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High-intensity ultrasonic waves in solids
High-intensity ultrasonic waves in solids: nonlinear dynamics and effects
4
A. Alippi La Sapienza Universita` di Roma, Rome, Italy
4.1
Introduction
The nonlinear dynamics of solids encompasses a very wide range of effects and conditions. Taken together with the additional considerations of field diffraction, energy dissipation, temperature dependent wave propagation, and even fluid behavior, which paradoxically may also be regarded as part of the subject matter, they present an overwhelming amount of information for consideration. The author has therefore chosen to limit the treatment to general considerations in analytical form. This takes into account the typical anisotropic nature of a solid and provides evidence for the phenomenology of nonlinear wave propagation as it appears in solid media, harmonic generation, wave number modulation, frequency mixing, and finite media resonance. Each section is followed by suggestions on the appropriate application in technological developments. However, the specific details of such applications are explored in the following chapters, as it is assumed that suggestions can be drawn from the analytical results when considering models for the development of these applications.
4.2
Fundamental nonlinear equations
4.2.1
Constitutive equations and equation of motion
The differential wave equation in a medium is derived from Newton’s equation of motion. This is applied to the mass contained in an infinitesimal volume, combined with the constitutive equation that describes the deformation undergone by the volume under a system of forces applied to it. This may generally be written as F¼
dðρvÞ and f ðXi Þ ¼ 0, dt
(4.1)
where F is the net force applied to the unit volume; ρ, the medium mass density; v, the velocity; Xi, the physical quantities describing the state of the medium; t, the time; f, a specific function of the parameters Xi, which define the state of the medium. The space coordinates will be taken as xi, and correspond to the position of the particles of the Power Ultrasonics. http://dx.doi.org/10.1016/B978-1-78242-028-6.00004-1 Copyright © 2015 Elsevier Ltd. All rights reserved.
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medium when no forces are applied to it or when the medium is at rest (Lagrangian coordinates). They will be used throughout to describe the particle motion, as the density in this case is a constant function equal to ρo, even where the actual value changes under the relative motion of the different parts. By using primed coordinates xi0 for the current ones (Eulerian coordinates), the relationships with the unprimed coordinates and displacements ui are given by (Landau and Lifchitz, 1967) x0i ¼ xi + ui ,
(4.2)
whose differentials are bound through dx0i ¼ dxi + dui ,
(4.3)
which permits the introduction of the strain tensor εij through evaluation of the squared differential elements lengths dl2 and dl0 2 dl0 ¼ ðdxi + dui Þðdxi + dui Þ ¼ dl2 + 2εij dxi dxj 2
with the strain εij given by εij ¼ 12
@ui @xj
(4.4)
@u l @ul + @xij + @u @xi @xj , with the usual notation summing
up the values of any index repeated within a term. The origin of the so-called geometrical nonlinearity term (@ul/@xi)(@ul/@xj) is present as it has to be added to the linear definition of the strain tensor ½[(@ui/@xj) + (@uj/@xi)]. This derives directly from consideration of the second-order term in the differential displacement vector u. In order to write the wave equation in terms of the variable u alone, the Newton’s equation, written as F5ρo
d2 u , dt2
(4.5)
should be combined with the constitutive equation, taking into account that F derives from the gradient of the stress tensor σ, which is bound to the strain tensor ε. The free energy (or potential energy) of the medium per unit volume may be written as a series of power terms in the strain tensor components, as W¼
1 1 Cijkl εij εkl + Cijklmn εij εkl εmn + 2! 3!
(4.6)
The strain/stress relation is therefore easily derived by taking the energy density as equal to the work done by the external stress on a unit volume or alternatively, that the stress is the strain derivative of the energy density σ ij ¼
@W ¼ Cijkl εkl + Cijklmn εkl εmn + @εij
(4.7)
Any component of the wave equation may now be written in terms of the displacement vector alone; particularly
High-intensity ultrasonic waves in solids
ρo
81
2 @ 2 ui @ uk @us @ 2 us ¼ C + ijkl @t2 @xj @xl @xk @xj @xl @um @ 2 uk @us @ 2 us @us @us @ 2 uk @us @ 2 us + +, + Cijklmn + + @xn @xj @xl @xm @xj @xl @xm @xn @xj @xl @xk @xj @xl (4.8)
where no other forces are taken as acting on the mass elements. This last condition is demanding, as internal friction is generally a determining factor in the behavior of solids. However, attenuation is ignored here, as the focus is on new evidence arising from nonlinearity. Equation (4.8) may then be formally written in a shorter form, by using Thurston (1984) notation ρo
@ 2 ui @ 2 uk @um + , ¼ C + M ijkl ijklmn @t2 @xj @xl @xn
(4.9)
where Mijklmn ¼ Cijklmn + Cijkn δkm :
(4.10)
δij being the Kronecker symbol, equal to 1 for i ¼ j, and otherwise equal to 0. It should be 6 Mjiklmn, although noted that no symmetry rules hold for tensor M, such that Mijklmn ¼ they hold for tensor C, in such a way that Cijklmn ¼ Cjiklmn and Cijklmn ¼ Cklijmn ¼ Cmnklij (Cowin and Mehrabadi, 1955; Lubarda, 1997). The symmetry rules drastically reduce the independent elastic tensor components from 34 ¼ 81 to 21 for the linear second-order elastic tensor Cijkl, and from 36 ¼ 729 to 56 for the third-order one Cijklmn. As may be seen from Equation (4.9), the introduction of tensor M permits the generation of nonlinear effects to be confined to its components, while Cijkl is bound to linear propagation. As previously stated, no account has been taken of the internal viscoelastic dissipation in the propagation media, introduced by Landau and Lifchitz (1967), by considering a time-dependent force, bound to the strain time derivatives @εij/@t. Nor has any thermal flux been considered, by implicitly assuming that solid particles undergo adiabatic transformations under elastic perturbation (Rienstra and Hirschberg, 2013). This assumption may easily be accepted where the frequencies are sufficiently high and no migration of heat is assumed to have taken place during a single wave period. In cases where the acoustic propagation may be assumed to be linear, or if the acoustic strain is sufficiently low, in any spatial direction three independent plane waves may propagate, whose eigenvectors are mutually orthogonal and are called quasi-longitudinal and quasi-transversal. These fall into the longitudinal and transversal eigenvalues, respectively, where the propagation direction continuously changes from that selected to a crystallographic axis of symmetry (Morse and Ingard, 1986). No general rule can be stated in the nonlinear case, as pure modes do not exist where different acoustic waves couple one with the other throughout the propagation. The terms in Equation (4.9), which are the product between two different components of the displacement vector, are responsible for the coupling between these components, or for the growth of one component resulting from the power drained from the other.
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4.2.2
Approximate analytical solutions
No analytical solutions are available for the differential nonlinear equation (4.9), even where only those terms of the expansion that are specifically reported in the truncated expression are considered. Numerical approaches are therefore usually applied or simplifying approximations adopted. A description of the latter case is given here, and the former is left to a different chapter. A perturbation method is usually adopted, which is a good approximation in cases of weak nonlinearity where the nonlinear terms are much lower than the linear ones (Ostrovsky and Gorshkov, 2000). Equation (4.9) may therefore be considered as being limited to the terms specifically reported there, with the assumption that the amplitude of the wave does not change greatly in respect of the ratio between the linear and nonlinear coefficients of the elastic tensors: Cijkl @um @xn Cijklmn
(4.11)
then search is done for plane wave solutions of the equation. As described in the previous section, if the effects of nonlinearity could be ignored for any selected direction of propagation defined by the direction cosines θi or wave vector k (k1, k2, k3), with ki ¼ k cos θi, three independent waves may propagate (sometimes erroneously called modes). These have normalized eigenvectors aα, one quasi-longitudinal and two quasi-transversal, identified by the index α. Where only a low level of nonlinearity can be taken into account, it may be assumed that one of the modes (which should be properly identified with the α index, not reported here for sake of clarity in the subsequent notation) is propagating freely as aðr, tÞ ¼ A cos ðωt krÞ
(4.12)
with A the amplitude vector of that mode, and generating three secondary waves bβ(r, t) (β ¼ 1, 2, 3) along the direction of propagation of the wave vector k0 β, whose amplitude Bβ is a slowing and varying function of the propagation distance r: β β bβ ðr, tÞ ¼Bβ ðr Þsin ω0 t k0 r φβ (4.13) with amplitude Bβ A and phase φβ; upper indexes β identify the mode of the generated wave. The solution of Equation (4.9) should then be the sum of linear and nonlinear terms a + bβ. By introducing these into Equation (4.9) and simplifying for the matching of terms that satisfy the linear equation, the following is obtained: ρo
@ 2 bβk @ 2 bβi 1 C ¼ Mijklmn kj kl kn Ak Am sin 2ðωt krÞ + higher order terms: ijkl 2 @t @xj @xl 2 (4.14)
A trial solution can be adopted for Bβ equal to Bβ ðr, tÞ ¼ f β ðr ÞDβ
(4.15)
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with Dβ being the normalized amplitude vector and f β(r) a slowly varying function of the modulus of the displacement vector r. This is evaluated by introducing the trial solution into Equation (4.14): β 4f β ðr Þ ρo ω2 Dβi + Cijkl Dβk kj kl sin ω0 t k0 r φβ 2 @f β β xj kl + xl kj cos ω0 t k0 r φβ ¼ Cijkl Dβk r @r 2 β @ f 1 @f β xj xl 0 0β β + sin ω t k r φ @r 2 r @r r 2 1 + Mijklmn kj kl kn Ak Am sin 2ðωt krÞ + 2
(4.16)
Solutions of Equation (4.16) are only possible if ω0 ¼ 2ω and k0 β ¼ 2k, because the identity should hold for any value of t and r. The generated waves are second harmonic waves of the fundamental mode, which should have the same velocity. Dispersion would prevent harmonic generation from being a cumulative effect along the propagation direction. This implies that the fundamental α mode can only generate β modes with wave numbers equal to twice that of the fundamental, presumably the same α mode or a degenerate one. It may also be seen that terms within parentheses in the first member of Equation (4.16) are equal to zero, as they correspond to the algebraic homogeneous system for the solution eigenvector of the linear equation. Therefore, the generated waves must correspond to the modes that are proper to the propagation direction of the fundamental: Dβ ∝ aβ. The second member should also be zero. As the second term within brackets can be assumed to be much smaller than the first for values of r greater than one wavelength, and where f β is a slowly varying function of r, it should give φβ ¼ π/2, so that the equation can be derived for function f β @f β Mijklmn ffi k j k l k n Ak Am : Dβk Cijkl kl cos ϑj + kj cos ϑl @r 4
(4.17)
Equation (4.17) for i ¼ 1, 2, 3 is a system in the three unknown functions f β that may be easily solved once the elastic coefficients are known for any direction of propagation for which the normalized eigenvectors Dβ have been previously solved from the linear system. Solutions are given by f β ðr Þ ¼
kDet 0 k r ¼ Γ βijklmn Ak Am r, kDet k
(4.18)
where kDet 0 k and kDetk are proper determinants of system (4.17), whose ratio equals ΓβijklmnAkAm. The results clearly show that where dispersion effects may be ignored, a slight nonlinearity in the propagation of a harmonic wave in a solid medium generates
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second harmonic waves that grow along the propagation direction proportional to the propagation distance and to the square of the fundamental wave amplitude. This is a general effect and is well known in the case of fluids where only longitudinal modes can propagate. The difference in the case of solids relies on the possible coupling between different modes that are bound to the nonvanishing value of the correct nonlinear coefficients. It should be recalled that notation for the fundamental wave variables was previously adopted with no index α of the mode involved. By considering the specific α mode of the fundamental wave, it may be seen from Equation (4.17) that each generated β mode is coupled to the fundamental through a selected set of coefficients Γ and depends upon the amplitude components of the fundamental. However, this condition depends on the matching of wave numbers between generating and generated waves. In the case of a change between the generated and the generating modes, it may therefore be practically consistent only for coupling between degenerate transversal modes where there is no effective dispersion in the propagation medium. The general nonlinear equation (4.9) has thus far been solved through the so-called perturbative method, which considers the linear solution as a first approximation and adds to it a small term to be determined successively. The solution may then be derived by limiting various terms in the equation to those of the first order. The method may be continued by considering the new solution as a better approximation and looking for a higher-order term. The higher-order terms in the equation are taken into account. It can therefore be qualitatively stated that each time a higher harmonic wave is involved at a low level of nonlinearity, it will grow along the propagation direction bound both to higher powers of the propagation distance and of the fundamental wave amplitude.
4.2.2.1
Applications
Second harmonic generation is probably the most commonly known effect of nonlinearity in wave propagation and is exploited in a variety of fields. In the case of the nonlinear sounding of materials, as typically produced in medical echography, the second harmonic propagation permits the resolution in the imaging process to be doubled or acoustic microscopy where enhanced resolution is attained via the contemporary effect of π phase change in the focal point of a convergent beam. Biological tissues are untypical materials that may be treated as liquids or lossy isotropic solids, and are a highly nonlinear acoustic media due to their irregular structure. Harmonic growth is therefore easily produced along the propagation direction. Tissue harmonic imaging has now become a standard technique. The acoustic focalized beam from an array transducer has a narrower beam cross section in respect to the fundamental frequency beam, and an even shorter beam waist zone; thus allowing a higher resolution in both the lateral and axial directions. Higher harmonic imaging is therefore produced to enhance these characteristics. Figure 4.1 is an example of such characteristics. Although the experiment was conducted in a water tank, the acoustic field of the beam from a transducer has been visualized at its fundamental frequency, second, third, and fourth harmonic, respectively. There is a visible decrease in the size of both the axial and the lateral focal regions that corresponds to an increase of the
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Figure 4.1 Imaging of an ultrasonic beam at the first four harmonic frequencies.
imaging resolution (Bouakaz and de Jong, 2003). There are, however, some drawbacks. The bandwidth and attenuation increase correspondingly with the relative additional electronic demands and energy dissipation effects. Figure 4.2 schematically represents the intensity of the fundamental wave and the second harmonic versus the propagation distance inside of the body. While the fundamental wave constantly decreases because of attenuation, the harmonic initially increases due to the energy drain from the fundamental and then decreases when the energy dissipation is higher than the energy drain. This sets an optimum distance for the use of second harmonic imaging, which is obviously dependent on the beam frequency. This effect is analogous to that which occurs in acoustic microscopy at much higher frequencies. A particular effect has been set into evidence that is bound to the focusing phenomena in wave propagation. The wave field undergoes a π-phase change when crossing the focal point and produces an equal phase shift between the fundamental wave and the second harmonic due to the different relative wavelengths. Therefore, apart from any attenuation effect, the second harmonic increases up to the focal point because of the synchronous power draining from the fundamental wave. From that point onward, the phase shift between the two waves causes the draining to occur π out of phase from that generated prior to this point; so, depleting the harmonic power, which returns to zero. Despite the generation and detection of waves at a given
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Figure 4.2 Schematic representation of the energy distribution versus distance of propagation for the fundamental wave and its first harmonic inside human body.
frequency, the overall effect is that the sounding of the medium in the focal plane of a converging lens is performed via the second harmonic. An increased resolution then follows because of the reduced wavelength (evidence first given by Cunningham and Quate, 1972).
4.2.3
Isotropic solids and wave number modulation
The general solution for the harmonic wave in solids is greatly simplified in isotropic solids where all directions in space are equivalent from the elastic standpoint. Therefore, the propagation direction may be considered as the direction of the coordinate axis x, so obtaining the following equation of motion: @2u @2u @u ρo 2 ffi 2 CL + ðCL + CNL Þ @t @x @x
(4.19)
for any mode component u of the wave, where CL is the linear elastic coefficient C1111, and CNL the nonlinear one C111111. For the second harmonic growing function f(x), f ðx Þ ffi
M 2 2 k Ax 4CL
(4.20)
with M ¼ CNL + CL is obtained. In the simplified case of an isotropic solid, it should be noted that the second harmonic generation obtained through the perturbation method is consistent with the model of propagation of a sinusoidal wave in a medium where the velocity—and thus the wave vector—is modulated by the perturbation induced by the wave itself. A solution of Equation (4.19) in the form a(x, t) ¼ A cos(ωt kx) may be assumed where a
High-intensity ultrasonic waves in solids
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and A are the nonvanishing components of the displacement and of the displacement amplitude, respectively. The wave velocity c is then locally assumed to be given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½CL + AkðCL + CNL Þsin ðωt kxÞ c¼ ρo
(4.21)
and the wave number to be correspondingly obtained, such that the final form of the wave is ðCL + CNL Þ aðx, tÞ ffi A sin ωt ko 1 Ako sin ðωt ko xÞ x 2CL
(4.22)
with ko, the unperturbed wave number, or aðx, tÞ ffi A sin ðωt ko xÞ +
ðCL + CNL Þ 2 2 A ko x sin 2ðωt ko xÞ, 4CL
(4.23)
where use has been made of the Bessel function series for the development of the double sine function. The solution is again obtained as the sum of a fundamental wave and a second harmonic whose amplitude grows linearly with the distance x, proportional to the square of both the amplitude and wave number of the fundamental, times the elastic parameter combination (CL + CNL)/(4CL), as previously obtained through the solution of the unknown growing function in Equation (4.20). The same procedure is currently used for fluids, where the equation of motion is formally the same (see, for instance, Blackstock et al., 1998).
4.2.3.1 Applications The reduction of indices in isotropic solids in respect to anisotropic solids and the corresponding existence of pure modes in the propagation of waves in any direction of space limit the drain of power toward harmonic waves to a single component. The result is a simplification of the phase evaluation of the generated wave in respect to the fundamental. This is clearly seen from Equation (4.23) and is partially responsible for the generation of shock waves, as in the case of fluids. The modulation of the wave velocity produced by the wave itself stands as a positive feedback effect of the energy aggregation in the wave profile, which gradually increases to a sawtooth shape and finally results in a shock (see Figure 4.3). This effect is exploited in clinical lithotripsy, which is the breaking up of stones anomalously grown within human organs, particularly the kidneys, saliva glands, or bladder. A high-intensity ultrasound pulse, a few microseconds long, is focused on the region of concern, where it may attain a positive pressure peak of about 50 MPa in a time that may be less than 5 ns. This is followed by a negative pressure peak around 10 MPa. The interaction of the highly nonlinear wave thus generated with the stone is complex, but the abrupt rise in the pressure value, followed by the negative waveform, induces a sequence of stresses
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f
Frequency
Time (depth) step 1
4f f
2f
3f
Frequency
Time (depth) step 2
Distance
Amplitude
Distance
Amplitude
Amplitude
Distance
4f 5f 6f f
2f 3f
Frequency
Time (depth) step 3
Figure 4.3 Deformation of wave profile and generation of harmonic components along the propagation direction.
inside the stone that breaks it into smaller pieces. Further, these may be subjected to the induction of stresses, or may be anatomically eliminated, depending upon their size. The effect of reflection from the stone surface also has to be taken into consideration, together with the corresponding change of pressure values at the interface between the medium of the organ and the stone. Isotropic solids are usually treated in the same way as liquids with the parameter B/A standing as the nonlinear combination parameters (CL + CNL)/(4CL) for the characterization of tissues in biomedicine or nondestructive evaluation in industrial applications. Granular materials, though highly anisotropic at smaller scales, usually present a statistical isotropy at larger scales. The presence of numerous discontinuity surfaces also make them highly nonlinear from the elastic standpoint, as the elastic response to compression differs greatly from that of dilation. The generation of harmonic waves is a common effect in acoustic propagation in granular media, as is the generation of harmonics from defects in homogeneous media.
4.3 4.3.1
Nonlinear effects in progressive and stationary waves Harmonic balance in progressive waves: dispersion and attenuation
The perturbation method described in the previous section assumes the nonlinear terms in the equation of motion to be an order of magnitude smaller than the linear one. Therefore, the solution could be assumed to be the linear one slightly “perturbed” by the nonlinear contribution. This approach may be extended, as previously stated, with the third and higher harmonics deduced from those of the lower order. This also feeds back into the lower-order harmonics, perturbing the previously obtained values in a never-ending process. This is the reason for the development of discontinuities in the generation of lower-frequency waves, as may occur in the bistability phenomenon of the fundamental wave (Lyakov et al., 1995).
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89
Where multiple coupling has to be taken into account, where energy flows from fundamental to higher harmonic waves and back again to the lower ones, a more general method can be adopted. This is termed harmonic balance and considers the solution of the equation to be given by a series of harmonic waves, each with its correct amplitude and phase. uβ ðr, tÞ ¼
X
j k Bβp ðr Þ sin p ωt kβ r φβp :
(4.24)
p
When such a solution is inserted into the truncated nonlinear wave equation (4.9), X h i β 2 β β B p sin p ωt k r φ ρo ω2 kjβ klβ Cijkl ip p p h i X X ¼ 2kjβ klβ knβ Cijkl Mijklmn p q Bβkðp + qÞ Bβmq ðp + qÞ2 q2 sin p ωt kβ r φβp + (4.25) is obtained. This is only satisfied for each time t and position r if every harmonic term with index p identically satisfies the equation
Gp+
2 2 + G ¼0 p
(4.26)
with β 2 G p ¼ Bip p
ρo ω2 kjβ kjβ Cijkl
1 + kjβ klβ knβ Mijklmn 2
X q
* + cos φβ p sin φβp
+ cos φβpq + φβq BβkðpqÞ Bβmq ðp qÞ2 q sin φβpq + φβq *
+! cos φβpq φβq + BβkðpqÞ Bβmq ðp + qÞ2 q ¼0 sin φβpq φβq *
(4.27)
the double sign in the G p terms corresponds to the upper and lower expressions reported within the symbolic brackets and separated by the symbolic fraction sign h—i. This is an infinite set of nonlinear algebraic equations in the unknown coefficients Bβip, defining the relative amplitudes of each harmonic wave and must be solved through numerical methods, by assuming that no strong nonlinearity is present in the phenomenon. In this case, a limited number of harmonic waves may be retained and summation of the trial function (4.24) limited to a finite value N. A set of correspondingly limited equations must be solved through the appropriate algorithms.
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Obviously, the balancing of harmonics can only be performed if all the wave numbers are multiples of the fundamental and are proportional to the corresponding circular frequency; that is, if the propagation of the waves is not affected by dispersion. A small degree of dispersion would prevent the generation of harmonics increasing to their steady value, because the local generation of any harmonic wave from its coupling with others would take place out of phase with what previously had been generated.
4.3.2
Frequency mixing
The consideration of phase matching leads us to the problem of the frequency mixing of waves. This occurs when two waves of different frequency give rise to a third, either with a sum frequency value or a different frequency. As in the case of harmonic generation (see Equation (4.12)), it may be assumed that two different frequency waves are propagating as a1 ðr, tÞ ¼ A1 cos ðω1 t k1 rÞ and a2 ðr, tÞ ¼ A2 cos ðω2 t k2 rÞ
(4.28)
and that they generate waves 0 bβ ðr, tÞ ¼Bβ ðr Þsin ω0 t kβ r φβ
(4.29)
with an obvious notation of the indexes and amplitudes Bβ A1,2. The conditions of phase matching and energy conservation require that k0 ¼ k1 k2 and ω0 ¼ ω1 ω2, where the upper sign corresponds to up-frequency and the lower one to downfrequency conversion. In the case of collinear propagation (k1 parallel to k2), that reduces to k0 ¼ k1 k2 and ω0 ¼ ω1 ω2. When replicating what has been developed for harmonic generation, with consideration of slight nonlinear effects, the generation of cross-product terms between the two interacting waves would be found (in addition to the case of second harmonic generation), whose growing functions f β have to satisfy the equation @fβ Dβk Cijkl ðk1l k2l Þcos ϑj + k1j k2j cos ϑl @r Mijklmn ffi k1n k2j k2l A1m A2k + k1j k1l k2n A1k A2m , 4
(4.30)
where care has been taken in introducing a double sign for the function f, depending on whether it refers to the up converted wave (k3 ¼ k1 + k2) or to the down converted one (k3 ¼ k1 k2). As seen in the previous section, Equation (4.30) greatly simplifies in the case of isotropic solids where a single coordinate x can be taken in place of the general displacement vector r. The solution would then reduce to
High-intensity ultrasonic waves in solids
fβ ffi
91
CL + CNL k1 k2 A1 A2 x: 4CL
(4.31)
Conservation of the linear momentum is satisfied in the dispersionless medium, which has been assumed here, as both energy (ℏω) and momentum (ℏω/c) are linearly related in the particle description. A small degree of dispersion could easily be taken into consideration and would add an oscillating factor sin(Δkx) to the solution, with Δk ¼ k3 (k1 k2), responsible for the flowing back and forth of energy between the generating and the generated waves. (For the effect in surface acoustic waves (SAWs), see Adamou et al., (1979)). However, the phase matching between harmonics may be seen from a different point of view when dealing with waves propagating into different directions. This is specific to solid-state acoustics, where anisotropy and multiple modes may produce phase matching conditions for waves propagating along different directions. In this case, phase matching conditions can be achieved with different mode propagating waves, provided that two equations are contemporaneously satisfied (Figure 4.4): 0
0
kγ ¼ kα1 kβ2 , ωγ ¼ ωα1 ωβ2 :
(4.32)
4.3.2.1 Applications Frequency mixing is exploited in signal processing techniques and in underwater or soil exploration. The need for processing high-frequency signals was greater in the Quasi-shear
Pure shear
Quasi-longitudinal
Figure 4.4 Slowness curves for the three acoustic modes propagating on the XY crystallographic plane of GaAs (cubic system) and wave vectors satisfying momentum (k1 + k2 ¼ k3) and frequency (ω1 + ω2 ¼ ω3, with ω1 ¼ ω2 ¼ ω3/2) matching conditions.
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past when electronics largely relied on analogue hardware and SAW devices were in common usage. This was particularly the case where convolution/correlation, integrals, Fourier transformations, and so on, were necessary for real time signal filtering processes, as in communicating through highly noisy environments or in image identification. The use of SAWs on solid or crystal substrates, typically quartz or LiNbO3, greatly improved the potential of devices because of the planar geometry of waves that limited undesired diffraction effects. The product between two signals was directly obtained through the nonlinear interaction of two waves propagating contra-directed, while integration was obtained either through acousto-optical interaction with light flashing on the interacting volume, or through appropriate electrode distribution on the propagation surface. Frequency mixing is also used for generating low-frequency waves for underwater or soil exploration. This utilizes transducers of limited size by means of down-conversion effect from the beating of two high-frequency waves of close frequency. In this way, as pointed out in the pioneering work of Westervelt (1963), the directivity of the transducer is bound to the short wavelengths of the high-frequency emission, while the generation of the long wavelength taking place in the cylindrical volume of interaction limits the heavy diffraction effect of low-frequency waves. The low-frequency wave can then propagate with low attenuation to reach deeply submerged objects, although with partial loss of definition characteristics.
4.3.3
Stationary waves: nonlinear sources
The excitation of elastic perturbations in a finite structure takes place according to a set of specific modes; that is, to a stationary distribution of strains varying sinusoidally in time with a spectrum of well-defined frequencies. The specific mode that is set into resonance within the structure depends on the overlapping integral between the spatial distribution of strains and the spatial distribution of stresses in the exciting force, in addition to the matching of the frequency values between the two distributions. A dissipative term causes the input energy acquired from the driving source to balance the loss of energy from the structure and to smooth out the response curve of the system at the resonance values of the frequency. The study of the spectral modes of a finite structure is facilitated by the assumption that there is no dissipation. This is achieved by considering a trial solution of the linear equation (4.9) with Mijklmn ¼ 0, and successively solving the space differential equation with the correct boundary conditions. As usual, the angular frequencies ωn will stand as the eigenvalues of the equation of motion. The insertion of nonlinear terms in the equation describing the elastic behavior of a finite structure subject to external driving forces will add the effect of coupling among different modes; thus exciting new modes in the structure that could not be directly excited by the driving force. In order to study the conditions for the onset of this effect, the equation must be written in its general form:
High-intensity ultrasonic waves in solids
ρo
@ 2 ui @um @ @ 2 uk ¼ F ð r Þcos ð Ωt Þ + C + M + η + i ijkl ijklmn ijkl @t2 @xn @t @xj @xl
93
(4.33)
with f(r, t) ¼ F(r)cos(Ωt), the driving force per unit volume and η the shear viscosity coefficient (Landau and Lifchitz, 1967, p. 191). Transient solutions will eventually fail, while a stationary trial solution uðr, tÞ ¼ UðrÞcos ðΩt + ψ Þ
(4.34)
can be inserted into Equation (4.33) to give Ω ρo Ui cos ðΩt + ψ Þ 2
ηijkl Ω sin ðΩt + ψ Þ
@Um cos ðΩt + ψ Þ Cijkl + Mijklmn @xn
@ 2 Uk ¼ Fi ðrÞcos Ωt: @xj @xl
(4.35)
The amplitude of displacement U(r) can be obtained by first separating, then squaring and adding, the sin Ωt and cos Ωt parts of the equation so as to be valid at any time t, giving
2 2 @Um @ 2 Uk @ 2 Uk Ω2 ρo Ui + Cijkl + Mijklmn + ηijkl Ω ¼ F2i ðrÞ: @xn @xj @xl @xj @xl
(4.36)
The phase can also be directly obtained through the separation of the sin Ωt from the cos Ωt parts and solving the sin Ωt part, which does not contain the forcing term F: @ 2 Uk @xj @xl : tan ψ ¼ @Um @ 2 Uk Ω2 ρo Ui + Cijkl + Mijklmn @xn @xj @xl ηijkl Ω
(4.37)
Where the viscosity term may be correctly ignored (η ¼ 0), the solution is greatly simplified, giving
@Um @ 2 Uk ¼ Fi , Ω ρo Ui + Cijkl + Mijklmn @xn @xj @xl ψ ¼ 0: 2
(4.38)
However, an external force with no dissipation would drive the system to an amplitude oscillation with an infinite value, so that the dissipationless case is better studied in the absence of external forces. If nonlinearity terms could also be ignored, the solution would further reduce to
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ω2n ρo Ui + Cijkl ψ ¼ 0:
@ 2 Uk ffi 0, @xj @xl
(4.39)
This case defines the eigenfrequencies ωn of the structure and the corresponding ˆ n(r). Prior to consideration of the nonlinearity terms, it should be remembered modes U that the viscosity would smooth out the amplitude of any excited mode as uðr, tÞ ¼ UðrÞcos ðΩt + ψ ÞexpðαtÞ
(4.40)
with the α term depending on frequency. A driving force would then simply compensate for the loss of energy caused by the internal dissipation of the structure. The effects of a small degree of nonlinearity are better understood by considering a stationary condition in which one of the proper modes at ωN remains at a constant amplitude UN, depending on the excitation value of that single frequency. A small perturbation is produced that corresponds to the energy flowing out from that mode, so that a trial solution may be given as uðr, tÞ ¼ UN ðrÞcos ωN t + εðr, tÞ
(4.41)
with ε UN. By inserting the solution into the nonlinear equation, taking into account that UN solves the linear equation and ignoring the higher-order terms, the following is obtained: 2 @ εi @ 2 εi @ 2 UNk @UNm 1 + cos ð2ωN tÞ C , ffi M ρo ijkl ijklmn @t2 @xj @xl @xj @xl @xn 2
(4.42)
which states that the perturbation term is drawing energy both from the dc term and from the time-second harmonic of the mode, which are excited. If the perturbation term ε is developed into Fourier series, according to εðr, tÞ ¼
X
Es ðrÞcos ðsωN t + φs Þ
(4.43)
s
and the harmonic balance is performed for the zero (s ¼ 0) and second harmonic (s ¼ 2) frequency, respectively, the following is obtained: Cijkl
@ 2 Eoi Mijklmn @ 2 UNk @UNm ¼ , @xj @xl 2 @xj @xl @xn
ρo ω2N E2i
Mijklmn @ 2 UNk @UNm @ 2 E2i ffi + Cijkl , @xj @xl 2 @xj @xl @xn
(4.44a)
(4.44b)
which shows how nonlinearity couples the principal mode to the others. Phase φo has been set equal to zero, with no loss of generality, because it simply modifies the value
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of the amplitude through the factor cos φo. For reasons of simplicity, if the excited term is set as equal to the first-order mode (ωN ¼ ω1), the nonlinear coupling of the modes may draw energy from the fundamental mode back to the dc term as well as to the second harmonic; thus producing a feedback process on the driving mode that makes for bistability. Equations (4.44a) and (4.44b) may be easily solved where monodimensional plane waves are set in an isotropic medium. By using the same notations already introduced in the case of progressive waves they reduce to C
@ 2 Eo M @ 2 U1 @U1 , ¼ @x2 2 @x2 @x
ρo ω22 E2
@ 2 E2 M @ 2 U1 @U1 : +C 2 ¼ @x 2 @x2 @x
(4.45a)
(4.45b)
Solutions of Equations (4.45a) and (4.45b) which satisfy the correct boundary conditions set for the structure must then be found. The possible growth of subharmonic waves in a finite structure from nonlinear interaction with the driven mode is of considerable interest. According to conservation laws, the rise of lower frequencies is only possible from the breaking down of one quantum state into two of lower frequency or, in terms of wave propagation, from the down-conversion of one higher frequency wave into two lower ones. This is a typical threshold phenomenon that may only take place if the energy of the generating wave is sufficiently high to overcome a specific, or threshold, level. Where that is the case, any perturbation, however small, occurring at the correct frequency in the structure will be amplified by the draining of energy from the fundamental and will grow up to a defined level. Both the threshold value of the fundamental mode and the regime level of the subharmonic depend upon the value of the dissipation term. In the limiting case that the system is not dissipative, the threshold level decreases to zero as any quantity of energy introduced into the subharmonic frequency wave will accumulate without loss and may grow indefinitely in a manner similar to the indefinite growth of the fundamental in a dissipationless system. The threshold effect may be evaluated in a manner parallel to that used for the growth of the harmonic waves. This is achieved by considering the general nonlinear equation (4.33) and supposing its solution to be given by a function V(r, t), which is comprehensive of both the solution of the corresponding linear equation (M 0) and all the higher frequency terms (harmonics). It may be supposed that an infinitesimal perturbation ε(r, t) at a frequency lower than that of the driving one—taken for example as Ω/2—is accidentally present in the system and added to the preexisting solution gives W ðr, tÞ ¼ V ðr, tÞ + εðr, tÞ:
(4.46)
Such a perturbation is bound to be extinguished, because of the dissipation term η, unless the nonlinear coupling between the waves drains sufficient energy from the
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fundamental to overcome the dissipation loss. In order to verify this condition, solution (4.46) may be placed into Equation (4.33) to give @ 2 ð V i + εi Þ @ ð Vm + ε m Þ @ @ 2 ð Vk + ε k Þ ¼Fi ðrÞcos ðΩtÞ+ Cijkl +Mijklmn +ηijkl + ρo @t2 @xn @t @xj @xl (4.47) and subsequently, ρo
@ 2 εi @ @ 2 εk @εm @ 2 Vk @Vm @ 2 εk + ¼ C + η + M + ijkl ijklmn ijkl @t2 @xn @xj @xl @xn @xj @xl @t @xj @xl (4.48)
by retaining only the highest order terms and considering that V solves the nonlinear equation. As in the perturbative cases previously examined, the equation thus derived is linear for the perturbation term ε, with a driving force given by the cross products of first and second derivatives of ε and V. These are periodical terms at frequencies Ω and Ω/2, producing driving terms at the sum and difference frequencies, 3Ω/2 and Ω/2. That means that a perturbation accidentally present in the system may produce a driving term which generates an oscillation at the same frequency with the same configuration as the initial one present in the system. If the generated wave amplitude is greater than the original, the perturbation may continuously increase its amplitude and attain a finite regime value much higher than the initial one. However, this condition is not easy to derive because the cross product generating the new oscillation is proportional to V and V is proportional to F and it could be qualitatively said that the threshold value of the driving amplitude Fth for the onset of the subharmonic wave is inversely proportional to the nonlinear parameter M. Once a subharmonic oscillation at Ω/2 is set up in a system where the driving term is greater than the threshold value, a successive subharmonic at Ω/4 could be generated via the same process, so that a new condition for a second threshold is satisfied and thus for a subharmonic at Ω/8 and Ω/16, and so on. When a subharmonic at Ω/4 has been produced, nonlinearity can generate an oscillation at 3Ω/4, through the beating of the Ω/4 frequency wave with that already generated at Ω/2 and, analogously, oscillations at 3Ω/8 by the beating of Ω/8 with Ω/4. Successive values of the threshold amplitudes should be evaluated, but their values rapidly converge to a value Fch, where all possible subharmonic oscillations may increase suddenly and drive the system into a chaotic behavior.
4.3.3.1
Applications
Musical instruments provide the best examples of finite systems where elastic perturbations undergo nonlinear interactions. They are also the earliest known objects expressly made for producing sound, yet the approach to their study is usually presented in terms of linear theory, though musical “forte” conditions drive almost
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any instrument deep into nonlinear conditions. Schematically, nonlinearity may derive either from the nonlinear coupling of the driving device (the bow hair in the case of violins, the reed in case of clarinets, etc.) or from the resonant system emitting the sound (strings, tube, plates, and so on) (see Fletcher et al., 1990). The focus here is on the second case, which has been theoretically described in the previous section. It is sufficient to note in regard to the first case, that it is mainly due to the onset of a positive resistance in the driving mechanism. At low amplitudes of oscillation in the linear regime, monodimensional structures such as strings or tubes present a spectral response of resonance frequencies, all of which are integral multiples of the lowest or fundamental frequency. At higher levels, the increased average time conditions of the stress usually increase the resonance frequencies, and the harmonicity of the spectral response may be lost. The nonlinearity mechanism preserves the integral relation among the different mode frequencies through the coupling of different modes. In the case of string instruments, for example, the coupling between the oscillations on the two possible planes makes the single point of the string move in a “rosetta”-type motion. The single component changes its oscillation period, but the periodical motion from the inner and outer parts of the rosetta does not. The conditions in multidimensional instruments such as cymbals and gongs are quite different, as mode locking is prevented and, following percussion, energy flows from lower to higher harmonics, producing a variable pitch or “multiphonic” sound. The results outlined above replicate at different bandwidths in the case of ultrasonic transducers, where nonlinearity drives the systems into motions that are rich in harmonic and subharmonic modes. These are usually difficult to analyze in detail. As the amplitude of the driving mode increases, a series of multiple harmonic modes appear at increasing intensities up to a point where the coupling between them cannot be easily followed. The appearance of subharmonic oscillations, mainly at half the fundamental frequency, is of great interest. When the corresponding threshold conditions are attained, successive period doubling will be present with the corresponding possible frequency coupling (three-quarters, three-eighths, and five-eighths of fundamental frequency, etc.) (see Figure 4.5). As the chaotic state is attained, the frequency spectrum enriches to a level where it returns to an almost linear condition and continues for a short interval of amplitude before returning to chaos. This behavior is common to chaotic systems and could be successfully used in case of high-power transducers where the chaotic conditions may damage transducers and affect the results of devices based on them (Cardoni et al., 2013).
4.4
Conclusions
The treatment of acoustic nonlinearities in solids presents a complex problem. The general anisotropy of materials combined with the various modes of propagation renders the formal approach a jungle of indices that must be carefully unraveled if the correct solution for individual cases is to be found. The general problem has been formulated here and various phenomena analyzed. These have been successively reduced
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−10
w
Amplitude (dB)
−20
~9 dB
w/2
w/8 ~15 dB
−30
w/2 + w/4 + w/8
w/4 −40
w/4 + w/8
w/2 + w/4 w/2 + w/8
−50 −60 −70 0
100
200
300 400 Frequency (kHz)
500
600
700
Figure 4.5 Spectrum of excited modes in a piezoelectric element, driven at fundamental frequency f ¼ 602.4 kHz, with evidence of subharmonic fractional modes (Bettucci et al., 2008).
to simplified forms (mainly to monodimensional cases for isotropic materials) to make clear the origin of the effects involved, while leaving the general form to further developments. The perturbation method has therefore been fully described where nonlinearity is considered as a recursive effect that slightly modifies the linear problem. This has been described in the case of progressive waves and stationary conditions. In the former, agreement with the model of self-modulation in the wave vector by the propagating wave itself has been described. Coupling between different modes and frequency mixing have also been discussed, together with the identification of the overall coupling coefficients and the feedback effect on the zero-order mode.
References Adamou, A., Alippi, A., Palma, A., Palmieri, L., Socino, G., 1979. Parametric mixing of surface acoustic waves. J. Appl. Phys. 50 (6), 4120–4127. Bettucci, A., Biagioni, A., D’Orazio,, A., Passeri, D., 2008. Periodic oscillations within the chaotic region in finite piezoelectric structures. In: Proceedings of the 18th International Symposium on Nonlinear Acoustics, Nonlinear Acoustics: Fundamentals and Applications, Stockholm, pp. 251–254. Blackstock, D.T., Hamilton, M.F., Pierce, A.D., 1998. Progressive waves in lossless and lossy fluids. In: Hamilton, M.F., Blackstock, D.T. (Eds.), Nonlinear Acoustics. Academic Press, San Diego. Bouakaz, A., de Jong, N., 2003. Native tissue imaging at superharmonic frequencies. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50 (5), 496–506.
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Cardoni, A., Riera, E., Gallego-Juarez, J.A., 2013. Nonlinear response in airborne piezoelectric transducer for power ultrasonics. In: 2013 International Congress on Ultrasonics (ICU 2013), Singapore. Cowin, S.C., Mehrabadi, M.M., 1955. Anisotropy symmetries of linear elasticity. Appl. Mech. Rev. 48, 247–285. Cunningham, J.A., Quate, C.F., 1972. High-resolution, high-contrast acoustic imaging. J. Phys. Colloq. 33 (C6), 42–47. Fletcher, N.H., Perrin, R., Legge, K.A., 1990. Nonlinearity and chaos in acoustics. Acoust. Aust. 18 (1), 9–13. Landau, L.D., Lifchitz, E.M., 1967. Theorie de l’e´lasticite´. MIR, Moscow. Lubarda, V.A., 1997. New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals. J. Mech. Phys. Solids 45 (4), 471–490. Lyakov, A., Proskuryakov, A.K., Shipilov, K.F., Umnova, O.V., 1995. Bistability and chaos in acoustic resonators. Ultrasonics 33 (1), 55–59. Morse, Ph.M., Ingard, K.U., 1986. Theoretical Acoustics. Princeton University Press, Princeton, NJ. Ostrovsky, L., Gorshkov, K., 2000. Perturbation theories for nonlinear waves. In: Christiansen, P.L., Sørensen, M.P., Scott, A.C. (Eds.), Nonlinear Science at the Dawn of the 21st Century. Springer, Berlin. Rienstra, S.W., Hirschberg, A., 2013. An Introduction to Acoustics. Eindhoven University Press, Eindhoven. Thurston, R.N., 1984. Waves in solids. In: Truesdell, C. (Ed.), Mechanics of Solids, vol. 4. Springer, Berlin, pp. 109–308. Westervelt, J., 1963. Parametric acoustic array. J. Acoust. Soc. Am. 35, 535–537.
4
A. Alippi La Sapienza Universita` di Roma, Rome, Italy
4.1
Introduction
The nonlinear dynamics of solids encompasses a very wide range of effects and conditions. Taken together with the additional considerations of field diffraction, energy dissipation, temperature dependent wave propagation, and even fluid behavior, which paradoxically may also be regarded as part of the subject matter, they present an overwhelming amount of information for consideration. The author has therefore chosen to limit the treatment to general considerations in analytical form. This takes into account the typical anisotropic nature of a solid and provides evidence for the phenomenology of nonlinear wave propagation as it appears in solid media, harmonic generation, wave number modulation, frequency mixing, and finite media resonance. Each section is followed by suggestions on the appropriate application in technological developments. However, the specific details of such applications are explored in the following chapters, as it is assumed that suggestions can be drawn from the analytical results when considering models for the development of these applications.
4.2
Fundamental nonlinear equations
4.2.1
Constitutive equations and equation of motion
The differential wave equation in a medium is derived from Newton’s equation of motion. This is applied to the mass contained in an infinitesimal volume, combined with the constitutive equation that describes the deformation undergone by the volume under a system of forces applied to it. This may generally be written as F¼
dðρvÞ and f ðXi Þ ¼ 0, dt
(4.1)
where F is the net force applied to the unit volume; ρ, the medium mass density; v, the velocity; Xi, the physical quantities describing the state of the medium; t, the time; f, a specific function of the parameters Xi, which define the state of the medium. The space coordinates will be taken as xi, and correspond to the position of the particles of the Power Ultrasonics. http://dx.doi.org/10.1016/B978-1-78242-028-6.00004-1 Copyright © 2015 Elsevier Ltd. All rights reserved.
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medium when no forces are applied to it or when the medium is at rest (Lagrangian coordinates). They will be used throughout to describe the particle motion, as the density in this case is a constant function equal to ρo, even where the actual value changes under the relative motion of the different parts. By using primed coordinates xi0 for the current ones (Eulerian coordinates), the relationships with the unprimed coordinates and displacements ui are given by (Landau and Lifchitz, 1967) x0i ¼ xi + ui ,
(4.2)
whose differentials are bound through dx0i ¼ dxi + dui ,
(4.3)
which permits the introduction of the strain tensor εij through evaluation of the squared differential elements lengths dl2 and dl0 2 dl0 ¼ ðdxi + dui Þðdxi + dui Þ ¼ dl2 + 2εij dxi dxj 2
with the strain εij given by εij ¼ 12
@ui @xj
(4.4)
@u l @ul + @xij + @u @xi @xj , with the usual notation summing
up the values of any index repeated within a term. The origin of the so-called geometrical nonlinearity term (@ul/@xi)(@ul/@xj) is present as it has to be added to the linear definition of the strain tensor ½[(@ui/@xj) + (@uj/@xi)]. This derives directly from consideration of the second-order term in the differential displacement vector u. In order to write the wave equation in terms of the variable u alone, the Newton’s equation, written as F5ρo
d2 u , dt2
(4.5)
should be combined with the constitutive equation, taking into account that F derives from the gradient of the stress tensor σ, which is bound to the strain tensor ε. The free energy (or potential energy) of the medium per unit volume may be written as a series of power terms in the strain tensor components, as W¼
1 1 Cijkl εij εkl + Cijklmn εij εkl εmn + 2! 3!
(4.6)
The strain/stress relation is therefore easily derived by taking the energy density as equal to the work done by the external stress on a unit volume or alternatively, that the stress is the strain derivative of the energy density σ ij ¼
@W ¼ Cijkl εkl + Cijklmn εkl εmn + @εij
(4.7)
Any component of the wave equation may now be written in terms of the displacement vector alone; particularly
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ρo
81
2 @ 2 ui @ uk @us @ 2 us ¼ C + ijkl @t2 @xj @xl @xk @xj @xl @um @ 2 uk @us @ 2 us @us @us @ 2 uk @us @ 2 us + +, + Cijklmn + + @xn @xj @xl @xm @xj @xl @xm @xn @xj @xl @xk @xj @xl (4.8)
where no other forces are taken as acting on the mass elements. This last condition is demanding, as internal friction is generally a determining factor in the behavior of solids. However, attenuation is ignored here, as the focus is on new evidence arising from nonlinearity. Equation (4.8) may then be formally written in a shorter form, by using Thurston (1984) notation ρo
@ 2 ui @ 2 uk @um + , ¼ C + M ijkl ijklmn @t2 @xj @xl @xn
(4.9)
where Mijklmn ¼ Cijklmn + Cijkn δkm :
(4.10)
δij being the Kronecker symbol, equal to 1 for i ¼ j, and otherwise equal to 0. It should be 6 Mjiklmn, although noted that no symmetry rules hold for tensor M, such that Mijklmn ¼ they hold for tensor C, in such a way that Cijklmn ¼ Cjiklmn and Cijklmn ¼ Cklijmn ¼ Cmnklij (Cowin and Mehrabadi, 1955; Lubarda, 1997). The symmetry rules drastically reduce the independent elastic tensor components from 34 ¼ 81 to 21 for the linear second-order elastic tensor Cijkl, and from 36 ¼ 729 to 56 for the third-order one Cijklmn. As may be seen from Equation (4.9), the introduction of tensor M permits the generation of nonlinear effects to be confined to its components, while Cijkl is bound to linear propagation. As previously stated, no account has been taken of the internal viscoelastic dissipation in the propagation media, introduced by Landau and Lifchitz (1967), by considering a time-dependent force, bound to the strain time derivatives @εij/@t. Nor has any thermal flux been considered, by implicitly assuming that solid particles undergo adiabatic transformations under elastic perturbation (Rienstra and Hirschberg, 2013). This assumption may easily be accepted where the frequencies are sufficiently high and no migration of heat is assumed to have taken place during a single wave period. In cases where the acoustic propagation may be assumed to be linear, or if the acoustic strain is sufficiently low, in any spatial direction three independent plane waves may propagate, whose eigenvectors are mutually orthogonal and are called quasi-longitudinal and quasi-transversal. These fall into the longitudinal and transversal eigenvalues, respectively, where the propagation direction continuously changes from that selected to a crystallographic axis of symmetry (Morse and Ingard, 1986). No general rule can be stated in the nonlinear case, as pure modes do not exist where different acoustic waves couple one with the other throughout the propagation. The terms in Equation (4.9), which are the product between two different components of the displacement vector, are responsible for the coupling between these components, or for the growth of one component resulting from the power drained from the other.
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4.2.2
Approximate analytical solutions
No analytical solutions are available for the differential nonlinear equation (4.9), even where only those terms of the expansion that are specifically reported in the truncated expression are considered. Numerical approaches are therefore usually applied or simplifying approximations adopted. A description of the latter case is given here, and the former is left to a different chapter. A perturbation method is usually adopted, which is a good approximation in cases of weak nonlinearity where the nonlinear terms are much lower than the linear ones (Ostrovsky and Gorshkov, 2000). Equation (4.9) may therefore be considered as being limited to the terms specifically reported there, with the assumption that the amplitude of the wave does not change greatly in respect of the ratio between the linear and nonlinear coefficients of the elastic tensors: Cijkl @um @xn Cijklmn
(4.11)
then search is done for plane wave solutions of the equation. As described in the previous section, if the effects of nonlinearity could be ignored for any selected direction of propagation defined by the direction cosines θi or wave vector k (k1, k2, k3), with ki ¼ k cos θi, three independent waves may propagate (sometimes erroneously called modes). These have normalized eigenvectors aα, one quasi-longitudinal and two quasi-transversal, identified by the index α. Where only a low level of nonlinearity can be taken into account, it may be assumed that one of the modes (which should be properly identified with the α index, not reported here for sake of clarity in the subsequent notation) is propagating freely as aðr, tÞ ¼ A cos ðωt krÞ
(4.12)
with A the amplitude vector of that mode, and generating three secondary waves bβ(r, t) (β ¼ 1, 2, 3) along the direction of propagation of the wave vector k0 β, whose amplitude Bβ is a slowing and varying function of the propagation distance r: β β bβ ðr, tÞ ¼Bβ ðr Þsin ω0 t k0 r φβ (4.13) with amplitude Bβ A and phase φβ; upper indexes β identify the mode of the generated wave. The solution of Equation (4.9) should then be the sum of linear and nonlinear terms a + bβ. By introducing these into Equation (4.9) and simplifying for the matching of terms that satisfy the linear equation, the following is obtained: ρo
@ 2 bβk @ 2 bβi 1 C ¼ Mijklmn kj kl kn Ak Am sin 2ðωt krÞ + higher order terms: ijkl 2 @t @xj @xl 2 (4.14)
A trial solution can be adopted for Bβ equal to Bβ ðr, tÞ ¼ f β ðr ÞDβ
(4.15)
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with Dβ being the normalized amplitude vector and f β(r) a slowly varying function of the modulus of the displacement vector r. This is evaluated by introducing the trial solution into Equation (4.14): β 4f β ðr Þ ρo ω2 Dβi + Cijkl Dβk kj kl sin ω0 t k0 r φβ 2 @f β β xj kl + xl kj cos ω0 t k0 r φβ ¼ Cijkl Dβk r @r 2 β @ f 1 @f β xj xl 0 0β β + sin ω t k r φ @r 2 r @r r 2 1 + Mijklmn kj kl kn Ak Am sin 2ðωt krÞ + 2
(4.16)
Solutions of Equation (4.16) are only possible if ω0 ¼ 2ω and k0 β ¼ 2k, because the identity should hold for any value of t and r. The generated waves are second harmonic waves of the fundamental mode, which should have the same velocity. Dispersion would prevent harmonic generation from being a cumulative effect along the propagation direction. This implies that the fundamental α mode can only generate β modes with wave numbers equal to twice that of the fundamental, presumably the same α mode or a degenerate one. It may also be seen that terms within parentheses in the first member of Equation (4.16) are equal to zero, as they correspond to the algebraic homogeneous system for the solution eigenvector of the linear equation. Therefore, the generated waves must correspond to the modes that are proper to the propagation direction of the fundamental: Dβ ∝ aβ. The second member should also be zero. As the second term within brackets can be assumed to be much smaller than the first for values of r greater than one wavelength, and where f β is a slowly varying function of r, it should give φβ ¼ π/2, so that the equation can be derived for function f β @f β Mijklmn ffi k j k l k n Ak Am : Dβk Cijkl kl cos ϑj + kj cos ϑl @r 4
(4.17)
Equation (4.17) for i ¼ 1, 2, 3 is a system in the three unknown functions f β that may be easily solved once the elastic coefficients are known for any direction of propagation for which the normalized eigenvectors Dβ have been previously solved from the linear system. Solutions are given by f β ðr Þ ¼
kDet 0 k r ¼ Γ βijklmn Ak Am r, kDet k
(4.18)
where kDet 0 k and kDetk are proper determinants of system (4.17), whose ratio equals ΓβijklmnAkAm. The results clearly show that where dispersion effects may be ignored, a slight nonlinearity in the propagation of a harmonic wave in a solid medium generates
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second harmonic waves that grow along the propagation direction proportional to the propagation distance and to the square of the fundamental wave amplitude. This is a general effect and is well known in the case of fluids where only longitudinal modes can propagate. The difference in the case of solids relies on the possible coupling between different modes that are bound to the nonvanishing value of the correct nonlinear coefficients. It should be recalled that notation for the fundamental wave variables was previously adopted with no index α of the mode involved. By considering the specific α mode of the fundamental wave, it may be seen from Equation (4.17) that each generated β mode is coupled to the fundamental through a selected set of coefficients Γ and depends upon the amplitude components of the fundamental. However, this condition depends on the matching of wave numbers between generating and generated waves. In the case of a change between the generated and the generating modes, it may therefore be practically consistent only for coupling between degenerate transversal modes where there is no effective dispersion in the propagation medium. The general nonlinear equation (4.9) has thus far been solved through the so-called perturbative method, which considers the linear solution as a first approximation and adds to it a small term to be determined successively. The solution may then be derived by limiting various terms in the equation to those of the first order. The method may be continued by considering the new solution as a better approximation and looking for a higher-order term. The higher-order terms in the equation are taken into account. It can therefore be qualitatively stated that each time a higher harmonic wave is involved at a low level of nonlinearity, it will grow along the propagation direction bound both to higher powers of the propagation distance and of the fundamental wave amplitude.
4.2.2.1
Applications
Second harmonic generation is probably the most commonly known effect of nonlinearity in wave propagation and is exploited in a variety of fields. In the case of the nonlinear sounding of materials, as typically produced in medical echography, the second harmonic propagation permits the resolution in the imaging process to be doubled or acoustic microscopy where enhanced resolution is attained via the contemporary effect of π phase change in the focal point of a convergent beam. Biological tissues are untypical materials that may be treated as liquids or lossy isotropic solids, and are a highly nonlinear acoustic media due to their irregular structure. Harmonic growth is therefore easily produced along the propagation direction. Tissue harmonic imaging has now become a standard technique. The acoustic focalized beam from an array transducer has a narrower beam cross section in respect to the fundamental frequency beam, and an even shorter beam waist zone; thus allowing a higher resolution in both the lateral and axial directions. Higher harmonic imaging is therefore produced to enhance these characteristics. Figure 4.1 is an example of such characteristics. Although the experiment was conducted in a water tank, the acoustic field of the beam from a transducer has been visualized at its fundamental frequency, second, third, and fourth harmonic, respectively. There is a visible decrease in the size of both the axial and the lateral focal regions that corresponds to an increase of the
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Figure 4.1 Imaging of an ultrasonic beam at the first four harmonic frequencies.
imaging resolution (Bouakaz and de Jong, 2003). There are, however, some drawbacks. The bandwidth and attenuation increase correspondingly with the relative additional electronic demands and energy dissipation effects. Figure 4.2 schematically represents the intensity of the fundamental wave and the second harmonic versus the propagation distance inside of the body. While the fundamental wave constantly decreases because of attenuation, the harmonic initially increases due to the energy drain from the fundamental and then decreases when the energy dissipation is higher than the energy drain. This sets an optimum distance for the use of second harmonic imaging, which is obviously dependent on the beam frequency. This effect is analogous to that which occurs in acoustic microscopy at much higher frequencies. A particular effect has been set into evidence that is bound to the focusing phenomena in wave propagation. The wave field undergoes a π-phase change when crossing the focal point and produces an equal phase shift between the fundamental wave and the second harmonic due to the different relative wavelengths. Therefore, apart from any attenuation effect, the second harmonic increases up to the focal point because of the synchronous power draining from the fundamental wave. From that point onward, the phase shift between the two waves causes the draining to occur π out of phase from that generated prior to this point; so, depleting the harmonic power, which returns to zero. Despite the generation and detection of waves at a given
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Figure 4.2 Schematic representation of the energy distribution versus distance of propagation for the fundamental wave and its first harmonic inside human body.
frequency, the overall effect is that the sounding of the medium in the focal plane of a converging lens is performed via the second harmonic. An increased resolution then follows because of the reduced wavelength (evidence first given by Cunningham and Quate, 1972).
4.2.3
Isotropic solids and wave number modulation
The general solution for the harmonic wave in solids is greatly simplified in isotropic solids where all directions in space are equivalent from the elastic standpoint. Therefore, the propagation direction may be considered as the direction of the coordinate axis x, so obtaining the following equation of motion: @2u @2u @u ρo 2 ffi 2 CL + ðCL + CNL Þ @t @x @x
(4.19)
for any mode component u of the wave, where CL is the linear elastic coefficient C1111, and CNL the nonlinear one C111111. For the second harmonic growing function f(x), f ðx Þ ffi
M 2 2 k Ax 4CL
(4.20)
with M ¼ CNL + CL is obtained. In the simplified case of an isotropic solid, it should be noted that the second harmonic generation obtained through the perturbation method is consistent with the model of propagation of a sinusoidal wave in a medium where the velocity—and thus the wave vector—is modulated by the perturbation induced by the wave itself. A solution of Equation (4.19) in the form a(x, t) ¼ A cos(ωt kx) may be assumed where a
High-intensity ultrasonic waves in solids
87
and A are the nonvanishing components of the displacement and of the displacement amplitude, respectively. The wave velocity c is then locally assumed to be given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½CL + AkðCL + CNL Þsin ðωt kxÞ c¼ ρo
(4.21)
and the wave number to be correspondingly obtained, such that the final form of the wave is ðCL + CNL Þ aðx, tÞ ffi A sin ωt ko 1 Ako sin ðωt ko xÞ x 2CL
(4.22)
with ko, the unperturbed wave number, or aðx, tÞ ffi A sin ðωt ko xÞ +
ðCL + CNL Þ 2 2 A ko x sin 2ðωt ko xÞ, 4CL
(4.23)
where use has been made of the Bessel function series for the development of the double sine function. The solution is again obtained as the sum of a fundamental wave and a second harmonic whose amplitude grows linearly with the distance x, proportional to the square of both the amplitude and wave number of the fundamental, times the elastic parameter combination (CL + CNL)/(4CL), as previously obtained through the solution of the unknown growing function in Equation (4.20). The same procedure is currently used for fluids, where the equation of motion is formally the same (see, for instance, Blackstock et al., 1998).
4.2.3.1 Applications The reduction of indices in isotropic solids in respect to anisotropic solids and the corresponding existence of pure modes in the propagation of waves in any direction of space limit the drain of power toward harmonic waves to a single component. The result is a simplification of the phase evaluation of the generated wave in respect to the fundamental. This is clearly seen from Equation (4.23) and is partially responsible for the generation of shock waves, as in the case of fluids. The modulation of the wave velocity produced by the wave itself stands as a positive feedback effect of the energy aggregation in the wave profile, which gradually increases to a sawtooth shape and finally results in a shock (see Figure 4.3). This effect is exploited in clinical lithotripsy, which is the breaking up of stones anomalously grown within human organs, particularly the kidneys, saliva glands, or bladder. A high-intensity ultrasound pulse, a few microseconds long, is focused on the region of concern, where it may attain a positive pressure peak of about 50 MPa in a time that may be less than 5 ns. This is followed by a negative pressure peak around 10 MPa. The interaction of the highly nonlinear wave thus generated with the stone is complex, but the abrupt rise in the pressure value, followed by the negative waveform, induces a sequence of stresses
88
Power Ultrasonics Velocity
f
Frequency
Time (depth) step 1
4f f
2f
3f
Frequency
Time (depth) step 2
Distance
Amplitude
Distance
Amplitude
Amplitude
Distance
4f 5f 6f f
2f 3f
Frequency
Time (depth) step 3
Figure 4.3 Deformation of wave profile and generation of harmonic components along the propagation direction.
inside the stone that breaks it into smaller pieces. Further, these may be subjected to the induction of stresses, or may be anatomically eliminated, depending upon their size. The effect of reflection from the stone surface also has to be taken into consideration, together with the corresponding change of pressure values at the interface between the medium of the organ and the stone. Isotropic solids are usually treated in the same way as liquids with the parameter B/A standing as the nonlinear combination parameters (CL + CNL)/(4CL) for the characterization of tissues in biomedicine or nondestructive evaluation in industrial applications. Granular materials, though highly anisotropic at smaller scales, usually present a statistical isotropy at larger scales. The presence of numerous discontinuity surfaces also make them highly nonlinear from the elastic standpoint, as the elastic response to compression differs greatly from that of dilation. The generation of harmonic waves is a common effect in acoustic propagation in granular media, as is the generation of harmonics from defects in homogeneous media.
4.3 4.3.1
Nonlinear effects in progressive and stationary waves Harmonic balance in progressive waves: dispersion and attenuation
The perturbation method described in the previous section assumes the nonlinear terms in the equation of motion to be an order of magnitude smaller than the linear one. Therefore, the solution could be assumed to be the linear one slightly “perturbed” by the nonlinear contribution. This approach may be extended, as previously stated, with the third and higher harmonics deduced from those of the lower order. This also feeds back into the lower-order harmonics, perturbing the previously obtained values in a never-ending process. This is the reason for the development of discontinuities in the generation of lower-frequency waves, as may occur in the bistability phenomenon of the fundamental wave (Lyakov et al., 1995).
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89
Where multiple coupling has to be taken into account, where energy flows from fundamental to higher harmonic waves and back again to the lower ones, a more general method can be adopted. This is termed harmonic balance and considers the solution of the equation to be given by a series of harmonic waves, each with its correct amplitude and phase. uβ ðr, tÞ ¼
X
j k Bβp ðr Þ sin p ωt kβ r φβp :
(4.24)
p
When such a solution is inserted into the truncated nonlinear wave equation (4.9), X h i β 2 β β B p sin p ωt k r φ ρo ω2 kjβ klβ Cijkl ip p p h i X X ¼ 2kjβ klβ knβ Cijkl Mijklmn p q Bβkðp + qÞ Bβmq ðp + qÞ2 q2 sin p ωt kβ r φβp + (4.25) is obtained. This is only satisfied for each time t and position r if every harmonic term with index p identically satisfies the equation
Gp+
2 2 + G ¼0 p
(4.26)
with β 2 G p ¼ Bip p
ρo ω2 kjβ kjβ Cijkl
1 + kjβ klβ knβ Mijklmn 2
X q
* + cos φβ p sin φβp
+ cos φβpq + φβq BβkðpqÞ Bβmq ðp qÞ2 q sin φβpq + φβq *
+! cos φβpq φβq + BβkðpqÞ Bβmq ðp + qÞ2 q ¼0 sin φβpq φβq *
(4.27)
the double sign in the G p terms corresponds to the upper and lower expressions reported within the symbolic brackets and separated by the symbolic fraction sign h—i. This is an infinite set of nonlinear algebraic equations in the unknown coefficients Bβip, defining the relative amplitudes of each harmonic wave and must be solved through numerical methods, by assuming that no strong nonlinearity is present in the phenomenon. In this case, a limited number of harmonic waves may be retained and summation of the trial function (4.24) limited to a finite value N. A set of correspondingly limited equations must be solved through the appropriate algorithms.
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Power Ultrasonics
Obviously, the balancing of harmonics can only be performed if all the wave numbers are multiples of the fundamental and are proportional to the corresponding circular frequency; that is, if the propagation of the waves is not affected by dispersion. A small degree of dispersion would prevent the generation of harmonics increasing to their steady value, because the local generation of any harmonic wave from its coupling with others would take place out of phase with what previously had been generated.
4.3.2
Frequency mixing
The consideration of phase matching leads us to the problem of the frequency mixing of waves. This occurs when two waves of different frequency give rise to a third, either with a sum frequency value or a different frequency. As in the case of harmonic generation (see Equation (4.12)), it may be assumed that two different frequency waves are propagating as a1 ðr, tÞ ¼ A1 cos ðω1 t k1 rÞ and a2 ðr, tÞ ¼ A2 cos ðω2 t k2 rÞ
(4.28)
and that they generate waves 0 bβ ðr, tÞ ¼Bβ ðr Þsin ω0 t kβ r φβ
(4.29)
with an obvious notation of the indexes and amplitudes Bβ A1,2. The conditions of phase matching and energy conservation require that k0 ¼ k1 k2 and ω0 ¼ ω1 ω2, where the upper sign corresponds to up-frequency and the lower one to downfrequency conversion. In the case of collinear propagation (k1 parallel to k2), that reduces to k0 ¼ k1 k2 and ω0 ¼ ω1 ω2. When replicating what has been developed for harmonic generation, with consideration of slight nonlinear effects, the generation of cross-product terms between the two interacting waves would be found (in addition to the case of second harmonic generation), whose growing functions f β have to satisfy the equation @fβ Dβk Cijkl ðk1l k2l Þcos ϑj + k1j k2j cos ϑl @r Mijklmn ffi k1n k2j k2l A1m A2k + k1j k1l k2n A1k A2m , 4
(4.30)
where care has been taken in introducing a double sign for the function f, depending on whether it refers to the up converted wave (k3 ¼ k1 + k2) or to the down converted one (k3 ¼ k1 k2). As seen in the previous section, Equation (4.30) greatly simplifies in the case of isotropic solids where a single coordinate x can be taken in place of the general displacement vector r. The solution would then reduce to
High-intensity ultrasonic waves in solids
fβ ffi
91
CL + CNL k1 k2 A1 A2 x: 4CL
(4.31)
Conservation of the linear momentum is satisfied in the dispersionless medium, which has been assumed here, as both energy (ℏω) and momentum (ℏω/c) are linearly related in the particle description. A small degree of dispersion could easily be taken into consideration and would add an oscillating factor sin(Δkx) to the solution, with Δk ¼ k3 (k1 k2), responsible for the flowing back and forth of energy between the generating and the generated waves. (For the effect in surface acoustic waves (SAWs), see Adamou et al., (1979)). However, the phase matching between harmonics may be seen from a different point of view when dealing with waves propagating into different directions. This is specific to solid-state acoustics, where anisotropy and multiple modes may produce phase matching conditions for waves propagating along different directions. In this case, phase matching conditions can be achieved with different mode propagating waves, provided that two equations are contemporaneously satisfied (Figure 4.4): 0
0
kγ ¼ kα1 kβ2 , ωγ ¼ ωα1 ωβ2 :
(4.32)
4.3.2.1 Applications Frequency mixing is exploited in signal processing techniques and in underwater or soil exploration. The need for processing high-frequency signals was greater in the Quasi-shear
Pure shear
Quasi-longitudinal
Figure 4.4 Slowness curves for the three acoustic modes propagating on the XY crystallographic plane of GaAs (cubic system) and wave vectors satisfying momentum (k1 + k2 ¼ k3) and frequency (ω1 + ω2 ¼ ω3, with ω1 ¼ ω2 ¼ ω3/2) matching conditions.
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Power Ultrasonics
past when electronics largely relied on analogue hardware and SAW devices were in common usage. This was particularly the case where convolution/correlation, integrals, Fourier transformations, and so on, were necessary for real time signal filtering processes, as in communicating through highly noisy environments or in image identification. The use of SAWs on solid or crystal substrates, typically quartz or LiNbO3, greatly improved the potential of devices because of the planar geometry of waves that limited undesired diffraction effects. The product between two signals was directly obtained through the nonlinear interaction of two waves propagating contra-directed, while integration was obtained either through acousto-optical interaction with light flashing on the interacting volume, or through appropriate electrode distribution on the propagation surface. Frequency mixing is also used for generating low-frequency waves for underwater or soil exploration. This utilizes transducers of limited size by means of down-conversion effect from the beating of two high-frequency waves of close frequency. In this way, as pointed out in the pioneering work of Westervelt (1963), the directivity of the transducer is bound to the short wavelengths of the high-frequency emission, while the generation of the long wavelength taking place in the cylindrical volume of interaction limits the heavy diffraction effect of low-frequency waves. The low-frequency wave can then propagate with low attenuation to reach deeply submerged objects, although with partial loss of definition characteristics.
4.3.3
Stationary waves: nonlinear sources
The excitation of elastic perturbations in a finite structure takes place according to a set of specific modes; that is, to a stationary distribution of strains varying sinusoidally in time with a spectrum of well-defined frequencies. The specific mode that is set into resonance within the structure depends on the overlapping integral between the spatial distribution of strains and the spatial distribution of stresses in the exciting force, in addition to the matching of the frequency values between the two distributions. A dissipative term causes the input energy acquired from the driving source to balance the loss of energy from the structure and to smooth out the response curve of the system at the resonance values of the frequency. The study of the spectral modes of a finite structure is facilitated by the assumption that there is no dissipation. This is achieved by considering a trial solution of the linear equation (4.9) with Mijklmn ¼ 0, and successively solving the space differential equation with the correct boundary conditions. As usual, the angular frequencies ωn will stand as the eigenvalues of the equation of motion. The insertion of nonlinear terms in the equation describing the elastic behavior of a finite structure subject to external driving forces will add the effect of coupling among different modes; thus exciting new modes in the structure that could not be directly excited by the driving force. In order to study the conditions for the onset of this effect, the equation must be written in its general form:
High-intensity ultrasonic waves in solids
ρo
@ 2 ui @um @ @ 2 uk ¼ F ð r Þcos ð Ωt Þ + C + M + η + i ijkl ijklmn ijkl @t2 @xn @t @xj @xl
93
(4.33)
with f(r, t) ¼ F(r)cos(Ωt), the driving force per unit volume and η the shear viscosity coefficient (Landau and Lifchitz, 1967, p. 191). Transient solutions will eventually fail, while a stationary trial solution uðr, tÞ ¼ UðrÞcos ðΩt + ψ Þ
(4.34)
can be inserted into Equation (4.33) to give Ω ρo Ui cos ðΩt + ψ Þ 2
ηijkl Ω sin ðΩt + ψ Þ
@Um cos ðΩt + ψ Þ Cijkl + Mijklmn @xn
@ 2 Uk ¼ Fi ðrÞcos Ωt: @xj @xl
(4.35)
The amplitude of displacement U(r) can be obtained by first separating, then squaring and adding, the sin Ωt and cos Ωt parts of the equation so as to be valid at any time t, giving
2 2 @Um @ 2 Uk @ 2 Uk Ω2 ρo Ui + Cijkl + Mijklmn + ηijkl Ω ¼ F2i ðrÞ: @xn @xj @xl @xj @xl
(4.36)
The phase can also be directly obtained through the separation of the sin Ωt from the cos Ωt parts and solving the sin Ωt part, which does not contain the forcing term F: @ 2 Uk @xj @xl : tan ψ ¼ @Um @ 2 Uk Ω2 ρo Ui + Cijkl + Mijklmn @xn @xj @xl ηijkl Ω
(4.37)
Where the viscosity term may be correctly ignored (η ¼ 0), the solution is greatly simplified, giving
@Um @ 2 Uk ¼ Fi , Ω ρo Ui + Cijkl + Mijklmn @xn @xj @xl ψ ¼ 0: 2
(4.38)
However, an external force with no dissipation would drive the system to an amplitude oscillation with an infinite value, so that the dissipationless case is better studied in the absence of external forces. If nonlinearity terms could also be ignored, the solution would further reduce to
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Power Ultrasonics
ω2n ρo Ui + Cijkl ψ ¼ 0:
@ 2 Uk ffi 0, @xj @xl
(4.39)
This case defines the eigenfrequencies ωn of the structure and the corresponding ˆ n(r). Prior to consideration of the nonlinearity terms, it should be remembered modes U that the viscosity would smooth out the amplitude of any excited mode as uðr, tÞ ¼ UðrÞcos ðΩt + ψ ÞexpðαtÞ
(4.40)
with the α term depending on frequency. A driving force would then simply compensate for the loss of energy caused by the internal dissipation of the structure. The effects of a small degree of nonlinearity are better understood by considering a stationary condition in which one of the proper modes at ωN remains at a constant amplitude UN, depending on the excitation value of that single frequency. A small perturbation is produced that corresponds to the energy flowing out from that mode, so that a trial solution may be given as uðr, tÞ ¼ UN ðrÞcos ωN t + εðr, tÞ
(4.41)
with ε UN. By inserting the solution into the nonlinear equation, taking into account that UN solves the linear equation and ignoring the higher-order terms, the following is obtained: 2 @ εi @ 2 εi @ 2 UNk @UNm 1 + cos ð2ωN tÞ C , ffi M ρo ijkl ijklmn @t2 @xj @xl @xj @xl @xn 2
(4.42)
which states that the perturbation term is drawing energy both from the dc term and from the time-second harmonic of the mode, which are excited. If the perturbation term ε is developed into Fourier series, according to εðr, tÞ ¼
X
Es ðrÞcos ðsωN t + φs Þ
(4.43)
s
and the harmonic balance is performed for the zero (s ¼ 0) and second harmonic (s ¼ 2) frequency, respectively, the following is obtained: Cijkl
@ 2 Eoi Mijklmn @ 2 UNk @UNm ¼ , @xj @xl 2 @xj @xl @xn
ρo ω2N E2i
Mijklmn @ 2 UNk @UNm @ 2 E2i ffi + Cijkl , @xj @xl 2 @xj @xl @xn
(4.44a)
(4.44b)
which shows how nonlinearity couples the principal mode to the others. Phase φo has been set equal to zero, with no loss of generality, because it simply modifies the value
High-intensity ultrasonic waves in solids
95
of the amplitude through the factor cos φo. For reasons of simplicity, if the excited term is set as equal to the first-order mode (ωN ¼ ω1), the nonlinear coupling of the modes may draw energy from the fundamental mode back to the dc term as well as to the second harmonic; thus producing a feedback process on the driving mode that makes for bistability. Equations (4.44a) and (4.44b) may be easily solved where monodimensional plane waves are set in an isotropic medium. By using the same notations already introduced in the case of progressive waves they reduce to C
@ 2 Eo M @ 2 U1 @U1 , ¼ @x2 2 @x2 @x
ρo ω22 E2
@ 2 E2 M @ 2 U1 @U1 : +C 2 ¼ @x 2 @x2 @x
(4.45a)
(4.45b)
Solutions of Equations (4.45a) and (4.45b) which satisfy the correct boundary conditions set for the structure must then be found. The possible growth of subharmonic waves in a finite structure from nonlinear interaction with the driven mode is of considerable interest. According to conservation laws, the rise of lower frequencies is only possible from the breaking down of one quantum state into two of lower frequency or, in terms of wave propagation, from the down-conversion of one higher frequency wave into two lower ones. This is a typical threshold phenomenon that may only take place if the energy of the generating wave is sufficiently high to overcome a specific, or threshold, level. Where that is the case, any perturbation, however small, occurring at the correct frequency in the structure will be amplified by the draining of energy from the fundamental and will grow up to a defined level. Both the threshold value of the fundamental mode and the regime level of the subharmonic depend upon the value of the dissipation term. In the limiting case that the system is not dissipative, the threshold level decreases to zero as any quantity of energy introduced into the subharmonic frequency wave will accumulate without loss and may grow indefinitely in a manner similar to the indefinite growth of the fundamental in a dissipationless system. The threshold effect may be evaluated in a manner parallel to that used for the growth of the harmonic waves. This is achieved by considering the general nonlinear equation (4.33) and supposing its solution to be given by a function V(r, t), which is comprehensive of both the solution of the corresponding linear equation (M 0) and all the higher frequency terms (harmonics). It may be supposed that an infinitesimal perturbation ε(r, t) at a frequency lower than that of the driving one—taken for example as Ω/2—is accidentally present in the system and added to the preexisting solution gives W ðr, tÞ ¼ V ðr, tÞ + εðr, tÞ:
(4.46)
Such a perturbation is bound to be extinguished, because of the dissipation term η, unless the nonlinear coupling between the waves drains sufficient energy from the
96
Power Ultrasonics
fundamental to overcome the dissipation loss. In order to verify this condition, solution (4.46) may be placed into Equation (4.33) to give @ 2 ð V i + εi Þ @ ð Vm + ε m Þ @ @ 2 ð Vk + ε k Þ ¼Fi ðrÞcos ðΩtÞ+ Cijkl +Mijklmn +ηijkl + ρo @t2 @xn @t @xj @xl (4.47) and subsequently, ρo
@ 2 εi @ @ 2 εk @εm @ 2 Vk @Vm @ 2 εk + ¼ C + η + M + ijkl ijklmn ijkl @t2 @xn @xj @xl @xn @xj @xl @t @xj @xl (4.48)
by retaining only the highest order terms and considering that V solves the nonlinear equation. As in the perturbative cases previously examined, the equation thus derived is linear for the perturbation term ε, with a driving force given by the cross products of first and second derivatives of ε and V. These are periodical terms at frequencies Ω and Ω/2, producing driving terms at the sum and difference frequencies, 3Ω/2 and Ω/2. That means that a perturbation accidentally present in the system may produce a driving term which generates an oscillation at the same frequency with the same configuration as the initial one present in the system. If the generated wave amplitude is greater than the original, the perturbation may continuously increase its amplitude and attain a finite regime value much higher than the initial one. However, this condition is not easy to derive because the cross product generating the new oscillation is proportional to V and V is proportional to F and it could be qualitatively said that the threshold value of the driving amplitude Fth for the onset of the subharmonic wave is inversely proportional to the nonlinear parameter M. Once a subharmonic oscillation at Ω/2 is set up in a system where the driving term is greater than the threshold value, a successive subharmonic at Ω/4 could be generated via the same process, so that a new condition for a second threshold is satisfied and thus for a subharmonic at Ω/8 and Ω/16, and so on. When a subharmonic at Ω/4 has been produced, nonlinearity can generate an oscillation at 3Ω/4, through the beating of the Ω/4 frequency wave with that already generated at Ω/2 and, analogously, oscillations at 3Ω/8 by the beating of Ω/8 with Ω/4. Successive values of the threshold amplitudes should be evaluated, but their values rapidly converge to a value Fch, where all possible subharmonic oscillations may increase suddenly and drive the system into a chaotic behavior.
4.3.3.1
Applications
Musical instruments provide the best examples of finite systems where elastic perturbations undergo nonlinear interactions. They are also the earliest known objects expressly made for producing sound, yet the approach to their study is usually presented in terms of linear theory, though musical “forte” conditions drive almost
High-intensity ultrasonic waves in solids
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any instrument deep into nonlinear conditions. Schematically, nonlinearity may derive either from the nonlinear coupling of the driving device (the bow hair in the case of violins, the reed in case of clarinets, etc.) or from the resonant system emitting the sound (strings, tube, plates, and so on) (see Fletcher et al., 1990). The focus here is on the second case, which has been theoretically described in the previous section. It is sufficient to note in regard to the first case, that it is mainly due to the onset of a positive resistance in the driving mechanism. At low amplitudes of oscillation in the linear regime, monodimensional structures such as strings or tubes present a spectral response of resonance frequencies, all of which are integral multiples of the lowest or fundamental frequency. At higher levels, the increased average time conditions of the stress usually increase the resonance frequencies, and the harmonicity of the spectral response may be lost. The nonlinearity mechanism preserves the integral relation among the different mode frequencies through the coupling of different modes. In the case of string instruments, for example, the coupling between the oscillations on the two possible planes makes the single point of the string move in a “rosetta”-type motion. The single component changes its oscillation period, but the periodical motion from the inner and outer parts of the rosetta does not. The conditions in multidimensional instruments such as cymbals and gongs are quite different, as mode locking is prevented and, following percussion, energy flows from lower to higher harmonics, producing a variable pitch or “multiphonic” sound. The results outlined above replicate at different bandwidths in the case of ultrasonic transducers, where nonlinearity drives the systems into motions that are rich in harmonic and subharmonic modes. These are usually difficult to analyze in detail. As the amplitude of the driving mode increases, a series of multiple harmonic modes appear at increasing intensities up to a point where the coupling between them cannot be easily followed. The appearance of subharmonic oscillations, mainly at half the fundamental frequency, is of great interest. When the corresponding threshold conditions are attained, successive period doubling will be present with the corresponding possible frequency coupling (three-quarters, three-eighths, and five-eighths of fundamental frequency, etc.) (see Figure 4.5). As the chaotic state is attained, the frequency spectrum enriches to a level where it returns to an almost linear condition and continues for a short interval of amplitude before returning to chaos. This behavior is common to chaotic systems and could be successfully used in case of high-power transducers where the chaotic conditions may damage transducers and affect the results of devices based on them (Cardoni et al., 2013).
4.4
Conclusions
The treatment of acoustic nonlinearities in solids presents a complex problem. The general anisotropy of materials combined with the various modes of propagation renders the formal approach a jungle of indices that must be carefully unraveled if the correct solution for individual cases is to be found. The general problem has been formulated here and various phenomena analyzed. These have been successively reduced
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Power Ultrasonics
−10
w
Amplitude (dB)
−20
~9 dB
w/2
w/8 ~15 dB
−30
w/2 + w/4 + w/8
w/4 −40
w/4 + w/8
w/2 + w/4 w/2 + w/8
−50 −60 −70 0
100
200
300 400 Frequency (kHz)
500
600
700
Figure 4.5 Spectrum of excited modes in a piezoelectric element, driven at fundamental frequency f ¼ 602.4 kHz, with evidence of subharmonic fractional modes (Bettucci et al., 2008).
to simplified forms (mainly to monodimensional cases for isotropic materials) to make clear the origin of the effects involved, while leaving the general form to further developments. The perturbation method has therefore been fully described where nonlinearity is considered as a recursive effect that slightly modifies the linear problem. This has been described in the case of progressive waves and stationary conditions. In the former, agreement with the model of self-modulation in the wave vector by the propagating wave itself has been described. Coupling between different modes and frequency mixing have also been discussed, together with the identification of the overall coupling coefficients and the feedback effect on the zero-order mode.
References Adamou, A., Alippi, A., Palma, A., Palmieri, L., Socino, G., 1979. Parametric mixing of surface acoustic waves. J. Appl. Phys. 50 (6), 4120–4127. Bettucci, A., Biagioni, A., D’Orazio,, A., Passeri, D., 2008. Periodic oscillations within the chaotic region in finite piezoelectric structures. In: Proceedings of the 18th International Symposium on Nonlinear Acoustics, Nonlinear Acoustics: Fundamentals and Applications, Stockholm, pp. 251–254. Blackstock, D.T., Hamilton, M.F., Pierce, A.D., 1998. Progressive waves in lossless and lossy fluids. In: Hamilton, M.F., Blackstock, D.T. (Eds.), Nonlinear Acoustics. Academic Press, San Diego. Bouakaz, A., de Jong, N., 2003. Native tissue imaging at superharmonic frequencies. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50 (5), 496–506.
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Cardoni, A., Riera, E., Gallego-Juarez, J.A., 2013. Nonlinear response in airborne piezoelectric transducer for power ultrasonics. In: 2013 International Congress on Ultrasonics (ICU 2013), Singapore. Cowin, S.C., Mehrabadi, M.M., 1955. Anisotropy symmetries of linear elasticity. Appl. Mech. Rev. 48, 247–285. Cunningham, J.A., Quate, C.F., 1972. High-resolution, high-contrast acoustic imaging. J. Phys. Colloq. 33 (C6), 42–47. Fletcher, N.H., Perrin, R., Legge, K.A., 1990. Nonlinearity and chaos in acoustics. Acoust. Aust. 18 (1), 9–13. Landau, L.D., Lifchitz, E.M., 1967. Theorie de l’e´lasticite´. MIR, Moscow. Lubarda, V.A., 1997. New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals. J. Mech. Phys. Solids 45 (4), 471–490. Lyakov, A., Proskuryakov, A.K., Shipilov, K.F., Umnova, O.V., 1995. Bistability and chaos in acoustic resonators. Ultrasonics 33 (1), 55–59. Morse, Ph.M., Ingard, K.U., 1986. Theoretical Acoustics. Princeton University Press, Princeton, NJ. Ostrovsky, L., Gorshkov, K., 2000. Perturbation theories for nonlinear waves. In: Christiansen, P.L., Sørensen, M.P., Scott, A.C. (Eds.), Nonlinear Science at the Dawn of the 21st Century. Springer, Berlin. Rienstra, S.W., Hirschberg, A., 2013. An Introduction to Acoustics. Eindhoven University Press, Eindhoven. Thurston, R.N., 1984. Waves in solids. In: Truesdell, C. (Ed.), Mechanics of Solids, vol. 4. Springer, Berlin, pp. 109–308. Westervelt, J., 1963. Parametric acoustic array. J. Acoust. Soc. Am. 35, 535–537.