal. As an introduction, we show that a modified scalar product leads to the orthogonality of modes, except the constant one. For vibrating systems, the product defined by Eq.共38兲 corresponds to the product with respect to the mass, a complement being the calculation of the prod-uct related to the stiffness共see, e.g., Meirovitch兲:20

⌳nm

⬘

=

冕

0 ᐉ d dxfn共x兲 d dxfm共x兲dx.

By integrating by parts, and using Eq.共39兲, this product, for n⫽m, is found to be equal to:

⌳nm

⬘

=关fn共x兲dxfm共x兲兴0ᐉ+

␻m2 c2⌳nm= −

␻n␻m c2 ⌳nm.

Therefore, the modes become orthogonal if we define a new product, as follows:

冕

0 ᐉ

冋

⳵xpn⳵xpm− 1 c2⳵tpn⳵tpm

册

t=0 dx =␦nmᐉ␻n 2 /c2, 共50兲 where pn= pn共x,t兲= fn共x兲exp共i␻nt兲 and similarly for index m. We remark that the modes pn and p_␯=共−1兲␮pn* are or-thogonal for this product. For the calculation of the

solu-tion from initial condisolu-tions, using Eq. 共37兲 at t=0, the following result is obtained:

冕

0 ᐉ

冋

d dxfn共x兲⳵xp共x,0兲 − i␻n c2 fn共x兲⳵tp共x,0兲

册

dx = An ␻n2 c2ᐉ. 共51兲 As a consequence, the initial conditions need to be written by using the derivatives of the function p共x,t兲 with respect to abscissa and time, respectively. Result 共48兲 can be checked by integrating by parts the first term of the integral. Never-theless, the product共50兲 is not useful for the constant mode, and the first method needs to be used 共see Subsection VI A 3兲. Moreover, this derivation does not prove that the product is a scalar product, and that the modes form a basis of the space of solutions of the problem. This will be done hereafter.

2. Riesz basis of the modes

Several works have been done by mathematicians con-cerning spectral operators when boundary conditions are not simple conditions, such as Neumann or Dirichlet conditions. We quote the work by Russell,21 Majda,22Lagnese,23Banks

et al.,24 Darmawijoyo and Van Horssen,7 and Cox and Zuazua.6 Rideau4 has treated the 1D case with a 共unique兲 resistive termination, giving explicitly a scalar product 共see also Ref. 5兲. We generalize his calculation using a similar method, by considering the wave equation with source in the following form:

⳵t␺共x,t兲 = A␺共x,t兲 +␾s共x,t兲, 共52兲

where␺共x,t兲=共p,v兲T, p andv /共␳c兲 being the acoustic

pres-sure and velocity, respectively. Operator A is: A =

冉

0 − c⳵x

− c⳵x 0

冊

, 共53兲

and boundary conditions are written as:

p共0,t兲 = −␨v共0,t兲 and p共ᐉ,t兲 =␨_ᐉv共ᐉ,t兲 " t. 共54兲

The family of eigenelements of A are found to satisfy: ␭npn共x兲 = − c⳵xvn共x兲; ␭nvn共x兲 = − c⳵xpn共x兲, 共55兲 thus

冉

pn共x兲 vn共x兲

冊

=

冉

cosh共␭nx/c +␩兲 − sinh共␭nx/c +␩兲

冊

共56兲 ␭n=共−␩−␩ᐉ+ in␲兲c/ᐉ = i␻n 共57兲

关see Eq. 共20兲兴. pn共x兲= fn共x兲 and ␭nare identical to the eigen-functions and eigenvalues found before. Nevertheless, the constant mode is eliminated 共except for the very particular case ␩= −␩_ᐉ兲, because the boundary conditions are slightly different: Eqs.共2兲 and 共3兲 are obtained by deriving Eq. 共54兲 with respect to t. In Eq.共56兲, the argument of the hyperbolic functions can be written as:

␣共x兲 = −␩ᐉrx/ᐉ +␩r共1 − x/ᐉ兲; 共59兲 ␤n共x兲 =␲关−␮_ᐉx/ᐉ +␮共1 − x/ᐉ兲兴/2 + n␲x/ᐉ. 共60兲 关See definitions 共10兲 and 共11兲兴. Denoting ␺n␣共x兲 =共pn共x兲,vn共x兲兲T, we show in the Appendix that the family of elements ␺n␣共x兲 is a Riesz basis, i.e., a complete basis of elements, which become orthogonal for the following scalar product: 具␺,␸典H␣=

冕

0 ᐉ ␸T*_G 2␣共x兲␺dx, 共61兲

where G_␣共x兲 =

冉

cosh␣共x兲 sinh␣共x兲

sinh␣共x兲 cosh␣共x兲

冊

. 共62兲

For a given vector ␺=共p,v兲T_{, the value of the scalar} product with eigenvector␺n␣ is found to be:

具␺,␺n␣典H␣=

冕

0

ᐉ

关p共x,t兲pn共x兲 − v共x,t兲vn共x兲兴dx. 共63兲 This is in accordance with the product 共50兲. A direct application of this result is the solution of Eq. 共52兲 with initial conditions ␺共x,0兲=关p共x,0兲,v共x,0兲兴T_{. The modal}

de-composition is uniquely determined as ␺共x,t兲

=兺nhn共t兲␺n␣共x兲 in the energy space H, and leads to the fol-lowing family of decoupled ordinary differential equations:

ᐉ关⳵thn−␭nhn兴 = 具␾s共x,t兲,␺n␣典H␣; 共64兲

hn共0兲 = 具␺共x,0兲,␺n␣典H␣. 共65兲

This result can be first applied to the calculation done in Sec. VI A. In order to find a solution␹共x,t兲 of the second-order equation without source, we denote ␺共x,t兲 =关⳵t␹共x,t兲,−c⳵x␹共x,t兲兴T, and obtain by integrating ␺共x,t兲 with respect to t: ␹共x,0兲 =

兺

n ␭n −1 hn共0兲pn共x兲 + A. 共66兲

Using Eqs.共65兲 and 共63兲, and replacing␹共x,t兲 by p共x,t兲,

pn共x兲 by fn共x兲, ␭nby i␻n, and␭n −1_h

n共0兲 by An, formula共51兲 is checked. Notice that coefficient A cannot be directly deter-mined with this method.

3. Example of the Green function

Similarly, the Green function can be calculated by using the previous result. In order for the unknown function to satisfy the boundary conditions 共54兲, or 共2兲 and 共3兲, it is convenient to define the following vectors:

␺共x,t兲 =

冉

⳵tg共x,t兲 − c⳵xg共x,t兲

冊

; ␾s共x,t兲 =

冉

c2␦共x − x0兲␦共t兲 0

冊

. 共67兲

The first row of Eq.共52兲 is Eq. 共1兲, while the second one comes from the definition of vector ␺. Using Eq. 共64兲, the solution is found to be:␺共x,t兲=兺n␺n␣共x兲hn共t兲, where

⳵thn−␭nhn= c2ᐉ−1pn共x0兲␦共t兲. 共68兲

The initial conditions for the Green function imply ␺共x,t兲=0 for t⬍0, therefore hn共t兲=0 for t⬍0. Thus, the solution of Eq.共68兲 is hn共t兲 = Ane␭ntH共t兲; An= c2ᐉ−1pn共x0兲. As a consequence,

冉

⳵tg共x,t兲 − c⳵xg共x,t兲

冊

=

兺

n An

冉

pn共x兲 vn共x兲

冊

e␭nt_H共t兲. 共69兲 Integrating the first row with respect to time leads to:

g共x,t兲 = H共t兲

兺

n

关Anpn共x兲e␭nt+ A共x兲兴. 共70兲 Derivating this expression with respect to x and using the second row of Eq.共69兲 leads to⳵xA共x兲=0, thus A is a stant, as expected. In order to deduce the value of this con-stant, we need the following result:

⳵tg共x,0兲 = p共x,0兲 = c2␦共x − x0兲. 共71兲

It is obtained by derivating the first row of Eq. 共69兲 with respect to time, and the second row of Eq.共69兲 with respect to abscissa, leading to p共x,0兲␦共t兲=c2_␦共x−x

0兲␦共t兲 关recall that

⳵t关F共t兲H共t兲兴=H共t兲⳵tF共t兲+F共0兲␦共t兲兴. The end of the calcula-tion is done in Sec. VI A 3, giving Eq. 共49兲, by replacing

p共x,0兲 by g共x,0兲 and taking into account that g共0,0兲

= g共ᐉ,0兲=0. We notice that the calculation is valid for both passive and active boundaries.

VII. EIGENMODES EXPANSION IN THE FREQUENCY DOMAIN: BIORTHOGONALITY

Frequency domain approach is very popular in acoustics 共see, e.g., Ref. 7兲, and leads to the use of biorthogonality 共see, e.g., Ref. 16, p. 884兲 of modes, except when the bound-ary impedances are imaginbound-ary, corresponding to nondissipa-tive boundaries: For that case, modes are orthogonal, and the Laplacian operator is self-adjoint. In this section, we limit the discussion to the Green function calculation, and succes-sively use the two above-used approaches: The second-order equation, then the system of two first-order equations, ignor-ing the constant mode. Because we are now in the Fourier domain, equations are ordinary differential equations, bior-thogonality theory ensuring the completeness of the modes family.

A. Solution of the second-order equation

1. Modal expansion

In order to calculate the inverse FT of G共x,␻兲, another solution is possible: The expansion of G共x,␻兲 in eigen-modes. This is done for a particular case by Filippi共Ref. 1 p. 58: This author considers another type of excitation instead of the Dirac function, thus uses the Laplace transform instead of the FT; notice that the constant mode is missing in this

work兲. We will see how this method leads to the same poles and residues that the direct method using the closed-form expression 共12兲. We are searching for the following expan-sion:

G共x,␻兲 =

兺

n

Gn共x,␻兲, 共72兲

where the eigenmodes Gn共x,␻兲 are solutions of:

关⳵xx2 +␪n2共␻兲/c2兴Gn共x,␻兲 = 0, 共73兲 and satisfy the boundary conditions 共7兲 and 共8兲. The key point is that eigenmodes Gn共x,␻兲 and eigenfrequencies ␪n共␻兲 depend on frequency ␻: This is due to the boundary conditions, which are of Robin type. Solutions of Eq. 共73兲 can be written as follows:

Gn共x,␻兲 = cosh共i␪n共␻兲x/c +␸n共␻兲兲, 共74兲 where ␪n共␻兲 and ␸n共␻兲 are given by the boundary condi-tions. Thus, they satisfy

␪n共␻兲tanh␸n共␻兲 =␻/␨; 共75兲

␪n共␻兲tanh共i␪n共␻兲ᐉ/c +␸n共␻兲兲 = −␻/␨ᐉ. 共76兲 Eliminating quantity␸n共␻兲, the eigenvalues are found to sat-isfy the following equation:

tanh共i␪n共␻兲ᐉ/c兲

冋

␪n共␻兲 + ␻2 ␪n共␻兲␨␨ᐉ

册

= −␻

冋

1 ␨+ 1 ␨ᐉ

册

. 共77兲 When␪n共␻兲 and␻are not simultaneously zero, this equation can be rewritten as e2i␪nᐉ/c₌

冋

␪n␨−␻ ␪n␨+␻

册冋

␪n␨_ᐉ−␻ ␪n␨_ᐉ+␻

册

. 共78兲

Calculation of all solutions of this equation is not nec-essary, only one of them being useful in the following. Op-erator D =⳵xx2 is formally equal to its adjoint D¯ , but the boundary conditions are different共conditions for D¯ are com-plex conjugate of conditions for D兲. Modes of D¯ are the complex conjugate of modes Gn共x,␻兲 关they are equal to modes Gn共x,␻兲 only if␨and␨ᐉare imaginary, because of the factor i in boundary conditions共7兲 and 共8兲兴. Thus, in general, operator D is not self-adjoint, and eigenmodes Gn共x,␻兲 and Gn共x,␻兲=Gn

*_{共x,␻兲 are biorthogonal 共see Ref. 16兲. The scalar}

product of modes Gn共x,␻兲 with Gm共x,␻兲 is simply given by:

冕

0 ᐉ Gm共x,␻兲Gn共x,␻兲dx = ⌫n␦nm, 共79兲 where ⌫n= ᐉ 2

冋

1 + c

sinh 2共i␪nᐉ/c +␸n兲 − sinh 2␸n

2iᐉ␪n

册

. 共80兲

Therefore modes Gn共x,␻兲 are orthogonal 关for the prod-uct共79兲兴 and fulfill the same boundary conditions as G共x,␻兲,

contrary to resonance modes fn共x兲 in Eq. 共29兲. Finally, the solution of Eq.共6兲 can be written as follows:

G共x,␻兲 = c2

兺

n

Gn共x,␻兲Gn共x0,␻兲

⌫n共␪n2共␻兲 −␻2兲

. 共81兲

2. Calculation of poles and residues

In order to calculate the inverse FT, the residue calculus will be used again. The only terms of the series contributing to poles verify:

␪n共␻兲 = ±␻. 共82兲

Looking at Eq.共78兲, it can be seen that these two solu-tions lead to the same equation for␻. Rewriting Eq.共78兲 by using Eqs. 共75兲 and 共76兲, the resonance modes frequencies are found to be solutions of Eq. 共20兲. Solutions ␻p of this equation are the nonzero poles of the integral in the inverse FT. Nevertheless, the pole ␻= 0 exists again, because the zero value satisfies Eq. 共82兲, the eigenvalue ␪n共␻兲=0 satis-fying Eq.共77兲.

It remains to calculate the residues. Starting with the poles␻p⫽0, we need to select in the series 共81兲 the terms involving poles. For a given␻p, there are two terms. How-ever, it appears that modes corresponding to␪nand −␪nare identical. As a consequence, only one term of the series con-tributes to the inverse FT: It will be denoted ␪p共␻兲. The corresponding residue is found by expanding Eq.共81兲 for␻ close to ␻p, as follows: G共x,␻兲 = c2 Gp共x,␻p兲Gp共x0,␻p兲 ⌫p共2␻p兲共␻p−␻兲

冉

1 −

冋

d d␻␪p共␻兲

册

_␻=␻ p

冊

.

A similar expression can be found in Filippi,1 which points out that Morse and Ingard共Ref. 7 p. 559兲 forgot the deriva-tive. The same error is found in Morse and Feshbach 共Ref. 16 p. 1347兲, with another error in the derivation of Eq. 共77兲: These authors treated the problem of a string with one non-rigid共and resistive兲 support.

Actually, the derivative of␪p共␻兲, denoted␪p

⬘

共␻兲 can be calculated analytically, as follows. Taking the derivative of Eq. 共78兲 with respect to␻, or more conveniently, taking the logarithmic derivative of Eqs. 共75兲 and 共76兲, the following results are obtained:

1 ␻p −␪p

⬘

␪p = 2␸p

⬘

sinh 2␸p = 2共i␪p

⬘

ᐉ/c +␸p

⬘

兲 sinh 2共i␪pᐉ/c +␸p兲 .

Thus, eliminating the derivative␸

⬘

_p, writing␪p=␻pand using Eq. 共80兲, it is found after some algebra:

1 −␪

⬘

p=ᐉ/2⌫p. 共83兲

Finally, for␻close to␻p: G共x,␻兲 = −c 2 ᐉ Gp共x,␻p兲Gp共x0,␻p兲 ␻p共␻−␻p兲 , 共84兲

Otherwise, for␻ close to 0, the solution ␪共␻兲 which is close to 0, solution of Eq.共82兲, satisfies the following equa-tion, deduced from Eq.共77兲:

关1 − ⌰2_{/3 + O共⌰}4_兲兴关⌰2_+⍀2_/共␨␨

ᐉ兲兴 = i⍀关␨−1+␨ᐉ−1兴,

where⌰=␪ᐉ/c and ⍀=␻ᐉ/c. Therefore, ⌰2 _{is of order⍀,}

and

␪2_{= i␻cᐉ}−1_关␨−1_+␨ ᐉ

−1_{兴 + O共␻}2_{兲. 共85兲}

Using Eq.共81兲, the residue for the pole␻= 0 is obtained, and Eq.共25兲 is confirmed. We conclude that the method of the expansion of orthogonal modes in the Fourier domain leads to the same result关Eq. 共26兲兴 than the “direct” method, but the derivation is more delicate.

B. System of two first-order equations We consider now the FT of Eq.共52兲:

i␻␺共x,␻兲 = A␺共x,␻兲 +␾s共x,␻兲, 共86兲

where ␺共x,␻兲=关P共x,␻兲,V共x,␻兲兴T. It is interesting that the boundary conditions are independent of frequency:

P共0,␻兲 = −␨V共0,␻兲; P共ᐉ,␻兲 =␨_ᐉV共ᐉ,␻兲. 共87兲

Eigenvalues ␭n and eigenvectors ␺n␣ of operator A are already known关see Eqs. 共56兲 and 共57兲兴. The Appendix shows that the adjoint operator of A, is A¯ =−A, and gives the boundary conditions for it. This formulation differs slightly from the work,9,10 considering a different operator, but the principle is identical: We notice that these authors treat the problem for more general operators and boundary condi-tions. Eigenvalues and eigenvectors of A¯ are solutions of:

A¯␺m␣=␭m␺m␣,

pm=␨vmfor x = 0 pm= −␨ᐉvmfor x =ᐉ.

Thus, the adjoint eigenvalue problem to be solved is the same as the direct one, by replacing c by −c,␩by −␩and␩_ᐉ by −␩_ᐉ. The eigenelements are thus found to be:

␺m␣=

冉

pm共x兲 vm共x兲

冊

=

冉

cosh共␭mx/c +␩兲 sinh共␭mx/c +␩兲

冊

, 共88兲 where ␭m= −共␩+␩ᐉ+ im␲兲c/ᐉ = ␭−m. 共89兲

Comparing with the family ␺n␣ 关Eq. 共56兲兴, there is a differ-ence in sign for the second row: We notice that Rideau3 made an error in the biorthogonal family. By construction, the biorthogonality relationship is ensured:

共␭n−␭m*兲 ⬍␺n␣, ␺m␣⬎ = 0. 共90兲

Using Eq. 共27兲, we remark that ␭n=␭m* implies m = −␯, as defined in Eq.共28兲. Therefore

具␺n␣,␺m␣典 =

冕

0

ᐉ

␺m␣T*␺n␣dx =共− 1兲␮ᐉ␦m,−␯. 共91兲 This latter relation enables one to perform a modal decom-position on the共␺n␣兲 family: But, contrarily to standard cases, the nth coefficient is not given by the scalar product with ␺n␣, but by the scalar product with␺−␯␣ _n关up to the normal-ization coefficient 共−1兲␮ᐉ兴. Notice that if ␩ and ␩_ᐉ are both real, ␭m=␭m*, and Eq. 共91兲 is obvious from the ex-pressions of eigenvalues and eigenvectors 共for this case, ␭n=␭m* implies n = m兲. For the general case, the scalar product can be written: 具␺n␣,␺m␣典=兰0ᐉcos关␤n共x兲+␤−m共x兲兴dx

=共−1兲␮兰₀ᐉcos关共␯+ m兲␲x /ᐉ兴dx. Comparison with Eq. 共A4兲

exhibits the difference between the two methods.

It remains to apply orthogonality to Eq.共86兲. We choose the case of the Green function关Eq. 共67兲兴, with the following result: G共x,␻兲 = −c 2 ᐉ

兺

n fn共x兲fx共x0兲 ␻共␻−␻n兲 . 共92兲

The calculation is easy, because ␻ndoes not depend on fre-quency. Comparison with Eq. 共29兲 exhibits a difference in the denominator, i.e., a factor␻instead of␻n, and, of course, the absence of constant mode. When returning to the time domain, all of the terms corresponding to ␻=␻n are identi-cal, and a constant mode is found for the pole ␻= 0, but again it is not possible to deduce it from orthogonality rela-tions, as in Sec. VI B. Nevertheless, because of the indepen-dence of the boundary conditions with respect to frequency, the calculation of the residues is much easier than for the second-order equation. For the same reason, the calculation in the time domain would be possible with the same modal decomposition, and this is a major difference with the meth-ods based upon the second-order equation.

VIII. CONCLUSION

The simple problem we have studied, which can be re-garded in particular as a radiation problem, exhibits interest-ing properties for the resonance modes: They are complex valued, and nonorthogonal for the simple product 共38兲 be-cause of the bounded character of the considered medium, but except the constant mode, they are orthogonal for a prod-uct modified in a proper way, and are a basis for the space of solutions. Second-order equations allow one to find the con-stant mode, while first-order systems of equations allow a more direct formulation of boundary impedances.

Thanks to the simplicity of the problem, the analytical treatment is possible with several methods, elucidating the relationship between them, which can be useful for more intricate problems共e.g., when damping is added to propaga-tion, or when boundary impedances involve a mass or a spring兲. No approximations are needed, the results are valid whatever the value of the terminal resistances. Active bound-aries can also be considered, thanks to a change in functions. We notice that an advantage of the frequency domain calcu-lations is the possibility of the treatment of an arbitrary de-pendence of the boundary conditions. For a dede-pendence␩共␻兲 and ␩_ᐉ共␻兲, Eq. 共29兲 remains valid by replacing ᐉ by 关ᐉ

− ic共␩

⬘

p+␩ᐉp

⬘

兲兴, where␩

⬘

p=共d␩/ d␻兲␻=␻_p, and similarly for␩ᐉ. This can be shown by generalizing Eq. 共22兲, or, with some algebra, using the modal expansion.

Finally, considering the problem of a stratified medium 共see Sec. II兲, it could be deduced in the field outside of the interval 关0,ᐉ兴. When terminations are passive, a result is, that modes tend to infinity when x tends to ±⬁. An interest-ing study has been done in Ref. 13, usinterest-ing biorthogonality and explaining the relation between the energy outside of the interval and the terms responsible of nonorthogonality in Eq. 共39兲.

ACKNOWLEDGMENTS

We would like to thank José Antunes, Patrick Ballard, Sergio Bellizzi, Michel Bruneau, Paul Filippi, Dominique Habault, Pierre-Olivier Mattei, and Vincent Pagneux for very fruitful discussions.

APPENDIX: PROOF OF THE COMPLETENESS OF THE EIGENELEMENTS OF OPERATOR A

The operator A defined by Eq.共53兲 is a differential op-erator defined on the energy space H = L2_{共0,ᐉ兲⫻L}2_{共0,ᐉ兲, it}

has a compact resolvent共cf Ref. 4, p. 191兲. Using the ordi-nary scalar product 具␺,␸典=兰₀ᐉ关pq*_+vw*_{兴dx, between}

␺共x,t兲=共p,v兲T

and ␸共x,t兲=共q,w兲T, the following result is obtained:

具A␺,␸典 + 具␺,A␸典 = − c关v共q*_+␨ ᐉw*兲兴x=ᐉ + c关v共q*_−␨w*_兲兴

x=0.

It is deduced that the adjoint operator of A is A¯ =−A 共we denote all quantities related to the adjoint problem with an overline兲, and on its domain, the following adjoint boundary conditions must be fulfilled:

q共0,t兲 =␨*w共0,t兲 and q共ᐉ,t兲 = −␨_ᐉ*w共ᐉ,t兲 " t.

关Here,␨=␨*_{, and␨}

ᐉ=␨ᐉ*; if␨is infinite, the boundary

condi-tions arev共0,t兲=0, and w共0,t兲=0, and similarly for

bound-ary x =ᐉ兴. Therefore, A is skew symmetric, but not skew

adjoint, because the domains of A and A¯ are different,

ex-cept if both ␨or ␨_ᐉ are either zero or infinite 共Dirichlet or Neumann conditions兲; notice that for a skew-adjoint opera-tor, the eigenvalues are purely imaginary. In order to find a new scalar product, we denote, from Eqs.共58兲–共60兲:

␺n␣共x兲 =

冉

e␣共x兲 e−␣共x兲

− e␣共x兲 e−␣共x兲

冊冉

ei␤n共x兲

e−i␤n共x兲

冊

.

In H, the standard scalar product 具␺n␣,␺␣p典H=兰₀ᐉ关pnpp* +vnvp

*_{兴dx does not vanish for n⫽p, except if␣共x兲=0. If we}

denote␺n

0_{共x兲, the functions corresponding to the latter case,}

it is possible to construct a new scalar product ensuring or-thogonality, in a similar way to Rideau.3 From Eq.共62兲, the following hyperbolic rotation is obtained:

␺n0共x兲 = G␣共x兲␺n␣共x兲. 共A1兲

We will now prove that the new product

具␺,␸典H␣=具G␣␺,G␣␸典H=

冕

0

ᐉ

␸T*_M

␣共x兲␺dx, 共A2兲

where M_␣共x兲=G_␣TG_␣, leads to the orthogonality of the modes. M_␣共x兲 is found to be equal to G_2␣. It is symmetrical and positive definite because

共储␺储H␣兲2=共储共p,v兲 T_储 H ␣_兲2 =

冕

0 ᐉ 关cosh关2␣共x兲兴共兩p兩2_+兩v兩2_兲 + 2 sinh关2␣共x兲兴Re共pv*_兲兴dx

can be rewritten as: 共储␺储H␣兲2=

1 2

冕

₀

ᐉ

关e2␣共x兲_{兩p + v兩}2_{+ e}−2␣共x兲_{兩p − v兩}2_兴dx.

Moreover, ␣共x兲 is a function varying monotonously from ␩r to −␩ᐉrwhen x increases from 0 toᐉ, and the fol-lowing bounds can be found for储␺储H␣:

c_␣储␺储H⬍ 储␺储H␣⬍ C␣⬍ 储␺储H, 共A3兲

where c_␣= e−␩˜ and C_␣= e␩˜, with␩˜ = sup关兩␩r兩, .兩␩ᐉr兩兴.

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0_兲

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冕

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冕

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