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<Text-field style="Author" layout="Author"><Font encoding="UTF-8">\302\251 2013 G.P.Chuiko,  </Font><Hyperlink linktarget="mailto:gp47@mail.ru" hyperlink="true"><Font style="Author" underline="true" foreground="[0,128,128]">gp47@mail.ru</Font></Hyperlink>  and S.I. Shiyan <Hyperlink linktarget="mailto:shyiansi@gmail.com" hyperlink="true"><Font style="Author" underline="true" foreground="[0,128,128]">shyiansi@gmail.com</Font></Hyperlink>,</Text-field>
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<Text-field style="Heading 2" layout="Heading 2">1. Model and problem formulation</Text-field></Title><Presentation-Block>
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<Text-field style="Text" layout="Normal">Model of fluid flow along the elastic channel with thin walls is used to describe the flow of blood in the large blood vessels  [1,2].  </Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred">Fig.1  The popular model of the arterial blood flow  </Text-field>
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<Text-field style="Text" layout="Normal">The development of a fluid flow in a thin elastic tube may be conditionally dividing into several relatively independent hydrodynamic processes[1,2]: </Text-field>
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<Text-field style="Ordered List 1" layout="Ordered List 1" spacebelow="6">Stationary transfer of fluid along the tube (channel); </Text-field>
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<Text-field style="Ordered List 1" layout="Ordered List 1" spacebelow="6">Spread of waves of pressure and deformation of the walls of the tube at a speed that significantly exceeds the stationary velocity of the fluid. These commonly referred as arterial pulse waves (APW);</Text-field>
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<Input>
<Text-field style="Ordered List 1" layout="Ordered List 1" spacebelow="6">Occurrence of self-excited oscillations (i.e flutter) because of the instability of the system &quot;stream - elastic walls&quot;</Text-field>
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<Text-field style="Text" layout="Ordered List 1" spacebelow="6" bullet="none">All of these phenomena can be described within the theoretical basis of hydrodynamics -   Navier-Stokes  and continuity equations.</Text-field>
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<Text-field style="Text" layout="Ordered List 1" spacebelow="6" bullet="none">In this paper, we are primarily focusing on the arterial pulse waves, more exactly even their dispersion. It is important because presence of this together with the non-linearity generates the conditions of existence of localized waveforms. Thus, we would like to obtain here the dispersion law for pulse waves with Maple.</Text-field>
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<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 2" layout="Heading 2">2. Dimensional analysis and the law of dispersion</Text-field></Title><Presentation-Block>
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<Text-field style="Text" layout="Normal">Some problems, like above outlined, were successfully solvable by methods of dimensional analysis [3].  </Text-field>
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<Text-field style="Text" layout="Normal">To do this, let  set the so-called leading control options (variables and constants), which significantly affect the spread of pulse waves.<Font encoding="UTF-8"> TTo do this, set the so-called lead control parameters (variables and constants), which significantly affect the spread of arterial pulse waves. The following control parameters can form the short list:  the cyclic frequency of the APW (\317\211), wavenumber (k), kinematic viscosity of blood (\316\274), a distinctive cross channel size (let it be \316\264 - effective diameter of the artery), blood density (\317\201) and effective modulus of elasticity of the arterial wall (C) [1,2].  An equation of state unites the selected factors:</Font> </Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>      (I) </Text-field>
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<Text-field style="Text" layout="Normal">Define the dimensions of control factors:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:with(Units):
AddDimension('vawe_number'='length^(-1)'):
AddDimension('elastic_modulus'='pressure'):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">map(GetDimension,['frequency','vawe_number','kinematic_viscosity','length','mass_density','elastic_modulus']);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NygqJEkldGltZUclKnByb3RlY3RlZEchIiIqJEknbGVuZ3RoR0YlRiYqJkYoIiIjRiRGJkYoKiZJJW1hc3NHNiIiIiJGKCEiJCooRixGLkYoRiZGJCEiIw==</Equation></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">We need only three fundamental units (length, time and mass: <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiVEYnRi9GMkY1LUYsNiVRIk1GJ0YvRjJGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiVEYnRi9GMkY1LUYsNiVRIk1GJ0YvRjJGOQ==</Equation>) to describe the dimensions of the six control parameters. It can be seen from the list of dimensions (2.1). Therefore, the equation of state (I) may be rewritten in terms of the three (6-3 = 3) dimensionless factors constructed from six primary control parameters <Font encoding="UTF-8">in accordance with the famous Bek\303\254ngem's theorem [3] . The unit dimension of these three factors </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> can be recorded by the dimensions of primary control parameters in the monomial form:</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>    (II)<Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation></Text-field>
<Text-field style="Text" layout="Normal">hare  <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> - are degree indicators.  One can form a system of linear equations for these indicators from the expression (II) </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eq1:=-r2+2*r3+r4-3*r5-r6=0;
Eq2:=-r1-r3-2*r6=0;
Eq3:=r5+r6=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTFHRigvLCxJI3IyR0YoISIiSSNyM0dGKCIiI0kjcjRHRigiIiJJI3I1R0YoISIkSSNyNkdGKEYxIiIhNyNGLg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTJHRigvLChJI3IxR0YoISIiSSNyM0dGKEYxSSNyNkdGKCEiIyIiITcjRi4=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRFcTNHRigvLCZJI3I1R0YoIiIiSSNyNkdGKEYxIiIhNyNGLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Get this system in relation to the three indicators that presenting via the other three:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">r:=solve([Eq1,Eq2,Eq3],[r2,r5,r6]);<Font foreground="[0,0,255]">#</Font> <Font foreground="[0,0,255]">Solutions of (2.2) for r2,r5,r6
</Font>assign(r);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoNyM3JS9JI3IyR0YoLChJI3IzR0YoIiIiSSNyNEdGKEY0SSNyMUdGKCEiIi9JI3I1R0YoLCZGNiNGNCIiI0YzRjsvSSNyNkdGKCwmRjYjRjdGPEYzRkA3I0Yu</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Lets substitute the solution (2.3) in the monomial expression (II)</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">omega^r1*k^r2*mu^(r3)*delta^r4*rho^r5*C^r6=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LyouKUkmb21lZ2FHNiJJI3IxR0YmIiIiKUkia0dGJiwoSSNyM0dGJkYoSSNyNEdGJkYoRichIiJGKClJI211R0YmRixGKClJJmRlbHRhR0YmRi1GKClJJHJob0dGJiwmRicjRigiIiNGLEY2RigpSSJDR0YmLCZGJyNGLkY3RixGO0YoRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">There the dimensions of parameters are noted simply by symbols of them, of course.
Lets apply the method of extremals [4] for unambiguous identification of the rest unknown indicators (<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>) . Taking the logarithm of the left side of the expression (2.4), we write it as a function of those indicators:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p:=(r1,r3,r4)-&gt;r1*log(omega)+(r3+r4-r1)*log(k)+(r3)*log(mu)+(r4)*log(delta)+((1/2)*r1+(1/2)*r3)*log(rho)+(-(1/2)*r1-(1/2)*r3)*log(C);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJwR0YoZio2JUkjcjFHRihJI3IzR0YoSSNyNEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLC4qJjkkIiIiLUkkbG9nR0YlNiNJJm9tZWdhR0YoRjlGOSomLCg5JUY5OSZGOUY4ISIiRjktRjs2I0kia0dGKEY5RjkqJkZARjktRjs2I0kjbXVHRihGOUY5KiZGQUY5LUY7NiNJJmRlbHRhR0YoRjlGOSomLCZGOCNGOSIiI0ZARlBGOS1GOzYjSSRyaG9HRihGOUY5KiYsJkY4I0ZCRlFGQEZXRjktRjs2I0kiQ0dGKEY5RjlGKEYoRig3I0Yu</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, cosequently find the derivatives of the function  (2,5) on its variables, looking for extreme</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq1:=D[1](p)(r1)=0;
eq2:=D[2](p)(r3)=0;
eq3:=D[3](p)(r4)=0;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigvLCotSSNsbkdGJTYjSSZvbWVnYUdGKCIiIi1GMTYjSSJrR0YoISIiLUYxNiNJJHJob0dGKCNGNCIiIy1GMTYjSSJDR0YoI0Y4Rj0iIiE3I0Yu</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTJHRigvLCotSSNsbkdGJTYjSSJrR0YoIiIiLUYxNiNJI211R0YoRjQtRjE2I0kkcmhvR0YoI0Y0IiIjLUYxNiNJIkNHRigjISIiRjwiIiE3I0Yu</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTNHRigvLCYtSSNsbkdGJTYjSSJrR0YoIiIiLUYxNiNJJmRlbHRhR0YoRjQiIiE3I0Yu</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">simplify(solve([eq1,eq2,eq3],[omega,mu,delta]));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NyM3JS9JJm9tZWdhRzYiKiYtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwmSSJDRzYiISIiKiZJJHJob0c2IiIiIkkjX1pHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiMiIiIvSSZsYWJlbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IkkkX0w0RzYiIiIiSSJrRzYiIiIiL0kjbXVHNiIqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCZJIkNHNiIhIiIqJkkkcmhvRzYiIiIiSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIi9JJmxhYmVsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSRfTDJHNiIiIiJJImtHNiIhIiIvSSZkZWx0YUc2IiokSSJrRzYiISIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solutions (2.7) are including the unknown expressions (<Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEnUm9vdE9mRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRkwvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Yv">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEnUm9vdE9mRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRkwvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Yv</Equation>  type). Let's be clear their contents:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">expr1:=RootOf(-C+rho*_Z^2, label = _L4):
expr2:=RootOf(-C+rho*_Z^2, label = _L2):
a:=allvalues(expr1);
b:=allvalues(expr2);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoNiQqJComSSRyaG9HRighIiJJIkNHRigiIiIjRjQiIiMsJEYvRjI3JEYvRjc=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJiR0YoNiQqJComSSRyaG9HRighIiJJIkNHRigiIiIjRjQiIiMsJEYvRjI3JEYvRjc=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Thus, the problem is just the selection of right sign for the well-known Moens-Korteweg velocity (2.8) [1,2].
Taking into account  (2.7) and (2.8), find the three dimensionless control parameters </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u1:=omega/(a[1]*k);
u2:=mu*k/b[1];
u3:=delta*k;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN1MUdGKCooSSZvbWVnYUdGKCIiIiomSSRyaG9HRighIiJJIkNHRihGMCNGMyIiI0kia0dGKEYzNyNGLg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN1MkdGKCooSSNtdUdGKCIiIkkia0dGKEYwKiZJJHJob0dGKCEiIkkiQ0dGKEYwI0Y0IiIjNyNGLg==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN1M0dGKComSSJrR0YoIiIiSSZkZWx0YUdGKEYwNyNGLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L67" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Now we can rewrite the equation of state for APW in such a form</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L641" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u1=f(u2,u3);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LyooSSZvbWVnYUc2IiIiIiomSSRyaG9HRiUhIiJJIkNHRiVGJiNGKSIiI0kia0dGJUYpLUkiZkdGJTYkKihJI211R0YlRiZGLUYmRidGKyomRi1GJkkmZGVsdGFHRiVGJg==</Equation></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">Equation (2.10) is actually the law of dispersion for the frequency of the pulse waves  since it is the connection between the frequency and wavenumber. Both dimensionless parameters, which the function  of right part (2.10) depends on, are  proportional to vawe-number. </Text-field>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L771" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">One of these dimensionless parameters <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUYvNiVRImtGJ0YyRjUtRkQ2LVEnJnNkb3Q7RidGPkZHRkpGTEZORlBGUkZUL0ZXUSYwLjBlbUYnL0ZaRlxvLUYvNiVRJyYjOTQ4O0YnL0YzRklGPi1GRDYtUSJ+RidGPkZHRkpGTEZORlBGUkZURltvRl1vRj4=</Equation> is associated with the forces that act on blood  due the changes of the channel crossing, or the so-called Bernoulli effects. Another parameter (<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>) describes the effect of viscous friction on the propagation of the APW.  </Text-field>
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<Input>
<Text-field style="Text" layout="Normal">Both dimensionless parameters are  proportional vawe-number, that allows us to consider their function as a function of this one variable.  Then the law of dispersion (2.11) can get such a compact form:</Text-field>
</Input>
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<Group view="code" labelreference="L643" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">omega(k)=a[1]*k*f(k);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">Ly1JJm9tZWdhRzYiNiNJImtHRiUqKComSSRyaG9HRiUhIiJJIkNHRiUiIiIjRi0iIiNGJ0YtLUkiZkdGJUYmRi0=</Equation></Text-field>
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</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 2" layout="Heading 2">3. Analysis and simplification of the result</Text-field></Title><Presentation-Block>
<Group labelreference="L638" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Just the dependence of the  function <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> on vawe-number, and so its difference from a constant, determines the non-linearity of the dispersion law (2.11). Opposite, the dispersion law becomes linear in the case  <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImtGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRJmNvbnN0RidGL0YyRj0=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImtGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRJmNvbnN0RidGL0YyRj0=</Equation>  , i.e. the phase and group velocities of APW coincide.</Text-field>
<Text-field style="Text" layout="Normal">It is easy to see that the ratio of two arguments of function <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>from  (2.10) is exactly equal to well-known unitless Reynold's number</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u3/u2=Rey;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LyooSSZkZWx0YUc2IiIiIkkjbXVHRiUhIiIqJkkkcmhvR0YlRihJIkNHRiVGJiNGJiIiI0kkUmV5R0Yl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L637" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">This number describes the contribution of the viscous forces in the current mode of the stream. 
Evaluations of APW (Moens-Korteweg) velosity as well as Reynolds number according to the data of [1,2] give:</Text-field>
</Input>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">a[1]=eval(a[1],[mu=4e-6,delta=2.4e-2,rho=1.06e3,C=5e4])*Unit('m/s');
Rey=eval(u3/u2,[mu=4e-6,delta=2.4e-2,rho=1.05e3,C=6e4]); </Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvKiQqJkkkcmhvR0YoISIiSSJDR0YoIiIiI0YyIiIjLCQtSSVVbml0R0YoNiMqJkkibUdGKEYySSJzR0YoRjAkIisoPkchb28hIio3Iy9GLSwkLUklVW5pdEc2JC9JK21vZHVsZW5hbWVHRihJJlVuaXRzR0YlRiY2IyomNiMuRjpGMjYjLkY7RjBGPA==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkUmV5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRLDQ1MzU1LjczNjc2RidGOUY5">L0kkUmV5RzYiJCIrd09kTlghIiY=</Equation></Text-field>
</Output>
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<Input>
<Text-field style="Text" layout="Normal">Hence, the  Moens-Korteweg velosity significantly exceeds the typical value for stationary blood flow (about 0.2 m/s), and the Reynolds number for the pulse waves is a few orders of magnitude greater than  unit.  Extremely small parameter <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIjxGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGQ0YrRj4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIjxGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGQ0YrRj4=</Equation> may be naglected in (2.10) according to the recommendations of the theory [3] . From the physical point of view it means choosing the model of ideal fluids for blood flow [1], neglecting the effects of viscous friction on the spread of IAPW.  On the basis of such considerations, the equation of state (2.10) can be rewritten in the following compact and simpler form in comparison with (I) as follows:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u1=f(u3);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRKCZvbWVnYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSZtc3FydEdGJDYjLUYsNigtRi82JVEiQ0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUYvNiVRJiZyaG87RidGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZOLyUpYmV2ZWxsZWRHRjQtSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmFvLUYvNiVRImtGJ0ZCRkRGNUZJRkxGT0ZRLUZUNi1RKSZlcXVhbHM7RidGNUZXRllGZW5GZ25GaW5GW29GXW8vRmBvUSwwLjI3Nzc3NzhlbUYnL0Zjb0ZbcC1GIzYmLUYvNiVRImZGJ0ZCRkQtRlQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNUZXRllGZW5GZ25GaW5GW29GXW9GX29GYm8tSShtZmVuY2VkR0YkNiQtRiM2JkZkb0ZTLUYvNiVRKCZkZWx0YTtGJ0YyRjVGNUY1RjVGNQ==">LyooSSZvbWVnYUc2IiIiIiomSSRyaG9HRiUhIiJJIkNHRiVGJiNGKSIiI0kia0dGJUYpLUkiZkdGJTYjKiZGLUYmSSZkZWx0YUdGJUYm</Equation></Text-field>
</Output>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L629" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Estimation of the APW length is provided in [2] :  <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation>, that corresponds to the experimental results [1]. This allows us to evaluate the factor <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdTNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdTNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn</Equation></Text-field>
</Input>
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<Group view="code" labelreference="L630" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">u3=evalf(eval(u3,[k=2*Pi/2.7,delta=2.4e-2]));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LyomSSJrRzYiIiIiSSZkZWx0YUdGJUYmJCIrM08wJmUmISM2</Equation></Text-field>
</Output>
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<Group labelreference="L631" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">So, this factor is visible smaller of unit and APW shall be taking as long waves. These waves would be considered as short only if the factor <Equation executable="false" style="2D Math" input-equation="" display="LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEjdTNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=">LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEjdTNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=</Equation> excedds the unit (at <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  Let us decompose the function of right side (3.3) nearly to zero point (<Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoayYjOTQ4O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLSZSaWdodEFycm93O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMEYnRjlGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoayYjOTQ4O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLSZSaWdodEFycm93O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMEYnRjlGOQ==</Equation>). Such decomposition is correct, if this function has a finite limit, and derivatives at this point [3].</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fs:=series(f(u), u=0,3):
fs:=convert(fs,polynom);
fs:=eval(fs,[f(0)=A,(D(f))(0)=B,((D@@2)(f))(0)=G]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmc0dGKCwoLUkiZkdGKDYjIiIhIiIiKiYtLUkiREdGJTYjRjBGMUYzSSJ1R0YoRjNGMyomLS0tSSNAQEdGJTYkRjciIiNGOEYxRjNGOUZAI0YzRkA3I0Yu</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmc0dGKCwoSSJBR0YoIiIiKiZJIkJHRihGMEkidUdGKEYwRjAqJkkiR0dGKEYwRjMiIiMjRjBGNjcjRi4=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L632" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Here <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkdGJ0YvRjItRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkxGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkdGJ0YvRjItRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkxGOQ==</Equation>- some digital unitless constants.
Now the dispersion law may be rewritten as follows</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">omega:=a[1]*k*eval(fs,[u=k*delta]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZvbWVnYUdGKCooKiZJJHJob0dGKCEiIkkiQ0dGKCIiIiNGMyIiI0kia0dGKEYzLChJIkFHRihGMyooSSJCR0YoRjNGNkYzSSZkZWx0YUdGKEYzRjMqKEkiR0dGKEYzRjZGNUY7RjVGNEYzNyNGLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L644" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">If function (3.3) has not only the finite limit  but an extrema too at zero pint,  then the constant <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=</Equation>. The dispersion law (3.6) would be one-to-one with that founded in [2] under such conditions and additional assumings: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">omega1:=eval(omega,[A=1,B=0,G=1/12]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdvbWVnYTFHRigqKComSSRyaG9HRighIiJJIkNHRigiIiIjRjMiIiNJImtHRihGMywmRjNGMyomRjZGNUkmZGVsdGFHRihGNSNGMyIjQ0YzNyNGLg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L645" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The dispersion law (3.7) is practically linear because second factor within brackets is much smaller of first (see (3.4)). 
The velosities of phase propagation as well as of energy transport are determined by (3.6):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">v[ph]:=omega/k;
v[gr]:=diff(omega,k);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkidkdGKDYjSSNwaEdGKComKiZJJHJob0dGKCEiIkkiQ0dGKCIiIiNGNiIiIywoSSJBR0YoRjYqKEkiQkdGKEY2SSJrR0YoRjZJJmRlbHRhR0YoRjZGNiooSSJHR0YoRjZGPUY4Rj5GOEY3RjY3I0Yx</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Let us illustrate the dependence of these velosities on vawenumber. Unitless velisities (3.8) were examinated as their ratio to Moens-Korteweg speed with following  assunings: <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>  </Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot([eval(v[ph]/sqrt(C/rho),[A=1,B=1/3,G=1/12,delta=2.4e-2]),eval(v[gr]/sqrt(C/rho),[A=1,B=1/3,G=1/12,delta=2.4e-2])], k=1.5..3.5,1..1.06,gridlines=true,axes=boxed,color=[red,blue],thickness=3,labels=['k','v'],labelfont=[helvetica,16],legend=[`v[ph]`,`v[gr]`],title=`Fig.2 Pulse vawe speeds (Korteveg-Moens speed units)<Font encoding="UTF-8">\134n</Font> vs vawe number (dispersion)`,font=[helvetica,16]);</Text-field>
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<Text-field style="Text" layout="Normal">Fig.2 shows, thet the velosities are a bit lower for the longer vawes with smaller vawenumbers and vice versa. Such dispersion is named us anomal in optics. The change of sign of <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2JC1GIzYnRistRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZYLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZobi1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGOS1GIzYkLUZebzYkUSIzRidGOUY5LyUubGluZXRoaWNrbmVzc0dGYG8vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGam8vJSliZXZlbGxlZEdGPUY5RjlGNS1GLDYjUSFGJ0Y5</Equation>is able to do this dispersion normal.</Text-field>
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<Text-field style="Heading 2" layout="Heading 2">4. Conclusions</Text-field></Title><Presentation-Block>
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<Text-field style="Text" layout="Normal">The pointed out dispersion of pulse waves, i.e. visible dependence of their speed on the wave-number (i.e and wave length), is arising just from  analysis of the dimensions. It means, that this effect  &quot;takes a place forever&quot;. This allows the soliton-like nature of APW as exact solutions of Navier-Stockes equation in principle,  especially taking into account well-known non-linearity of them. Dispersion and non-linearity can interact to produce permanent and localized wave forms [5] and pulse waves are their examples.</Text-field>
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<Text-field style="Heading 2" layout="Heading 2">5. References</Text-field></Title><Presentation-Block>
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<Text-field style="Text" layout="Normal">[1]  T.J. Predley. The Fluid Mechanics of Large Blood Vessels. Cambridge University Press. Cambridge, 1980.-401 pp.</Text-field>
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<Text-field style="Text" layout="Normal">[2]  A.N. Volobuev. A flow of fluid in the elastic-wall tubes.// Uspechi fiz. nauk. <Font encoding="UTF-8">1995.- v.165, \342\204\2262, p. 177-186 (in Russian)</Font></Text-field>
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<Text-field style="Text" layout="Normal">[3]  G.I. Barenblatt. Scaling, Self-similarity and Intermediate Asymptotics. Cambridge University Press. Cambridge, 1996 .</Text-field>
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<Text-field style="Text" layout="Normal">[4] E.R. Smolyakov. Extrema principle in dimensional theory and new fubdamental physical constants.// Proc. of Institute of System Analysis of Russian Acad. Sci., <Font encoding="UTF-8">2008.\342\200\224 \342\204\22633, p. 78-95. </Font>(in Russian)</Text-field>
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<Text-field style="Text" layout="Normal">[5] P. G. Drazin, ; R. S. Johnson,. Solitons: an introduction (2nd ed.). Cambridge University Press. 1989, ISBN 0-521-33655-4.</Text-field>
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<Text-field style="Text" layout="Normal"><Font italic="true">Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.</Font></Text-field>
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