CUBICALLY NONLINEAR VERSUS QUADRATICALLY NONLINEAR ELASTIC WAVES:
MAIN WAVE EFFECTS
C. Cattani1 and Ya. Ya. Rushchitskii2 UDK 539.3 + 524.145
This paper is a review of studies on quadratically and cubically nonlinear elastic waves in elastic
materials. The main methods for analysis of the wave equations are demonstrated. The main wave
phenomena are described. The disproportion between the achievements in the analyses of quadratically
and cubically nonlinear waves is pointed out—cubically nonlinear waves have been studied much less.
Keywords: quadratically and cubically nonlinear elastic waves, main wave effects, wave interaction, wave
triplets and quadruples, theoretical and computer analysis
1. Introduction. Themain goal of the present review is to establish the level of knowledge on elastic nonlinear waves in
materials. It primarily includes various wave effects that have already been detected and described and problems and effects that
have not been studied at all or adequately. The present study may be regarded as a development of [14]. As in [14], we consider
nonlinear elastic waves within the framework of the modern mechanics of materials with internal structure. The authors call the
reader’s attention to the great interest in nanotechnologies shown for the last ten years. Therefore, we may divide the mechanics
of materials into macro-, meso-, micro-, and nanomechanics and reinterpret many results on the micromechanics of composite
materials in terms of nanomechanics. Thus, the mechanics of materials is still a developing area of science; and waves occupy a
significant place in this area.
Because the overwhelming majority of investigations on elastic nonlinear waves are concerned with either quadratically nonlinear or cubically nonlinear waves and these latter waves are much less explored, we will chose the scheme of
exposition announced in the title—cubically nonlinear waves versus quadratically nonlinear waves.
We prefer to consider this review as the second part of the review [14]. This relieves us of introducing many preliminary
notions and facts. We will have to do with hyperelastic materials only and will express the Murnaghan elastic potential [19, 21,
23, 55] in terms of the Green strain tensor ε ik ,
W A Bik mm ik ik im km ik mm( ) ( ) ( ) ( )ε λ ε µ ε ε ε ε ε ε= + + + +1
2
1
3
12 2 2
33C mm( )ε , (1.1)
or in terms of the first algebraic invariants I k of the tensor ε ik ,
W I I I I I AI BI I CI ( , , )1 2 3 12 2 3 1 2 13
1
2
1
3
1
3= + + + +λ µ , (1.2)
where I I I ik ik ik 1 22
33= = =tr tr tr ( ), [( ) ] , [( ) ]ε ε ε ; λ and µ are the Lamé elastic constants (constants of the second order); and
A, B, and C are the Murnaghan elastic constants (constants of the third order).
International Applied Mechanics, Vol. 39, No. 12, 2003
It is common practice to start studying plane polarized waves and corresponding wave equations with the
above-mentioned pioneering papers. This procedure is described in [14] for the equations of motion in the classical theory of
elasticity. Recall the main assumption: plane waves propagate along the abscissa axis,
{ }ru u x t k = ( , )1 . (1.11)
Substituting (1.11) into the well-known Christoffel equation, we obtain one longitudinal plane wave ( P -wave) and two
transverse plane waves (horizontally polarized wave (SH -wave) and vertically polarized wave (SV -wave)). The correspondinglinear equations have the form
u x t u x t u x t 1 1 1( , ) ( , ) ( , )* **= + ( ) ( ) ( )= − −+
−u kx t x
N u k k x t 0 1
10 2
12
1
1
8 22cos cos ( )* *ω
λ µω . (2.8)
This long-standing result displays many features of quadratically nonlinear waves in materials, including two important
effects: self-interaction of waves and generation of the second harmonic.
Both effects are well manifested in the evolution of the initially harmonic profile. This evolution was the subject of computer simulation in [81, 83].
The main goal of the computer simulation was to estimate the ranges of frequencies and amplitudes where the evolution
of the wave profile is most intensive and, hence, can be detected well. The simulation has revealed new elements in the evolution
pattern. Let the profile evolve in four stages:
Stage 1. The initial cosine profile tilts downward at a constant angle, i.e., the maximum positive values decrease and
maximum negative values increase.
Stage 2. The peaks of the profile get lower, gradually forming a plateau. Then the middle part of the plateau begins to
sag, producing two humps. The frequency of reoccurrence of the same profile is equal to the initial frequency of oscillation.
Stage 3. Preserving the prior period, the profile takes an increasingly pronounced two-humped shape until its trough
touches the abscissa axis.
Stage 4. As the trough goes farther down, the profile becomes nearly harmonic, having the frequency of the secondharmonic and unequal rises and falls (the first rise is high, the first fall is roughly half the first rise, the second rise is slightly
higher than the first one, and the second fall is roughly twice as high as the first fall). Thus, gradually changing, the profile of the
first harmonic turns into that of the second harmonic, i.e., we observe a transition of one harmonic into another.
Remark 2.5. The computer analysis involves six physical constants (density, two Lamé constants, and three Murnaghan
constants) for eighteen granular composite materials. These materials are denoted by KM , where K = 1 corresponds to steel
granules and polystyrene matrix; K = 2, to copper granules and polystyrene matrix; K = 3, to copper granules and tungsten
matrix; K = 4, to copper granules and molybdenum matrix; K = 5, to tungsten granules and aluminum matrix; and K = 6, to
tungsten granules and molybdenum matrix. There are three modifications for each of the K materials. These modifications,
labeled by M , differ by the volume fractions of granules, c1, and matrix, c2: M = 1 corresponds to c1 = 0.2 and c2 = 0.8; M = 2, to
c1 = 0.4 and c2 = 0.6; and M = 3, to c1 = 0.6 and c2 = 0.4. The physical constants are given in [55, Table 5.2].
The results of the computer simulations (numerous plots) were used to analyze three most typical materials: 11, 41, and62. The frequency was varied within the ultrasonic range from relatively small values (at which visible distortions of the profile
appeared only after a large number of oscillations) to relatively large values (at which the profile evolution started to be observed
after the second or third oscillation). The amplitude was varied from 0.05 mm (small displacements) to 0.5 mm (not very large
displacements). In all the experiments, the maximum amplitude of the distorted wave was exactly half as much again the initial
amplitude. This phenomenon stems from the convention adopted in nonlinear wave theory [27, 91] that the initial hypotheses of
the method of successive approximations are invalid for large amplitudes.
All the plots were grouped so as to represent different stages of the profile evolution. On all the plots, the abscissa is the
distance x in meters and the ordinate is the displacement amplitude u1 in millimeters. Time is fixed. In what follows, we will
discuss only the plots for the material 11.
The first group of plots is where the initial amplitude is the same, u10 = 0.1 mm, and the frequencies vary. Figure 1
demonstrates the first stage of profile evolution (the profile tilts downward). This effect is more or less clear for differentmaterials and can be observed well for the material 11, whose wave phase velocity is relatively low, v11 =1.848⋅103 m/sec. This
plot corresponds to the smallest studied frequency of 10 kHz.
Figure 2 demonstrates the first three stages (developed evolution). This plot corresponds to a frequency of 50 kHz.
Given a small initial amplitude, we conclude that all the three stages can be seen at a distance of 2 m.
Figure 3 represents a frequency of 100 kHz, which is extremely high for the material 11 (here a significant distortion is
observed to the right of the beginning of wave propagation).
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where w m1 and w m3 are newly generated waves. Remark 2.6. These waves can be or cannot be synchronized with the basic wave. Two waves are synchronized (in
phase) if they have the same characteristics in the phase plane. The phenomenon of generation of new waves can be considered to
consist of the phenomenon of generation of a new harmonic and the phenomenon of generation of new composite waves. The
new harmonic is understood as follows: an initially sinusoidal wave,
u x t u k x t 1 1 0 1( , ) cos ( )= −ω , (2.19)
propagating through a nonlinear medium generates new longitudinal waves with phases multiple of that from (2.19):
u m k x t m N m1 0 1cos ( ) ( )− ∈ω . (2.20)
These waves are phase synchronized with the basic wave (2.19). When wave (2.19) generates its transverse harmonics,they combine with the basic transverse wave producing waves modulated in space, which are synchronized neither with the
initial longitudinal wave nor with the initial transverse wave. They are said to be composite waves.
The third problem assumes that both longitudinal and transverse waves are initially excited. The following two
equations have to be solved:
( )ρ λ µu u N u u N u utt xx xx x xx x1 1 1 1 1 2 3 32,**
,**
,*
,*
,*
,*− + = + , (2.21)
( )ρ µu u N u u u utt xx xx x xx x3 3 2 3 1 1 3,**
,**
,*
,*
,*
,*− = + . (2.22)
The solution for the longitudinal wave consists of two parts: (2.8) and (2.17). The transverse wave is the sum of a
classical linear transverse wave (2.12) and a composite wave with a small period of modulation
( )u x t N u u
vk k x3
2 1 0 3 0
33 3 1
1
2** ( , )
( )sin= −
ω
ρ ph
×+
− ++
( ) ( )
( ) ( ) ( )cos
v v
v v vk
12
32
14
12
32 3
3 2
1
23
ph ph
ph ph ph
( ) ( )k x t v
v vk k x1
32
12
32 3 12
1
2−
−−
−
ω( )
( ) ( )cos
ph
ph ph
. (2.23)
Thus, the above three problems of nonlinear acoustics (solved by the method of successive approximations) allow us to
describe a series of nonlinear wave phenomena, which can be united as those associated with the basic nonlinear phenomenon of
wave interaction.
Let us look at what is going on when the medium through which waves propagate is orthotropic [36, 55]. If the elastic potential has the form
( )W c c A Biimm mm ikik ik ik i im ik mk i ik = + − + +1
21
1
32 2 2ε δ ε ε ε ε ε ε mm i mmC +
1
33ε , (2.24)
then plane waves propagating along the axes of symmetry are described by equations similar to (3.16)–(3.18) from [14]. For
example, the basic system of equations for quadratically nonlinear plane polarized waves propagating along the Ox1-axis is
ρ u c u N u u N u u N tt 1 1111 1 11 11
1 1 1 1 1 21
2 11 2 1 3, ,( )
, ,( )
, ,− = + + ( ), ,
13 11 3 1u u , (2.25)
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Let us turn back to the main nonlinear wave effect—interaction of waves. This effect can be divided into two ones:interaction of waves with equal frequencies and interaction of waves with different frequencies.
All the previous problems studied the interaction of waves with equal frequencies. It was also additionally assumed that
waves propagate in the same direction. Lifting these two restrictions complicates the problem. An example is the
synchronization of waves in a quadratically nonlinear medium. No more than three waves can be synchronized in this medium;
three synchronized waves are called a wave triplet. What is most important is that two noncollinear waves with all parameters
(amplitude, frequency, and polarization) different are initially excited [20, 21, 55, 94]:
( ) ( ) ( )r r r r r r ru x x x t A t k r A t k r 1 2 3 1 0 1 1 2 0 2 2, , , cos cos= − + −ω ω , (2.28)
where the wave vectorsrk α may form an arbitrary angle; the amplitude vectors
r Aα0 may be polarized differently—they may be
both parallel and perpendicular to the propagation direction; andrr OX = → , X x x x= ( , , )1 2 3 , is the radius-vector of the point X .
The second approximation follows from the solution of some inhomogeneous vector Lamé equation. According to this
solution, there is some special, resonance-like case where the energies of the given two waves are fully pumped into a new, third
wave. This wave has the frequency
ω ω ω3 1 2= ± (2.29)
and the wave number
r r rk k k 3 1 2= ± . (2.30)
Physics calls (2.29) and (2.30) resonance conditions; and three waves related by these conditions are called a wave
triplet or just a triplet.
There are six different combinations of wave constituents in triplets
L L L L L T ( ) ( ) ( ), ( ) ( ) ( ) ,ω ω ω ω ω ω1 2 3 1 2 3+ = + =
L T L L T T ( ) ( ) ( ), ( ) ( ) ( ) ,ω ω ω ω ω ω1 2 3 1 2 3+ = + =
T T T T T L( ) ( ) ( ), ( ) ( ) ( ) ,ω ω ω ω ω ω1 2 3 1 2 3+ = + = (2.31)
where L corresponds to the longitudinal wave and T , to the transverse wave.
The third wave (similarly to the first and second ones) can have two different polarizations.
Also we can obtain special restrictions on the angle between the two primary waves—the triplet does not exist for all
values of the angle. The frequency ranges of two primary waves cannot be arbitrary. Some problems arise in calculating theamplitude of the third wave. That is why the method of slowly varying amplitudes is additionally used.
Next few pages will be devoted to problems for quadratically nonlinear waves solved by the method of slowly varying
amplitudes.
Let us again turn to the article [14] where the methods of successive approximations and slowly varying amplitudes
were described and compared, with the remark that they supplement each other.
We start with the evolution equations [14] (they are written for longitudinal waves, denoted by the subscript “1”)
( ),( ) A A A ei k k k x
11 1 1 12 1313 12 11= − −σ , ( ),
( ) A A A ei k k k x12 1 2 11 13
13 12 11= − −σ ,
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is considered a measure of phase matching. The condition ∆k =0 is called the phase index matching condition. When ∆k =0, the
behavior of waves can be explained by an example. Let the third wave be not initially excited, A13 0( ) =0. By the hypothesis that
amplitudes vary slowly, the amplitudes A11 and A12 may be assumed constant over a not very long distance. Then only the third
equation in (2.32) can easily be solved:
[ ]{ } A x N k k A A x13 1 1 2 11 122 2( ) / ( )= +λ µ . (2.34)
Thus, under full (frequency and space) synchronism, the amplitude of the third wave is directly proportional to the
distance traveled. This situation is similar to a resonance. From the assumed balance of energy of the triplet, it follows that the
energies of the two primary waves are pumped into the third wave. This example shows that both the method of successive
approximations and the method of slowly varying amplitudes include the full synchronism conditions as a necessary conditionfor the existence of a triplet.
Energy analysis is very useful in the theory of triplets. For elastic triplets, the Manley–Rowe relations introduced in
nonlinear optics [27, 55, 91] can be written as
σ σ σ σ σ2 11 1 12 1 3 12 2 13 2 3~
( )~
( ) ,~
( )~
( ) ,~
N x N x C N x N x C N − = + = 11 1 13 3( )~
( ) x N x C + =σ , (2.35)
where~
( ) | | N x A A Am m m m1 1 1 12= = are the intensities of waves. Formulas (2.35) turned out to be a unique tool for solving
problems of parametric amplification of waves and related problems.
Remark 2.7. Relations (2.35) help to look at triplets from an unexpected point of view. It is easy to obtain from (2.35) the
energy balance law in the form ω σm m mm
A12
1
3
=
∑ =const. This formula may be regarded as an equation for a general ellipsoid with
semiaxes ω ω2 3 , ω ω1 3 , and 1 1 2/ ω ω in the three-dimensional space of variables A11, A12, and A13. The surfaces of
constant energy will be general ellipsoids; and the phase trajectories are the intersections of the ellipsoid with the cylindrical
surfaces defined by the Manley–Rowe relations
σ σ3 112
1 132 A A+ = const, σ σ3 12
22 13
2 A A+ = const, σ σ2 112
1 122 A A− = const .
In the neighborhood of the A11- and A12-axes, the trajectories have the form of ellipses, i.e., under small variations the
amplitudes of the two primary waves oscillate weakly, and the process is stable.
However, trajectories that reach the third axis have a different property—two waves receive energy from the third wave.
In other words, the third wave breaks down.
Thus, the three waves interact by two scenarios: (i) two initially given waves put out their energy to the third wave andthen practically cease to exist and (ii) one initially given wave breaks down into two new waves and then practically cease to
exist.
The last issue to consider in this subsection is the problem of interaction of two waves in a special formulation. Consider
two coupled waves entering a medium or light guide. Initially one wave is more powerful (a pumping wave with a high
amplitude) and the other wave is weak (a control signal with a very small amplitude). The parameters of the two waves are fixed
at some finite distance from the entry (at the exit). Under certain conditions, a very small change in the control signal causes an
abrupt change in the intensities of the waves on leaving the medium; in particular, the strong wave can switch from its frequency
to the frequency of the control signal and back (that is why this problem is called the self-switching problem).
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Thus, we have found the analytical solution to the problem on the interaction of two waves. It remains to discuss the
wave self-switching phenomena—our interest is with the intensities and phase difference at the exit of the medium—at a
distance l from the entry.
The most useful information can be derived from the cubic equation ) ) I I Gsign sign( )1 2 2− − = 0, following from (2.42),
and its roots
( ) ) ) I I sign sign( ) ( ) sin ( ),1
20 0≈ ϕ)
m )
I I sign sign( , ) ( ) sin ( ) .2 3 1 0 0≈ ϕ (2.43)
Here two cases are very interesting:
A. The initial intensity of the signal wave is zero, ) I sign ( )0 = 0. Then
) ) I I l sign pum( ) ( )ξ = =1 0 . Thus, all of the energy of
the pumping wave goes to the signal wave. Here we observe that the first harmonic enters the medium and turns into the second
harmonic on leaving the medium. The wave switches from the first to the second harmonic.
B. The initial intensity of the signal wave is very small, yet not zero. Let us calculate the normalized intensity of the
signal wave that leaves the medium, supposing that the strong wave hardly supplies energy to the signal wave:
) I
U
U l sign ( )ξ ≈−+
<<
1
12 1,
U I e l ≈ ≈4 0 0 12 2 )sign ( )sin ( )ϕ ξ ( )→ ≈ << ≈− )
)
I I
el l
sign
sign( )
( )
sin ( )sin ( )ξ
ϕϕξ64 0
01 0 1
22 . (2.44)
Here the situation is different. If the signal wave is of very low intensity, then the strong wave travels through the medium almost
changeless. At the exit of the medium, we observe a strong pumping wave with frequency ω and a weak signal wave. If the
intensity of the signal wave is zero, then the signal wave will have the frequency 2ω at the exit. This phenomenon is called
self-switching of waves.
Similar results were obtained in [64] for a more complicated medium, a two-phase elastic mixture.
2.2. Harmonic (Periodic) Waves in the Microstructural Theory of the Second Order. We start with the basic system of
wave equations (1.31)–(1.33). First, let us compare systems (1.15)–(1.17) and (1.31)–(1.33) and formulate the main distinction.
Formally speaking, each of the three equations (1.15)–(1.17) corresponds to the coupled system of two equations (1.31)–(1.33).
From the wave point of view, this fact implies wave doubling. The wave pattern in a mixture is richer; therefore, introducingnonlinearity complicates the mechanism of wave interaction.
Most of the results obtained in the microstructural theory of the first order can be extended to the microstructural theory
of the second order.
Let us analyze these results according to the scheme proposed in Section 2.1.
First, consider problems solved by the method of successive approximations, beginning with the three standard
problems of nonlinear acoustics. The formulations of all the problems are classical, except that the medium of propagation is a
mixture and that this medium is dispersive.
The first problem assumes solving the coupled system [14, 55]
( ) ( )ρ λ µ λ µ βααα
α αα δ α
u u u u utt xx xx1 1 3 3 1 12 2,( )
,( )
,( ) ( )− + − + − −( )1 1 1 1
( ) ( ),
( ),
( )δ α α α= N u u xx x (2.45)
with the boundary conditions u t u t o1 10( ) ( )
( , ) cosα α ω= .
In the second approximation, the solution is
u x t C e l k C i k x t 1 1 1 1 1
1 1( )** ( ) ( ) ( )* ( )
( , ) ( )( )*α α ω δ δα
= +− − e i k x t − −( )( )*
1 1δ
ω
+ +
− x
M S l k ik c C u e i1
11 1 2 1 0 1
2ωαδ
δ δδ
δ δ( )
( )* ( )* ( ) ( ) ( ) ( ) ( )( )* ( )*
k x t i k x t x
M S C e1 1 1
1
11
2δ αω
ααα ωω− − −+
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The first and second terms in (2.46) constitute the solution in the linear approximation, i.e., they are two harmonic
waves (two modes) with different wave numbers and other properties known from the linear theory of mixtures. The third and
fourth terms describe the second harmonics of the first and second modes, respectively. These harmonics are generated by the
modes independently. Attention should be focused on the dependence of the amplitudes of the second harmonics on the
amplitudes of the initial impulses and frequencies, which is more complicated than in a liquid or an elastic body, and also on the
fact that they are no longer real, but complex numbers. Finally, the fifth terms describe a new complex longitudinal wave, which
is due to the nonlinear interaction of two different modes. The wave numbers k 11( )*
and k 12( )*
are always different, the difference
increasing as the cut-off frequency is approached from the right. The new wave describes spatial beating with the beating
frequency depending only on the parameters of the linear problem. The spatial modulation period also depends on the parameters
of the linear problem.
Remark 2.9. This last wave should be discussed individually. It describes a new microstructural
phenomenon—interaction of two different modes of the same longitudinal wave. Such an interaction is characteristic of
two-phase mixtures. It would also be manifested with transverse waves and has to be taken into account in studying the full wave
pattern. The fact should be considered that the second mode is stopped at low frequencies. Therefore, the new wave arises at high
frequencies (they are supersonic frequencies for many real composites).The second standard problem for mixtures is more interesting than the corresponding classical problem. This problem
implies solving the following systems in the second approximation:
ρ λ µ λ µααα
α αα δ
u u utt xx xx1 1 3 3 12 2,* * ( )
,* * ( )
,** (
( ) ( )− + − + ( )) **( ) **( ) ( ),
*( ),
*( )− − =β α δ α α αu u N u u xx x1 1 2 3 3 , (2.47)
ρ µ µ βααα
αα δ α
u u u u utt xx xx3 3 3 3 3,* * ( )
,* * ( )
,**( ) **( )− − − −( )3 0
**( )δ = . (2.48)
The solution of (2.48) for the boundary conditions u t u t 3 30( ) ( )
( , ) cosα α
ω= init and u t u t 2 10 0 0( ) ( )
( , ) ( , )α α
= = and zero initial
conditions is well known [55]:
[ ] ( )u u e l k u ei k x t i k x
3 30
3 3 30
3 1 3**( ) ( ) ( ) ( )
( ) ( )α α ω δ δα δ
= +− + − [ ]130 3 3 3 3
3 31
+ =−
−
ω α α δ δ
δt
uu l k u
l k ,
( )
(
( )( ) ( ) ( )
( )
init init
) ( )( )
l k 3 3δ . (2.49)
Remark 2.10. The second harmonic is not generated here—the wave does not interact with itself. Also, amplitudes are
not equal to the initial amplitude. This property is characteristic of mixtures.
Let us dwell on system (2.47). First, it is necessary to calculate the nonlinear right-hand side from the known linear
solution (2.49). System (2.47) may be solved by selecting similar functions and is represented as a superposition of five waves:
[ ] ( )u x t u e l k u ei k x t i k
1 1 10
1 10
1 1 1( ) ( ) ( ) ( )
( , )( ) (α α ω δ δα
= +− + − [ ] [ ]δ αωα
ω) ( ) x t i k x t
iS e1 3 11
2+ − ++
[ ] [ ]+ +− + − + +iS e iS ei k x t i k k x t 2
23
3 1 31
32
1α ω α ω
δ( ) ( ) ( )
( ) . (2.50)
The first two waves in (2.50) are two modes of linear longitudinal waves. The third and fourth waves are in fact the
second harmonics for the previous two waves. The fifth wave is of greatest interest.
Remark 2.11. The fifth wave represents a new effect not described in classical nonlinear theory—interaction of two
different modes of the same wave. This effect is due to the microstructural nature of mixture theory.
Remark 2.12. In composites, the new synchronism 2 1 31
32
k k k ( ) ( ) ( )α ≈ + should result in small amplitudes, i.e., the wave
numbers in the frequency range where the two modes exist are of orders from 1 to 10 –2 m –1. As a rule, there is no synchronism
and no nonlinear effects are superimposed.
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The solution for transverse waves of system (2.52), (2.53) has the form
[ ] ( )u x t A e l k A ei k x t i
3 1 3 3 3 33 1**( ) ( ) *( ) ( )
( , )*( )α α ω δ δα
= +− + − [ ]k x t 3 1*( )δ
ω+
+
− x
M S l k k c A u e
i1
11 3 3 3 1 3 0 3
ωα
α αα
α α~( ) ~*( ) *( ) ( ) ( ) [ ]( )
*( ) *( )k k x t 1 3 2
α α ω+ − [ ]+ − + − x
M S A u e
i k k x t 1
11 3 0 3
21 3ω
αα α ωα α~ ( ) ( ) ( )
*( ) *( )
+ +
x
M S l k ik c A u e
1
12 3 3 3 2 3 0 3
ωδ
δ δδ
δ δ~( ) ~*( ) *( ) ( ) ( ) [ ]− + −i k k x t ( )
*( ) *( )1 3 2δ δ ω [ ]+ − + − x
M S A u e
i k k x t 1
12 3 0 3
21 3ωδ
δ δ ωδ δ~ ( ) ( ) ( )*( ) *( )
. (2.55)
It consists of six waves. The first two waves are a normal linear two-modal representation of a transverse wave in a
two-phase mixture. The other waves are second harmonics with wave numbers that are linear combinations of wave numbers of
various modes.
Remark 2.16. What composite waves have in common is that their amplitudes depend proportionally on the wave
numbers. Note that the amplitudes of the self-generated second harmonics contain squared wave numbers in the numerators.
Transition to high frequencies increases the effect of self-generation.
Let us now consider features of the interaction of waves with different frequencies. The main problem here is the triplet
problem. It should be pointed out that 21 types of triplets may hypothetically exist in a two-phase mixture:
T T L L L T ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ), ( ) ( ) (11
22
13
11
22
1ω ω ω ω ω+ = + = ω 3 ) ,
L T L( ) ( ) ( )( ) ( ) ( ) ,21
12
13ω ω ω+ = and so on. (2.56)
The superscript indicates the mode number.
Using the classical procedure is not trivial for a mixture, since, in contrast to the classical nonlinear procedure where
wave vectors are constant, these vectors are functions of frequency. The distribution coefficients of the amplitude matrix depend
on the frequency too. A major difficulty also arises in calculating the inverse Fourier transform.
The resonance conditions have the form [55]
ω ω ω1 2 3 0± + = , (2.57)
k r k k l t ( )[( )]( )
( )
( )
( )
(( ) ( )
ακ
δδκ
δω ω ω− ± − −1 2
01 1 21
1
2
r r r δω2
2 0)
( ) = . (2.58)
Thus [55], various triplets are possible in solid mixtures (composite materials).
Let us now analyze some results obtained by the method of slowly varying amplitudes.
The basic assumption of the method deals with nonlinearity: it is weak during wave motion and the wave amplitude
varies slightly for a time equal to one period (or over a distance equal to the wavelength). Also, we will extend the spectrum of
nonlinear properties. For this purpose, we will use the first variant of the Murnaghan potential, i.e., we additionally take into
account the nonlinear interaction between phases by incorporating three new constants A3, B3, and C 3 into the Murnaghan
potential.
Let us consider a longitudinal wave in a two-phase mixture to demonstrate features of the wave pattern.
First, the mode exists in both phases of the mixture simultaneously. If the first longitudinal mode is selected as a participant of interaction, then a harmonic wave propagate in the first phase of the mixture and a similar wave, in the second
phase. In the case of the second mode, a different pair of waves with different amplitudes propagates through the mixture. Thus,
an individual mode participates in interaction. Let us narrow the analysis to longitudinal waves alone. They do not generate
transverse waves during the interaction, which simplifies the description.
If the first mode is chosen as a participant of interaction, then a wave with an arbitrary amplitude and a
frequency-dependent wave number (the mixture is a dispersive medium) [ ]u x t A e
i k x t 1
11
11
1( ) ( ) ( )
( , )( )
= −ω ω propagates in the first
phase. The second phase conveys the same wave but with a different amplitude, [ ]u x t l A e
i k x t 1
21
21
11
1( ) ( ) ( ) ( )
( , ) ( )( )
= −ω ω ω. This
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amplitude depends on the frequency. When the second mode is chosen as a participant of interaction, then the situation will be
similar.
Consider the basic system for longitudinal waves [55]
ρ λ µ λ µ βααα
α αα δ
u u u utt xx xx1 1 3 3 1 12 2,( )
,( )
,( ) (
( ) ( )− + − + − ( )α δ) ( )− u1 = + N u u N u u xx x xx x1 1 1 13
1 1( )
,( )
,( ) ( )
,( )
,( )
.α α α δ δ
(2.59)
We will restrict ourselves to the interaction of three waves. First, let us assume that the synchronism conditions are
satisfied in terms of frequencies. Next we select one of the six possible triplets: “1st harmonic 1st mode + 1st harmonic 1st mode =2nd harmonic 2nd mode.” Under frequency synchronism, the shortened equations (one equation for each phase) break down into
three evolution equations (three for each phase). The spatialsynchronism condition has the form k k k 111
1 122
2 131
3( ) ( ) ( )
( ) ( ) ( )ω ω ω+ = .
Remark 2.17. Because of the dispersivity of solid mixtures, the wave numbers are no longer constants. If the wave
synchronism conditions are satisfied, then the evolution equations assume the classical form (we write one of them),
( )( )( ) ( ) ( ) ( )* ( )*
A A A A A132
13 111
122
111
122′ = −σ , (2.60)
where the coefficients for the first phase σ1k and for the second phase σ2k are different. Compare, for example σ13 ,
σ
ω ω
λ µ λ µ13
11
2 12
3 11
13
3 3 2 22 2=
+
+ + +
l l N N ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) l
k k
k k k 1
23
12
2 12
3
11
31
2
2 1
2
( )
( ) ( )
( )
( ) (
( )
( ) ( )
( ) ( )ω
ω ω
ω ω +[ ])
( )ω 3 ,
and σ23 ,
σω ω
λ µ λ µ23
11
2 12
3 12
13
2 2 3 32 2=
+
+ + +
l l N N ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) l
k k
k k k
12
3
12
2 12
3
11
3
12
2 12
( )
( ) ( )
( )
( ) (
( )
( ) ( )
( )( )
ω
ω ω
ωω +[ ])
( )ω 3 .
They are also different from the classical one. The classical coefficients are constant, whereas the others are functions of
the frequencies of all the waves.
The spatial synchronism condition has the form
k k k 11
1 12
2 13
3 0( ) ( ) ( )( ) ( ) ( )ω ω ω+ + = .
Let us now examine the evolution equations. Formally, they have a classical form. However, different phases will have
three different evolution equations with frequency-dependent coefficients. The situation is similar with the Manley–Rowe
relations. Three integrals (two independent and the third one derives from the first two) can easily be written for the evolution
equations. Different phases are described by three relations with different coefficients. Also, each triple of waves obeys the law
of conservation of energy
( )ω σ ω ω ωk k k k
A( , , )( )
1 2 3 11 2
1
3
=∑ = const. (2.61)
Remark 2.18. The energy of the mixture is distributed in two ways: (i) from wave to wave within each triplet (in each phase) and (ii) from mode to mode in case of change in the excitation frequency.
The Manley–Rowe equations make it possible to investigate some interesting physical effects associated with triplet
evolution in space. Let us consider some nonclassical cases such as the interaction of the first mode with itself, resulting in
generation of the second mode. Here the classical procedure is used. It is supposed that the initial amplitude of the third wave
A132
0( )
( ) is zero and that the amplitudes A111( )
and A121( )
of the primary waves are constant. In establishing the synchronism
condition, it is assumed that ω ω ω1 2= = and ω ω3 2= . The solutions of the evolution equations are
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It is then possible to calculate the evolution of the amplitude of the second harmonic in both phases.Thus, the first mode generates the second harmonic of the second mode. In different phases, the amplitudes of the new
wave increase differently. This is a new, theoretically revealed effect.
Remark 2.19. This situation is common for mixtures. Considering a specific triplet in a two-phase mixture actually
implies considering two triplets with similar synchronism conditions and different amplitudes (i.e., the relative energies carried
by the wave constituents are different). Such triplets exist separately in the phases of the mixture.
Remark 2.20. Two similar harmonics of the first mode generate the second harmonic not only in the second mode but
also in the first one.
Let us consider the case where two waves enter the medium and one (first) of them carries a major portion of energy.
Then
~
( )
~
( ),
~
( )( ) ( ) ( )
N N N 11
1
12
1
13
1
0 0 0 0>> = . (2.64)
The third Manley–Rowe relation has the form
σ σ σ3 111
1 1 131
1 3 111
30~
( )~
( )~
( )( ) ( ) ( )
N x N x N C − = = ; (2.65)
therefore,~
( )( )
N x131
1 can increase only at the cost of ~
( )( )
N x111
1 . However, according to the first Manley–Rowe relation
σ σ2 111
1 1 121
1 1~
( )~
( )( ) ( )
N x N x C − = , (2.66)
such an increase can occur only when~
( )( )
N x121
1 decreases. On the other hand, the second Manley–Rowe relation must hold too:
σ σ σ3 121
1 2 131
1 3 121
20~
( )~
( )~
( )( ) ( ) ( )
N x N x N C − = = , (2.67)
and C 2 is small compared with C 3. Thus,~
( )( )
N x131
can increase by no more than C N 2 3 121
0= σ~
( )( )
. But it is small.
If the first and second waves in the triplet are of low frequency and the third one is of high frequency, then we obtain an
interesting result well known in nonlinear physics: the energy of low-frequency waves cannot be transferred to high-frequency
waves. This looks like an increase in the signal frequency, i.e., the low-frequency signal wave (first wave) interacts with the
strong idler wave (second wave), forming a new high-frequency pumping wave (third wave). The energy of the new wave is low,
and the effect is weak.
Remark 2.21. The distinction between the classical triplet and the triplet in a two-phase mixture is in the facts that the
energy is redistributed between the allied triplets in the phases of the mixture and that these two triplets exist simultaneously.
Let us now examine the case where a high-frequency wave as a constituent of a triplet entering a medium carries a major
portion of the triplet energy,~
( )~
( )( ) ( )
N N 131
111
0 0>> ,~
( )( )
N 121
0 . It then follows from the first and third Manley–Rowe formulas
( )~( )
~( )
~( ) ,
~( )
( ) ( ) ( ) ( ) N x N N x N x12
11
2
313
113
11 11
110= −
σ
σ( )= −
σ
σ1
313
113
110
~( )
~( )
( ) ( ) N N x (2.68)
that both~
( )( )
N x111
1 and~
( )( )
N x121
1 can increase considerably and simultaneously. Consequently, the energy of the high-frequency
wave may be pumped over to two different low-frequency waves. This phenomenon of parametric oscillations, known in
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nonlinear physics [27, 91, 92, 95], is called the break-down instability of a triplet, since it reflects the ability of a high-frequency
wave to break down into two low-frequency waves.
The next two sections are devoted to solitary (aperiodic) waves. We will briefly discuss a few principal results for
quadratically nonlinear waves. For cubically nonlinear waves, such results are absent yet.
2.3. Solitary (Aperiodic) Waves by the Microstructural Theory of the First Order. We start with the basic system of
wave equations (1.15)–(1.17). Let us define a solitary wave first. A solitary plane wave is meant an aperiodic wave whose profile
concentrates within a finite domain outside which the amplitude of the wave is either zero or almost zero. The finite domain is
called the bottom of the profile and its length is called the bottom length.Our goal is to show that classic quadratically nonlinear theory permits analyzing the interaction of not only harmonic
(periodic) waves with sinusoidal profiles but also aperiodic (simple) waves with arbitrary smooth initial profiles.
Remark 2.22. A simple wave describes the distortion of the initial profile, and this distortion is due to the nonlinear
dependence of the phase velocity on the phase. A similar dependence for the phase velocity is incorporated into the definition of
dispersive periodic waves: the frequency and wave number appearing in the dispersion equation are in nonlinear relationship.
Therefore, the phase velocity depends nonlinearly on the frequency. This nonlinear dependence is the cause of the distortion of a
pulse formed from waves with close frequencies. It is obvious that these mechanisms are similar.
We will further consider plane waves and two standard cases of their excitation. Let us begin with the first standard
case. This is about one longitudinal wave entering a medium. In this case, the wave phenomena are described by one equation
(2.1). It is convenient to rearrange it as follows:
u c u utt L x xx12
1 11 0, , ,( )− + =α , (2.69)
where [ ]c L = +( ) /λ µ ρ2 is the phase velocity of a longitudinal wave in a linear elastic body and α λ µ= +[ / ( )] N 1 2 is a
nondimensional quantity.
The subsequent analysis is devoted to a wave of longitudinal displacement. We will deviate from the classic harmonic
solution and will search for the solution in the form of a simple wave—an initial disturbance propagating in space and varying in
time. This disturbance is a finite longitudinal displacement whose initial profile,
u x F x( , ) ( )0 1= , (2.70)
is a twice differentiable continuous function.
Remark 2.23. The above-posed problem on the motion of a pulse may be considered as a model problem for a
semiinfinite rod: a longitudinal displacement, disturbance, initially generated at the end of the rod within a small bounded area
moves as a longitudinal wave along the rod.
The solution of Eq. (2.69) is sought in the form of the D’Alembert wave with an unknown variable phase velocity v ph
u x t F x v t ( , ) ( )= −1 ph . (2.71)
Remark 2.24. The phase velocity v ph may be treated as a local velocity at a point x and at time t . In this case, the analysis
of solution (2.71) agrees with the well-known procedure for simple waves. The procedure used below agrees also with
Lighthill’s description of Riemann’s waves [55].
In the case being considered, the formula v c u x ph = +1 11 α , is valid, whence it follows that the phase velocity depends
nonlinearly on the solution, which means that the wave has all the attributes of a simple wave.
Instead of the exact solution (2.71), we may use an approximate formula accurate to within the squares of the small
nonlinear terms αu x1, :
( )u x t F x c u t x1 1 1 1
1 21( , ) ,
/= − +
α ( )( ) ( ) ( ) ( )[ ]≈ − − ≈ − − ′ − F x c t c tu F x c t c t F x c t x1 1 1 1 1 1 1 1 1
21 2 1 2/ /,α α
( ) [ ] ( )= − = −u x t c t u x t u u x t x1 1 1
2
10
11 2 1 2lin lin lin( , ) / ( , ) ( , ) /,α ( ) [ ]c t u u x t x1 10 2
1
2α , ( , )lin , (2.72)
where u10 is the maximum amplitude in the initial profile and u x t 1
lin ( , ) is the linear solution normalized to u10 .
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Remark 2.25. The first term in Eq. (2.76) describes a linear harmonic wave, which is also called the first harmonic. The
second term also describes a linear harmonic wave, which is the second harmonic. This is the way the generation of the second
harmonic is described and this is the main feature of solution (2.76). However, there is one more important feature associated
with the form of the amplitude of the second harmonic. This amplitude is proportional to the squared wave number k 12 , the
squared initial amplitude a0
2 , and the spatial coordinate x1
. The presence of the spatial coordinate is responsible for the distortion
of the initial profile. Such a distortion is the main nonlinear effect in this case and in the whole theory of nonlinear waves.
Let us now address solution (2.75). Its structure is the same as that of the classical solution (2.76). The first term on the
right-hand side describes a wave that is the initial bell-shaped profile moving without distortions with a constant phase velocity
equal to the velocity of longitudinal waves in a linear elastic body.
Thus, the first and second harmonics for the bell-shaped wave are e x c t − −( ) /12 2 and e x c t − −( )1
2, respectively. They both
are simple waves in the sense of D’Alembert since their velocity is constant. Thus, the effect of generation of the new simple
wave—second harmonic—is described using the above approach. Computer simulation of wave profile evolution was discussed
in [55].
The above-described interaction of longitudinal simple plane waves is not an exception. It must be manifested with all
types of waves. We will demonstrate this for simple transverse plane waves. Consider the second standard problem. Recall that it
is about transverse oscillation motion only. In this problem, wave motion is described by system (2.9), (2.10). This system issolved by the method of successive approximations, assuming smallness of the nonlinear terms in the second approximation. In
the first approximation, the transverse motion is linear and the solution assumes the form of a simple D’Alembert wave
u x t F F x c t 3 3 3 3 3( , ) ( ) ( )= ≡ −τ . (2.77)
The wave is simple in the sense of D’Alembert and its initial profile u x F x3 30( , ) ( )= propagates without distortions with
a constant velocity equal to that of classical transverse waves in an elastic medium, c3.
In the first approximation, the longitudinal wave does not interact with the transverse wave and it is not excited. The
second approximation for the transverse wave agrees with the first approximation, i.e., the simple wave has the basic property of
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the harmonic wave—it does not generate new transverse waves. However, it generates a new longitudinal wave, as follows from
Eqs. (2.9) and (2.10).
Let us take representation (2.77) into account and write Eq. (2.9) in the form
( ) [ ]{ }u c u N F tt xx1 12
1 1 3 32
3, , / ( ) /− = ∂ ∂ρ τ τ . (2.78)
It is easy to show that the partial solution of this equation with a known right-hand side has the form of a simple wave
[ ]u x t n1 2 3 3 3 1( , ) ( ) ( )= −Φ Φτ τ ,
( ) [ ]τ ρ τ θ θτ
1 1 2 2 32
12
3 3 32
0
2
3
= − = − = ∫ x c t n N c c F d , / ( ) , ( ) ( )Φ .
Thus, the new simple D’Alembert longitudinal wave is the difference between two simple D’Alembert longitudinal
waves propagating with different phase velocities c1 and c3.
2.4. Solitary (Aperiodic) Waves by the Microstructural Theory of the Second Order. We start with the basic system of
wave equations (1.31)–(1.33). A problem of simple waves in a microstructural medium such as a mixture arises quite naturally
once this problem is successfully solved in the classical formulation. However, there are some restrictions in this problem. The
first restriction is that the basic equations of mixture theory do not admit D’Alembert wave solutions.Let us discuss this problem. First recall that the equations of plane linear waves in a mixture are the direct generalization
of the classical Klein–Gordon equation. Both equations describe similar effects. The first of them is the microstructural
dispersion of harmonic waves.
From the mathematical standpoint, the first restriction is due to the presence of the term − −βα δ
( )( ) ( )
u uk k .
In all cases, harmonic solutions follow from the property of the exponential function whereby all its derivatives are
expressed in terms of the function itself. Then the question logically arises: Will a pulse whose initial profile is not arbitrary, in
contrast to the D’Alembert solution, but is described by some function with predefined properties preserve its shape while
propagating? Or, to put it differently, is there a solution in the form of a simple wave whose profile is defined by the initial
function?
Let us first outline this problem as a whole. Assume that, according to the above-mentioned property, the second
derivative of the function is expressed in terms of the function itself. This condition is not exotic. Some classes of special
functions, such as
the Chebyshov–Hermite function ′′ + + − =ψ ψ n n z n z z ( ) ( ) ( )1 2 02 ,
the Whittaker function ( ) ( )[ ]{ }′′ + − + + − =w z z w1 4 1 4 02 2/ / /λ / µ , and
the Mathieu function ′′+ − =w p q x w( cos )2 2 0,
possess such a property. All the three equations have a similar form:
′′ = F z f z F z ( ) ( ) ( ), (2.79)
f is a known function.
If we select the initial pulse in the form of the function F(x) from Eq. (2.79) and search for the solution in the form of a
simple wave with the same profile F z ( ) ( z x v z t = − ph ( ) ), then the nonclassical terms − −β α δ( )( ) ( )u uk k do not hamper the
transformation of the basic equations into the dispersion equations. In this case, however, the variability of the phase velocities
will be restricted tv z o′ =( ) ( )1 , which is the price we pay for using this approach.
Let the initial pulse be given by u x A x( ) ( )( , ) ( )α α ψ 0 0= . We will search for the solution in the form
u x t A z A e z 1 0
22( ) ( ) ( ) /( , ) ( )α α αψ = ≡ − , (2.80)
where z x v z t = − ( ) is the phase and v z ( ) is the phase velocity. The final solution has the following form [55]:
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In this case, the quadratically nonlinear term ( ), , N u u1 1 1 1 1 1 on the right-hand side is responsible for the
above-mentioned basic effect in the theory of nonlinear waves. The additional cubically nonlinear term N u u3 1 11 1 12
, ,( ) in
Eq. (3.4) is responsible for the generation of third harmonics. Thus, we have a new possibility to analyze the generation of third
harmonics.
Possibility 2. Possibility 2 is the most natural special case of Possibility 1. We can analyze how the progress (either in
time or in space) of the second and third harmonics taken separately affects the evolution of the initial harmonic profile of the
longitudinal wave. Results of this analysis can be compared at this stage, but later they should be considered together.
Possibility 3. In the second standard problem of the quadratically nonlinear wave-interaction analysis, a transverse
vertical wave is initially excited and new longitudinal waves are generated (transverse waves cannot be generated). Introducing
cubic nonlinearity endows this problem with new features. The term N u u4 2 11 2 12
,*
,*( ) in Eq. (3.5) describing the propagation of a
transverse wave guarantees that in the approach of elastic potential a transverse horizontal wave will generate its own third
harmonics. It is also important that in the classical quadratic approach where the term N u u4 2 11 2 12
,*
,*( ) is absent, the initially
excited transverse wave in the form of the first harmonic does not generate other harmonics beside itself, and, therefore, the
second standard problem is not interesting in the context of transverse waves.
Possibility 4. A new problem can be formulated within the framework of approach (3.4)–(3.6). We will call it the fourth
standard problem. Here a certain (say, vertical) transverse wave is initially excited, and then it generates another (horizontal)
transverse wave. Thus, it becomes possible to describe the new effect of energy pumping from one transverse wave to another.
The next two possibilities are associated with the method of slowly varying amplitudes. Let us return to Eqs. (3.1)–(3.3)and apply this method (which involves long calculations). Here we face many possibilities, of which we will mention two. Note
that using Eqs. (3.1)–(3.3), one can obtain and analyze the shortened and evolution equations only for the specific case of cubic
nonlinearity.
Possibility 5. Using the evolution equations, we can study the interaction of four waves or the so-called wave quadruple
problem. There are various possibilities for the choice of four elastic waves, including the case of two pumping waves, one signal
wave, and one idler wave, which is similar to the parametric amplification problem.
Possibility 6. Using the evolution equations with cubic nonlinearity, we can analyze the self-switching problem (it has
been considered above for quadratically nonlinear elastic waves). Here it is also possible to describe the frequency switching
mechanism for elastic waves whereby the frequency changes by a factor of three.
Each of the six possibilities can be treated as a particular development of the nonlinear model constructed here, on the
one hand, and as a theoretical prediction of new wave effects, on the other hand.Thus, there are many new possibilities for the analysis of nonlinear waves.
Let us start with the first possibility and consider the first standard problem. We neglect some nonlinear terms on the
right-hand side of the initial wave equation Eq. (3.4):
( )ρ λ µu u N u utt 1 1 11 3 1 11 1 122,
**,
**,
*,
*( )− + = . (3.7)
The first approximation is still linear and can be described by formula (2.2). The second approximation can be obtained
by the procedure for quadratic nonlinearity:
( ) ( ) ( )u x t x N
u k k x t 13
10 3
13
1
1
8 2** * *( , ) sin=
+
−
λ µω ( ) ( ) ( )−
+
−
1
24 23
310 3
13
1 x N
u k k x t λ µ
ω* *sin , (3.8)
and the full solution is expressed as u x t u x t u x t 1 1 1( , ) ( , ) ( , )* **= + .
The plots below show how the progress of the third harmonic affects the evolution of the longitudinal wave profile.
There is some similarity between the influence analyses in this and the quadratically nonlinear cases. First, the same three
materials were found out to be the most characteristic. We divide all the plots into two sets (with one parameter fixed and another
varied), each set consisting of three groups. The essential distinction from the quadratically nonlinear case is that appreciable
influence at reasonable distances cannot be observed for small strains (small initial amplitudes) and low frequencies.
The transition from small to finite strains is possible with the Murnaghan potential, since it was originally introduced for
finite strains.
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The last plot corresponds to an initial amplitude five times greater than 0.1 mm (Fig. 14). Similarly to Figs. 3, 7, and 13,
the materials 11 and 62 represent a limiting case of evolution, which means that the wave profile becomes strongly distorted after
several initial oscillations (developed evolution due to the progress of the third harmonic for a fixed initial frequency).
Thus, a ten-fold increase in the amplitude (from 0.05 to 0.5 mm) results in the manifestation of all the stages of
evolution at rather small distances for the material 62.
In contrast to the second harmonic, the progress of the thirdharmonic has a different effect on the evolution (the patterns
of wave profile evolution are essentially dissimilar) and the resulting influence is more sensitive to changes in the initial
amplitude than to changes in the initial frequency. The third harmonic of a longitudinal wave can be detected at higher
frequencies and higher amplitudes than the second harmonics.
Remark 3.1. Cubic and quadratic nonlinearities are both present. Therefore, the observed second and third harmonics of a longitudinal wave cannot be separated (or can be separated conditionally). The third harmonic can be experimentally detected
only in combination with the second one. The computer simulation revealed that it is possible to detect second harmonics whose
initial frequencies and amplitudes much lower than those needed to detect the third harmonic. In some materials (e.g., the
materials 11 and 62), when the second harmonic is fully mature, the third harmonic has not even begun to develop yet. However,
the material 41 is an example of materials in which both harmonics affect the evolution simultaneously. This is, however,
observed within a range where the second harmonic already dominates and the third harmonic still develops.
Remark 3.2. The above-mentioned difficulties with separation of the effects of the second and third harmonics for
longitudinal waves are absent for transverse waves. In this case, the second harmonic is not excited, and we observe only the
third harmonic. This enables us to combine theoretical and experimental studies.
1391
Fig. 10
0.1
0 20.2 20.4 20.6 20.8 21
Fig. 11
0 0.6 0.7 0.8 0.9
0.1
Fig. 12
0.15
0 0.05 0.1 0.15 0.2 0.25
8/3/2019 001 Cattani and Rushchitskii Int App Mechanics 2003 39 12
Let us now analyze a typical problem for cubically nonlinear waves using the method of slowly varying amplitudes. Let
only a longitudinal wave be initially excited and let it do not generate transverse waves.
We will exclude from the analysis the effect of quadratic nonlinearity on the interaction of longitudinal waves,
assuming that this problem has been studied well. Thus, we will analyze the self-interaction and interaction of longitudinal
waves with one another. These effects are described by only one equation (2.9)
( )ρ λ µu u N u utt 1 1 11 3 1 11 1 122, , , ,( )− + = . (3.9)
Remark 3.3. It is useful to note that so-called four-wave interactions are most interesting from the physics standpoint in
the wave interaction analysis for cubically nonlinear media. Just as three-wave interactions in quadratically nonlinear media
imply that wave triplets occur, four-wave interactions in cubically nonlinear media imply that wave quadruplets can form under certain conditions. Therefore, an analysis of four-wave interactions in an elastic medium (according to Eq. (3.9)) seems
promising.
General conditions for effective interactionof waves-participants are matching of frequencies and wave vectors [82, 88]:
ω ω ω ω4 1 2 3=± ± ± , (3.10)
r r r rk k k k 4 1 2 3= ± ± ± . (3.11)
The scheme of interaction according to the general conditions (3.10) and (3.11) is usually interpreted in optics [88] as
the generation of a new wave by three different pumping waves. This scheme can be implemented in an infinite number of ways,
1392
Fig. 13
0.04
0 3.1 3.2 3.3 3.4 3.5
Fig. 14
0 50.2 50.4 50.6 50.8 51
0.1
Fig. 15
1.5
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175
8/3/2019 001 Cattani and Rushchitskii Int App Mechanics 2003 39 12
31. Ya. Ya. Rushchitskii and I. A. Ostrakov, “Distortion of a plane harmonic wave propagating in a composite material,”
Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 11, 51–54 (1991).
32. Ya. Ya. Rushchitskii, “Interaction of elastic waves in a two-phase material,” Prikl. Mekh., 28, No. 5, 13–21 (1992).
33. Ya. Ya. Rushchitskii and E. V. Savel’eva, “Evolution of harmonic wave propagating in a two-phase material,” Prikl.
Mekh., 28, No. 9, 42–46 (1992).
34. Ya. Ya. Rushchitskii and V. O. Bilyi, “On longitudinal and transverse affinor Green’s functions and existence of wave
triplets in multiphase media,” in: Abstracts of Papers Read at Int. Math. Conf. dedicated to Academic P. Kravchuk ,
Kiev–Lutsk, Ukraine (1992), p. 184.35. Ya. Ya. Rushchitskii, “Interaction of compression and shear waves in a composite material with nonlinearly elastic
components in the microstructure,” Prikl. Mekh., 29, No. 4, 18–26 (1993).
36. Ya. Ya. Rushchitskii, “Nonlinear plane waves in an orthotropic body,” Dop. Akad. Nauk Ukrainy, No. 12, 60–62 (1993).
37. J. J. Rushchitsky, “Analytical investigation of wave triplets in a multi-phase medium,” in: Abstracts of Papers Read at
the 1st European and 13th Int. Conf. on Nonlinear Vibrations, Hamburg–Harburg, Germany (1993), p. 138.
38. Ya. Ya. Rushchitskii, “Resonance interaction of nonlinear waves in two-phase mixtures,” Prikl. Mekh., 30, No. 5, 32–41
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39. J. J. Rushchitsky, “The combinational scattering of sound on sound in multi-phase solids,” in: Abstracts of Lectures Read
at the 26th Symp. on Vibrations in Physical Systems, Poznan, Poland (1994), pp. 124–125.
40. Ya. Ya. Rushchitskii and V. O. Bilyi, “Toward the existence of certain type of combinational scattering of sound on
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42. J. J. Rushchitsky, “Interaction of elastic waves in an isotropic multi-phase materials,” J. Theor. Appl. Mech., Bulgarian
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43. J. J. Rushchitsky, “Wave triplets in two-phase composites,” Mech. Comp. Mater ., 31, No. 5, 660–670 (1995).
44. Ya. Ya. Rushchitskii, “Analysis of wave interaction in two-phase materials within the framework of the micro-structural
theory of mixtures,” in: Abstracts of Papers Read at the Int. Sci. Colloq. on Nonlinear Dynamics of Solids with
Microstructure [in Russian], Nizhnii Novgorod, Russia (1995), pp. 21–22.
45. Ya. Ya. Rushchitskii, “Waves and the direct piezoelectric effect in two-phase piezoelectrics,” Prikl. Mekh., 31, No. 10,
42–48 (1995).
46. J. J. Rushchitsky, “Lax–Nelson problem on the direct piezoelectric effect in two-phase materials,” ZAMM (Special issuefor lectures of ICIAM-95), 52, No. 1, 411–414 (1996).
47. Ya. Ya. Rushchitskii, E. V. Savel’eva, and A. P. Kovalenko, “Self-generation of transverse waves in hyperelastic
media,” Prikl. Mekh., 32, No. 6, 30–38 (1996).
48. J. J. Rushchitsky, “New properties of transverse waves in quadratically nonlinear elastic materials (non-classic results in
classic theory),” in: Abstracts of Lectures Read at the 26th Symp. on Vibrations in Physical Systems , Poznan, Poland
(1996), pp. 244–245.
49. Ya. Ya. Rushchitskii, “Three-wave interaction and generation of the second harmonic in single-phase and two-phase
56. Ya. Ya. Rushchitskii and I. M. Khotenko, “Nonlinear plane harmonic waves in two-phase piezoelectric powders,” Dop.
Acad. Nauk Ukrainy, No. 10, 60–66 (1998).
57. Ya. Ya. Rushchitski and S. I. Tsurpal, Waves in Materials with Microstructure [in Russian], Inst. Mekh., Kiev (1998).
58. J. J. Rushchitsky, “Interaction of waves in solid mixtures,” Appl. Mech. Rev., 52, No. 2, 35–74 (1999).
59. Ya. Ya. Rushchitskii, “On the classification of elastic waves,” Int. Appl. Mech., 35, No. 11, 1104–1110 (1999).
60. J. J. Rushchitsky, “Theoretical and computer modeling of non-linearly deformed piezoelectric powders,” Int. J. Nonlin.
Sci. Phys. Simul ., No. 2, 202–213 (2000).
61. Ya. Ya. Rushchitskii, “Extension of the microstructural theory of two-phase mixtures to composite materials,” Int. Appl. Mech., 36, No. 5, 586–614 (2000).
62. J. J. Rushchitsky, “Phenomenon of self-switching waves: statement, shortened and evolution equations,” in: Abstracts of
Lectures Read at the 19th Symp. on Vibrations in Physical Systems, Poznan, Poland (2000), pp. 202–203.
63. J. J. Rushchitsky, “Novel solitary waves in elastic materials: existence and nonlinear interaction,” in: Abstract Book of
the ICTAM 2000, Chicago, USA (2000), p. 165.
64. Ya. Ya. Rushchitskii, “Novel solitary waves in elastic materials: existence and nonlinear interaction,” Int. Appl. Mech.,
37, No. 7, 921–928 (2001).
65. Ya. Ya. Rushchitskii, “Self-switching of plane waves in hyperelastic materials,” in: Abstracts of Lectures Read at the 8th
All-Russian Congr. on Applied and Theoretical Mechanics [in Russian], Perm, Russia (2001), p. 56.
66. Ya. Ya. Rushchitskii, “Self-switching of waves in materials,” Int. Appl. Mech., 37, No. 11, 1492–1498 (2001).
67. J. J. Rushchitsky and S. V. Sinchilo, “Energy of plane harmonic waves in composite materials,” in: Abstracts of Papers Read at the 1st SIAM-EMS Conf. on Applied Mathematics in Our Changing World , Berlin, Germany (2001), p. 46.
68. J. J. Rushchitsky, “Computer-aided analysis of the new type of waves in elastic materials,” in: Book of Abstracts of the
5th World Congr. on Computational Mechanics, Vol. 1, Vienna, Austria, (2002), p. 286.
69. Ya. Ya. Rushchitskii, I. N. Khotenko, and E. V. Savelieva, “Plane shear waves in piezopowders: unexpected new
phenomenon—the generation of the third harmonics,” in: Book of Abstracts of the 30th Int. Summer School “ Advanced
Problems in Mechanics”, St. Petersburg, Russia (2002), p. 84.
70. J. J. Rushchitsky, “Self-switching of displacement waves in elastic nonlinearly deformed materials,” Comptes Rendus de
l’Academie des Sciences, Serie IIb Mecanique, 330, No. 2, 175–180 (2002).
71. J. J. Rushchitsky, “Some new aspects of the plane wave second harmonics generation in elastic bodies,” in: Abstracts of
Papers Read at the 20th Symp. on Vibrations in Physical Systems , Poznan, Poland (2002), 252–253.
72. J. J. Rushchitsky and E. V. Terletska, “Computer modelling of the plane solitary wave evolution (the micro-structurecharacteristic size and the wave bottom interaction),” in: Abstracts of Papers Read at the 20th Symp. on Vibrations in
Physical Systems, Poznan, Poland (2002), pp. 254–255.
73. J. J. Rushchitsky, I. N. Khotenko, and E. V. Savelieva, “Some new aspects of the plane wave second harmonics
generation in elastic bodies,” in: Abstracts of Papers Read at the 20th Symp. on Vibrations in Physical Systems , Poznan,
Poland (2002), pp. 256–257.
74. J. J. Rushchitsky and C. Cattani, “Nonlinear acoustic waves in materials: retrospect and some new lines of
development,” in: Abstracts of Workshop on Potential Flows and Complex Analysis, Kiev, Ukraine (2002), p. 16.
75. Ya. Ya. Rushchitskii and C. Cattani, “Generation of the third harmonics by plane waves in Murnaghan materials,” Int.
Appl. Mech., 38, No. 12, 1482–1487 (2002).
76. Ya. Ya. Rushchitskii and C. Cattani, “The subharmonic resonance and second harmonic of a plane wave in nonlinearly
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material and the trough length of a solitary wave on its evolution,” Int. Appl. Mech., 39, No. 2, 197–202 (2003).
78. J. J. Rushchitsky and C. Cattani, “The third harmonics generation and wave quadruples,” in: Abstracts of GAMM Annual
Conf ., Padua, Italy (2003), p. 234.
79. Ya. Ya. Rushchitskii, “Cubic nonlinearity in the theory of waves in elastic materials,” in: Abstracts of Papers Read at the
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84. J. J. Rushchitsky and C. Cattani, “Shortened and evolution equations of interaction of plane cubically nonlinear elastic
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85. J. J. Rushchitsky, C. Cattani, and S. V. Sinchilo, “Comparative analysis of evolution of the elastic harmonic wave profile
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86. J. J. Rushchitsky and I. N. Khotenko, “Cubically nonlinear waves in piezoelastic materials,” Int. Appl. Mech., 40 (2004)
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87. J. J. Rushchitsky, C. Cattani, and E. V. Terletska, “Evolution of a solitary wave in a microstructural
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88. J. J. Rushchitsky, I. N. Khotenko, E. V. Savel’eva, and S. V. Sinchilo, “Cubically nonlinear waves in piezocomposites—computer simulation of new interaction effects,” Int. Appl. Mech., 40 (2004) (in print).
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