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Theory of non-linear wave propagation with application to the interaction and inverse problems
THEORY WITH
OF NON-LINEAR WAVE PROPAGATION APPLICATION TO THE INTERACTION AND INVERSE PROBLEMS? J.
IdtutC
ENGELBRECHT
ofcybernetics,Lenin Avc 10,200104TdIiq
Estonian S.S.R.,U.S.S.R.
(Received 30 September 1976) Abstract-The propagation of non-linear deformation waves in a dissipative medium is descrfbcd by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical mod& are derived from the theories of thermoeiasticity or viscoelasticity taking into account the geometric and physical non-hncarities and dispersion. On the basis of cikonal equations for the associated linear problem the transport equations of the ntb order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are anafyz4. Conditions are also studied when the interaction of non-linear waves does not take ptace. The inverse problem of determining materials properties according to pulse shape changes is discuss&.
1. INTRODUCTION
In recent years asymptotic methods have been developed for the analysis of problems in nonlinear wave propagation. Among the others, the methods based on the analysis of wave front kinematics correspond more exactly to the nature of the wave propagation process. For linear problems this leads to ray method which have played an important role in geometrical optics, geometrical acoustics etc. The main results in this field may be found in the publications of Lewis [ 11, Hayes [2] and Babich and Bouldyrev [3]. Using the ideas of the ray method for the non-linear problems, the main attention must be devoted to the choice of ray curves and to the construction of the set of transport equations. An excellent review on this subject is given by Germaine [4] and Taniuti and his co-workers [S-7]. Such a practical approach to the wave propagation problems leads to the describing of interesting physical phenomena in plasma physics [S-S], non-linear acoustics [9-J,deformation wave theory [IO] etc. In this paper an attempt is made to derive a general three-dimensional theory of non-linear deformation wave propagation following the basic ideas of Germaine [S] and Taniuti [S). In Part 2 the general problems are analysed. As the mathematical models are usually the mixed hyperbolic-parabolic quasilinear systems with weak parabolic part, the associated hyperbolic linear system can be evaluated by a trivial expansion. On the basis of the eikonal equation for this hyperbolic system the phase velocities are determined and then the transport equations of nth order (n = 1,2,. . .) along the correspondent bicharacteristics are obtained taking into account all effects of higher order. The transport equations govern the amplitude factor when the structure of the wave vector is determined by the zero-th approximation which in the simpler cases gives the dependence on the correspondent eigenvectors. In the multidimensional case the method of matched separation of initial equation is proposed that permits one to obtain the transport equations taking into account the diffractional expansion of a bounded impulse. On the basis of general theory, the exceptional cases are analysed. First, the transport equations &e obtained for plane, cylindrical and spherical one~mension~ waves. In the case of a nonhomogeneous media the second order effects due to non-linearity and nonhomogeneity are compared. In Part 3, the interaction problems are discussed. The principle of multiwave systems generalizes the usual ray method and permits one to analyze the interaction in head-on t PreMntut at the
14th Int. Ccmgr. of T~#~~al
and Applied Mechunics, De@
(1976). 189
Titc Netherlands.
September
J. ENGELBRECHT
190
collision of deformation waves and in reflexion process from boundaries and interfaces. The transport equations for incident and reflected pulses are of the same form the difference being only in the coefficient of the non-linear term. The contact conditions for the non-linear transport equations of the first order, following the same order of expansion, are linear. In Part 4, the inverse problems are formulated and analysed. First the simplest inverse problem in the case of one-dimensiohal non-linear and dissipative medium is formulated. The transport equation of the first order in this case is the Burgers equation and the inverse problem leads to determining of the parameter r by known solutions at two points’of space. This parameter r, as the measure of the importance of non-linearity to that of dissipation, gives the additional information about the properties of the medium. Some more complicated inverse problems of acoustodiagnostics of layered medium are further analysed. Part 5 consists of some conclusions. 2. GENERAL
THEORY
We consider a system which can be written with the summation convention in the form %!+ I at
AK -$$
+ E 1
B;f
P=2
dpU qx=yqxJy
(2.1)
+ H = O
where U = IlUil!, AK = Ila~ll, e! i.j =
1,2 ,..., n;
a,fl=
0,1,2,3;
= ll&$l!, K = 1,2,3;
H = l!Ml r+s
= p 2
2
Here X“ are space coordinates (further, mainly Lagrangian ones are used), X0 =. t, U is a column vector of field Variables, Eis a small parameter. The matrices AK. B;/ and the vector H in general, depend on the XK and U(X’, t) being the functions of geometrical and physical parameters of media. Further we pay attention mainly to the homogeneous media, therefore we set AK = A’(U), B$ = B;:(U), H = H(XK, U). (2.2) In section 2.3, the problems of nonhomogeneous media are briefly discussed. The asymptotic analysis of the system (2.1) with the following initial and boundary conditions (2.3 1 U(XK, t)lr=to = 0, U(XK, t)l, = cp(Xl) where S is a determined contour, forms the main content of this paper. The following assumptions are made [ 1l] : (i) The terms in equation (2.1) are smooth functions of U. (ii) It is possible to develop the vectors U, H and matrices AK, B;f into the power series in a small parameter u = u,+&u,+ .... B;f = B$,+EB;{~(U),
AK = A;+eA:(U)+ ... . H = H,+&H,+ . . .
(2.4)
(iii) The eigenvalues of At are real, distinct and at least one of them is non-degenerate. No special attention is paid to the class of equations to which the method is applicable, however the case when the eigenspace does not comprise any invariant subspace is meant. The equation (2.1), in general, presents a mixed hyperbolic-parabolic system with infinite velocities and the ray method cannot be used for the system at once. Introducing the formal expansion (2.4) in (2.1), one gets for the first approximation
z-duo + . at
AK
au0
odXK+Ho=O.
(2.5)
On the basis of assumptions made above, the equation (2.5) is hyperbolic. Further, equation (2.5) is called the linear associated system. For the associated equation the usual scheme of the ray method may be used. The wave front is determined by the eikonal equation t = (p(XK)
(2.6)
Theory of non-linear wave propagation
191
when the amplitude function is determined by the transport equations along the correspondent rays. In this case the transport equations are ordinary linear dilTerentia1 equations [3] and the solution is expressed as a sum of simple waves. The equation (2.17) (further called a basic system) describes a more complicated process. The wave vector U(T) describes a progressive wave, i.e. “there exists a family of propagating surfaces & = t - (p(XK) such that the magnitude of the rate of change of U(P) or of its derivatives is small compared with the magnitude of the rate of change of U(X’) when XR is kept fixed” [4]. Thevariable t is the rate of the distance from the wave front (when t = 0, then t = r&X=)). The main principle for finding the asymptotic solution of the basic equation can now be formulated as follows : the transport equations of the basic system in terms of the amplitude factor along the correspondent rays are constructed on the basis of wave front kinematics of the associated system. The main principle needs some additional remarks: 1. As the associated and basic systems differ in the order O(E)then the estimations @((E”), where m is a real number are used. 2. For making use of the definition of a progressive wave, the variable 5 is introduced among a new set of independent variables. 3. If bounded pulses in multidimensional medium are analysed then the method of matched separation of initial equations together with the method of stretched coordinates is used. The principle and the remarks suggest introduction of new independent variables through the relations < =&+&Xx)),
TVz ?+l*m(“)X.
(2.7)
Here v has three values from the set {0,1,2,3} in accordance with the character of the boundary conditions. The power k governs the character of the process-k = - 1, leads to best asymptotics for short pulses, k = 0 describes the whole process, etc. The power m(v) gives the order of the change rate along corresponding coordinates. Further we shall analyse separately two main cases. 2.1 Equal rates in all directions. Without loss of generality we assume m(v) = 0 and analyse the process in the far field (k = 0). Hence substituting the series (2.4) in equation (2.1) and performing the transformation (2.7) yields the set of equations :
(2.8)
+ MofUo)+H,
=0
(2.9) (2.10)
etc. Here
(2.11) (2.12)
delta. For the case of simplicity it is assumed that <,, u O(e). From (2.8) and (2.11) one finds
and 6,, is Kronecker
UO NLM Vol. 12.No. 4-B
=
ao(t,
r”)m
(2.13)
J. ENGELBRECHT
192
where a,, is the amplitude factor and m(Y)is the structure factor. If H, = 0 then (2.14)
ull = a,(<, r” )r
where r is the right eigenvector of At for p,r. Substituting (2.13) or (2.14) in (2.9) and multiplying equation (2.9) by the normalized eigenvector I of A$ for rp, we get
aa0
aa0
ao,dt’+alao,5 Here ao,,,al,
azpl
a3 are constr
+
C a2,p=2
apa, g-P
+
a3ao
=
(2.15)
0.
‘s given by the relations a,,* = I(lb,, + AtS,,,)r al = lAf(r)rcp, azp = B%(5,)%,)S~ uj = IH,(r).
The equation (2.15) is the transport equation of the first order along the corresponding ray. This general form contains four independent variables and .the transport equation is no longer an ordinary differential equation as in the linear case but a partial differential equation. The equation (2.15) must be solved with the initial condition
(2.16)
a,(& r” )Is = 1$(4 f f Is.
Next, using the same procedure from the equation (2.10) yields the transport equation of the second order that allows one to determine the amplitude factor a, for U,. Tb solution of the equation (2.15) plays the role of a source for the transport equation of the second order. In general, the procedure may be continued for constructing the transport equations of the nth order, every equation of nth order depends on the solution of the equation of (n - 1)th order. However, it must be pointed out, that for correct non-linear transport equations of second order and higher, the accuracy of constitutive equations must be correspondingly high. So for example, to get the correct non-linear transport equation of second order the constitutive equations must contain cubic terms of deformation tensors, etc. According to this scheme, the correspondence between linear and non-linear problems is quite clear. The zero-th approximation gives in both cases the eikonal equation and the next approximation and higher give the transport equations for basic equations. On this stage all the effects of second order non-linear, dissipative, dispersive etc. effects are taken into account. Let us consider for the example the viscoelastic one-dimensional media [12]. With dimensionless variables 5 = A,t - X’, f1 = &X1, I, = 1 we get the transport equation of the first and second order correspondingly -
$+a,,,%
$+b,
f? + b,, x
a%, a<
a%,
x2
+ az2 F
= 0 (2.17)
+ b,a, +b,
= 0
etc. Here ao, a, are amplitude factors of U. and U, correspondingly, coefficients.
and Ui, bi(ao) are
2.2 D@erent rates in various directions This case must be analysed for the bounded pulse. Let the pulse be generated in the direction Xi. Then, according to the idea of the method of stretched coordinates we introduce 5 = Ek(t_(P(XK)),
ri = ak+i+m(i)Xi. rv = ek+l+m(v)X.
(2.18)
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193
Besides, we assume that wave vector U may be separated in parts V = I(vkl!= I!ukll, k = 1,2,. . . ,q
u=v+w,
W=
IjWkI!
=
J!UjlJ,
j=q+l,...,n
(2.19)
and the expansion V=V,+&V,+...,
w = &‘(Wo+&W,+ . ..)
(2.20)
is possible. The equation (2.1) yields
av
av
aw
aw
Ix+pK@
apv
+ ’ pT2 ‘ifa(xya(xry +c=o
+QKS
(2.21)
IpRKs
+ SKf$
+D=O + c E:,8 a(x=ya(xs)s E
4-=2
where PK, QK, RK, SK - GF/, E;! are matrices, and C, D are vectors. Substituting (2.20) in the equations (2.21) and performing the transformation (2.18) to the ray coordinates we get a complicated system for V and W, consisting of the estimations 0()(~‘), [(a”), l$s’). Dependent on those estimations, on the order of cpi, cpu, and on the value of matrices the coefficients of this system lead to various transport equations. Further we shall analyze the basic case. Let the components of vector V be generated on the boundary S. Due to coupling the components of vector W will be generated in the course of time. From the zero-th approximation : (2.22) VI3 = a,(& tl‘)r. Then from the first approximation
the operator (2.23)
may be separated. If the structure of the first approximation does not allow the generation of W then the operator ML (2.23) describes the propagation of the pulse V, in the course of all time and we have (2.24)
1Mo = 0.
This is exactly the transport equation of the first order for the case of the plane front. In other cases, we search for certain m and t that allow one to obtain the transport equation in the form (2.25) where $ = (7") 7i,t;>.The equation (2.25) is the generalized transport equation of the first order that takes into the account the diffractional expansion in the directions Xj,j # i. The character of the equation (2.25) depends on the structure of the operator Mb. The equation (2.25) for the special case of propagation of acoustic waves is liquids was derived in [ 133 and is called the Zabolotskaya-Khokhlov equation [9]. Let us consider for example two-dimensional medium in Cartesian coordinates. For a liquid we use the Navier-Stokes equations [9], for a solid medium-the model of non-linear viscoelastic theory [12]. The transport equation of the first order for bounded pulse in the Xl-direction yields
a da,
@1g + ‘la052 ag +
b’a, 1
az2 2 at
d2a,
=
'5 q72)2
(2.26)
where non-dimensional coordinates < = t-X’, r1 = &X1,7' = smXZare used, and a,, az2, as coefficients. The equation (2.26) is obtained for the liquid with m = *, t = 3; for the solid
are
194
J. ENGELBRECHT
medium with m = j, t = 4. It shows that in a solid medium the change rate in the transverse direction X2 is greater than in the case of the liquid. 2.3. Exceptional cases 2.3.1. One dimensional plane waves. In this case the transport
equation of the first order
takes the form (2.27) that is known in many publications [4-l 11. This transport equation may lead to the Burgers equation, to the Korteweg-de Vries equation, etc. The solution of the equation (2.27) for the Cauchy problem is investigated by many authors [5,8,9, 141 and many interesting physical effects are described. For deformation waves in solids the equation (2.27) is obtained in [ 10, 15, 161, for viscoelastic rods and plates [lo], for a viscoelastic rod taking into account the geometrical dispersion [15], and for thermoelastic and Maxwell-type viscoelastic media [ 161. One-dimensional longitudinal and transverse waves are analysed in [ 173. 2.3.2. One-dimensional non-planar waves. The transport equation of the first order for the viscoelastic medium yields dct,
n
da0
~+~~O+vo~+a22~=0
azCr,
(2.28)
where n = 1,2 for cylindrical and spherical waves respectively. Such processes in thermoelastic and viscoelastic media are analysed in [ 181. 2.3.3. Non-homogeneous medium. In this case the basic equation (2.1) needs an additional term to describe more general class of problems and has the form
Iau dt+AKy d”;+ c B;! E
aw
dF
a(x=ya(xfly +KMaxX +f~='
(2.29)
where the vector F(XK) is the known function of space coordinates characterizing the nonhomogeneity of the medium, and AX = AK(XM,U), B;! = B;!(X”,U). The associated equation now gives the velocities of the wavefronts as functions of space coordinates i.e. the rays form the curves in space. Further we restrict ourselves to one-dimensional processes. Performing the transformation (2.30) where 1,(X’) is the velocity of a certain wave we get using Uo = x0(5, r)r the transport equation of the first order aa0
at
+
+ a,cr, + ad(T) = 0 ((2,ao+a(T))$$! + 1 a2p-Pa, w
(2.31)
p=2
wherea,,a 22, a3 are constants and a(r), a,(z) are functions of r. A similar equation is obtained in [6]. The transport equation is a complicated one and needs further analysis. However, some conclusion can be made on the basis of shock wave analysis in solid media and numerical calculations : (i) Non-linearity predominates over nonhomogeneity when la,aol >>la(r)1 and nonhomogeneity predominates over non-linearity when la,czol CCIa(s (ii) Waves with different amplitudes propagate differently in the nonhomogeneous nonlinear medium. (iii) The formation of a shock wave is possible when alao + a(t) > 0. (iv) If the velocity i, is increasing (decreasing) then the dissipative effects decrease (increase).
195
Theory of non-linear wave propagation
To sum up, the effects of the second order due to non-linearity and nonhomogeneity of a medium are of the same order and need to be taken into account simultaneously. The numerical calculations for the equation (2.31) show that the steepening of the wave profile for certain amplitudes can be caused by nonhomogeneity at even a greater rate than by nonlinearity. The calculations were done for weak change of elastic moduli of the second order and it has demonstrated the validity of the conclusion (i). As the coefficient a(t usually a monotonous function of z (or initially of space coordinate) then there exists a certain region where the nonlinearity may be a dominating factor only in the beginning of the process if a(7) is an increasing function, or from a certain space point if a(7) is a decreasing function. In the case of weak change in cross section of a circular bar with constant elastic moduli taking into account the influence of geometrical dispersion, the transport equation of the first order leads to the nonhomogeneous Korteweg-de Vries equation with variable dissipation. It follows then that the formation of a shock profile (without dispersion effects) is not dependent on change of the cross-section. 2.3.4. Geometrical background. The geometrical background of the change of independent variables is as follows. The new variable [ is chosen so that it is the measure from the front of the associated system. According to physical reasons it has bounded values or changes very slowly. The measure of the movement in space is characterized by one of the space coordinates that is perpendicular to the surface of the action. For the two-dimensional case this is illustrated in Fig. 1, where Fig la shows the ray tube in the initial coordinates t, X’, X2 and Fig. lb shows the same in new coordinates {, 7l, 7’. When the values of coordinates t, X1
r2/
Fig. 1.
became arbitrary large with X2 changing moderately then from the new coordinates only 7l will become arbitrarily large, < will be bounded and 72 will change moderately. This will provide a certain convenience when using asymptotic methods, especially in the case of high derivatives near the front. In distinction from linear problems the wave profile is changing in the course of propagation due to non-linear effects. 3. INTERACTION
OF
NON-LINEAR
WAVES
The principle of superposition does not hold for non-linear waves therefore the processes of head-on collision, of reflexion from boundaries, etc., need special attention in non-linear theory. According to the principles of the ray method, the generalization with regard to
196
J. ENGELBRECHT
multiwave systems gives the best results [7, 191. We shall present here the main results for one-dimensional waves and apply the method for the analysis of deformation waves. Following [7, 191 we introduce the stretched variables 5j
=
&k[ijt-X'-&~j(X',t)],
5
=
(3.1)
Ek+"X'
for the basic equation (2.1) in one-dimensional form. In (3.1) ij are the eigenvalues of the matrice AA;and #](X’, t) are the phase functions that take into account the variation in wave velocities due to the interactions. Then the wave vector U, is expanded in the series of proper eigenvectors of the matrice Ah (3.2)
u* = x ai(5i, rNi*
Every amplitude factor ai is determined by its own transport equation of the first order: Jai %Oz
da, +
apa.
+
%laiz
C ai2p I
+
at!
U,i”i
=
0
(3.3)
while the phase functions are determined by the relation: 4i =
c
liAt(rj)ri(i.i-i.j)-'
(3.4)
j#i
where 8, is an arbitrary function determined from boundary conditions (usually It follows from (3.3) and (3.4) that amplitude functions may be superposed with account the phase changes. It is easily seen that in the linear case A: = 0 therefore superposition principle holds in the sense of the word. The investigation of the head-on collision of two longitudinal waves in viscoelastic media leads to the following conclusions:
13~= 0). taking into 4i = 0 and non-linear
(9 The greater the amplitude and the length of a bounded pulse, the more the phase velocity of another bounded pulse changes in head-on collision with the first one. (ii) The qualitative change of the phase velocity depends on the character of the medium and is governed by physical and geometrical non-linearities. 1. In the head-on collision of pressure pulses the velocities of both waves increase when(l+mo)~Oanddecreasewhen(l+mo) 0. (iii) If the condition :, aid5 = 0 s0 is fulfilled then the non-linear interaction is of higher order and does not change any phase velocity of aj, j # i. 1. Sine-pulses with complete number of waves do not change the phase velocities after head-on collision. 2. Solitons always interact with each other. Here m. = (v, + v2 -t va)(i + 2~)~ ’ is the relation of third and second order elastic moduli in one-dimensional process. For metals (1 + mo) < 0 always [ 151. In a liquid the relation (1 + m,) must be changed to -(y + 1) where y is the adiabate. As y > 0 then it is obvious that the main conclusions for metals and liquids coincide. The next problem is the interaction of the incident and reflected waves in the course of reflexion. Applying the method given above for analyzing the one-dimensional wave propagation in a layer we get the transport equation for all incident and reflected pulses in a form + afak = 0
(3.5)
Theory of non-linear wave propagation
197
where u: is a sign alternative coefficient. The independent variables are given by formulae (3.6)
x: = x11,,‘, x;
= (X1 -2X,)43,
x:
= (Xl +2X,)&i,.
..
where X, is the thickness of the layer, Arnris the wave velocity according to the associated equation, k = 0,1,2,. . . is the number of the pulse (k = 0 corresponds to the pulse generated at X1 = 0, k = 1 corresponds to the first reflected pulse at Xi = X, etc.). For this non-linear transport equation the boundary and contact conditions are linear. This means, that the nonlinear effects due to propagation are more significant compared to those due to non-linear contact conditions and interaction between every incident and reflected pulse. The transport equations of second order take into account the non-linearity of boundarity conditions while the interaction of every incident and reflected pulse may be determined by the method described above for the head-on collision. In this case we conclude the following:
(9 The greater the amplitude and the length of an incident pulse, the more the phase velocity changes in the reflexion process. (ii) The qualitative change of the phase velocity depends on the properties of the medium and on the character of the boundary condition : 1. The length of the pulse does not change in the case of an acoustically rigid boundary. 2. If (1 + m,-J > 0, then the length of the compression pulse increases and the length of the rarefaction pulse decreases in the case of an acoustically soft boundary. 3. If (1 + me) < 0 then the length of the compression pulse decreases and the length of the rarefaction pulse increases in the case of an acoustically soft boundary. (iii) In the case of the reflexion from an interface between the elastic layers the conclusions for the acoustically rigid boundary are correct, if K < 1 and those for the acoustically soft boundary are correct if K > 1. Here K is the ratio of acoustical impedances of the (n + 1)th and nth layers. In the case of successive reflexion in a layer with acoustically soft boundaries the (iv) length of the pulse changes monotonically. The same conclusions may be drawn for the multidimensional bounded pulse. It follows from the first conclusion that the influence of interaction of short pulses is small and can be neglected in many cases.
4. INVERSE
PROBLEMS
Every pulse (deformation wave) propagating in a non-linear and dissipative (and dispersive) medium will be distorted in the course of time. The propagation problems have been discussed by many authors, for example [8, 9, 14, IS]. The exact solution of corresponding transport equations is possible only for a few cases but the asymptotic formulae have been developed for a large class of problems [14] that allows one to find an asymptotic solution for a known equation with known boundary conditions. The corresponding inverse problem can be formulated as follows: knowing the form of the transport equation and solutions at least in two points of space-time region the coefficients of the transport equation are searched for. The distortion of a generated pulse permits one to determine the coefficients of the transport equation that allows one to get information about the physical and geometrical properties of the medium We shall formulate now the simplest inverse problem. In the one-dimensional viscoelastic medium the transport equation of the first order is the well-known Burgers’ equation. In normalized form we write : (4.1)
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J.ENGELBIWHT
where
Here x,-is the maximum amplitude, r‘eis the characteristic length of the incident pulse, and the parameter I is a measure of the importance of non-linearity relative to that of dissipation. The direct problem for the equation (4.1) is to find the solution with initial condition (4.2)
81dZ.7, = 81(r)
The exact solution needs the Cole-Hopf transformation [20] that permits one to write the solution in the form of the improper integral. The inverse problem includes the finding the parameter I knowing the solutions BL,
= B&)9
L,,
= 820
(4.3)
This leads to integral equation with regard to the parameter I. As the exact solution of this equation is not known, for the case of incident sine-pulse the asymptotic solutions ofCole, Fay, improved Fay etc. [14] may be used. In this case the amplitudes of higher harmonics give the most important information. The distortion nomograms for various amplitudes and frequencies seem to be best for a practical use. With the aid of the inverse problem of the transport equation (4.1) it is possible to take into account not only the usual attenuation due to the dissipative effects but also the extra attenuation as a result of non-linear effects over the normal small amplitude attenuation. The inverse problem for direct propagation gives the information about the parameter I for the given medium, more exactly about the material parameter G = - I(r,crm,3- 1 = at (&)- ‘. The solution of model inverse problems permits one to analyse more complicated cases of acoustodiagnostics. That leads to the construction of various reflected and refracted pulses. The ray method solves the indicated problem in a very elegant way. Let us consider the simplest onedimensional problem of construction of reflected and refracted pulses from an interface between two media A and B, when wave propagation in both media is described by the equation (2.1). According to the linear associated equation we have for the incident pulse U,, and for the reflected pulse U,, the independent variables (3.6) and for the refracted pulse Ugr r B*=t-
?s, = &(X-X,).
x’ - (1 -&l’)Xe, i,El
(4.4)
The transport equations of the first order for all those pulses are of the form (3.5) (read index k as ik, where i = A, B). As was shown in Part 3, the boundary conditions for non-linear transport equation are linear and read for the reflected pulse a2l72A=0= - La1(5fAI
T,),
T1 = sXO
(4.5)
and for the refracted pulse &L,,=O = wAea1(418,T1)
(4.6)
where K-
VAB =-,
K+l
1
WAB=-, 2K K+l
K=--.PBCB PAC.4
The same procedure can be followed in more complicated problems with more interfaces. One of such problems is shown in Fig. 2 with two layers A and B and a half-space C. The inverse problem of acoustodiaghostics for this particular case consists of the determination of the properties of the layer B and/or of the half-space C, making use of reflected pulses U,, and UAB,. The transport equations for all incident reflected and refracted pulses can be determined and the direct problem is solved easily. If all the layers and half-space are described by non-linear theory with viscous effects then all the transport equations of the first order take the form of Burgers’ equation. Following the model inverse problem, the extra
Theoryof non-linearwavepropagation
199
1
4 .:I .;: <
4 ::: 2): 5:
/
w /
/ -x’
L
*I
x0
-I
C
Fig.2. information may be obtained for this particular case. As a matter of fact the information available from the registrated pulses is as follows:
6) The incident pulse =
~,41L~=O
A,JI,4,0).
(4.7)
(ii) The first reflected pulse from the interface between the layers A and B UAIIX’=O
=
B&At)
= b,A,k,,“,,$,&)
t1 = 2x,.
(4.8)
(iii) The second reflected pulse from the interface between the layer B and half-space C, that is refracted into the layer A ~MI1IX’=O= BAslijAB,(t)= t,
=
WBAI/BC WAB A 1k Al k El k I#2k ABl $ ABI
(4.9
2X0 + 2X;k; I.
Here k, is the normalized velocity in the layer B; eA2, tiAB, are the functional dependances of the pulse shape, kiL, k,jb in’= A, B; k = 1,2,. . . are the attenuation coefficients due to viscous and non-linear effekts, and t;, i = 1,2 is the pulse entering time. The analysis of formulae (4.7), (4.8), (4.9) gives the possibility to determine: (i) The impedance of the layer B, taking into account the attenuation of the incident signal. (ii) The additional parameter Ts (or GB). (iii) The impedance of the half-space C. The last cases (ii) and (iii) need some additional conditions. When the impedances of layers A and B are equal then the reflected pulse UA2 corresponding to the correctness of the transport equations of the first order equals zero and only one reflected signal reaches the point X1 = 0. The transport equations of second order give the corresponding signal not equal to zero, but having order O(E). However, if the attenuation properties of the layers A and B differ, then the reflected pulse from the interface
200
J. ENGELBRECHT
between layer B and half-space C permits one to detect the layer B according to the change in the reflected pulse amplitude. It must be pointed out that in the inverse problems of acoustodiagnostics the dissipation plays a very important role for a quantitatively correct solution and must be taken into account also in linear problems. The non-linearity from its side gives new information about the properties of a medium. 5. CONCLUSIONS The transport equations permit one to find asymptotic values of the amplitude factor according to corresponding boundary conditions. The wave vector is then determined usually as a series of corresponding eigenvectors and in the case of the multi-wave method it contains a sum of all types of waves. The main results for one-dimensional problems can easily be generalized for multidimensional problems and so the interaction of pulses and the inverse problems can be analysed principally in the same way. As far as is known to the author, the solution of the inverse problem in non-linear and nonhomogeneous medium in spite of its practical importance is not known. The general theory of non-linear waves for special cases of plasma physics, fluid dynamics etc. was developed under the name of “the reductive perturbation method” [S-7,11,19]. It seems that the concept of ray techniques, since it is the background of the method, must be more clearly underlined. The notion of a moving frame only cannot give the basic ideas of the ray method that includes the eikonal equations and the set of transport equations. Therefore the basic idea of the ray method must be always underlined in asymptotic techniques of this type C4]. REFERENCES 1. R. M. Lewis, Asymptotic theory of wave propagation. Archs ration. Mech. Analysis 20, 191-250 (1965). 2. W. D. Hayes, Kinematic wave theory. Proc. R. Sot. A320,209-226 (1970). 3. V. M. Babich and V. S. Bouldyrev, Asymptotic Methods in Short Wave Diffraction Problems. Nauka, Moscow (1972) (In Russian). 4. P. Getmaine, ~~ress~~e B’uues. Pp. 1l-30. fber. DGLR (1971) KoIn ( 1972). 5. T. Taniuti, Reductive perturbation method and far fields of wave equations. Prog. theor. Phys. Suppl. No. 55, l-35 (1975). 6. N. Asano, Wave propagations in nonuniform media. Ibid. 52-79 (1974). 7. M. Oikawa, N. Yajima, Generalization of the reductive perturbation method to multiwave systems. Ibid. 36-51 (1974). 8. V. I. Karpman. Non-Linear Waves in Dispersive Media. Nauka. Moscow (1973) (in Russian). 9. D. V. Roudenko and S. I. Soluvan. Tbeoreticai Foundutions of Non-Linear Acoustics. Nauka, Moscow (1975). (in Russian). 10. A. Sedov, G. A. Nariboli, V&co-elastic waves by the use of wave-front theory. fnt. J. Non-Linear Mech. 6, 615-624 (1971).
J. phys. Sot. Japan 11. T. Taniuti, C.-C. Wei, Reductive perturbation method in non-linear wave propagation-I. 24,941-946 (1968). 12. C. Beevers and J. Engelbrecht, Coastitutive rate-dependent theory of thermo-viscoelasticity. ENS k’ Teud.‘Akad. Toim., Fiiiisika-Matemaatika (Proc. Est. Acad. Sci., Phvs. Matem.), X393-400 (1975). 13. E. A. iaboiotskaya and R. V. Khokhlov. Quasiplanar waves in.non-linear acoustics of bounded puises. Sourer J. Acoust. 15,40-47 (1969) (in Russian). 14. D. T. Blackstock, Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves. J. ucoust. Sot. Am. 36,534-542
(1964).
15. J. K. Engelbrecht, The mathematical models of shock waves in elastic bodies. Proc. Symg. Non-linear and Thermal Eficrs in Transient Wave Propagation, Tallinn, Gorky-Tallinn. Gorky University Press, Vol. I(l973). 16. V. G. Karnaukhov, On finite amplitude viscoelastic wave propagation. Appi. Mech. (Kieo) 9.36-44 (1973) (in Russian). 17. J. K. Engelbrecht, Non-linear dilatational and transverse waves in a half-space. Theor. uppl. Me&. (So_fiaf,5, 77-86 (1974) (in Russian). 18. J. K. Engelbrecht, Model equations of non-linear thermoelasticity. Selected Problems of Applied Mechanics, pp. 731-737. Nauka, Moscow (1974) (in Russian). 19. T. Tatsumi and H. Tokunaga, One-dimensional shock turbulence in a compressible fluid. J. Fluid Mech. 65, 581-601(1974). 20. J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics. Q. uppl. Math. 9-225-236 (1951).
Theory of non-linear wave propagation Re'sumeOn discute la propagation d'ondes de &formation non lineaires dans un milieu dissipatif par une th6orie anymptotique unifi6e utilisant la cin&natique du front d'onde et les concepts de propagation des ondes. Les modales math&natiques partent des thgories de la thermoBlasticitd ou de la visco6lasticit6 en prenant en compte les non li&aritCs gdom&riques et physiques ainsi que On obtient les 6quations de transport du lp,dispersion. nleme ordre a partir des hquations d'Eikon pour le probl& lin6aire associe. On propose pour le cas multidimensionnel la m6thode de separation parallele des Equations initiales. On analyse les probl&nes d'interaction qui apparaissent dans les collisions frontales et dans la reflexion aux limites ou interfaces. On etudie Bgalement les conditions o‘u l'interaction d'ondes non lineaires ne se produit pas. On discute le probl&ne inverse de.diterminer les propri6te's des matiriaux selon les changemnts de form des pulsations.
Zusamnenfassunq: Die Ausbreitung nichtlinearer Verformungswellen in einem dissipativen Material wird durch eine vereinheitlichte asymptotische Theorie beschrieben , wobei die Wellenfrontkinematik und der Begriff der progressiven Wellen benutzt werden. Due mathematischen Modelle werden von der Theorien der Thetmoelastizitat oder Viskoelastizitat unter Berucksichtigung geometrischer und physikalischer Nichtlinearitaten und Dispersion hergeleitet. Basierend auf den eikonischen Gleicehungen des zugehorigen linearen Problems werden die Transportgleichungen n-ter Ordnung aufgestelqt. Fur den mehrdimensionalen Fall wird die Plethode der angepassten Trennung der ursprunglichen Gleichungen vorgeschlagen. Die Wechselwirkungen fur direkte Zusammenstosse und fur Reflektionen von Bersrenzungen oder Zwischenschichten werden untersucht. Weiterhin werden Bedingungen untersucht, unter denen Wechselwirkung nichtlinearer Wellen unterbleibt. Das umgekehrte Problem, d.h. die Bestimmung von Stoffeigenschaften auf Grund der Formanderungen der Impulse, wird diskutiert.
201
OF NON-LINEAR WAVE PROPAGATION APPLICATION TO THE INTERACTION AND INVERSE PROBLEMS? J.
IdtutC
ENGELBRECHT
ofcybernetics,Lenin Avc 10,200104TdIiq
Estonian S.S.R.,U.S.S.R.
(Received 30 September 1976) Abstract-The propagation of non-linear deformation waves in a dissipative medium is descrfbcd by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical mod& are derived from the theories of thermoeiasticity or viscoelasticity taking into account the geometric and physical non-hncarities and dispersion. On the basis of cikonal equations for the associated linear problem the transport equations of the ntb order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are anafyz4. Conditions are also studied when the interaction of non-linear waves does not take ptace. The inverse problem of determining materials properties according to pulse shape changes is discuss&.
1. INTRODUCTION
In recent years asymptotic methods have been developed for the analysis of problems in nonlinear wave propagation. Among the others, the methods based on the analysis of wave front kinematics correspond more exactly to the nature of the wave propagation process. For linear problems this leads to ray method which have played an important role in geometrical optics, geometrical acoustics etc. The main results in this field may be found in the publications of Lewis [ 11, Hayes [2] and Babich and Bouldyrev [3]. Using the ideas of the ray method for the non-linear problems, the main attention must be devoted to the choice of ray curves and to the construction of the set of transport equations. An excellent review on this subject is given by Germaine [4] and Taniuti and his co-workers [S-7]. Such a practical approach to the wave propagation problems leads to the describing of interesting physical phenomena in plasma physics [S-S], non-linear acoustics [9-J,deformation wave theory [IO] etc. In this paper an attempt is made to derive a general three-dimensional theory of non-linear deformation wave propagation following the basic ideas of Germaine [S] and Taniuti [S). In Part 2 the general problems are analysed. As the mathematical models are usually the mixed hyperbolic-parabolic quasilinear systems with weak parabolic part, the associated hyperbolic linear system can be evaluated by a trivial expansion. On the basis of the eikonal equation for this hyperbolic system the phase velocities are determined and then the transport equations of nth order (n = 1,2,. . .) along the correspondent bicharacteristics are obtained taking into account all effects of higher order. The transport equations govern the amplitude factor when the structure of the wave vector is determined by the zero-th approximation which in the simpler cases gives the dependence on the correspondent eigenvectors. In the multidimensional case the method of matched separation of initial equation is proposed that permits one to obtain the transport equations taking into account the diffractional expansion of a bounded impulse. On the basis of general theory, the exceptional cases are analysed. First, the transport equations &e obtained for plane, cylindrical and spherical one~mension~ waves. In the case of a nonhomogeneous media the second order effects due to non-linearity and nonhomogeneity are compared. In Part 3, the interaction problems are discussed. The principle of multiwave systems generalizes the usual ray method and permits one to analyze the interaction in head-on t PreMntut at the
14th Int. Ccmgr. of T~#~~al
and Applied Mechunics, De@
(1976). 189
Titc Netherlands.
September
J. ENGELBRECHT
190
collision of deformation waves and in reflexion process from boundaries and interfaces. The transport equations for incident and reflected pulses are of the same form the difference being only in the coefficient of the non-linear term. The contact conditions for the non-linear transport equations of the first order, following the same order of expansion, are linear. In Part 4, the inverse problems are formulated and analysed. First the simplest inverse problem in the case of one-dimensiohal non-linear and dissipative medium is formulated. The transport equation of the first order in this case is the Burgers equation and the inverse problem leads to determining of the parameter r by known solutions at two points’of space. This parameter r, as the measure of the importance of non-linearity to that of dissipation, gives the additional information about the properties of the medium. Some more complicated inverse problems of acoustodiagnostics of layered medium are further analysed. Part 5 consists of some conclusions. 2. GENERAL
THEORY
We consider a system which can be written with the summation convention in the form %!+ I at
AK -$$
+ E 1
B;f
P=2
dpU qx=yqxJy
(2.1)
+ H = O
where U = IlUil!, AK = Ila~ll, e! i.j =
1,2 ,..., n;
a,fl=
0,1,2,3;
= ll&$l!, K = 1,2,3;
H = l!Ml r+s
= p 2
2
Here X“ are space coordinates (further, mainly Lagrangian ones are used), X0 =. t, U is a column vector of field Variables, Eis a small parameter. The matrices AK. B;/ and the vector H in general, depend on the XK and U(X’, t) being the functions of geometrical and physical parameters of media. Further we pay attention mainly to the homogeneous media, therefore we set AK = A’(U), B$ = B;:(U), H = H(XK, U). (2.2) In section 2.3, the problems of nonhomogeneous media are briefly discussed. The asymptotic analysis of the system (2.1) with the following initial and boundary conditions (2.3 1 U(XK, t)lr=to = 0, U(XK, t)l, = cp(Xl) where S is a determined contour, forms the main content of this paper. The following assumptions are made [ 1l] : (i) The terms in equation (2.1) are smooth functions of U. (ii) It is possible to develop the vectors U, H and matrices AK, B;f into the power series in a small parameter u = u,+&u,+ .... B;f = B$,+EB;{~(U),
AK = A;+eA:(U)+ ... . H = H,+&H,+ . . .
(2.4)
(iii) The eigenvalues of At are real, distinct and at least one of them is non-degenerate. No special attention is paid to the class of equations to which the method is applicable, however the case when the eigenspace does not comprise any invariant subspace is meant. The equation (2.1), in general, presents a mixed hyperbolic-parabolic system with infinite velocities and the ray method cannot be used for the system at once. Introducing the formal expansion (2.4) in (2.1), one gets for the first approximation
z-duo + . at
AK
au0
odXK+Ho=O.
(2.5)
On the basis of assumptions made above, the equation (2.5) is hyperbolic. Further, equation (2.5) is called the linear associated system. For the associated equation the usual scheme of the ray method may be used. The wave front is determined by the eikonal equation t = (p(XK)
(2.6)
Theory of non-linear wave propagation
191
when the amplitude function is determined by the transport equations along the correspondent rays. In this case the transport equations are ordinary linear dilTerentia1 equations [3] and the solution is expressed as a sum of simple waves. The equation (2.17) (further called a basic system) describes a more complicated process. The wave vector U(T) describes a progressive wave, i.e. “there exists a family of propagating surfaces & = t - (p(XK) such that the magnitude of the rate of change of U(P) or of its derivatives is small compared with the magnitude of the rate of change of U(X’) when XR is kept fixed” [4]. Thevariable t is the rate of the distance from the wave front (when t = 0, then t = r&X=)). The main principle for finding the asymptotic solution of the basic equation can now be formulated as follows : the transport equations of the basic system in terms of the amplitude factor along the correspondent rays are constructed on the basis of wave front kinematics of the associated system. The main principle needs some additional remarks: 1. As the associated and basic systems differ in the order O(E)then the estimations @((E”), where m is a real number are used. 2. For making use of the definition of a progressive wave, the variable 5 is introduced among a new set of independent variables. 3. If bounded pulses in multidimensional medium are analysed then the method of matched separation of initial equations together with the method of stretched coordinates is used. The principle and the remarks suggest introduction of new independent variables through the relations < =&+&Xx)),
TVz ?+l*m(“)X.
(2.7)
Here v has three values from the set {0,1,2,3} in accordance with the character of the boundary conditions. The power k governs the character of the process-k = - 1, leads to best asymptotics for short pulses, k = 0 describes the whole process, etc. The power m(v) gives the order of the change rate along corresponding coordinates. Further we shall analyse separately two main cases. 2.1 Equal rates in all directions. Without loss of generality we assume m(v) = 0 and analyse the process in the far field (k = 0). Hence substituting the series (2.4) in equation (2.1) and performing the transformation (2.7) yields the set of equations :
(2.8)
+ MofUo)+H,
=0
(2.9) (2.10)
etc. Here
(2.11) (2.12)
delta. For the case of simplicity it is assumed that <,, u O(e). From (2.8) and (2.11) one finds
and 6,, is Kronecker
UO NLM Vol. 12.No. 4-B
=
ao(t,
r”)m
(2.13)
J. ENGELBRECHT
192
where a,, is the amplitude factor and m(Y)is the structure factor. If H, = 0 then (2.14)
ull = a,(<, r” )r
where r is the right eigenvector of At for p,r. Substituting (2.13) or (2.14) in (2.9) and multiplying equation (2.9) by the normalized eigenvector I of A$ for rp, we get
aa0
aa0
ao,dt’+alao,5 Here ao,,,al,
azpl
a3 are constr
+
C a2,p=2
apa, g-P
+
a3ao
=
(2.15)
0.
‘s given by the relations a,,* = I(lb,, + AtS,,,)r al = lAf(r)rcp, azp = B%(5,)%,)S~ uj = IH,(r).
The equation (2.15) is the transport equation of the first order along the corresponding ray. This general form contains four independent variables and .the transport equation is no longer an ordinary differential equation as in the linear case but a partial differential equation. The equation (2.15) must be solved with the initial condition
(2.16)
a,(& r” )Is = 1$(4 f f Is.
Next, using the same procedure from the equation (2.10) yields the transport equation of the second order that allows one to determine the amplitude factor a, for U,. Tb solution of the equation (2.15) plays the role of a source for the transport equation of the second order. In general, the procedure may be continued for constructing the transport equations of the nth order, every equation of nth order depends on the solution of the equation of (n - 1)th order. However, it must be pointed out, that for correct non-linear transport equations of second order and higher, the accuracy of constitutive equations must be correspondingly high. So for example, to get the correct non-linear transport equation of second order the constitutive equations must contain cubic terms of deformation tensors, etc. According to this scheme, the correspondence between linear and non-linear problems is quite clear. The zero-th approximation gives in both cases the eikonal equation and the next approximation and higher give the transport equations for basic equations. On this stage all the effects of second order non-linear, dissipative, dispersive etc. effects are taken into account. Let us consider for the example the viscoelastic one-dimensional media [12]. With dimensionless variables 5 = A,t - X’, f1 = &X1, I, = 1 we get the transport equation of the first and second order correspondingly -
$+a,,,%
$+b,
f? + b,, x
a%, a<
a%,
x2
+ az2 F
= 0 (2.17)
+ b,a, +b,
= 0
etc. Here ao, a, are amplitude factors of U. and U, correspondingly, coefficients.
and Ui, bi(ao) are
2.2 D@erent rates in various directions This case must be analysed for the bounded pulse. Let the pulse be generated in the direction Xi. Then, according to the idea of the method of stretched coordinates we introduce 5 = Ek(t_(P(XK)),
ri = ak+i+m(i)Xi. rv = ek+l+m(v)X.
(2.18)
Theoryof non-linearwavepropagation
193
Besides, we assume that wave vector U may be separated in parts V = I(vkl!= I!ukll, k = 1,2,. . . ,q
u=v+w,
W=
IjWkI!
=
J!UjlJ,
j=q+l,...,n
(2.19)
and the expansion V=V,+&V,+...,
w = &‘(Wo+&W,+ . ..)
(2.20)
is possible. The equation (2.1) yields
av
av
aw
aw
Ix+pK@
apv
+ ’ pT2 ‘ifa(xya(xry +c=o
+QKS
(2.21)
IpRKs
+ SKf$
+D=O + c E:,8 a(x=ya(xs)s E
4-=2
where PK, QK, RK, SK - GF/, E;! are matrices, and C, D are vectors. Substituting (2.20) in the equations (2.21) and performing the transformation (2.18) to the ray coordinates we get a complicated system for V and W, consisting of the estimations 0()(~‘), [(a”), l$s’). Dependent on those estimations, on the order of cpi, cpu, and on the value of matrices the coefficients of this system lead to various transport equations. Further we shall analyze the basic case. Let the components of vector V be generated on the boundary S. Due to coupling the components of vector W will be generated in the course of time. From the zero-th approximation : (2.22) VI3 = a,(& tl‘)r. Then from the first approximation
the operator (2.23)
may be separated. If the structure of the first approximation does not allow the generation of W then the operator ML (2.23) describes the propagation of the pulse V, in the course of all time and we have (2.24)
1Mo = 0.
This is exactly the transport equation of the first order for the case of the plane front. In other cases, we search for certain m and t that allow one to obtain the transport equation in the form (2.25) where $ = (7") 7i,t;>.The equation (2.25) is the generalized transport equation of the first order that takes into the account the diffractional expansion in the directions Xj,j # i. The character of the equation (2.25) depends on the structure of the operator Mb. The equation (2.25) for the special case of propagation of acoustic waves is liquids was derived in [ 133 and is called the Zabolotskaya-Khokhlov equation [9]. Let us consider for example two-dimensional medium in Cartesian coordinates. For a liquid we use the Navier-Stokes equations [9], for a solid medium-the model of non-linear viscoelastic theory [12]. The transport equation of the first order for bounded pulse in the Xl-direction yields
a da,
@1g + ‘la052 ag +
b’a, 1
az2 2 at
d2a,
=
'5 q72)2
(2.26)
where non-dimensional coordinates < = t-X’, r1 = &X1,7' = smXZare used, and a,, az2, as coefficients. The equation (2.26) is obtained for the liquid with m = *, t = 3; for the solid
are
194
J. ENGELBRECHT
medium with m = j, t = 4. It shows that in a solid medium the change rate in the transverse direction X2 is greater than in the case of the liquid. 2.3. Exceptional cases 2.3.1. One dimensional plane waves. In this case the transport
equation of the first order
takes the form (2.27) that is known in many publications [4-l 11. This transport equation may lead to the Burgers equation, to the Korteweg-de Vries equation, etc. The solution of the equation (2.27) for the Cauchy problem is investigated by many authors [5,8,9, 141 and many interesting physical effects are described. For deformation waves in solids the equation (2.27) is obtained in [ 10, 15, 161, for viscoelastic rods and plates [lo], for a viscoelastic rod taking into account the geometrical dispersion [15], and for thermoelastic and Maxwell-type viscoelastic media [ 161. One-dimensional longitudinal and transverse waves are analysed in [ 173. 2.3.2. One-dimensional non-planar waves. The transport equation of the first order for the viscoelastic medium yields dct,
n
da0
~+~~O+vo~+a22~=0
azCr,
(2.28)
where n = 1,2 for cylindrical and spherical waves respectively. Such processes in thermoelastic and viscoelastic media are analysed in [ 181. 2.3.3. Non-homogeneous medium. In this case the basic equation (2.1) needs an additional term to describe more general class of problems and has the form
Iau dt+AKy d”;+ c B;! E
aw
dF
a(x=ya(xfly +KMaxX +f~='
(2.29)
where the vector F(XK) is the known function of space coordinates characterizing the nonhomogeneity of the medium, and AX = AK(XM,U), B;! = B;!(X”,U). The associated equation now gives the velocities of the wavefronts as functions of space coordinates i.e. the rays form the curves in space. Further we restrict ourselves to one-dimensional processes. Performing the transformation (2.30) where 1,(X’) is the velocity of a certain wave we get using Uo = x0(5, r)r the transport equation of the first order aa0
at
+
+ a,cr, + ad(T) = 0 ((2,ao+a(T))$$! + 1 a2p-Pa, w
(2.31)
p=2
wherea,,a 22, a3 are constants and a(r), a,(z) are functions of r. A similar equation is obtained in [6]. The transport equation is a complicated one and needs further analysis. However, some conclusion can be made on the basis of shock wave analysis in solid media and numerical calculations : (i) Non-linearity predominates over nonhomogeneity when la,aol >>la(r)1 and nonhomogeneity predominates over non-linearity when la,czol CCIa(s (ii) Waves with different amplitudes propagate differently in the nonhomogeneous nonlinear medium. (iii) The formation of a shock wave is possible when alao + a(t) > 0. (iv) If the velocity i, is increasing (decreasing) then the dissipative effects decrease (increase).
195
Theory of non-linear wave propagation
To sum up, the effects of the second order due to non-linearity and nonhomogeneity of a medium are of the same order and need to be taken into account simultaneously. The numerical calculations for the equation (2.31) show that the steepening of the wave profile for certain amplitudes can be caused by nonhomogeneity at even a greater rate than by nonlinearity. The calculations were done for weak change of elastic moduli of the second order and it has demonstrated the validity of the conclusion (i). As the coefficient a(t usually a monotonous function of z (or initially of space coordinate) then there exists a certain region where the nonlinearity may be a dominating factor only in the beginning of the process if a(7) is an increasing function, or from a certain space point if a(7) is a decreasing function. In the case of weak change in cross section of a circular bar with constant elastic moduli taking into account the influence of geometrical dispersion, the transport equation of the first order leads to the nonhomogeneous Korteweg-de Vries equation with variable dissipation. It follows then that the formation of a shock profile (without dispersion effects) is not dependent on change of the cross-section. 2.3.4. Geometrical background. The geometrical background of the change of independent variables is as follows. The new variable [ is chosen so that it is the measure from the front of the associated system. According to physical reasons it has bounded values or changes very slowly. The measure of the movement in space is characterized by one of the space coordinates that is perpendicular to the surface of the action. For the two-dimensional case this is illustrated in Fig. 1, where Fig la shows the ray tube in the initial coordinates t, X’, X2 and Fig. lb shows the same in new coordinates {, 7l, 7’. When the values of coordinates t, X1
r2/
Fig. 1.
became arbitrary large with X2 changing moderately then from the new coordinates only 7l will become arbitrarily large, < will be bounded and 72 will change moderately. This will provide a certain convenience when using asymptotic methods, especially in the case of high derivatives near the front. In distinction from linear problems the wave profile is changing in the course of propagation due to non-linear effects. 3. INTERACTION
OF
NON-LINEAR
WAVES
The principle of superposition does not hold for non-linear waves therefore the processes of head-on collision, of reflexion from boundaries, etc., need special attention in non-linear theory. According to the principles of the ray method, the generalization with regard to
196
J. ENGELBRECHT
multiwave systems gives the best results [7, 191. We shall present here the main results for one-dimensional waves and apply the method for the analysis of deformation waves. Following [7, 191 we introduce the stretched variables 5j
=
&k[ijt-X'-&~j(X',t)],
5
=
(3.1)
Ek+"X'
for the basic equation (2.1) in one-dimensional form. In (3.1) ij are the eigenvalues of the matrice AA;and #](X’, t) are the phase functions that take into account the variation in wave velocities due to the interactions. Then the wave vector U, is expanded in the series of proper eigenvectors of the matrice Ah (3.2)
u* = x ai(5i, rNi*
Every amplitude factor ai is determined by its own transport equation of the first order: Jai %Oz
da, +
apa.
+
%laiz
C ai2p I
+
at!
U,i”i
=
0
(3.3)
while the phase functions are determined by the relation: 4i =
c
liAt(rj)ri(i.i-i.j)-'
(3.4)
j#i
where 8, is an arbitrary function determined from boundary conditions (usually It follows from (3.3) and (3.4) that amplitude functions may be superposed with account the phase changes. It is easily seen that in the linear case A: = 0 therefore superposition principle holds in the sense of the word. The investigation of the head-on collision of two longitudinal waves in viscoelastic media leads to the following conclusions:
13~= 0). taking into 4i = 0 and non-linear
(9 The greater the amplitude and the length of a bounded pulse, the more the phase velocity of another bounded pulse changes in head-on collision with the first one. (ii) The qualitative change of the phase velocity depends on the character of the medium and is governed by physical and geometrical non-linearities. 1. In the head-on collision of pressure pulses the velocities of both waves increase when(l+mo)~Oanddecreasewhen(l+mo)
(3.5)
Theory of non-linear wave propagation
197
where u: is a sign alternative coefficient. The independent variables are given by formulae (3.6)
x: = x11,,‘, x;
= (X1 -2X,)43,
x:
= (Xl +2X,)&i,.
..
where X, is the thickness of the layer, Arnris the wave velocity according to the associated equation, k = 0,1,2,. . . is the number of the pulse (k = 0 corresponds to the pulse generated at X1 = 0, k = 1 corresponds to the first reflected pulse at Xi = X, etc.). For this non-linear transport equation the boundary and contact conditions are linear. This means, that the nonlinear effects due to propagation are more significant compared to those due to non-linear contact conditions and interaction between every incident and reflected pulse. The transport equations of second order take into account the non-linearity of boundarity conditions while the interaction of every incident and reflected pulse may be determined by the method described above for the head-on collision. In this case we conclude the following:
(9 The greater the amplitude and the length of an incident pulse, the more the phase velocity changes in the reflexion process. (ii) The qualitative change of the phase velocity depends on the properties of the medium and on the character of the boundary condition : 1. The length of the pulse does not change in the case of an acoustically rigid boundary. 2. If (1 + m,-J > 0, then the length of the compression pulse increases and the length of the rarefaction pulse decreases in the case of an acoustically soft boundary. 3. If (1 + me) < 0 then the length of the compression pulse decreases and the length of the rarefaction pulse increases in the case of an acoustically soft boundary. (iii) In the case of the reflexion from an interface between the elastic layers the conclusions for the acoustically rigid boundary are correct, if K < 1 and those for the acoustically soft boundary are correct if K > 1. Here K is the ratio of acoustical impedances of the (n + 1)th and nth layers. In the case of successive reflexion in a layer with acoustically soft boundaries the (iv) length of the pulse changes monotonically. The same conclusions may be drawn for the multidimensional bounded pulse. It follows from the first conclusion that the influence of interaction of short pulses is small and can be neglected in many cases.
4. INVERSE
PROBLEMS
Every pulse (deformation wave) propagating in a non-linear and dissipative (and dispersive) medium will be distorted in the course of time. The propagation problems have been discussed by many authors, for example [8, 9, 14, IS]. The exact solution of corresponding transport equations is possible only for a few cases but the asymptotic formulae have been developed for a large class of problems [14] that allows one to find an asymptotic solution for a known equation with known boundary conditions. The corresponding inverse problem can be formulated as follows: knowing the form of the transport equation and solutions at least in two points of space-time region the coefficients of the transport equation are searched for. The distortion of a generated pulse permits one to determine the coefficients of the transport equation that allows one to get information about the physical and geometrical properties of the medium We shall formulate now the simplest inverse problem. In the one-dimensional viscoelastic medium the transport equation of the first order is the well-known Burgers’ equation. In normalized form we write : (4.1)
198
J.ENGELBIWHT
where
Here x,-is the maximum amplitude, r‘eis the characteristic length of the incident pulse, and the parameter I is a measure of the importance of non-linearity relative to that of dissipation. The direct problem for the equation (4.1) is to find the solution with initial condition (4.2)
81dZ.7, = 81(r)
The exact solution needs the Cole-Hopf transformation [20] that permits one to write the solution in the form of the improper integral. The inverse problem includes the finding the parameter I knowing the solutions BL,
= B&)9
L,,
= 820
(4.3)
This leads to integral equation with regard to the parameter I. As the exact solution of this equation is not known, for the case of incident sine-pulse the asymptotic solutions ofCole, Fay, improved Fay etc. [14] may be used. In this case the amplitudes of higher harmonics give the most important information. The distortion nomograms for various amplitudes and frequencies seem to be best for a practical use. With the aid of the inverse problem of the transport equation (4.1) it is possible to take into account not only the usual attenuation due to the dissipative effects but also the extra attenuation as a result of non-linear effects over the normal small amplitude attenuation. The inverse problem for direct propagation gives the information about the parameter I for the given medium, more exactly about the material parameter G = - I(r,crm,3- 1 = at (&)- ‘. The solution of model inverse problems permits one to analyse more complicated cases of acoustodiagnostics. That leads to the construction of various reflected and refracted pulses. The ray method solves the indicated problem in a very elegant way. Let us consider the simplest onedimensional problem of construction of reflected and refracted pulses from an interface between two media A and B, when wave propagation in both media is described by the equation (2.1). According to the linear associated equation we have for the incident pulse U,, and for the reflected pulse U,, the independent variables (3.6) and for the refracted pulse Ugr r B*=t-
?s, = &(X-X,).
x’ - (1 -&l’)Xe, i,El
(4.4)
The transport equations of the first order for all those pulses are of the form (3.5) (read index k as ik, where i = A, B). As was shown in Part 3, the boundary conditions for non-linear transport equation are linear and read for the reflected pulse a2l72A=0= - La1(5fAI
T,),
T1 = sXO
(4.5)
and for the refracted pulse &L,,=O = wAea1(418,T1)
(4.6)
where K-
VAB =-,
K+l
1
WAB=-, 2K K+l
K=--.PBCB PAC.4
The same procedure can be followed in more complicated problems with more interfaces. One of such problems is shown in Fig. 2 with two layers A and B and a half-space C. The inverse problem of acoustodiaghostics for this particular case consists of the determination of the properties of the layer B and/or of the half-space C, making use of reflected pulses U,, and UAB,. The transport equations for all incident reflected and refracted pulses can be determined and the direct problem is solved easily. If all the layers and half-space are described by non-linear theory with viscous effects then all the transport equations of the first order take the form of Burgers’ equation. Following the model inverse problem, the extra
Theoryof non-linearwavepropagation
199
1
4 .:I .;: <
4 ::: 2): 5:
/
w /
/ -x’
L
*I
x0
-I
C
Fig.2. information may be obtained for this particular case. As a matter of fact the information available from the registrated pulses is as follows:
6) The incident pulse =
~,41L~=O
A,JI,4,0).
(4.7)
(ii) The first reflected pulse from the interface between the layers A and B UAIIX’=O
=
B&At)
= b,A,k,,“,,$,&)
t1 = 2x,.
(4.8)
(iii) The second reflected pulse from the interface between the layer B and half-space C, that is refracted into the layer A ~MI1IX’=O= BAslijAB,(t)= t,
=
WBAI/BC WAB A 1k Al k El k I#2k ABl $ ABI
(4.9
2X0 + 2X;k; I.
Here k, is the normalized velocity in the layer B; eA2, tiAB, are the functional dependances of the pulse shape, kiL, k,jb in’= A, B; k = 1,2,. . . are the attenuation coefficients due to viscous and non-linear effekts, and t;, i = 1,2 is the pulse entering time. The analysis of formulae (4.7), (4.8), (4.9) gives the possibility to determine: (i) The impedance of the layer B, taking into account the attenuation of the incident signal. (ii) The additional parameter Ts (or GB). (iii) The impedance of the half-space C. The last cases (ii) and (iii) need some additional conditions. When the impedances of layers A and B are equal then the reflected pulse UA2 corresponding to the correctness of the transport equations of the first order equals zero and only one reflected signal reaches the point X1 = 0. The transport equations of second order give the corresponding signal not equal to zero, but having order O(E). However, if the attenuation properties of the layers A and B differ, then the reflected pulse from the interface
200
J. ENGELBRECHT
between layer B and half-space C permits one to detect the layer B according to the change in the reflected pulse amplitude. It must be pointed out that in the inverse problems of acoustodiagnostics the dissipation plays a very important role for a quantitatively correct solution and must be taken into account also in linear problems. The non-linearity from its side gives new information about the properties of a medium. 5. CONCLUSIONS The transport equations permit one to find asymptotic values of the amplitude factor according to corresponding boundary conditions. The wave vector is then determined usually as a series of corresponding eigenvectors and in the case of the multi-wave method it contains a sum of all types of waves. The main results for one-dimensional problems can easily be generalized for multidimensional problems and so the interaction of pulses and the inverse problems can be analysed principally in the same way. As far as is known to the author, the solution of the inverse problem in non-linear and nonhomogeneous medium in spite of its practical importance is not known. The general theory of non-linear waves for special cases of plasma physics, fluid dynamics etc. was developed under the name of “the reductive perturbation method” [S-7,11,19]. It seems that the concept of ray techniques, since it is the background of the method, must be more clearly underlined. The notion of a moving frame only cannot give the basic ideas of the ray method that includes the eikonal equations and the set of transport equations. Therefore the basic idea of the ray method must be always underlined in asymptotic techniques of this type C4]. REFERENCES 1. R. M. Lewis, Asymptotic theory of wave propagation. Archs ration. Mech. Analysis 20, 191-250 (1965). 2. W. D. Hayes, Kinematic wave theory. Proc. R. Sot. A320,209-226 (1970). 3. V. M. Babich and V. S. Bouldyrev, Asymptotic Methods in Short Wave Diffraction Problems. Nauka, Moscow (1972) (In Russian). 4. P. Getmaine, ~~ress~~e B’uues. Pp. 1l-30. fber. DGLR (1971) KoIn ( 1972). 5. T. Taniuti, Reductive perturbation method and far fields of wave equations. Prog. theor. Phys. Suppl. No. 55, l-35 (1975). 6. N. Asano, Wave propagations in nonuniform media. Ibid. 52-79 (1974). 7. M. Oikawa, N. Yajima, Generalization of the reductive perturbation method to multiwave systems. Ibid. 36-51 (1974). 8. V. I. Karpman. Non-Linear Waves in Dispersive Media. Nauka. Moscow (1973) (in Russian). 9. D. V. Roudenko and S. I. Soluvan. Tbeoreticai Foundutions of Non-Linear Acoustics. Nauka, Moscow (1975). (in Russian). 10. A. Sedov, G. A. Nariboli, V&co-elastic waves by the use of wave-front theory. fnt. J. Non-Linear Mech. 6, 615-624 (1971).
J. phys. Sot. Japan 11. T. Taniuti, C.-C. Wei, Reductive perturbation method in non-linear wave propagation-I. 24,941-946 (1968). 12. C. Beevers and J. Engelbrecht, Coastitutive rate-dependent theory of thermo-viscoelasticity. ENS k’ Teud.‘Akad. Toim., Fiiiisika-Matemaatika (Proc. Est. Acad. Sci., Phvs. Matem.), X393-400 (1975). 13. E. A. iaboiotskaya and R. V. Khokhlov. Quasiplanar waves in.non-linear acoustics of bounded puises. Sourer J. Acoust. 15,40-47 (1969) (in Russian). 14. D. T. Blackstock, Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves. J. ucoust. Sot. Am. 36,534-542
(1964).
15. J. K. Engelbrecht, The mathematical models of shock waves in elastic bodies. Proc. Symg. Non-linear and Thermal Eficrs in Transient Wave Propagation, Tallinn, Gorky-Tallinn. Gorky University Press, Vol. I(l973). 16. V. G. Karnaukhov, On finite amplitude viscoelastic wave propagation. Appi. Mech. (Kieo) 9.36-44 (1973) (in Russian). 17. J. K. Engelbrecht, Non-linear dilatational and transverse waves in a half-space. Theor. uppl. Me&. (So_fiaf,5, 77-86 (1974) (in Russian). 18. J. K. Engelbrecht, Model equations of non-linear thermoelasticity. Selected Problems of Applied Mechanics, pp. 731-737. Nauka, Moscow (1974) (in Russian). 19. T. Tatsumi and H. Tokunaga, One-dimensional shock turbulence in a compressible fluid. J. Fluid Mech. 65, 581-601(1974). 20. J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics. Q. uppl. Math. 9-225-236 (1951).
Theory of non-linear wave propagation Re'sumeOn discute la propagation d'ondes de &formation non lineaires dans un milieu dissipatif par une th6orie anymptotique unifi6e utilisant la cin&natique du front d'onde et les concepts de propagation des ondes. Les modales math&natiques partent des thgories de la thermoBlasticitd ou de la visco6lasticit6 en prenant en compte les non li&aritCs gdom&riques et physiques ainsi que On obtient les 6quations de transport du lp,dispersion. nleme ordre a partir des hquations d'Eikon pour le probl& lin6aire associe. On propose pour le cas multidimensionnel la m6thode de separation parallele des Equations initiales. On analyse les probl&nes d'interaction qui apparaissent dans les collisions frontales et dans la reflexion aux limites ou interfaces. On etudie Bgalement les conditions o‘u l'interaction d'ondes non lineaires ne se produit pas. On discute le probl&ne inverse de.diterminer les propri6te's des matiriaux selon les changemnts de form des pulsations.
Zusamnenfassunq: Die Ausbreitung nichtlinearer Verformungswellen in einem dissipativen Material wird durch eine vereinheitlichte asymptotische Theorie beschrieben , wobei die Wellenfrontkinematik und der Begriff der progressiven Wellen benutzt werden. Due mathematischen Modelle werden von der Theorien der Thetmoelastizitat oder Viskoelastizitat unter Berucksichtigung geometrischer und physikalischer Nichtlinearitaten und Dispersion hergeleitet. Basierend auf den eikonischen Gleicehungen des zugehorigen linearen Problems werden die Transportgleichungen n-ter Ordnung aufgestelqt. Fur den mehrdimensionalen Fall wird die Plethode der angepassten Trennung der ursprunglichen Gleichungen vorgeschlagen. Die Wechselwirkungen fur direkte Zusammenstosse und fur Reflektionen von Bersrenzungen oder Zwischenschichten werden untersucht. Weiterhin werden Bedingungen untersucht, unter denen Wechselwirkung nichtlinearer Wellen unterbleibt. Das umgekehrte Problem, d.h. die Bestimmung von Stoffeigenschaften auf Grund der Formanderungen der Impulse, wird diskutiert.
201