Ultrasonic waveguides—A physical approach

ULTRASONIC WAVEGUIDES -A PHYSICAL APPROACH by M. REDWOOD*

Guided waves are often found in ultrasonics, since they appear in any system in which it is not possible to ignore the effect of those boundaries parallel to the direction of propagation. Many modes of propagation are possible in such waveguides and these are usually classified into broad groups loosely called %ompressional,” %hear,” “torsional” and ‘Vlexural.” An understanding of practical problems is frequently aided if a mode is thought of as one or more plane waves of a simple type (compressional or shear) travelling a zigzag path along the guide by successive reflections at the side boundaries. This physical approach is often helpful in understanding the propagation of continuous waves, and often essential to an understanding of the propagation of pulses. This method of interpreting modes is discussed here, and illustrated by applying it to many of the well known modes of propagation of guided waves in solids and fluids

T he study

of guided ultrasonic waves is important for two reasons. Firstly, in some systems guided waves are deliberately used to prevent loss of energy. Familiar examples are provided by the several types of ultrasonic delay line; the wire line operating with torsional waves, the fused quartz plate using shear waves, and the mercury line using compressional waves. In these, long paths are required for ~ximum delay, and to minimize the loss of energy, the signal is confined between the side boundaries of the line; the ultraso~c waves are “guided.” A second impo~nt reason for stud~ng guided ultrasonic waves is that it is often not possible to avoid a guided wave, ~thou~ avoidance would be desirable. For example, much current research in ultrasonics is concerned with the measurement of the a~enuation or the velocity in single crystals of solids, among them many metals and such materials as diamond, ruby, and quartz. Here, only small specimens are available and the side boundaries often have a considerable effect on the characteristics of propagation. Ideally one would like propagation to take place as in an unbounded medium but almost always the side boundaries have some influence and often the waves must be regarded as entirely guided; it is difficult to make a clear distinction between the two types of propagation. An understanding of some of the properties of guided waves is important, therefore, to those using ultrasonic waves. The following discussion attempts to survey the subject, using a physical rather than a mathematical approach. The rigorous way of treating these problems involves mathematical analysis; the appropriate wave equation for the fluid or solid of the waveguide must first be selected, and it is then necessary to find solutions to this * Queen MaryCollege, Universityof London

differential equation which satisfy the boundary conditions for the guide, The velocities of propagation and the characteristic pressures, stresses or particle displacements corresponding to these solutions then follow. This method (which is often tedious and sometimes very difficult) leads us to a large number of modes of propagation, some well known, others less so. In solids, for example, we find broad groups loosely called compressional (in which the Young’s Modulus mode is included), shear, torsional, and flexural. Many of these modes have been the subjects of extensive theoretical and experimental investigation. A considerable insight into the physical nature of these waveguide modes may be obtained, however, ~thout any knowledge of the detailed mathemati~l derivations, if it is realized that each mode may be thought of in terms of plane waves travelling a zigzag path along the waveguide by successive reflections at the boundaries. It is this subject that will be discussed here, since it is often useful in interpreting the observed behaviour of waves in practical ultrasonic problems. The mathematical treatment of all the systems considered in the following discussion, and many others, together with a more extensive description than is possible here, will be found in the author’s book.l FLUIDWAVEGUIDES The jluid plate

The principIes of this subject are best illustrated by application to the simplest possible waveguide. To this end the fluid waveguide will be discussed in some detail before proceeding to the solid, which is of greater practical importance. The fluid waveguide might consist of an air-filled cylindrical tube carrying acoustic or low frequency ultrasonic waves, or

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at higher ultrasonic frequencies it might be a mercury-filled tube [Fig. I(b)]. Systems of circular cross-section require cylindrical coordinates for analysis and do not afford the simplest illustration of the methods to be described; to obtain an even simpler system we consider the plate waveguide. This consists of a uniform layer of fluid sandwiched between two very large flat planes, which form the boundaries [Fig. l(u)]. It will be shown later that propagation in any one waveguide mode in such a system may be regarded as the passage of a plane wave by a zigzag path, as illustrated in Fig. I(c).

Any other boundary conditions will lie between the two extremes of the rigid boundary with zero motion and the free boundary with zero pressure. Such intermediate cases are difficult to analyse since in them, unlike the two extremes, energy is transferred to the walls and consequently is propagated along the walls as well as through the fluid. In almost all ultrasonic applications the conditions are found to be sufficiently close to one or other of the extremes for these to be used in analysis. The fluid plate with free boundaries will be examined first.

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Boundary conditions

Fluid plate withfree boundaries: characteristics of modes

Boundary conditions must first be discussed. There are two extremes to these conditions since the boundaries may be entirely free or they may be completely rigid. For example, an approximation to a rigid boundary is found in the airfilled tube, since a heavy metal tube will move negligibly when the plane waves travelling through the air inside the tube impinge on the wall. Free boundary conditions may be illustrated by a mercury-filled tube. If the mercury does not wet the (roughened) wall of the tube in which it is placed it appears that the trapped layer of air allows the boundary of the mercury to behave at ultrasonic frequencies as if it were free, that is, to move unhindered through the very small distance that it must do when struck by an ultrasonic wave. Here the pressure is very near to zero at the boundary.

The mathematical solution to this problem predicts an infinite number of possible modes of propagation.2 The first two of these are shown in Fig. 2. Here the principal characteristics of each mode are illustrated by showing the velocity as a function of frequency and also the pressure distribution, p, over the cross-section. For the first mode the distribution shows at all frequencies a sinusoidal variation between the walls; zero pressure at the walls because of the boundary conditions there, and maximum pressure at the centre ; one half-cycle of a sinusoidal variation altogether. The phase velocity v of this mode is always greater than that of compressional waves in an unbounded fluid, c, and the group velocity u is always less. Interpretation of these velocities follows in the next section. At a certain frequency f, when the wavelength in an unbounded

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medium, h (= c/f), is twice the distance D between the boundaries, the group velocity falls to zero while the phase velocity rises to infinity. At lower frequencies than this the mode is highly attenuated and is described as “evanescent ;” in theory no propagation of energy is possible at frequencies below this cut-off. Interpretation

medium by the factor l/sin 8, i.e. v = c/sin 8. The geometry of the system gives 0 in terms of h; cos 0 = h/2D for this mode. The guide wavelength h,, the distance between points of the same phase measured along the guide wall, is similarly increased ; A. = h/sin 8. It will be seen that as 0 becomes smaller h, and v increase until at cut-off both tend to infinity. Phase velocity is the velocity of a point of intersection, and involves no consideration of energy. To deal with energy we must consider the group velocity, u, which may be derived from Fig. 3. The energy may be thought of as progressing by a zigzag path. From point X to point Y it travels with velocity c, taking a time D/c cos 8, but in this time only progresses by D tan 0 along the guide. Hence the group velocity u is (D tan 0) t (D/c cos 0) = c sin 8. This is always less than the velocity in an unbounded fluid and falls to zero at cut-off when 0 = 0”.

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The interpretation in terms of zigzagging plane waves enables us to understand these features. Consider the phase velocity curve first. At higher frequencies, where the wavelength is short, the situation is that illustrated in Fig. 2(a). The plane wave follows a zigzag path because of reflections at the boundaries, and the positions of the zeros of the plane wavefronts at one instant of time are shown. These zeros are X/2 apart. The plane wave travels this zigzag path at the velocity it would have in an unbounded medium, c, and has uniform pressure over its wavefronts; the combined wavefront of superimposed incident and reflected plane waves shows, however, the sinusoidal pressure variation of the mode, with zeros at the walls and a maximum at the centre of the guide. The angle 8 must of course be such that in the combined pattern, zeros do coincide at both walls. Consider a longer wave [Fig. 2(b)]. To achieve the same pattern a change in 0 is needed, and as the wavelength increases 0 must be made smaller and smaller. Finally when h/2 = D, f3= 0 and the wave cuts off, i.e. does not progress at all [Fig. 2(c)]. To understand the significance of group and phase velocities, consider Fig. 3. The phase velocity is the rate at which the point of intersection of a wavefront with the wall progresses along that wall. It will be seen from Fig. 3(a) that while the plane wavefront moves with velocity c over a distance X/2 in the direction inclined at 0 to the normal to the wall, the intersection moves h/2 sin 0; thus the phase velocity v is greater than the velocity in an unbounded

The mode described is the first of an infinite series. The second is also illustrated in Fig. 2 and the similarity to the first mode should be noted. There are two essential differences. Cut-off occurs at a higher frequency, given by h = D rather than h = 20, and the pattern of plane wavefronts combines in such a fashion as to produce two halfcycles of a sinusoidal function rather than the single half-cycle of mode 1. This is achieved through 0 being smaller for the same h. [Compare Figs. 2(a) and 2(d)]. For this second mode, cos 8 = 2(X/2D), cut-off occurs at 2(h/2D) = 1, and again v = c/sin 0 and u = c sin 8. For the third mode of the series, cos 0 = 3(h/2D), and so on; in general for the mth mode, cos 0 = m(h/2D), where m is an integer. Note that odd integers are associated with symmetrical pressure distribution, even integers with asymmetric distributions. Cylindrical systems

The system described in the preceding discussion was that of the plate waveguide, and this is probably found less frequently in practice than the cylindrical tube. A similar approach in terms of plane waves may be made for the tube, however. A mathematical analysis of the cylinder with free boundaries leads to curves very similar to those of Fig. 2, though the sine functions of rectangular coordinates must be replaced by the Bessel functions of cylindrical coordinates.3 Two waves no longer suffice in the plane wave interpretation, an infinite number of reflecting waves being necessary because the boundary is cylindrical, but with each of these it is still possible to associate a single angle 8. Excitation

of several modes

It is now possible to make some general comments about the practical application of such a waveguide with free boundaries, in the form of either a plate or a cylinder. Suppose it is excited by some form of piezoelectric transducer. It is generally. assumed that such a transducer vibrates with uniform amplitude over its whole area, i.e., forms the ideal piston source, and in many applications this is approximately true. At low frequencies, when h > 20, all modes are evanescent, and true propagation cannot take place, the waves being severely attenuated within a relatively short distance of the source. At frequencies for which 20 > h > D

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unattenuated propagation in the first mode is possible and takes place at velocities given by the curves of Fig. 2. Whatever the actual pressure distribution at the source, that at a distance will always correspond to the sinusoidal half-cycle of Fig. 2(u) or 2(b). At frequencies for which D > X > 2013, two modes are possible, at frequencies for which 2013 > X > D/2 three modes are possible, and so on. The nature of the source determines the relative amplitude of each mode when the frequency is such that more than one mode is possible. For example a source vibrating to produce a pressure such as is illustrated in Fig. 2(a) would excite only the first mode,

Fig. 4. Dispersion

of a step function

while a pressure as in Fig. 2(d) would excite only the second mode. Any other distribution will, in general, produce a combination of modes. Fourier methods may be used to calculate the relative proportions of each mode if required.4 For example, a uniform source in a plate produces the first, third, fifth, seventh modes, etc., in relative proportions 1 .1.1 :lv etc. (Incidentally this demonstrates a general ruie: Zat a symmetrical source excites only symmetrical modes.) Mode interference is an important factor if the frequency is such that more than one mode of propagation is possible; it will be appreciated that if several modes are excited, each travels at a different phase velocity. Although at the source all modes are in phase, some modes may be in antiphase with others at other points along the guide. For example, suppose two modes are excited, in relative proportions 1 : a. Since their phase velocities differ, the maximum pressure on the axis of the guide will vary between 1 + 4 and 1 - 6, with the distance between the points of maximum and minimum depending on the actual numerical difference in velocities and on the frequency. In practice the effect is considerably complicated by the larger number of modes excited and by the nature of the detector used in observations, since this usually occupies a considerable part of the cross-sectional area and integrates to record the average pressure over its area. Basically, however, mode interference still shows itself as a variation in signal amplitude with distance from the source.5 Mode interference is important since it may lead to errors in measuring ultrasonic absorption, loss of signal due to phase interference being misinterpreted as loss due to intrinsic attenuation. In solids (when it is present at all) it shows itself in much the same way as described here, the predominant characteristic being a variation in signal amplitude with distance.

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Dispersion in pulse propagation

Mode interference is not the only source of signal distortion in a waveguide, however. So far, attention has been restricted to continuous wave propagation, but pulses being propagated in a waveguide also suffer distortion through dispersion. From Fourier analysis it is known that a pulse may be represented by a series of continuous waves covering a spectrum of frequencies. Even if the pulse is launched in a single mode it will not be propagated without change of shape since its different frequency components will travel with different group (and phase) velocities. High frequency components will travel fastest and appear at the detector first, low frequency components travel more slowly and arrive last; if they fall below the cut-off frequency they may not arrive at all. The pulse will usually be elongated, or “dispersed,” in this process. How much dispersion will occur may often be judged from a knowledge of the Fourier spectrum of the pulse; a pulse represented by a wide band of frequencies (such as a short duration unidirectional impulse) will be distorted to a much greater extent than a pulse represented by a narrow band of frequencies (such as a wave train containing, say 100 sinusoidal oscillations). In the fluid waveguide considered previously, the latter may suffer negligible distortion, while the former may change shape considerably. Fig. 4 illustrates a typical change of shape through dispersion, here the dispersion of a step function propagated in a single mode in a fluid waveguide with free boundaries.

Fig. 5. Propagation in fluid plate with rigid boundaries p = Pressure distribution v = Phase velocity * = Group velocity c = Velocity in an unbounded medium (i) Plane wave mode (ii) Second mode (iii) Third mode

Fluid plate with rigid boundaries

It remains to discuss the other extreme boundary condition, that of rigid walls. This is difficult to achieve in practical ultrasonics, though it is important in acoustic propagation in air-filled tubes. Here we have a new feature (Fig. 5). A mode of constant velocity is possible, with phase and group velocities independent of frequency and equal to c, the velocity in an unbounded fluid. There is no cut-off in this mode, and the pressure is uniform over the cross-section of the guide. A true “plane-wave” mode is thus possible in

fluid plates and cylinders with rigid boundaries. This was not possible with the free boundaries of the previous waveguide since the pressure there was of necessity zero at the walls, while here it is not the pressure that must be zero but the particle velocity normal to the boundary walls. (This can of course be zero without it being necessary for the particle velocity in the direction of propagation to be zero also.) This plane-wave mode is of considerable practical importance in acoustics, since it allows distortionless transmission of signals. In ultrasonics it appears to be more difficult to achieve; most liquids in tubes appear to travel in modes whose characteristics are those of the free boundary type. Higher modes are also possible in this type of guide; they again show an integral number of half-cycles in their characteristic pressure distributions and may again be as though composed of zigzagging plane waves. SOLIDS

Boundary conditions

Propagation in solids is much more complicated than propagation in fluids. In a fluid, only a compressional wave is possible, while in unbounded solids both compressional and transverse (shear) waves are possible. In addition, when a boundary is present, the conditions it imposes are

reflection of a plane wave at the surface of a solid (Fig. 6). The simplest problem (type I reflection) is that of a transverse wave in which the particle motion of the wave is parallel to the surface and at right angles to the direction of propagation. Here, reflection takes place at an angle equal to the angle of incidence and without loss of amplitude. In the reflection of a compressional wave-a wave in which the particle motion is in the direction of propagation -two waves are generated (type II, Fig. 6). One of these is a compressional wave travelling with velocity c, at an angle BC,one a transverse wave travelling with velocity ct at an angle 13,. The transverse wave differs, however, in relation to the boundary, from that of type I in that it is “vertically polarized” while the other was “horizontally polarized,” that is the particle motion is now in the vertical plane, as signified in Fig. 6 by the short line perpendicular to the direction of propagation. The angles 8, and Bt are related through c&in 8, = c&in Bc i.e. the phase velocities of the two waves along the boundary are equal. The relation between the amplitudes of the three waves is complex, being markedly dependent both on the angle 8, and on the ratio c,/cr, which is closely related to Poisson’s ratio.s Type III reflection concerns a vertically polarized transverse wave. Here part of the energy in the incident transverse wave is sometimes converted into a compressional wave. This is not always so, however. The angles are again related by c&in 8, = c,/sin et and it is possible for sin 8, to become unity for an angle tit of about 30” to 40”, depending on Poisson’s ratio. For Bt greater than this, the reflected compressional wave disappears, and all the energy in the incident transverse wave reappears in the reflected transverse wave (type IIIA, Fig. 6). Type I: Shear and torsional modes

II:

Fig. 6. Reflection of a plane wave at a free solid boundary I Transverse wave II Compressional wave III Vertically polarized transverse wave, 0,<90’ IIIA Vertically polarized transverse wave, B, =90”

more complicated; in a fluid with a free boundary, the pressure there must be zero, while at a free solid surface not only must the stress normal to the surface be zero but also the shear stress at the surface. It is not surprising, then, that many more modes of propagation are possible in the solid than in the fluid waveguide. Before commencing on a description of these we shall classify the possible types of

It is now possible to examine the various modes of propagation in a solid and to interpret them in terms of plane waves, by means of the approach developed in discussing the fluid plate. Type I is indubitably the simplest, and this reflection of a horizontally polarized transverse wave has much in common with propagation in a fluid plate with rigid boundaries. For propagation in a large flat solid plate we find the characteristics illustrated in Fig. 7. Three modes are shown. First, and most important, a non-dispersive plane-wave mode is possible, in which the amplitude of the shear vibration is uniform over the cross-section of the guide, and the velocity of propagation is independent of frequency. The horizontally polarized shear wave here travels parallel to the surface. Extensive use is made of this mode in fused quartz plate delay lines as no distortion of any signal occurs. In theory it requires an infinitely wide plate, but in practice it is found sufficient to use merely a wide plate. Higher modes are again possible and these are dispersive but they will not normally be excited by a source vibrating with uniform amplitude over the crosssection; such a source may be closely approximated to by a Y-cut quartz transducer. Turning from rectangular to cylindrical coordinates, we find another important ultrasonic application in the torsional delay line (Fig. 7). This again possesses a nondispersive mode and the similarity of this to its counterpart in the plate will be observed. The particle motion illustrated by arrows is again parallel to the boundary, though in this instance the amplitude is proportional to the distance from the centre, rather than being uniform over the crosssection; although non-dispersive, this is not a true planevLTkksoNIcs/April-June

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wave mode. Again, higher torsional modes are possible and their excitation will depend on the source configuration, as in previous discussions.’ Types II and III: compressional, flexural and high frequency shear modes

Type II and type III reflections may be dealt with together, for in continuous wave propagation the incident and reflected waves of each type must be the same, as is illustrated by type IIIB of Fig. 8. This is a combination of types II and III. Type IIIA is again possible as a special case. First we must distinguish between symmetrical and asymmetrical modes. Consider first the symmetrical modes. One of these, though dispersive, has no cut-off frequency; it is normally known as the “Young’s modulus mode” and it is the mode commonly observed in attempts to propagate compressional waves of low frequency in bars or plates. The other modes show cut-off frequencies. In the interpretation of these modes in terms of plane waves, it is useful to remember that the compressional waves of type IIIB reflection only exist when the phase velocity of the mode is greater than c. and that the transverse waves only exist when the phase velocity is greater than ct. Hence type IIIB reflection of plane waves is a valid interpretation of the higher modes when their phase velocity is greater than co. The corresponding particle displacement in the direction of

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Fig. 8. Compressional and flexural modes IIIA Vertically polarized transverse wave IIIB Plane waves v>c,

propagation varies with frequency (unlike the fluid), but typically shows a double oscillatory characteristic as sketched in Fig. 8, one part resulting from the pair of compressional waves, one part from the pair of transverse waves. For phase velocities between ce and ct the vertically polarized transverse waves of type IIIA dominate the behaviour. Because of the direction of their particle motion, they can of course provide a considerable compressional component in the direction of propagation, and at low frequencies this is almost uniform [Fig. S(ii)]; the Young’s modulus mode can be nearly plane, but never precisely so, and it is also dispersive. The higher modes may likewise be thought of as consisting of transverse waves in this region. An interesting problem arises because propagation in the Young’s modulus mode is possible at a phase velocity below that of transverse waves, since this phase velocity tends asymptotically to that of Rayleigh surface waves, cII, at high frequencies. The disturbance in this region becomes predominantly a surface disturbance, as is shown in Fig. 8(G). It is suggested that this might be identified with the compressional wave of type IIIA. Although the transverse wave under the conditions of type IIIA is reflected without loss of amplitude, theory shows the existence of a surface disturbance corresponding to what was the compressional wave, a disturbance the amplitude of which diminishes exponentially with distance from the surface. This theory, however,

only applies to one surface. With the presence of another surface in the immediate vicinity it appears likely that this disturbance is affected in such a way that it can carry energy and it then behaves similarly to the well known Rayleigh surface wave, whose velocity it closely follows. It must be emphasized that this interpretation is only suggested as the correct one, and for confirmation a detailed mathematical study would be necessary. One feature remains to be discussed; the behaviour of the asymmetric modes, illustrated in Fig. 8. The mode without a cut-off is the well known “flexure” mode of a plate, whose asymmetrical displacement is largest at the surface. It would again appear that interpretation of this mode must be made in terms of surface disturbances rather than reflecting plane waves, though confirmation of this awaits a mathematical study of the problem. The higher asymmetric modes are rather simpler, as they correspond to vertically polarized transverse waves (in combination with compressional waves at the higher phase velocities). These higher modes (where phase velocities tend to cl) are important at high frequencies in the propagation of ultrasonic shear waves in plates, in which the particle motion is predominantly perpendicular to the surface, and also in cylinders excited by circular Y-cut quartz transducers.s Pulse propagation in solidr

As a typical example of the value of the type of approach described here, we shall consider a problem of practical importance in the measurement of attenuation in small solid specimens. In this work, a pulse whose frequency components lie in a narrow band is generally used, a typical pulse being of a few microseconds duration and containing some 10 to 100 sinusoidal oscillations. With a narrow band of frequencies it might be thought that the problem can be dealt with in terms of continuous wave theory. With a pulse of shear waves this is often so, the upper flexure modes of Fig. 8 being excited. If a pulse of compressional waves is used, however, (generated by a X-cut quartz transducer) none of the modes previously described corresponds to the observed characteristics of the propagation. The reason for this may be seen by considering again the reflection of plane waves. With a truly continuous compressional wave, the boundary condition of type IIIB will be set up, but with a pulse these conditions may not be achieved. In an attempt to launch a compressional wave, transverse waves are immediately generated at the boundaries and these travel to the opposite boundaries, where they are partially reconverted to compressional waves (Fig. 9). With truly continuous waves, all these plane waves mingle to give the boundary conditions depicted in type IIIB, and the corresponding modes of Fig. 8. In pulse propagation, however, the time taken by the transverse waves in crossing from one boundary to the other may be greater than the duration of the applied pulse, and if this is so, the compressional signal travels with continuous loss of energy to the transverse wave; it never regains this energy from reconversion of the transverse wave at the opposite boundary, since this comes too late to rejoin the initial pulse. It is characteristic of this type of propagation that a single pulse develops into a series of pulses as shown in Fig. 9. Clearly, unless the possibility of these is recognized and allowed for, considerable errors are possible in attenuation measurements, since much of the loss of amplitude of the compressional pulse may be due to this conversion by reflection,

(9

Fig. 9. Propagation of a compressional pulse of narrow frequency spectrum in a solid waveguide (i) Original pulse (ii) and (iii) Pulses generated by reflection at side

boundaries

rather than to intrinsic absorption in the material of the guide. Recognition of the different boundary conditions permits these unusual modes to be investigated theoretically; it is found that the first compressional pulse is propagated in modes with velocities remarkably similar to those in the fluid of Fig. 2, but with a frequency dependent loss of amplitude.g Mode interference also commonly accompanies this propagation. This trouble does not occur to the same extent with pulses of transverse waves, since these are propagated without loss by conversion, in waves of either type I or type IIIA, and at high frequencies this allows pulse propagation in the modes of Fig. 7 or Fig. 8 at velocities very close to c,. Mode interference is sometimes found again here, however, particularly in cylinders, since no mode corresponds to a uniform amplitude of vibration. Finally, in discussing pulse propagation in solids, the transmission of very short pulses must be mentioned. Here the frequency spectrum is so broad that the physical methods outlined here are of little help. With a continuous wave or a narrowband pulse it is possible to associate a single angle 0 with the plane-wave interpretation. With a wide spectrum of frequencies, however, a wide band of angles is necessary and this complicates the problem to such an extent that it is desirable to use entirely different methods to reconstruct the pulse shape after its different frequency components have travelled with the different velocities given by the mode curves. In the relatively simple problem of a step travelling in the Young’s modulus mode a distortion similar to that of Fig. 4 appears, but in many other modes dispersion produces very severe distortion. A survey of the important work in this subject will be found elsewhere.lO RJZERENCES 1. REDWOOD,

(1960). 2. ibid., 3. ibid., 4. ibid., 5. ibid., 6. ibid., 7. ibid., 8. ibid., 9. ibid., 10. ibid.,

M.,

“Mechanical

waveguides,”

Pergamon,

Oxford

Chapter 3. p.70. p.77. p.80. p.26. p.151. p.152. p.190. p.208.

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