Vol.2, No.4, 298-306 (2010) Natural Science
Copyright © 2010 SciRes. OPEN ACCESS
Wave processes-fundamental basis for modern high
Viktor Sergeevich Krutikov
Institute of Pulse Processes and Technologies at NAS of Ukraine, Nikolaev, Ukraine; iipt@iipt.com.ua
Received 28 December 2009; Revised 21 January 2010; accepted 16 February 2010.
Problems of moving boundaries, moving per-
meable boundaries, questions of control over
wave processes are fundamental physical prob-
lems (acc. to V.L.Ginzburg) that exist for a long
time from the moment of the wave equation
emergence, for over three hundred years. This
paper for the first time states a brief, but clear
and quite integral disclosure of the author's
approaches, and also a physical essence of
analytical methods of functions evaluation of
wave processes control - the basic processes of
the Nature and the natural sciences, character-
istic for all objects of the surrounding world
without exception and able to occur only in the
regions with moving and moving permeable
boundaries. Absolutely immovable boundaries
do not exist in the nature. Certain examples
which are fundamental in theoretical physics of
spherical, cylindrical and flat waves, including
the waves induced by dilation of the final length
cylinder, demonstrate physical, mathematical
and engineering lucidity and simplicity (the so-
lution comes to a quadratic equation), and, there-
fore, the practical value of definition of control
function for the predetermined (based on engi-
neering requirements) functions of effect. This
paper is designated for a wide range of scientific
readers, with aim to render to the reader first of
all the physical sense of the studied phenome-
non, to show the novelty that it has introduced
in the development of the corresponding direc-
tion, to show that the way of the research (it is
more important than the result) has not arisen
“out of nothing”, and the gained results are only
“a stone which cost him a whole life” (H.Poin-
Keywords: Waves; Mobility; Permeability;
Boundary; Control; Inverse Problems
“No recipes and prescriptions exist guiding how to move
in an unknown sphere. Steps are taken by the method of
attempts and mistakes. The winner will be the master of
a better intuition and ability to solve a complicated task.
However, it seems that luck and chance are of no less
significance, unless we speak of such giants as Einstein.”
(V.L. Ginsburg, Nobel Prize winner) [1].
The “technologies” of getting scientific results, the
logic of scientific discoveries are particularly individual.
These questions are not solved exhaustively and for the
present it is impossible to teach it. Special analysis
shows that only a small percentage (about two) of all
defended Ph.D. and doctoral theses contains a scientific
novelty. The same small percent-about two and a half–is
observed among the successful businessmen among the
enormous quantity of those who carry on business in
America. For all seeming triviality, the stated questions
have a great scientific and applied significance. There-
fore, it is rightful to describe and enumerate at one time
not only scientific results, but also the technology, the
logic of scientific discoveries which are especially rele-
vant for wave processes and immediately relate to the
theme of this paper. At present, one has just to accumu-
late and comprehend an individual philosophy of each
It is important to get an outstanding, great result, and
it is more important to conceive the ways of its acquisi-
tion. These questions interested many people a long time
ago, for example, Leibniz. The works of Henri Poincar
“Science and hypothesis”, “Value of science”, “Science
and method”, “Last thoughts”, “On science” [2], etc. are
devoted to examination of cognition routs in mathemat-
ics, mechanics, physics. It seems that a lot of researchers,
including Poincar, got scientific results at first, and then,
looking back at their and not only their own traversed
1The American Biographical Institute USA and International Bio-
graphical Centre Cambridge CB2 3QP England, whose question-
naires contain the paragraph "Individual philosophy of success"
and whose staff is more than 14000 of highly experienced and
high-paid experts and specialized scientific institutions are engaged
with it
see Great Minds of 21st Centur
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
path, they comprehended it. This process is essential and
productive. Some basic ideas are selected and extracted
to the epigraphs with similar purpose. It is worth to pay
attention to how harmoniously they blend with the se-
quence of operations as for the problem-solving of
moving permeable boundaries and the problems of wave
processes control. It is the evidence of many things and
first of that is the knowledge put in them which helps to
get a new knowledge, new truths.
This article is not a systematic statement on the given
question. The purpose of this publication is an opportu-
nity kindly rendered by the editors of journal of Natural
Science (NS) to share some considerations connected
with attempts to solve the most difficult problems of
mathematical physics which existed from the moment of
wave equation creation. Scientific results for the solution
to physical problems of moving boundaries (MB), mov-
ing permeable boundaries (MPB) and wave processes
control are rather explicitly stated in the following stud-
ies: V. S. Krutikov: Technical Physics Letters 1988, 1989,
1990, 1999, 2000, 2003, 2003, 2005, 2005 (Chief editor
Zh.I. Alferov); Doklady Physics 1993, 1999, 2006; Dok-
lady Mathematics 1999; Acoustic Physics 1996; Applied
Mathematics and Mechanics 1991; Izvestiya Russian Acad-
emy of Sciences МТТ 1992, etc. This would serve the
continuation of accumulation of information as for the
comprehension of cognition ways by the personal exam-
ple of complicated scientific problems solving.
2.1. Mathematical Models
Wave process is one of the most important forms of sub-
stance motion. To some extent, wave movements are
inherent to all objects of the material world without ex-
ception. Wave processes are the fundamental basis for
the development of natural science, modern techniques
and high technologies. Therefore, the problems-solving
of mathematical physics with its most complicated
problems of moving boundaries (MB), moving perme-
able boundaries (MPB) and the problems of wave proc-
esses control acquire great importance. Such problems
for the regions with moving boundaries are complicated
and little-studied [3].
Both linear and non-linear wave processes are inten-
sively studied in electrodynamics, plasma physics, optics,
fluid dynamics, acoustics, etc. The mechanisms of dis-
turbance propagation naturally differ greatly from each
other. The difference of physical mechanisms which
realize a wave process leads to various different features
in equation systems. However, there is often no neces-
sity to analyze the initial often complicated equations
systems in order to understand the most fundamental
phenomena characteristic for the waves of different na-
ture (interference, diffraction, dispersion, reverberation,
refraction, dispersion, etc). As a rule, elementary effects
are described with the help of simple and, therefore,
universal mathematic models. Hence it is obviously pos-
sible to make a conclusion that the purpose of the re-
search has to be not only the composition of especially
complicated systems of differential equations in partial
derivatives and their solution with the help of powerful
computers, but also the reduction of task solution to the
simplest mathematical model and acquisition of an ana-
lytical solution with the determination of its (model)
validity limits [4,5].
It is a wave equation that serves such a universal
mathematical model describing a lot of physical proc-
esses. It is also necessary to take it into account that due
to some factors, the direct considering of all factors de-
termining the complex phenomenon is a rather difficult
task, and the task solution is achieved by combining the
data on simpler model problems discovered at the analy-
sis stage. Such models are studied by various methods
with outstanding analytical methods of solution which
give the most precious results of an absolute character.
At the same time, physical experiment has a criterion
value, and the potent side of numerical methods of solu-
tion is their efficiency. This statement enhances the role
and shows the importance of solutions of a wave equa-
tion in the regions with moving boundaries, as rather
often the simplest model problems lead directly to it.
However, even if the process under consideration is
described with the help of a linear equation, the presence
of moving boundaries makes the problem substantially
nonlinear, the sum of two solutions is not a solution, and
the method of superposition is inapplicable. The essen-
tial nonlinearity of the problem of moving boundaries
for parabolic equations known as the problem of Stephen
is analyzed in the works of G.A. Grinberg; for the wave
equation it is shown in the works of the author [6,7]. It
was the explanation why there were no methods of an
exact analytical solution to such tasks. Exact solutions of
this sort of problems retrieved mainly due to successful
conjectures are known only for some particular kind of
boundary conditions. Regarding the wave equation, this
is the only one J. Taylor’s solution (1946) [8] of a par-
ticular kind of a direct problem of sphere expansion with
steady speed in a compressible medium. Direct problem
when the conditions are specified at the moving bound-
ary, inverse problem when the additional conditions are
specified at a fixed point of wave zone, it is necessary to
determine the functions under consideration in other
points, including the near-field zone and the surface of
moving boundaries. At the same time, the law of bound-
ary moving is unknown and should be determined; and it
can be non-linear. The presence of permeability of mov-
ing boundaries amends thеsе concepts, new con-
cepts–compound additional conditions–appear [9,10]. Let
us give a definition to some terms: wave zone, near-field
region, moving boundaries surface. While describing the
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
processes in terms of mathematics due to the wave equa-
tion, and when the additional conditions are defined cor-
rectly, the following cases are distinguished: а wave
zone, when the given function (e.g. the pressure function)
is determined through Langrangе- Cauchy linear integral;
b – near-field region – the given function (of pressure) is
determined through Langrangе- Cauchy linear integral; c
– the predetermined function on the moving boundary
surface is determined through Langrangе-Cauchy nonlin-
ear integral (nonlinear additional clause), at the same time
nonlinear law of radius change of the moving surface of
the boundary, which is known beforehand, is included in
compound arguments of these functions on the moving
boundary. Nonlinear additional clauses are the first nonlin-
earity; the presence of moving boundaries is the second
nonlinearity; if these two nonlinearities exist, the prob-
lem is supposed to be twice nonlinear.
Interaction of arguments means the following: the
functions of an additional clause with one sort of argu-
ments are put in an interim decision having another sort
of arguments. As a result, we get the solution with the
third sort of the argument.
The researches had to accept various assumptions: to
change the boundary conditions, to transfer the boundary
conditions to fixed boundaries or to replace the actions
of moving boundaries by the system of peculiarities.
This leads to the limited nature of decisions, and some-
times to unacceptable results. It is indicative that the
solution to the wave equation was got by d’Alember
(1747): Cauchy problem-initial conditions are known,
boundary conditions are missing, and the functions form
that depended on boundary conditions remained un-
known. In a known summarizing of a general method of
terminal integral transformations (Koshlyakov N.S.,
Grinberg G.А.) for direct problems, “momentary” ei-
genfunctions expansion is used. However, it leads to the
solution to the infinite system of first-order differential
equations. Traditional approaches appeared to be unac-
ceptable for the solution to the moving boundaries prob-
lem, and it was necessary to seek for the new ones.
2.2. Interaction of Complicated Non-Linear
The basis for mathematic physics is three main equations:
heat conduction equation, wave and Laplace’s equation.
The largest quantity of processes are described with the
help of wave equations.
Before it could be solved only numerically, that is in
the regions with moving boundaries. Let us ask our-
selves the following question: why the main equation of
mathematical physics was not solved, that is the wave
equation in the regions with moving boundaries. In the
author’s opinion, there are two reasons. The first one is
that only the direct problems were solved (G. Taylor),
which resulted in insuperable mathematical difficulties
under voluntary boundary conditions. The second one is
that the fundamental fact of all wave phenomena in
compressible spheres of disturbances propagation with
finite speed was not taken into account. The author
managed to overcome theses difficulties in the following
way. First of all, let us examine the fact of disturbances
propagation in a compressible sphere with the finite
speed which is directly connected with the concept of a
lag, and in its mathematical describing it is connected
with the concepts of compound arguments, the quantity
of compound arguments, and with the interaction of
compound nonlinear arguments. It is impossible to get
accurate analytical solutions of inverse and direct wave
problems in the regions with moving boundaries without
the comprehension of these concepts and their applica-
tion. Let disturbance appears at some point of time on
the surface of finite size of initial radius, for example, on
the surface of sphere with the source of its expansion.
Waves generated in this way are used in different tech-
nologies. Then in the point of wave zone in some dis-
tance from the center of the sphere the disturbance ap-
pears not in the moment of beginning of wave expansion
but after some dead time, which is equal to the quotient
from the division of distance from the source surface of
the sphere to the point of wave zone by the speed of dis-
turbance expansion according to the certain medium [6].
Thus, logical considerations would give us some more
varieties of arguments with lags which interact when
mathematical operations are carried out. There appears
to be five such arguments for the wave problem of sphere
expansion (see [6]). There are sixteen of them for the
problem with two moving boundaries. It is significant
that this information is got without the solution to the
equation itself. Having determined all varieties of com-
pound arguments with the lags, it is natural to try to find
that succession of mathematical operations in solving an
equation for wave problems which keeps all kinds of
found arguments. For the first time, such a succession
was found by the author and named as the methods of
inverse problems with regard for the interaction of
nonlinear arguments. This is the ground to the second
reason. At the same time it is meant that it is impossible
to solve the pointed problems just with the help of the
methods of inverse problems. These problems are also
necessarily solved with regard for a great number of
compound arguments, including nonlinear ones with
various lags and also taking into consideration the fact
that they interact.
Here the following question may appear: having de-
termined, for example five varieties of arguments with
lags, on what ground, on what is the confidence based
that all these kinds of arguments would correspond the
internal structure of wave equation?
Such a confidence appears already if one examines
d’Alember’s solution (of Cauchy problem) of wave equa-
tion. There already exists, though the only one, variety
of lag [11]. It is possible to become firmly convinced in
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
it when an accurate analytical development of wave
equation in the regions with moving boundaries with all
found varieties of arguments, the substitution of which
turns the left side of the wave equation into zero. It is
significant that for the first time in mathematical physics
the author determined all compound arguments, and the
fact of their interaction. It allowed fro the first time to
create the methods of inverse problems with regard to
the interaction of nonlinear arguments and for the first
time it allowed to get accurate analytical developments
of wave problems with moving boundaries and moving
permeable boundaries.
As we can see in this case, the truth inherent to the
nature itself is the fact of disturbance expansion in con-
tinuum with the finite speed in which mathematical de-
scription gave grounds to various arguments with delays.
2.3. Deeply Hidden Mathematical Truths
The author understands this in the following way. On the
one hand, the study of the done by other people is meant.
Historical, consecutive, according the chronology of
events, approach to the study of the material is based on
it. On the other hand, it is necessary to begin to gather
information for reflection gradually while studying
something unknown and solving more simple particular
problems. Since it is clear that particular solutions bear
the marks of accurate development and help to compre-
hend the ways of its receipt. Such an accumulation can
be carried out by the generations of scientists. It can be
also done and by the only one scientist in the course of
his whole life. For example, Gauss K.F. worked in such
a way. As is well known, Gauss was an incomparable
calculator and, just as other outstanding arithmeticians
did, he usually got his new results from extensive nu-
merical calculations which helped him to notice new,
deeply concealed mathematical truths, the proofs of
which he got quite often only in the result of painstaking,
enduring, sometimes long-term work. The succession of
accumulation of information is also observed in tech-
nique. For instance, stitching: awl - threads - needle with
an eye - sewing machine. Or another example: wheel -
axle – axle with blades - screw propeller - turbine.
As it turned out, in solving the problems of moving
and moving permeable boundaries, both of these meth-
ods are examined, but in the main the example of Gauss
is closer to what has really happened [6].
2.4. What was Done for the First Time in
Solving of Problems of Moving and
Moving Permeable Boundaries
The author considers the main result to be the fact that
for the first time the new non-traditional author’s method
for solving the problem of moving boundaries of
mathematical physics equations was proposed and de-
vised—these are the methods of inverse problems with
regard for the interaction of nonlinear arguments. The
new method of moving boundaries problem solving, just
as the wave equation itself lie outside the scope of use in
the problems which describe pulse processes. It let to get
universal analytical solutions of intricate nonlinear pro-
blems. They are suitable for inverse and direct problems
for a wave equation with one and two moving bounda-
ries, one of them moves and another one is fixed [13];
with nonlinear conditions and moving boundaries [12];
with nonlinear conditions at moving boundaries. Two
latter cases are twice nonlinear problems. At the same
time the laws of change of boundary moving speed, data
of initial radiuses and displacements may be voluntary.
Received results reflect the wave processes of various
physical natures. It is necessary to point out that the
name of the nontraditional approach to the solution to
moving boundaries equations of mathematical - the
methods of inverse problems involving interaction of
nonlinear arguments - reflects physical and mathematical
essence of the author's method very successfully and
exactly [6,10,14].
The prospects of the developed method considerable:
analytically direct and inverse problems of special diffi-
culty with permeable (radiant) moving boundaries are
posed and solved [6,10,15], the laws of change of per-
meability speed and speed of boundary moving may be
voluntary. Such problems in mathematical physics were
not examined. The developed system is a methodologi-
cal basis for the study of the influence of new, similar
off-center boundary conditions. At the same time, the
solution to twice-nonlinear problems is brought to the
algebraic equation solving. Trustworthiness of findings
is proved by the comparison with the results of experi-
ment and solutions of more complicated equations and
systems of nonlinear equations with the help of known
methods (characteristics, dimensions and similarity, small
parameter etc.), the correctness of current inverse prob-
lems is shown. Substitution of received solutions into the
wave equation turns its left side in zero. Developed ana-
lytical methods of compound wave problems solution
may be used to give solution to new problems of theo-
retical fluid mechanics, theoretical physics and mathe-
matical physics.
2.5. Wave Processes Control. The Mobility
and Permeability of Moving Boundaries
as the Principle of Control
To the author’s mind, the control is possible when the
study of the process reached certain level and when all
possible control consequences are conceived. The pro-
posed and developed method of solution to moving per-
meable boundaries problems of mathematical physics
equations allowed to proceed from the processes of cog-
nition of wave phenomena to more complex and impor-
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
tant processes of wave phenomena control.
The wave equation serves as the mathematical model
of a great number of physical processes, the necessity of
their control appears, as a rule, simultaneously with the
study of these phenomena. The abilities of control are
determined both by the availability of exact analytical
solutions of inverse wave problems (control problems),
and to a greater extent — by the formulating of mathe-
matical models, which let to describe practically all pos-
sible variety, all conceivable wave fields in a theoretical
way. Wave equation realized in the regions with moving
and moving permeable boundaries is such an important
mathematical model.
As is well known, control, in the general case, is the
function of organized systems of various nature (bio-
logical, physical, technical, social etc.), which describes
the preservation of their definite structure, maintenance
of activity regulations, realization of the program, and
purposes of the activity. The task of control is the deter-
mination of control function. In the context of the ex-
amined mathematical model under control is understood
the following: the first one – velocity function (moving
and permeable moving boundary) or the second – pres-
sure function (at the moving boundary). The purpose of
investigations at the control problems is the definition of
control functions of the first and second cases, which
supply the receipt of adjusted wave regions of velocity
and pressure in necessary points, including the near-field
region and moving surfaces of boundaries (where the
experimental determination passes with difficulty or
impossible), and also their purposeful change during
certain period of time.
The main, basic concepts which explain properties
and possibilities (to control) of named mathematical
model, are mobility and permeability of moving bounda-
ries, what is the control principle by definition.
The methods of inverse problems with regard for the
interaction of nonlinear arguments allowed to get exact
analytical solutions of inverse wave problems with
moving boundaries. In inverse problems solving the in-
vestigated pressure function is defined, reasoning from
technological necessities, the pressure function (or ve-
locity) under consideration in the point of wave zone.
The received solutions determine the examined functions
in any points and at moving boundary. The knowledge of
pressure and velocity functions at moving boundaries is
the knowledge of control functions of wave processes;
thereby the control problem becomes decided. In appli-
cations, using the energy balance equations, which con-
nect pressure and velocity at moving boundary of plasma
piston with the quantity of input energy in the channel,
we get accurate analytical dependences. They let to
know the law of input energy in plasma channel of dis-
charge or laser impulse, for example, for the receipt of
adjusted pressure and velocity wave fields in a com-
pressible medium [16].
Thus, we see that it is possible to control wave proc-
esses in an accurate way only on reaching the definite
level of knowledge of the process. In this case, the
knowledge of exact analytical dependences of control
functions of the first and second cases (velocity and
pressure functions at moving boundaries). Without this
exact knowledge of control functions, the control itself
turns into the series of attempts to determine the wishful
(required) thing through attempts and mistakes, mostly
accidentally. Frequently this leads to time and funds
losses, and even lives. As is well known, especially
tragic are the attempts to implement such a control in a
social sphere.
As we see, the conclusions concerning the possibility
of wave processes control also have a more general
character. They also contain information (signs) on other
more complicated control processes, including the ones
of a social character. At present, there is no possibility to
define control functions with the help of such complex
systems, because of their extreme difficulty. Their study
may take place on the stage of examination of simpler
Registration and use of permeability (radiation) of
moving boundaries are greatly significant for the receipt
of the adjusted forms of examined functions that is the
control problems solution. Such problems belong to
complex unexplored essentially nonlinear class of prob-
lems of mathematical physics. Mathematical formulating
and involvement of permeability of moving boundaries
are executed at first by the author [6,7,15].
Received solutions are fit for inverse and direct prob-
lems for voluntary values of initial radius, displacements,
laws of change of rate of movement and permeability of
boundaries. Such problems appear, in particular, in fluid
dynamics, seism acoustics, at examination of a dynamite
source, for instance, of electric discharge, laser pulse etc.
in fluid, which has the temperature close to critical one.
The effect of dynamite source results in intense vapori-
zation from the moving surface of plasma cavity. In this
case wave phenomena under research may be described
with the help of mathematical model with a moving
permeable boundary.
It should be noted that the most complex and unex-
plored is the definition of control function for the wave
equation with moving boundaries and nonlinear condi-
tions at moving boundary. At the same time, the law of
motion of moving boundaries is unknown, is to be de-
fined, and may be nonlinear. The problems with nonlin-
ear additional conditions and moving boundaries are
twice nonlinear problems and they are of great interest,
their applied importance is so great that they become in
the number of the issues of the day of mathematics,
physics, and mechanics. Such problems in mathematical
physics were not examined. The methods of inverse
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
problems with due regard for interaction of nonlinear
arguments allowed to get analytical solutions of theses
twice nonlinear wave problems, and their solution is for
the first time reduced to the solution to the algebraic
quadratic equation [13].
The definition of validity limits of physical and
mathematical models and their solutions is the require-
ment of completeness of every elaboration. The wave
equation is the mathematical model of many physical
processes. However for all that it is necessary to take
into account that the validity limits of its solutions may
be different in every particular case, for every process
under investigation. The absence of exact analytical so-
lutions for wave equations with moving boundary and
moving permeable boundary did not allow to define its
validity limits in impulse fluid dynamics and acoustics,
for example, to present day. These two disciplines, two
scientific trends began before the year 1717 – the time of
creation of wave equation (Taylor Brook). These limits
were vague and were system according the pressure
from several hundreds to several thousands atmospheres.
Such a dispersion in dozens of times is intolerable. Exact
analytical solutions, received with the help of inverse
problems methods with regard for the interaction of
nonlinear arguments, allowed to define the validity lim-
its of wave equation with moving boundaries и moving
permeable boundary in impulse fluid dynamics and
acoustics for the first time and unambiguously [6,7]. It
was carried out in a nontraditional form - not according
the pressure, what cannot be produced unambiguously,
but according the velocity of boundary moving in com-
pressible medium.
These limits are also defined for the cases of moving
permeable boundaries, what has great scientific and ap-
plied significance.
Thus, we come to a conclusion that the logic of inves-
tigations lies basically in the following sequence of pri-
orities. Victory, achieved in search of truth, heads the list
of all achievements. Naturally, the preference is given
not to knowledge hoarding, but to sincere, outstanding
search for the new knowledge. Mathematical models and
their solutions are built on basis of the truths inherent to
the nature itself, which is highly important. Gathering
and using both the information of the ancestors and the
one got in the process of particular problems solution in
a purposeful way, to create new untraditional methods.
New methods are the tool used to accept an infinite
number of valuable results.
A non-traditional approach to wave problems solution
in the regions with moving boundaries – the methods of
inverse problems with the regard of nonlinear arguments
interaction is fundamentally new in the development of
mathematical physics. It allowed to solve not only direct
wave problems with moving boundaries, but also inverse
problems which were not solved before; system and
solve complex substantially nonlinear problems with
moving permeable boundary – both direct and inverse;
to solve analytically twice nonlinear problems (with
moving boundaries and nonlinear additional conditions);
to reduce the solutions of twice nonlinear problems to
algebraic equation. It allows to solve the new classes of
problems of theoretical fluid mechanics and theoretical
physics, mathematical physics.
Moreover, the method allowed making a big qualita-
tive step forward: to pass from the learning of wave
phenomena to more important and complicated wave
phenomena control process. It is shown that the correct
wave processes control is possible only when certain
level of knowledge of the process is achieved–the
knowledge of exact analytical dependence of control
3.1. Some Aspects (Illustrating) of the
Received Results Application
Pulse processes may serve as an illustration of applica-
tion of received results. Electric discharge, laser pulse,
explosion of charge and premixed gases, a blow against
the surface etc. in compressible media belong to them.
They are one of the essential principles of modern high
technologies including informational ones (geophysics,
geoacoustics etc., mineral exploration overland and at
sea etc.). Many outstanding scientists were engaged in
the theory of pulse processes, amidst them are the fol-
lowing: Sedov, L.I., Lavrentiev, М.А., Christianovich,
S.А., Kochin, N.Е., Shemyakin, Е.I., Landau, L.D., Okun,
I.Z., Yakovlev, Y.S., Baum, F.S., Stanyukovich, К.P.,
Laurentiev, М.М., Alekseev, А.S, Korobeynikov, V.P.,
Naugolnych, К.А., Roy, N.А., Lyamshev, L.М., Kurant,
R., Fridrichs, К., Chariton, Y.B., Zeldovich, M.A., Rozh-
destvensky, B.L., Yanenko, N.N., Lighthill and many
Characteristic feature of pulse processes in com-
pressible media is the presence of moving boundaries of
phase division plasma-fluid, gas-fluid (and also of mov-
ing explosions, barriers etc.). Taking into account the
influence of this peculiarity and also of the initial radius
value is necessary while studying the plasma of electric
discharge channel, laser pulse etc. in fluid, while study-
ing wave processes, including the ones of the near-field
region of the expanding boundary of plasma cavity, the
solution to the problems of pulse processes control;
modeling of disruption (breakdown) of a spark gap (be-
ing a separate complex problem) and in many other
cases, for example, at materials surface cleaning and
treatment, punching, fragmentation, investigations of
behavior of bubbles in a two-phase media, etc.; growth
of wear-resistance and corrosion resistance, influence on
crystallizing alloy, interaction with plates and jackets
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
and many others. Hence we see that there is no such a
technology on base of pulse processes, where the main
equation of mathematical physics – wave with moving
boundaries – would not be used at the heart of the proc-
esses. There will also be other technologies based on
wave processes, but wave equation with moving bounda-
ries would also be at their heart. This is an imperishable
value wave equation with moving boundaries, and its
exact solutions, which were at first acquired with the
help of inverse problems methods with regard of nonlin-
ear arguments interaction.
It is necessary to pay attention to known important
circumstances. Technologies developed on basis of ex-
perimental data without support of fundamental theo-
retical elaborations have narrow orientation, cannot be
widely used, are not replicate, that is expensive experi-
ments will be again necessary for another allied tech-
nologies, a considerable waste of time and funds. How-
ever, for all that rather often when experiments are made
the physical sense of phenomena is not comprehensible
and studied enough because of their great difficulty. For
instance, physics of perturbation effects on crystallizing
metal, development physics and “healing” of cracks
formed during pulse processing etc., with all following
negative consequence. Besides, nowadays, as a rule,
workable technologies do not find manufacturing appli-
cation and become out-of-date because of an economical
grave condition. Minor number of high technologies and
industrial standards are used in modern manufacture. As
soon as the results of fundamental researches do not be-
come out-of-date, are of imperishable value and will
always be wanted for the elaboration of new high tech-
nologies both now and then. Therefore, it is evident that
fundamental researches should have priorities for elabo-
rations and financing, it is the most efficient and eco-
nomically profitable, advantageous and purposeful way.
In the presence of high technological culture some
countries spend dozens of milliards of dollars on the
creation of their fundamental science.
3.2. New Ways of Use of Particular Wave
Processes in Particular High
A lot of high technologies border on art, for example,
cooking, formulation of drugs, wine making, preparation
of paints, “Greek fire”, damask, the technologies of
treatment etc., many products of defense technology,
“technology” of performance of musical compositions,
creation of artistic masterpieces, “technologies” of get-
ting of new scientific truths, discoveries etc. Coming in
the form of complete product even to the country with
high technological culture, the products of high tech-
nologies cannot be reproduced for a long time, some-
times never at all. Yet, some of them cannot be described
with the help of physico-mathematical models.
At the same time, it does not require proves that mod-
ern high technologies are impossible without mathe-
matical physics. They are the concentration of scientific
discoveries of many scientists, sometimes of the whole
generations, engineering and technological develop-
ments. One of the main physical phenomena of modern
higher technologies based on pulse processes is the mo-
tion of moving boundaries of the division of plasma-
fluid phase, gas-fluid, which generate shock waves, per-
turbation waves, media waves etc., used in various tech-
nologies, at which mathematical description we come to
the problems of moving permeable boundaries. Among
the described pulse processes, underwater electric explo-
sion is the most powerful controlled generator of cavita-
tion phenomena, which exceeds the possibilities of me-
chanic oscillators. Recently it was established that un-
derwater electric explosion in its acoustic spectrum gen-
erates not only low-frequency vibrations, but also ultra-
sound ones to the extend of a hundred kHz.
Here are some examples of new methods of use of
wave processes, induced by electric discharge in fluid:
the explosion of thin coal conductors (fibers) - the re-
ceipt of fullerenes, necessary for manufacture, for exam-
ple, of special radio technical products; the receipt of
metals oxides, which find their application in the pro-
duction of paint, structural ceramics, used, for example,
in mechanical engineering, covering of spacecrafts of
nonexpendable usage, the production of superconducting
materials (superconductor) etc.; discharge-pulse tech-
nology of pure decomposition of various radio materials,
namely of phosphor; the receipt of high quality flax fi-
bers with the purpose of their application in textile
manufacture and at production of medicinal cotton, and
also of other strategic materials; breaking up of gravel,
cement blocks, wastes of porcelain manufacture etc. in a
wide spectrum from lumpy breaking up of non- metallic
materials to ultrafine pounding; moulding refinement–
removal of core sand mixture from mouldings; electro-
blasting diameter extension (extension of size), ends
fixity of tubes in tube plates of heat-exchange appara-
tuses by means of electroblasting diameter extension
(extension of size); sputter-ion processing of oil and wa-
ter wells on purpose of rising of their flow rate (increase
of oil recovery of plates etc.); protection of young fish
from getting to water supply points of water supply sys-
tems; stamping of goods from plate stock; the creation of
acoustic resistance devises in seas and oceans; the crea-
tion of generative probing signal, used at mineral explo-
ration; catalysts preparation and intensification of cata-
lytic processes; the preparation device of subsided rocks
for the construction of buildings and erections; electro-
blasting technology of water conditioning and steriliza-
tion of sewage by means of generation of high-level cavi-
tation processes; layer-by-layer sputter-ion sparing release
of high-radiation lavas safety fourth block Chernobyl
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
nuclear power-station from depositions and lots of others.
See works [17] and [18] to learn in details about these
and other technologies having no world analogs on basis
of electric discharge in fluid.
The significance of the solution to moving permeable
boundaries problems and wave processes control prob-
lems is not limited by their usage for the elaboration of
new technologies. New developed approaches to the
creation of analytical methods of complex wave prob-
lems solution, including twice nonlinear, can be used for
the creation of new methods of mathematical physics to
solve the problems within the spheres of theoretical fluid
mechanics and theoretical physics.
The content of paragraphs 1-5 which contain the in-
vestigations development descriptions and mentioned in
epigraphs, composes the necessary conditions of getting
the new truths. It is evident that non-observance of the
only one of them is impossible. From the author’s point
of view, sufficient conditions are in individual philoso-
phy of success.
General concepts, views, theories, laws and principles
should be considered of the first importance [19]. To this
extent, the most significant ones are the wave processes
and the questions regarding their control. This article is
devised for a wide circle of scientific readers with aim to
render to them first of all the physical ground of the
examined phenomena, and also to manifest the novelties
which it brought into the development of a respective
tendency, and that the way of the investigation did not
emerge “out of nothing”, and the retrieved results are
only “the stone which cost him a whole life” [2]. This
study is the synopsis of the information covering au-
thor’s publications.
Furthermore, it is necessary to mention the following.
Astonishing exceptional number of applications of the
wave equation “from the waves in the oceans of water,
air and ether”, as Russell [20] would tell, to the waves,
describing the elementary particles. Nowadays the wave
equation became so customary and usual, that nobody is
surprised at its effectiveness any longer. However, if one
tries to comprehend in one's mind everything that has
been done with the help of this equation, simply to
imagine what a wealth of natural phenomena is hiding
behind such a simple formula, the epithets “astonishing”
and “extraordinary” would seem inappropriate. Once, an
eminent contemporary physicist wrote a popular article
“On the incomprehensible effectiveness of mathematics
in natural sciences”. [21] There is surely something in-
comprehensible in the effectiveness of the wave equation,
no matter what everything-can-explain people say [20].
Evidently, now the effectiveness of the wave equation
rises repeatedly due to the presence of exact analytical
solutions of inverse (and direct) wave problems in the
regions with MB and MPB, which were for the first time
received by the methods of inverse problems involving
interaction of nonlinear arguments. So far, the simple
truth is that neither measurement, nor experiment and
observation are possible without a respective theoretical
scheme [22]. At development of such schemes a signifi-
cant role is played by control functions described above.
Without these control functions it is impossible to de-
velop such schemes. Moreover, in case when a success-
ful model of physical phenomenon is designed, i.e. the
model which allows to make accurate calculations and
predictions, then the mathematical structure of the model
itself reveals new sides of these phenomena. As the re-
sult, the “studies of an internal structure of a model” can
change and broaden our notions of physical phenomena
[23] Vol.1, p.7.
The following statement brings surety. Wave equation
and its accurate analytical solutions are devised on basis
of the phenomena adherent to the nature itself, the phe-
nomena of agitations propagation in continuum with a
final velocity. It is evident in terms of physics that wave
phenomena – the basic phenomena characteristic for all
objects without exception – reflection in a wave equation,
“a marvelous basic equation of mathematical physics”.
Other equations, e.g. parabolic, etc. are “unphysical”,
since they describe the agitations which propagate in-
stantaneously [24].
A wonderful correspondence of mathematical lan-
guage to the laws of physics [21] is actually a real gift.
But it is not a mystery and we can and are able and dig-
nified to accept it, which is demonstrated on the example
of the fore-mentioned physical problems solution. Opti-
mism is bred from an enormous role of mathematical
methods for the solution to physical problems, e.g. the
known scheme of three steps [25] for the solution to
various problems: the problem of any kind comes to
mathematical problem; mathematical problem of any
kind comes to algebraic problem; any algebraic problem
comes to solution to one single equation. This is demon-
strated by the formulas, e.g. stated in work [13].
It is worth citing the optimistic words of d’Alember
instead of conclusion: work, work, absolute comprehen-
sion comes later. As we see, it took three hundred years
to understand the essence to take into consideration the
influence of mobility and permeability of moving
boundaries on wave processes [7,9,10,14,26].
[1] Ginsburg, V.L. (1985) O fizike i astrofizike. Physics and
Astrophysics, 400, 100.
[2] Poincar, H. (1983) O nauke. Science, 270, 218.
V. S. Krutikov / Natural Science 2 (2010) 298-306
Copyright © 2010 SciRes. OPEN ACCESS
[3] Tikhonov, A.N. and Samarskiy, А.А. (1972) Uravneniya
matematicheskoy fiziki Equations of Mathematical
Physics, 735.
[4] Lavrent’yev, M.M. (1964) On one inverse problem for a
wave equation. Doklady Akademii Nauk SSSR, 157(3),
[5] Lavrent’yev, M.А., Shabat, B.V. (1973) Problems of
hydrodynamics and their mathematical models. Nauka,
[6] Krutikov, V.S. (1985) Odnomernye zadachi mekhaniki
sploshnoi sredy s podvizhnymi granitsami (One-dimensional
problems in continuum mechanics with moving bound-
aries). Kiev, Izdatel'stvo Naukova Dumka, 125, 128.
[7] Krutikov, V.S. (1996) Validity Limits of solutions to the
wave equation for regions with moving permeable
boundaries in impulse fluid dynamics and acoustics.
Acoustical Physics, 42(4), 471-477.
[8] Taylor, G. (1946) The airwave surrounding an expanding
sphere. Proceedings of the Royal Society A, Mathematical
and Physical Sciences, London, 186(1006), 273-292.
[9] Krutikov, V.S. (1999) Waves surrounding an expanding
permeable cylinder in a compressible medium. Doklady
Physics, 44(10), 674-677.
[10] Krutikov, V.S. (1993) Wave phenomena with finite dis-
placements of permeable boundaries. Doklady Akademii
Nauk, 333(4), 512-514.
[11] Isakovich, M.A. (1973). Obschaya akustika. Fundamen-
tal Acoustics, 495.
[12] Krutikov, V.S. (1992) Interaction of weak shock waves
with a spherical shell involving motion of the boundaries.
Mekhanika Tverdogo Tela, 2, 178-186.
[13] Krutikov, V.S. (1991) A Solution to the Inverse problem
for the wave equation with nonlinear conditions in re-
gions with moving boundaries. Prikladnaia Matematika i
Mekhanika, 55(6), 1058-1062.
[14] Krutikov, V.S. (1999) A new approach to solution to in-
verse problems for a wave equation in domains with
moving boundaries. Doklady Mathematics, 59(1), 10-13.
[15] Krutikov, V.S. (1999) Waves surrounding an expanding
permeable cylinder in a compressible medium. Doklady
Physics, 44(10), 674-677.
[16] Krutikov, V.S. (2003) An exact analytical solution to the
inverse problem for a plasma cylinder expanding in a
compressible medium. Technical Physics Letters, 29(12),
[17] Gulyi, G.A. (1990) Scientific fundamentals of pulse-
discharge technologies. Naukova Dumka, Kiev, 208.
[18] Shvets, I.S. (2002) 40th Anniversary of the institute of
pulse processes and technologies at NAS of Ukraine.
Science and production. Theory, Experiment, Practice of
Electrodischarge Technologies, 4, 3-6.
[19] Ginsburg, V.L. (2000) Physics: Past, present, and future.
Priroda, 3.
[20] Philippov, A.T. (1990) Mnogolikiy soliton. Nauka, 297.
[21] Wigner, E. (1968) On Incomprehensible efficiency of
mathematics in natural sciences. Uspekhi Phizicheskikh
Nauk, 94(3), 540.
[22] Prigozhin, I. and Stengers I. (1986) Order out of Chaos.
Bantam Dell Publishing Group, Moskow, 364.
[23] Reed, M. and Simon, B. (1977-1982) Methods of con-
temporary mathematical physics. Springer, Moscow, 1-4.
[24] Krutikov, V.S. (1995) Development of the method and
solutions of pulse problems of continuum mechanics
with moving boundaries. Dissertation by the Doctor of
Physico-mathematical Sciences, Institute of Geophysics
NAS of Ukraine, Kiev, 343.
[25] Poya, D. (1970) Mathematical invention. Nauka, 45.
[26] Krutikov, V.S. (2006) On an inverse problem for the
wave equation in regions with mobile boundaries and an
iteration method for the determination of control func-
tions. Doklady Physics, 51(1), 1-5.