Group velocity and acoustical resonances

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Group velocity and acoustical resonances

Gérard Quentin, André Derem, Bernard Poirée

To cite this version:

Gérard Quentin, André Derem, Bernard Poirée. Group velocity and acoustical resonances. Journal

de Physique, 1989, 50 (14), pp.1943-1952. �10.1051/jphys:0198900500140194300�. �jpa-00211038�

1943

Group velocity and acoustical resonances Gérard Quentin (1), André Derem (2) and Bernard Poirée (3)

(1) Groupe ^de Physique des Solides de l’Ecole Normale Supérieure, Université Paris 7, 75251 Paris Cedex 05, ^France

(2) ^{21 rue de l’abbé} Gridel, Nancy, ^France (3) D.R.E.T., SDR, G6, Paris, ^France

(Reçu ^{le 29} juillet 1988, révisé le 14 mars 1989, accepté ^{le 31 mars} 1989)

Résumé.

Après ^avoir présenté quelques notions fondamentales se rapportant ^au concept ^de vitesse de groupe, ^{nous montrons} comment, dans le cadre de la diffusion acoustique, ^ce concept peut s’avérer très utile au niveau de l’interprétation qui ^{s’attache aux} trajectoires ^de Regge. ^De fait, ^en acoustique, ^il apparaît que la vitesse de groupe ^est reliée à l’espacement des résonances dues à un type donné d’onde de surface, ^comme aussi à la pente ^{de la} trajectoire ^de Regge

associée à chacune des ondes de surface. Le même concept peut également intervenir pour

l’interprétation ^du rayonnement acoustique. ^{Enfin, nous} terminons par quelques remarques ^sur les rapports ^entre résonances, trajectoires ^de Regge ^et vitesse de groupe ^en physique ^{nucléaire et}

en physique ^des particules élémentaires.

Abstract.

After a summary of ^some fundamental notions concerning the group velocity concept, ^{we show} how, in the field of acoustic scattering, ^this concept may be very useful for the

interpretation ^of Regge trajectories. ^{As a matter of} fact, ⁱⁿ Acoustics, it appears that the group

velocity ^{of a} given ^{surface wave} is related to the frequency spacing ^between standing ^wave

resonances for the same wave and also to the slope ^{of its} Regge trajectory. ^{This same} concept ^is useful in the interpretation of acoustic radiation. Finally, ^we present ^some remarks about the

relationships ^between resonances, Regge trajectories ^{and group} velocity in nuclear physics ^and elementary particle physics.

J. Phys. ^{France 50} (1989) ^{1943-1952 15 JUILLET 1989,}

Classification

Physics ^Abstracts

43.30 - 43.35

14.00

1. Introduction.

Although ^the concept of group velocity ^{was known} earlier, its first development ^{can be}

attributed to Hamilton (1839) and Stokes (1876). ^The principle ^of stationary phase ^{was later}

used by ^Kelvin (1887) ^to generalize the notion of group velocity. ^These preliminary ^{works are} thoroughly reviewed in a monograph by ^Havelock [1] and in the book by ^Brillouin [2]. ^{Both in} Electromagnetism ^{and in} Acoustics, during the last few decades, ^{the use of} quasi ^continuous

waves has tended to be replaced by ^{the use} of wideband signals ^{in the form of short} pulses ^or frequency ^modulated signals. ^In analysing the transmission of such signals ⁱⁿ dispersive media, the group velocity ^{is often more} useful than the phase velocity [3-8]. ^As will be shown in the present paper, group velocity ^is equally ^{useful in} studying resonances of dispersive

waves.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500140194300

2. The group velocity.

We shall first consider a unidimensional model valid for an isotropic medium. The group

velocity ^{was first defined as the} velocity ^{of the} envelope of beats between two plane ^waves

with wave vectors (ko ^and ko ⁺ dk) very close ^{to one} another :

is the phase velocity.

If the dispersion equation ^{of a} given wave D ( w , k )

^{0 is known, the group} velocity ^{can be}

calculated through implicit derivation :

The concept of group velocity ^was later extended to a wave packet ^defined by ^a spectrum ^of

wave vectors. In the unidimensional case, all the wave vectors are supposed ^{to have the same}

direction and amplitudes very close ^{to one} another. Under general assumptions, ^the stationary phase method also leads to the definition given ^above [1].

The group velocity ^can be also deduced from the knowledge of the variation of the phase velocity C cp ^{relative to the wave vector k.} They ^{are related to one another} through ^a Legendre

transform

In three dimensions, ^{for an} homogeneous, ^non dissipative medium the relationship ^between phase and group velocities has been established by Hayes [9] :

where ’0 is the unit 3 x 3 matrix and Q ^{indicates a tensor} product. ^The underlined quantities

are the vectors and k is the unit vector in the direction of k.

3. Résonance scattering theory. S ^{Matrix and} Regge trajectories ^{in acoustics.}

Resonance scattering theory ^{is now} well known and widely used in the field of Acoustics. A

complete presentation ^{of this} theory and related experiments ^can be found in the book

La Diffusion Acoustique » [10].

This scattering theory ^was developed ^{in the United States} by ^{the team of Überall} [11, ^12, 13] and in France by ^{one of the} present ^authors (A.D.) [10, ^14, 15]. ^{We shall} give, ^{in this}

paper, ^{a brief} overview concerning ^acoustic scattering by ^a cylindrical target immersed inside

a fluid.

In ^a modal description ^of scattering, ^{the use} of the S-matrix formalism, applied ^first by Flax, Dragonette ^{and Überall} [13] ^to Acoustics, ^{allows us to} write the individual amplitudes

as

1945

where n is the index of normal modes of vibration (n

integer) and xi the dimensionless

frequency :

x1 - k1 a

kl

^{wave number} inside the fluid (6)

a

radius of the cylinder

and

The amplitude TJO) corresponds ^{to the} non-resonant component of the mode and

T,,* ^{to its resonant} component. ^{When a resonance effect occurs in} T,*, ^{the resonant} part ^{can be} written, using ^the Breit-Wigner formula, ^as

Successive resonances occur for each mode with index n corresponding ^{to different families} (or types) ^of resonances (denoted by ^{index f in} (8)), the value 1 being attributed to the second

index f for the first resonance, the value 2 for the second one and so on. Each

(n, f) ^{resonance has a} frequency x1 - xnl ^{and a width} Tnt. ^After using ^the Sommerfeld- Watson transform where the index n becomes a complex variable v, ^{we obtain the}

characteristic equation :

It can be shown that, ^{if we call} D,,(xl) the denominator of the amplitude Tn(XI) (Eq. (5)),

the function D ( v , x1 ) is, simply, ^the analytical continuation with respect ^{to n of the}

Dn(x1) polynomial family. ^{We call v}

Vt(XI) ^the complex ^{roots of the} characteristic

equation (9). ^{Each of these roots} is associated with a surface wave circumnavigating ^the cylinder [12, ^14, 15].

We consider the so-called

elastic

surface waves, that is the waves mainly supported by

the solid cylinder. ^{A direct} relationship between elastic surface waves and resonances of the scatterer has been established in the following ^form. Every time the condition

is satisfied, ^the frequency xi is almost identical with the frequency Xnf appearing ⁱⁿ (8). ^{It is,} then, necessary ^to attribute the second index 1 = 1, 2, ^{3... to} the different elastic surface wave

modes in a well defined order.

When xi varies, ^{the roots} vl (x1 ) ^follow trajectories ^{in the} complex plane ^{v. The real} part ^of

a trajectory, plotted ^versus xl, is called ^a Regge trajectory by analogy with usual practice ⁱⁿ elementary particle theory. Consequently, ^each Regge trajectory appears in ^a plane (Re (ve), xl). ^{It has a} well defined index i and each time Re ( vl ) equals ^as integer,

condition (10) ^{states that} a resonance exists at the corresponding frequency xl.

o

The wave numbers kl of the surface waves are given by

A resonance in a given vibration mode n occurs when

We may express this fact otherwise. At ^a (n, l) ^resonance frequency the circumference of the cylinder ^is equal ^{to n} wavelengths of the surface wave. It can also be shown that the width

r nl ^{of the resonance is} given [10] by

and consequently ^can be deduced from the imaginary part ^{of the root} vl (xl), taking ^into

account the local slope ^{of the} Regge trajectory.

4. Group velocity ^{and resonance} scattering theory.

In this paper ^we shall restrict ourselves to the case of scatterers with axial symmetry. ^{The first} paragraphs ^{deal with} cylindrical scatterers and paragraph 4.5 extends the results to spherical

scatterers.

4.1 PULSE INSONIFICATION AND INTERPRETATION OF THE RESULTS.

In order to study ^the scattering ^from targets one can use either long ultrasonic pulses (narrow band) ^{or short} pulses (wide band). ^{The main} advantages of the second technique is the better precision ^{in the}

measurements of the arrival times of echoes and also the possibility ^of using sophisticated

methods of signal processing [7, 8]. ^In figure ^{1 we have} plotted ^one typical signal ^scattered

from a thin elastic cylindrical metal shell insonified by ^a short ultrasonic pulse. ^{In this} figure,

Fig. ^1. - Typical signal backscattered from a thin duraluminium shell filled with air and immersed in water. The scatterer is insonified by ^a short ultrasonic pulse ^{and the} specular echo has been omitted. The observed séries of echoes corresponds ^to successive roundtrips of the surface wave along ^the

circumference of the target (see ^Refs. [7, 8]).

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only ^one series of echoes is seen. These echoes are periodic ^{in time} [7] ^and correspond ^{to a} given circumferential wave propagating around the shell. This signal ^is observed in the

backscattering geometry where the ultrasonic transducer acts both as transmitter and

receiver, and each echo corresponds ^{to a} given ^{number of} roundtrips ^{of the wave around the}

shell. The time separation ^{At between two} successive echoes can be related to the radius a of the shell and the group velocity Cg of the surface wave :

The radius a is the mean radius of the shell if the wave propagates inside the shell (pseudo-

Lamb wave for example) ^or the external radius if the wave is an interface wave (Franz, Rayleigh ^or Scholte-Stoneley ^{wave for} example).

4.2 RESONANCE SPACING AND GROUP VELOCITY.

If the scattered signal is studied in the

frequency ^domain through ^{a Fourier} transform, the echoes give ^{rise to} a resonance spectrum

(Fig. 2). ^The frequencies and widths of these resonances are predicted theoretically by ^the

o

resonance scattering theory ^outlined in section 3. If we call kn ^{the real} part ^{of the wave vector}

o

k,,, ^formula (12) ^can be written as

The next resonance will be given by

and the separation between these two resonances in terms of wave number is simply

The experimentally ^measured quantity ^{is the} spacing âf between successive resonances

Fig. ^2.

Resonances spectrum deduced from the signal ^of figure ¹ using ^a Fourier transform. The

frequency unit is the reduced frequency (kl a ) ^where k1 ^{is the wave number} inside the surrounding ^fluid

and a the radius of the shell (see ^Refs. [7, 8]).

and we deduce from (17) ^and (18) ^{the exact formula}

If the change ^{in the wave} number is small between two successive resonances this formula

can be expressed ^{in terms} of group velocity ^as

In this case the spacing ^{between two} resonances of a particular ^{surface wave is} proportional ^to

the group velocity ^{of this wave. Such a} property ^{leads to a} very simple experimental ^method

of measurement of group velocity ^{from the resonance} spectrum. ^This method has been

successfully ^{used to} study ^the scattering of ultrasonic waves in thin shells [7, 8]. The results obtained agree very well with the ^{exact resonance} scattering theory.

4.3 REGGE TRAJECTORIES AND GROUP VELOCITY.

In the brief outline of the resonance

scattering theory given in section 3 we explained that in the field of acoustical scattering, Regge trajectories ^are constructed in the plane (Re (v), Xl) . ^{We now} study ^a geometrical interpretation ^{of the} phase and group velocities from the Regge trajectories. ^Each trajectory corresponds ^{to a} given ^{kind i} of surface wave (Rayleigh, Whispering Gallery ^1, Whispering Gallery 2...) and for each such wave the n-th order resonance satisfies the condition

and corresponds ^{in the} Regge trajectory plane ^to the abscissa

where C -, ^{is the} phase velocity and C is the sound velocity for the medium in which the scatterer is immersed. Thus

The geometrical meaning is clear. The nondimensional quantity CIC,, ^{is equal to the slope}

of the straight ^line joining ^the origin of coordinates to the point (n, (X1)n) ^{on the} Regge trajectory ^that corresponds ^{to the resonance} studied. This fact is known but a similar

interpretation ^{can also be} given for the group velocity. ^{We can calculate as} follows the slope ^p

of the Regge trajectory ^{at the same} point (n, (x1)n ) :

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This shows that the nondimensional quantity C/Cg ^{is equal to the local slopep of the}

Regge trajectory ^{at the resonance} point (n, (X1)n) under consideration.

Consequently ^the phase and group velocities have ^a very simple geometrical interpretations

related to both the dispersion ^{curves and the} Regge trajectories n (k, a ) (Fig. 3).

Fig. ^3.

Comparison between the geometrical interpretation ^of phase and group velocities from the

dispersion ^curve and from the Regge trajectory : a) dispersion ^{curve ;} b) Regge trajectory : ^at any

resonance frequency n ^is equal ^{to an} integer ^and corresponds ^{to the ratio of} the circumference of the scatterer to the wavelength of the surface wave. The Regge trajectory ^is plotted ^versus the reduced

frequency, ^{as is the} spectrum ⁱⁿ figure ^2.

4.4 RESONANCE WIDTH AND GROUP VELOCITY.

The width of the resonances of circumfe- rential waves is related to radiation from these waves into the surrounding ^medium. Usually

in resonance scattering theory the width l’ is expressed in units of kl ^a. Experimentally ^the

attenuation of the wave circumnavigating ^{the scatterer} is measured as yNp/rad. ^As given

above by equation (13) ^{the two} quantities ^{are related} through ^{the formula :}

where v denotes the complex value of the integer ^{number n} used in the preceding paragraphs

and Re (v) ^is exactly n ^{for its} integer ^{values. That is, the} quantity in the denominator is

simply ^the slope ^{of the} Regge trajectory. Using the result of paragraph ^{4.3 we obtain in terms}

of group velocity ^an interpretation of the connection between resonance width and wave

attenuation by leakage into the fluid :

4.5 SPHERICAL RESONATORS. - For spherical targets with radius a, the complex ^wave number kn corresponding ^{to a} circumferential wave has a real part ^{which is} given by

The same explanation ^{as in} paragraph ^{3.2 can be} given ^{and leads to the same} expression ^for

the spacing ^{in the} frequency domain between two resonances :

Concerning Regge trajectories ^for spherical scatterers, ^{we obtain :}

Consequently, ^{the same} geometrical interpretation ^{as for} cylindrical targets can no more be

given ^{for the} phase velocity. Nevertheless the group velocity remains very simply ^{related to}

the slope ^{of the} Regge trajectories through ^{the same formula as for} cylindrical scatterers :

5. Group velocity ^and Regge trajectories ⁱⁿ quantum scattering theory.

In the field of elementary particle physics, Regge trajectories ^{in a} scattering process ^are

usually associated with exchanged particles (or ^field quanta) between the particles ^inter- acting [16]. Accordingly Regge trajectories play ^a fundamental part ^{in the} analysis ^of scattering ^reaction properties. ^{For that reason, a} simple physical signification ^{for the} slope ^of

these trajectories ^has constantly ^been sought. Although ^{if it is} likely ^{that this} slope is related

to the interaction strength [17], ^no elementary interpretation ^{has ever been} given.

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It may be remarked that intuitive explanations ^{which are} introduced in quantum physics usually ^start with ideas belonging ^{to classical} physics. ^{In the} study ^of electromagnetic phenomena, ^Watson poles (also named Franz poles) have been known for a long ^{time. In} electromagnetic ^wave scattering ^these poles play exactly ^{the same role as} Regge poles ^{in a} quantum scattering process. Over ^a period of time these poles have become directly related,

in electromagnetism, ^to the surface waves circumnavigating ^{a scatterer} [18]. Nevertheless it is in acoustics, ^and only ^{in the recent} years, that ^an immediate link has been established between surface waves and acoustical resonances of the target [13]. Consequently ^this physical picture ^{which is} available, ^{can be} completely organized around the concept ^{of the} Regge trajectory.

Surface waves are not a part ^and parcel of traditional quantum mechanics, although Regge poles ^have long ^{been a} part ^{of common} parlance in nuclear physics ^{and in} elementary particle physics. ^A certain evolution began, however, in 1978 when Farhan, George ^{and Überall} [19]

noticed that surface wave phenomenon ^{could be} usefully introduced into nuclear physics.

Circumferential nuclear tidal waves, having phase velocities that vary with the incident energy, could then be generated ^{in a} nuclear reaction. The collective multipole ^resonances

arise in this picture ^{from a} single nuclear surface wave, the latter

being ^a property ^{of the} medium of nuclear matter that carries the wave » [19]. ^The resonances are produced ^when phase matching ^of the surface waves occurs over the nuclear circumference. At this stage ^{the ’} relationship ^{between a} Regge pole, ^{a surface wave and the} resonances of a target ^{has become} clear in nuclear physics.

In this way, ^an important step ^{has been} taken, ^{but our} simple physical interpretation ^{of the} slope ^{of the} Regge trajectory ^{was not till now} put forward in nuclear or elementary particles physics. ^{In this} regard ^{we should} point ^{out that the} concept of group velocity ^{is of real} interest, only ^{when one works} with incident signals having ^{a non zero} bandwidth. Now in the quantum theory ^of scattering, it has become the rule to treat the incident beam as if it were

made up of rigorously monokinetical particles (see ^Newton [17] pp. 143 sqq.). ^Such being ^the

case, even though ^the Regge trajectory actually ^{is a} function of energy, the relationship ^with

group velocity ^is necessarily ^lost.

6. Conclusion.

In this paper ^we have shown that for a cylindrical ^{or a} spherical ^{scatterer the} spacing ^{in the} frequency ^{domain of two} successive resonances of a given circumferential wave is very simply

related to the group velocity ^{of this wave. We have also} given ^a geometrical interpretation ^of

the local slope ^{of the} Regge trajectories ^{in terms} of group velocity ^{and we deduced a} very

simple relationship between the width of a resonance and the attenuation of the wave which radiates into the fluid where the ^scatterer is immersed.

At the end of this paper ^we discussed briefly ^the possibility ^of using ^{the concrete notions of}

group velocity and surface wave in other fields of physics ^where Regge trajectories ^{are in}

common use.

We think that many physical phenomena concerning resonances and scattering may receive

a better and sometimes simpler interpretation ^{in terms} of group velocity ^{rather than in terms}

of phase velocity ^{and will} try ^{to extend our work to such} phenomena.

Acknowledgement.

We ^{want to} thank Professor B. Auld for critical reading ^{of the} manuscript and very useful discussions about the subject. ^{We want also to thank} M. Talmant. It is during ^{her Thesis at}

Paris 7 University ^{that the} separation between successive resonances has been related to

group velocity.

References

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[2] ^{BRILLOUIN L., Wave} Propagation ^and Group Velocity (Academic Press) ^1960.

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