# Vanishing Waves on Closed Intervals and Propagating Short-Range Phenomena

Authors: Ghiocel Toma; Flavia Doboga
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

This study presents mathematical aspects of wave equation
considered on closed space intervals. It is shown that a solution of this
equation can be represented by a certain superposition of traveling waves
with null values for the amplitude and for the time derivatives of the resulting
wave in the endpoints of this interval. Supplementary aspects
connected with the possible existence of initial conditions for a secondorder
differential system describing the amplitude of these localized oscillations
are also studied, and requirements necessary for establishing
a certain propagation direction for the wave (rejecting the possibility of
reverse radiation) are also presented. Then it is shown that these aspects
can be extended to a set of adjacent closed space intervals, by considering
that a certain traveling wave propagating from an endpoint to the
other can be defined on each space interval and a specific mathematical
law (which can be approximated by a differential equation) describes the
amplitude of these localized traveling waves as related to the space coordinates
corresponding to the middle point of the interval. Using specific
differential equations, it is shown that the existence of such propagating
law for the amplitude of localized oscillations can generate periodical
patterns and can explain fracture phenomena inside materials as well.


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## Body

## 1. Introduction

Test-functions (which differ to zero only on a limited
interval and have continuous derivatives of any order on the whole real axis)
are widely used in the mathematical theory of distributions and in Fourier
analysis of wavelets. Yet such test-functions, similar to the Dirac functions,
cannot be generated by a differential equation. The existence of such an
equation of evolution, beginning to act at an initial moment of time, would
imply the necessity for a derivative of certain order to make a jump at this
initial moment of time from the zero value to a nonzero value. But this aspect
is in contradiction with the property of test-functions to have continuous
derivatives of any order on the whole real axis, represented in this case by
the time axis. So it results that an ideal test-function cannot be generated by
a differential equation (see also [1]); the analysis has to be restricted at possibilities
of generating practical test-functions (functions similar to test-functions,
but having a finite number of continuous derivatives on the whole real axis)
useful for wavelets analysis. Due to the exact form of the derivatives of
test-functions, we cannot apply derivative free algorithms [2] or algorithms which can
change in time [3].
Starting from the exact mathematical expressions of a certain test-function and
of its derivatives, we must use specific differential equations for generating
such practical test-functions.This aspect is connected with causal aspects of
generating apparently acausal pulses as solutions of the wave equation,
presented in
[4]. Such
test-functions, considered at the macroscopic scale (that does
not mean Dirac-functions), can represent solutions for certain
equations in mathematical physics (an example being the wave-equation). The
main consequence of this aspect consists in the possibility of certain pulses
to appear as solutions of the wave-equation under initial null conditions for
the function and for all its derivatives and without any free-term (a
source-term) to exist. In order to prove the possibility of appearing acausal
pulses as solutions of the wave-equation (not determined by the initial
conditions or by some external forces) we begin by writing the
wave-equation(1.1)∂2ϕ∂x2−1v2∂2ϕ∂t2=0for a free string defined on the
length interval (0,l) (an open set), where ϕ represents the amplitude of the string
oscillations and v represents the velocity of the waves inside
the string medium. At the initial moment of time (the zero moment) the
amplitude ϕ together with all its derivatives of first and
second orders is equal to zero. From the mathematical theory of the
wave-equation, we know that any solution of this equation must be a
superposition of a direct wave and of a reverse wave. For the beginning, we
will restrict our analysis at direct waves by considering a supposed extension
of the string on the whole Ox axis, ϕ being defined by the function(1.2)ϕ(τ)={exp⁡(1(x−vt+1)2−1)for|x−vt+1|<1,0for|x−vt+1|≥1,where t≥0.
This function for the extended string satisfies the wave-equation (being a
function of x−vt, a direct wave). It is a continuous function,
having continuous partial derivatives of any order for x∈(−∞,∞) and for t≥0.
For x∈(0,l) (the real string) the amplitude ϕ and all its derivatives are equal to zero at
the zero moment of time, as required by the initial null conditions for the
real string (nonzero values appearing only for x∈(−2,0) for t=0,
while on this interval |x−vt+1|=|x+1|<1). We can notice that for t=0 the amplitude ϕ and its partial derivatives differ to zero
only on a finite space interval, this being a property of the functions defined
on a compact set (test-functions). But the argument of the exponential function
is x−vt; this implies that the positive amplitude
existing on the length interval (−2,0) at the zero moment of time will move along the
Ox axis in the direction x=+∞.
So at some time moments tk after the zero moment, a nonzero amplitude ϕ will appear inside the string, propagating
from one edge to the other. It can be noticed that the pulse passes through the
real string and at a certain time moment tfin (when the pulse existing at the zero moment of
time on the length interval (−2,0) has moved into the length interval (l,l+2)) its action upon the real string ceases. We
must point the fact that the limit points x=0 and x=l are not considered to belong to the string;
but this is in accordance with the rigorous definition of derivatives (for this
limit points cannot be defined as derivatives
related to any direction around them).This point of space (the limit of the open space
interval considered) is very important for our analysis, while we will extend
the study to closed space intervals. Considering small space intervals around
the points of space where the sources of the generated field are situated
(e.g., the case of electrical charges generating the electromagnetic field), it
will be shown that causal aspects require the logical existence of a certain
causal chain for transmitting interaction from one point of space to another,
which can be represented by mathematical functions which
vanish (its amplitude and all its derivatives) in
certain points of space. From this point of space, an informational connection
for transmitting the wave further could be considered (instead of a
transmission based on certain derivatives of the wave). Thus a kind of granular
approach for propagation along a certain axis can be considered suitable for
application in quantum theory. As an important consequence, some directions of
propagation for the generated wave will appear and the possibility of reverse
radiation will be rejected. Moreover, specific applications for other
propagating phenomena involving the generation of some spatial periodical
patterns or an increasing amplitude of oscillations along a certain spatial
axis can be also analyzed by this mathematical model.


## 2. Utility of Test-Functions in Mathematical Physics for Half-Closed Space Intervals

If we extend our analysis to half-closed intervals by
adding one endpoint of the space interval to the previously studied open
intervals (e.g., by adding the pointx=0 to the open interval (0,l)), we should take into account the fact that a
complete mathematical analysis usually implies the use of a certain function f(t) defined at the limit of the working space
interval (the point of space x=0,
in the previous example). Some other supplementary functions can be met in
mathematical physics.The use of such supplementary functions defined on the
limit of the half-closed interval could appear as a possible explanation for
the problem of generating acausal pulses as solutions of the wave equation on
bounded open intervals. The acausal pulse presented in the previous paragraph
(similar to wavelets) traveling along theOx axis requires a certain nonzero
function of time f0(t) for the amplitude of the pulse for the limit
of the interval x=0.
It could be argued that the complete mathematical problem of generating acausal
pulses for null initial conditions on this interval and for null function f0(t) corresponding to function ϕ (the pulse amplitude) at this endpoint of the
interval (x=0, resp.) would reject the possibility of
appearing the acausal pulse presented in the previous paragraph. The acausal
pulse ϕ previously presented implies nonzero values
for f0 at certain time moments, which represents a
contradiction with the requirement for this function f0 to present null values at any time moment. By
an intuitive approach, null external sources would imply null values for
function f0 and (as a consequence) null values for the
pulse amplitude ϕ.Yet it can be easily shown that the problem of
generating acausal pulses on half-closed intervals cannot be rejected by using
supplementary requirements for certain functionsf(t) defined at one limit of such bounded space
intervals. Let us simply suppose that instead of function(2.1)ϕ(τ)={exp⁡(1(x−vt+1)2−1)for|x−vt+1|<1,0for|x−vt+1|≥1presented in the previous
paragraph we must take into consideration two functions ϕ0 and ϕl defined as(2.2)ϕ0(τ)={exp⁡(1(x−vt+m)2−1)for|x−vt+m|<1,0for|x−vt+m|≥1,ϕl(τ)={−exp⁡(1(x+vt−m)2−1)for|x−vt+m|<1,0for|x+vt−m|≥1,with m selected as m>0,m−1>l (so as both functions ϕ0 and ϕl to have nonzero values outside the real string
and asymmetrical as related to the point of space x=0.
While function ϕ0 corresponds to a direct wave (its argument
being (x−vt)) and ϕl corresponds to a reverse wave (its argument
being (x+vt)) it results that both functions ϕ0 and ϕl arrive at the same space
origin x=0,
the sum of these two external pulses being null all the time (functions ϕ0 and ϕl being asymmetrical, ϕ0=−ϕl) at any moment of time. So by requiring that ϕ(t)=0 for x=0 (the left limit of a half-closed interval [0,l)) we cannot reject the mathematical
possibility of the appearance of an acausal pulse
on a half-closed interval.A possible mathematical explanation for this aspect
consists in the fact that we have used a reverse wave (an acausal pulse)
propagating fromx=∞ toward x=−∞,
which is first received at the right limit x=l of the half-closed interval [0,l) before arriving at the point of space x=0.
It can be argued that in case of a closed space interval [0,l], we should consider the complete mathematical
problem, consisting of two functions f0(t),fl(t) corresponding to both limits of the working
space intervals (the points of space x=0 and x=l). But in fact the wave equation corresponds
to a physical model valid in the three-dimensional space, under the
form(2.3)∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2−1v2∂2ϕ∂t2=0and the one-dimensional model
previously used is just an approximation. Moreover, the source of the field is
considered at a microscopic scale (e.g., quantum particles like electrons for
the case of the electromagnetic field) and the emitted field for such
elementary particles presents a spherical symmetry. Transforming the previous
equation in polar coordinates and supposing that the function ϕ depends only on r (the distance from the source of the field to
the point of space where this emitted field is received), it results
that(2.4)∂2U∂r2−1v2∂2U∂t2=0,where(2.5)U=rφ.An analysis of the field emitted
from the point of space r=0 (the source) toward a point of space r=r0 (where the field is received) should be
performed on the space interval (0,r] (a half-closed interval); the point of space r=0 cannot be included in the working interval as
long as the solution ϕ(r) for the field is obtained by dividing the
solution U(r) of the previous equation (in spherical
coordinates) through r (the denominator of the solution ϕ being zero, some supplementary aspects
connected to the limit of functions should be added, but still without
considering a function of time as condition for the space origin). This can be
put in correspondence with the previously presented case of an acausal pulse
defined on [0,l) if we consider that (as a rule) (a) the
endpoint where the function ϕ(t) is not defined represents the source of the
field (a round bracket being added, while it cannot be considered as part of
the working interval) and (b) the endpoint where the function φ vanishes represents a point of space where the
propagating phenomenon is recreated (by reflection or by interaction with
different particles, for the case of optical waves), a square bracket being
added. The endpoint represented by square bracket (where the wave vanishes) can
be considered as a source for the field propagating in a next space interval
after an interaction, and so on.Thus an asymmetry in the required methods for
analyzing phenomena appears. Moreover, for the appearance of a certain direction for the transmission of interaction (from one
space interval to another), it results that the possibility of retroradiation
(a reverse wave generated by points of space where a direct wave has arrived)
should be rejected (a memory of previous phenomena is determining the direction
of propagation).


## 3. Applications for Closed Space Intervals: Applications in Quantum Physics

The pulse presented in the previous paragraph is in
fact a traveling wave propagating fromx=∞ toward x=0 and back which vanishes at the point of space x=0 due to a kind of reflection. Yet we can extend
our analysis by considering a subsequent reflection of this pulse at the limit
point x=l and so on. Thus a resulting traveling wave can
be considered inside the closed space interval [0,l] with null values at the endpoints x=0,x=l at any time moment after the first reflection.At first sight, this localized oscillation is not
useful for our mathematical analysis of acausal pulses. It does not correspond
to initial null conditions on the closed bounded space interval[0,l] and to null time functions defined at the
endpoints x=0,x=l (while the traveling wave should already exist
inside this interval when null conditions for the endpoints at any subsequent
time moment are added). Yet we must take into consideration the fact that in
quantum physics the operators corresponding to creation and annihilation of
particles are obtained (in a heuristic manner) starting from an analysis of
electromagnetic field performed on bounded space intervals and extended to
unbounded intervals by simply replacing the space limits for a set of such
intervals with infinite values [5].
However, the previously mentioned analysis on bounded intervals makes use of
stationary waves which cannot be taken into consideration when a space limit
equals ±∞ (no reflection can appear). This logical
contradiction can be avoided if any extended space interval is considered as a
sum of adjacent small space intervals with specific localized oscillations
defined on each of them.Supposing that a localized oscillation is generated on
a certain limited space interval by an external force or by a received
wave-train, we can consider that subsequent oscillations are generated on
adjacent space intervals (as in the case of spherical waves) due to a kind of
informational connection existing on the boundaries of these intervals. A
mathematical connection described by wave-equation cannot be taken into
consideration any more, and thus the previous model of causal chain
corresponding to a sequence:changes
in the value of partial derivatives as related to space coordinates imply
changes in the partial derivatives of the amplitude as related to time, which
further imply changes in the value of the function, should be replaced by a
step-by-step transmission of interaction starting from an initial half-closed
interval (e.g., its open limit corresponding to the source of the field) to
adjacent space intervals. This corresponds to a granular aspect of space
suitable for applications in quantum physics, where the generation and
annihilation of quantum particles should be considered on limited space-time
intervals (asymmetrical pulses could be also used [6]). A specific physical
quantity (corresponding to the amplitude of localized oscillations) is
transmitted from one space interval to another, according to a certain
mathematical law.


## 4. Dynamical Spatial Generation of Structural Patterns

We will continue the study by presenting properties of
spatial linear systems described by a certain physical quantity generated by a
differential equation. This quantity can be represented by internal electric or
magnetic field inside the material or by similar physical quantities, and
corresponds to the amplitude of localized oscillations previously mentioned. A
specific mathematical law which can be approximated by a differential equation
generates this quantity considering as input the spatial alternating variations
of a certain internal parameter. As a consequence, specific spatial linear
variations of the corresponding physical quantity appear. In case of very
short-range variations of this internal parameter, systems described by a
differential equation able to generate a practical test-function [1] exhibit an output which
appears to an external observer under the form of two distinct envelopes. These
can be considered as two distinct structural patterns located in the same
material along a certain linear axis. This aspect differs
from the oscillations of unstable type second-order systems
studied using difference equations [7] or advanced differential equations [8], and they
differ also from the previous studies of the same
author [9] where the
frequency response of such systems to alternating inputs was studied (in
conjunction with the ergodic hypothesis). For our purpose, we will use the
function(4.1)φ(x)={exp⁡(1x2−1)ifx∈(−1,1),0otherwise,which is a test-function on [−1,1].
For a small value of the numerator of the exponent, a rectangular shape of the
output is obtained. An example is the case of the function(4.2)φ(x)={exp⁡(0.1x2−1)ifx∈(−1,1),0otherwise.Using the expression ofφ(x) and of its derivatives of first and second
orders, a differential equation which admits as solution the function φ corresponding to a certain physical quantity
can be obtained. However, a test-function cannot be the solution of a
differential equation. Such an equation of evolution implies a jump at the
initial space point for a derivative of certain order, and test-function must
possess continuous derivatives of any order on the whole real axis. So it
results that a differential equation which admits a test-function φ as solution can generate only a practical
test-function f similar to φ,
but having a finite number of continuous derivatives on the real Ox axis. In order to do this, we must add initial
conditions for the function f (generated by the differential equation) and
for some of its derivatives f(1),
and/or f(2) and so on equal to the values of the
test-function φ and of some of its derivatives φ(1),
and/or φ(2) and so on at an initial space point xin very close to the beginning of the working
spatial interval. This can be written under the form(4.3)fxin=φxin,fxin(1)=φxin(1),and/orfxin(2)=φxin(2),and so on.If we want to generate spatial practical
test-functionsf which are symmetrical as related to the middle
of the working spatial interval, we can choose as space origin for the Ox axis the middle of this interval, and so it
results that the function f should be invariant under the transformation(4.4)x→−x.Functions invariant under this
transformation can be written in the form f(x2) (similar to aspects presented in [1]) and so the form of a
general second-order differential equation generating this kind of functions
should be(4.5)a2(x2)d2fd(x2)2+a1(x2)dfdx2+a0(x2)f=0.However, for studying the generation of structural
patterns on such a working interval, we must add a free term corresponding to
the cause for the variations of the external observable physical quantity.
Thus, a model for generating a practical test-function using as input the
internal parameteru=u(x),x∈[−1,1],
is(4.6)a2(x2)d2fd(x2)2+a1(x2)dfdx2+a0(x2)f=usubject to(4.7)lim⁡x→±1fk(x)=0fork=0,1,…,n,which are the boundary
conditions of a practical test-function. For u represented by alternating functions, we
should notice periodical variations of the external observable physical
quantity f.According to the previous considerations for the form
of a differential equation invariant at the transformation(4.8)x→−x,a first-order system can be
written under the form(4.9)dfd(x2)=f+uwhich converts
to(4.10)dfdx=2xf+2xurepresenting a first-order
dynamical system. For a periodical input (corresponding to the internal
parameter) u=sin⁡10x,
numerical simulations performed using Runge-Kutta functions in Matlab present
an output of an irregular shape (Figure 1) not suitable for joining together
the outputs for a set of adjoining linear intervals (the value of f at the end of the interval differs in a
significant manner to the value of f at the beginning of the interval). A better
form for the physical quantity f is obtained for variations of the internal
parameter described by the equation u=cos10x.
In this case, the output is symmetrical as related to the middle of the
interval (as can be noticed in Figure 2) and the results obtained on each
interval can be joined together on the whole linear spatial axis, without any
discontinuities to appear. The resulting output would be represented by a sum
of two great spatial oscillations (one at the end of an interval and another
one at the beginning of the next interval) and two small spatial oscillations
(around the middle of the next interval).Figure 1
f versus distance for first-order system, input u=sin(10x).Figure 2
f versus distance for first-order system, input u=cos(10x).Similar results are obtained for an undamped dynamical
system first order, represented by(4.11)dfd(x2)=uwhich is equivalent
to(4.12)dfdx=2xu.When the internal parameter presents very short-range
variations, some new structural patterns can be noticed. Considering an
alternating input of the formu=sin⁡(100x),
it results in an observable physical quantity f represented in Figure 3; for an alternating
cosine input represented by u=cos(100x),
it results in the output f represented in Figure 4. Studying these two
graphics, we can notice the presence of two distinct envelopes. Their shape
depends on the phase of the input alternating component (the internal
parameter), as related to the space origin. At first sight, an external
observer could notice two distinct functions f inside the same material, along the Ox axis. These can be considered as two distinct
structural patterns located in the same material, generated by a short-range
alternating internal parameter u through a certain differential equation (invariant
at the transformation x→−x).Figure 3
f versus distance for first-order system, input u=sin⁡(100x).Figure 4
f versus distance for first-order system, input u=cos(100x).


## 5. Aspects Connected with Short-Range Breaking Phenomena

For simulating the generation of specific deformations
inside a material medium under the action of external forces, it can be
considered that some short wavelength vibrations appear in the area where the
force acts. Usually the corresponding deformation is simulated inside the
material medium, using linear differential equations or equations with partial
derivatives (similar to the wave equation or to the equation of diffusion). Yet
such linear equations cannot explain the distance between the space area where
the external force acts and the space area where fracture phenomena appear.
Using differential equations of higher order, some slow variations of
deformation along a certain direction could be obtained. Due to the fact that
the mathematical model should explain the sharp deformations at a certain
distance of the point of space where the force acts (leading to fracture
phenomena), some different types of differential equations must be studied. For
this reason, our study has taken into consideration some dynamical equations
able to generate practical test-functions (similar to wavelets) [1] and delayed pulses (when a
free term which corresponds to an external pulse is added) [10] for justifying fracture
phenomena appearing in a certain material medium. It is considered that an
external force (described by a short wavelength sine function multiplied by a
Gaussian function) acts upon the material medium in a certain area. As a
consequence, some localized vibrations (corresponding to localized oscillations
on closed space intervals presented in the previous paragraphs) appear. These localized
oscillations are transmitted from one space interval to another according to a
certain mathematical law which puts into correspondence the amplitude of these
local vibrations to spatial coordinates.Using a specific differential equation (able to generate
symmetrical functions for a null free term) for describing the generation of
the corresponding deformation along an axis inside the material medium, it
results that a significant deformation appears at a certain distance. This
significant deformation justifies the fracture phenomena, while the inner
structure of the material cannot allow significant sharp deformations without
breaking. The main problem is represented by the search of an adequate free
termu(x) able to justify fracture phenomena. We start
by using a constant free term, using an equation as(5.1)f(2)=0.6x4−0.36x2−0.2(x2−1)4f+u(x),where u(x) represents the external force (supposed to be
constant in a first approximation on the working space interval (−1,1)). The deformation f(x) is supposed to be first time generated by the
external force at the limit x=−1 of the working interval and then (according to
the differential equation) it generates the corresponding deformation along the
whole working interval, with the external constant force u acting in a continuous manner upon the
material. The deformation generated by such a constant force u should be symmetrical as related to origin 0 (the previous differential equations being
valid on the space interval (−1,1) with initial null conditions for f(x) at the initial point of space x=−1). The property of symmetry previously
mentioned is justified by invariance properties of this type of differential equations
[1]. However, even for u(x)=1 (the most simple external force acting upon
the material which is symmetrical as related to space origin 0) numerical simulations in Matlab present an
asymmetry of the output signal, justified by numerical errors (see Figure 5).
But numerical simulations present also a slow varying deformation along the
axis, with no spatial oscillations; thus the fracture phenomenon cannot be
explained.Figure 5
Deformation generated by an external constant force.A similar shape of the output can be noticed for an
input represented by a Gaussian external force, acting around the point of
spacex=−0.9 and having a width ten times smaller than the
working period—similar to the use of a Gaussian modulated signal for
generating delayed pulses [10]. In such a case the differential equation generating
the deformation along the working interval is represented by(5.2)f(2)=0.6x4−0.36x2−0.2(x2−1)4f+exp⁡−(x+0.9)2(0.01)2and the corresponding output is
represented in Figure 6.Figure 6
Deformation generated by an external Gaussian
(localized) force.So we must extend our search for adequate mathematical
models, and we will try a free termu(x) represented by(5.3)u(x)=exp⁡(−(x+0.9)2(0.01)2)sin⁡104x.This mathematical expression
describes an external force represented by a Gaussian multiplied by a sine
function with short wavelength, being considered that the applied force is
transformed by the surface of the material into a set of alternating internal
efforts with very short wavelength (similar to a localized vibration).The corresponding output is represented in Figure7.
It can be noticed that we have finally obtained a sharp deformation appearing
at a certain distance between the point of space where the external (modulated
Gaussian) force acts and the point of space where the sharp deformation
appears. Moreover, the sharp deformation appears as an alternating function
localized on a very short spatial interval. It is quite obvious that such a
deformation cannot be allowed by the inner structure of the material, leading
to fracture phenomena. This simulation explains also the fact that the fracture
point is usually situated at a certain distance from the point where the
external force is applied (as can be noticed studying the deformation presented
in Figure 7 generated by the internal efforts u(x) presented in Figure 8)Figure 7
Deformation generated by a
modulated Gaussian internal effort.Figure 8
Modulated
Gaussian internal effort by a sine function.For the case when the Gaussian input is modulated by a
cosine function, which means that(5.4)u(t)=exp⁡(−(τ+0.9)2(0.01)2)cos104τ,we obtain an output represented
by a slowly varying function, without alternate deformation. So a cosine
modulation of a Gaussian input is not suitable for simulating fracture
phenomena appearing at a certain distance from the point where the external
force acts.We must point the fact that such localized alternating
deformations generated by systems working on a limited interval and situated at
a certain distance from the point where the external force acts differ to
wavelets resulting from PDE equations (see [11]) and to propagating wavelets through dispersive media
[12], while the shape
of the resulting deformation is not symmetrical as related to Ox axis (its mean
value differs to zero). However, a multiscale analysis of such pulses should be
performed for explaining the complex fracture phenomena in an extended area and
for justifying why a certain direction for generating deformation has to be
chosen.


## 6. Conclusions

This study has shown that some solutions of the wave
equation for half-closed space interval are
considered around the point of space where the sources of the
generated field are situated (e.g., the case of electrical charges generating
the electromagnetic field). These solutions can be mathematically represented
by vanishing waves corresponding to a superposition of traveling
test-functions. Then some properties of spatial linear systems described by a
certain physical quantity (generated by a differential equation) are studied.
This quantity can be represented by internal electric or magnetic field inside
the material or by similar physical quantities, and corresponds to the
amplitude of localized oscillations previously mentioned. A specific
mathematical law which can be approximated by a differential equation generates
this quantity considering as input the spatial alternating variations of this
internal parameter. As a consequence, specific spatial linear variations of the
corresponding physical quantity appear. Finally, a specific differential
equation (able to generate symmetrical functions for a null free term) is used
for describing the generation of the corresponding deformation along an axis
inside the material medium. Numerical simulations have shown that a significant
deformation appears at a certain distance. This deformation justifies the
fracture phenomena, while the inner structure of the material cannot allow
significant sharp deformations without breaking.



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