Difference between revisions of "Mittag-Leffler Expansion for the Floating Elastic Plate Dispersion Relation"
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Define a sequence of square contours <math>C_{n}</math>, square with its four corners at | Define a sequence of square contours <math>C_{n}</math>, square with its four corners at | ||
<math>\epsilon_{n}-\mathrm{i}\epsilon_{n}</math>, <math>\epsilon_{n}+\mathrm{i} | <math>\epsilon_{n}-\mathrm{i}\epsilon_{n}</math>, <math>\epsilon_{n}+\mathrm{i} | ||
− | \epsilon_{n}<math>, < | + | \epsilon_{n}</math>, <math>-\epsilon_{n}+\mathrm{i}\epsilon_{n}</math> and |
− | <math>-\epsilon_{n}-\mathrm{i}\epsilon_{n}</math>, where <math>\epsilon_{n}=\left( | + | <math>-\epsilon_{n}-\mathrm{i}\epsilon_{n}</math>, |
− | n+\frac{1}{2}\right) \pi/H,\,n=N,N+1,...<math>. We start by showing that < | + | where <math>\epsilon_{n}=\left( |
− | \hat{w}\left( \gamma\right) \right| <math> is bounded on any < | + | n+\frac{1}{2}\right) \pi/H,\,n=N,N+1,...</math>. We start by showing that <math>\left| |
+ | \hat{w}\left( \gamma\right) \right| </math> is bounded on any <math>C_{n}</math> in order to | ||
follow the proof of Mittag-Leffler expansion given in the previous subsection. | follow the proof of Mittag-Leffler expansion given in the previous subsection. | ||
For the sake of simplicity, write <math>u=1-m\omega^{2}</math>. When <math>Im | For the sake of simplicity, write <math>u=1-m\omega^{2}</math>. When <math>Im | ||
− | \gamma<math> is large the poles of < | + | \gamma</math> is large the poles of <math>\hat{w}</math> are almost <math>\pm\mathrm{i} |
− | n\pi/H.<math> In fact, the poles < | + | n\pi/H.</math> In fact, the poles <math>\left\{ \mathrm{i}q_{n}\right\} |
− | _{n=1,2,...}<math>, < | + | _{n=1,2,...}</math>, <math>q_{n}\in\mathbf{R}</math> of <math>\hat{w}</math> satisfy |
<center><math> | <center><math> | ||
\frac{1}{\left( q_{n}+u\right) q_{n}}=\tan\left( q_{n}H\right) , | \frac{1}{\left( q_{n}+u\right) q_{n}}=\tan\left( q_{n}H\right) , | ||
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large <math>N</math>, the contour <math>C_{n}</math> is always a certain distance away from the | large <math>N</math>, the contour <math>C_{n}</math> is always a certain distance away from the | ||
poles for any <math>n\geq N</math>. We prove the boundedness of <math>\left| \hat{w}\right| | poles for any <math>n\geq N</math>. We prove the boundedness of <math>\left| \hat{w}\right| | ||
− | <math> by showing that < | + | </math> by showing that <math>\left| \hat{w}\left( x+\mathrm{i}y\right) |
− | \right| <math> is bounded for < | + | \right| </math> is bounded for <math>y=\pm\epsilon_{n}</math>, <math>n=N,N+1,...</math>, and |
<math>x,y\in\mathbf{R}</math>, and then for <math>x=\pm\epsilon_{n},\,n=N,N+1,...,</math> | <math>x,y\in\mathbf{R}</math>, and then for <math>x=\pm\epsilon_{n},\,n=N,N+1,...,</math> | ||
<math>y\in\left[ -\epsilon_{n},\epsilon_{n}\right] </math>. | <math>y\in\left[ -\epsilon_{n},\epsilon_{n}\right] </math>. | ||
Line 293: | Line 294: | ||
</math></center> | </math></center> | ||
Note that the summation on the first line is over all poles of <math>\hat{w}\left( | Note that the summation on the first line is over all poles of <math>\hat{w}\left( | ||
− | \gamma\right) <math>. Note that < | + | \gamma\right) </math>. Note that <math>R\left( q\right) =-R\left( -q\right) </math>, since |
<math>\hat{w}\left( \gamma\right) </math> is an even function and | <math>\hat{w}\left( \gamma\right) </math> is an even function and | ||
<center><math> | <center><math> | ||
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the function <math>\hat{w}\left( \gamma\right) \gamma</math> which has the same | the function <math>\hat{w}\left( \gamma\right) \gamma</math> which has the same | ||
analytic properties and poles as the function <math>\hat{w}</math> and residues <math>R\left( | analytic properties and poles as the function <math>\hat{w}</math> and residues <math>R\left( | ||
− | q\right) q<math> at < | + | q\right) q</math> at <math>\gamma=q</math>. Hence, <math>\hat{w}\left( \gamma\right) \gamma</math> is |
expanded as, | expanded as, | ||
<center><math> | <center><math> |
Revision as of 03:36, 24 June 2006
Introduction
We derive here the important results that the Dispersion Relation for a Floating Elastic Plate can be written in the following form using the Mittag-Leffler expansion
Mittag-Leffler expansion
Consider a function that is regular in the whole plane except at isolated points. A set of points is called isolated if there exists an open disk around each point that contains none other of the isolated points. Such a function is known as fractional function. We show that a fractional function that has an infinite number of poles can be expressed by infinite series of polynomials (Carrier, Krook and Pearson carrier).
Let [math]\displaystyle{ f\left( \gamma\right) }[/math] be a fractional function that has an infinite number of poles. We note that a number of poles that are situated within a bounded region is always finite since the set of poles does not have limit-points. Indeed, if there is a limit-point [math]\displaystyle{ \gamma=c }[/math] then any small circle with centre at [math]\displaystyle{ \gamma=c }[/math] would contain an infinite number of poles. Once we have a finite number of poles in a confined part of the plane we can number them in the order of their non-decreasing moduli, so that denoting the poles by [math]\displaystyle{ a_{i} }[/math] we have
where [math]\displaystyle{ \left| a_{i}\right| \rightarrow\infty }[/math] as [math]\displaystyle{ i\rightarrow\infty }[/math]. At every pole [math]\displaystyle{ \gamma=a_{i} }[/math] the function [math]\displaystyle{ f\left( \gamma\right) }[/math] will have a definite infinite part, which will be a polynomial with respect to the argument [math]\displaystyle{ 1/\left( \gamma-a_{i}\right) }[/math] without the constant term. We denote this polynomial term by
We show that the fractional function [math]\displaystyle{ f\left( \gamma\right) }[/math] can be represented by a simple infinite series of [math]\displaystyle{ G_{i} }[/math] by making certain additional assumptions. Suppose that a sequence of closed contours [math]\displaystyle{ C_{n} }[/math] which surround the origin exists and satisfies following conditions.
{\it {\bf Condition 1.}{ None of poles of }[math]\displaystyle{ f\left( \gamma\right) }[/math] are on the contours [math]\displaystyle{ C_{n},\,n=1,2,3,... }[/math]
{\bf Condition 2. }{ Every contour }[math]\displaystyle{ C_{n} }[/math] { lies inside the contour }[math]\displaystyle{ C_{n+1} }[/math].
{\bf Condition 3. } {Let }[math]\displaystyle{ l_{n} }[/math] { be length of the contour }[math]\displaystyle{ C_{n} }[/math] {\ and }[math]\displaystyle{ \delta_{n} }[/math] {\ be its shortest distance from the origin then }[math]\displaystyle{ \delta_{n}\rightarrow\infty }[/math] { as }[math]\displaystyle{ n\rightarrow\infty }[/math] {, i.e., the contours }</math>C_{n}</math> { widen indefinitely in all directions as }[math]\displaystyle{ n }[/math] {\ increases.}
{\bf Condition 4.}{ A positive number }[math]\displaystyle{ m }[/math]\it{\ exists such that }
}
We now suppose that given such a sequence of contours, there exists a positive number [math]\displaystyle{ M, }[/math] such that on any contour [math]\displaystyle{ C_{n} }[/math] our fractional function [math]\displaystyle{ f\left(\gamma\right) }[/math] satisfies [math]\displaystyle{ \left| f\left( \gamma\right) \right| \leq M }[/math]. Consider the integral
where the point [math]\displaystyle{ \gamma }[/math] lies inside [math]\displaystyle{ C_{n} }[/math] and is other than [math]\displaystyle{ a_{i} }[/math] (the poles inside [math]\displaystyle{ C_{n}. }[/math]) We also consider the sum of the polynomials ((ap-4)) for the poles [math]\displaystyle{ a_{i} }[/math], inside [math]\displaystyle{ C_{n} }[/math],
The integrand of ((ap-6)) has a pole [math]\displaystyle{ \gamma^{\prime}=\gamma }[/math] and poles [math]\displaystyle{ \gamma^{\prime}=a_{i} }[/math]. We can calculate the residue at the pole [math]\displaystyle{ \gamma^{\prime}=\gamma }[/math] by
The residues at the poles [math]\displaystyle{ \gamma^{\prime}=a_{i} }[/math] are, by the definition ((ap-8)), the same as the residues of the function
We note that all poles of this function are situated inside [math]\displaystyle{ C_{n} }[/math]. We now show that the sum of residues of function ((ap-7)) at the poles [math]\displaystyle{ a_{i} }[/math] is
Since the definition of [math]\displaystyle{ \omega_{n} }[/math] and [math]\displaystyle{ G_{i} }[/math] is a polynomial of [math]\displaystyle{ 1/\left( \gamma-a_{i}\right) , }[/math] the order of the denominator of function ((ap-7)) is at least two units higher than that of the numerator of function ((ap-7)). Hence, for a circle with a sufficiently large radius [math]\displaystyle{ R }[/math], we have
The LHS of this does not change as the radius [math]\displaystyle{ R }[/math] increases, and the RHS[math]\displaystyle{ \rightarrow0 }[/math] as [math]\displaystyle{ R\rightarrow\infty }[/math]. Indeed,
and the term [math]\displaystyle{ \left| \cdot\right| }[/math] tends to zero as [math]\displaystyle{ R\rightarrow\infty }[/math]. Thus, the sum of residues at poles within a finite distance is zero. Since we know that the residue of ((ap-7)) at [math]\displaystyle{ \gamma^{\prime}=\gamma }[/math] is [math]\displaystyle{ \omega_{n}\left( \gamma\right) }[/math], the sum of the rest is formula ((ap-9)). Thus, we have an expression for the integral ((ap-6)),
Also, when [math]\displaystyle{ \gamma=0 }[/math] we have
Subtracting Eqn.~((ap-10)) from Eqn.~((ap-11)) gives
We now prove that the integrand on the LHS of this expression tends to zero as [math]\displaystyle{ n\rightarrow\infty }[/math]. Since, [math]\displaystyle{ \left| \gamma^{\prime}\right| \geq\delta _{n},\,\,\left| \gamma^{\prime}-\gamma\right| \geq\left| \gamma^{\prime }\right| -\left| \gamma\right| \geq\delta_{n}-\left| \gamma\right| , }[/math] we have
Since [math]\displaystyle{ \delta_{n}\rightarrow\infty }[/math] as [math]\displaystyle{ n\rightarrow\infty }[/math] and {\bf condition 4}, the integral in inequality ((ap-13)) tends to zero as [math]\displaystyle{ n }[/math] increases.
Finally, we have formula for [math]\displaystyle{ f\left( \gamma\right) }[/math],
Since, the contour [math]\displaystyle{ C_{n} }[/math] will widen indefinitely as [math]\displaystyle{ n }[/math] increases, the second term is a sum over all poles, so we have [math]\displaystyle{ f\left( \gamma\right) }[/math] in the form of an infinite series
For the expansion formula of [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math], the polynomial term ((ap-4)) is
Expansion of the Dispersion Relation for a Floating Elastic Plate
Now we show that the function [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math] satisfies the conditions for the Mittag-Leffler expansion.
Define a sequence of square contours [math]\displaystyle{ C_{n} }[/math], square with its four corners at [math]\displaystyle{ \epsilon_{n}-\mathrm{i}\epsilon_{n} }[/math], [math]\displaystyle{ \epsilon_{n}+\mathrm{i} \epsilon_{n} }[/math], [math]\displaystyle{ -\epsilon_{n}+\mathrm{i}\epsilon_{n} }[/math] and [math]\displaystyle{ -\epsilon_{n}-\mathrm{i}\epsilon_{n} }[/math], where [math]\displaystyle{ \epsilon_{n}=\left( n+\frac{1}{2}\right) \pi/H,\,n=N,N+1,... }[/math]. We start by showing that [math]\displaystyle{ \left| \hat{w}\left( \gamma\right) \right| }[/math] is bounded on any [math]\displaystyle{ C_{n} }[/math] in order to follow the proof of Mittag-Leffler expansion given in the previous subsection.
For the sake of simplicity, write [math]\displaystyle{ u=1-m\omega^{2} }[/math]. When [math]\displaystyle{ Im \gamma }[/math] is large the poles of [math]\displaystyle{ \hat{w} }[/math] are almost [math]\displaystyle{ \pm\mathrm{i} n\pi/H. }[/math] In fact, the poles [math]\displaystyle{ \left\{ \mathrm{i}q_{n}\right\} _{n=1,2,...} }[/math], [math]\displaystyle{ q_{n}\in\mathbf{R} }[/math] of [math]\displaystyle{ \hat{w} }[/math] satisfy
so [math]\displaystyle{ \gamma_{n}\rightarrow\pm n\pi/H }[/math] as [math]\displaystyle{ n }[/math] increases. Thus, by choosing a large [math]\displaystyle{ N }[/math], the contour [math]\displaystyle{ C_{n} }[/math] is always a certain distance away from the poles for any [math]\displaystyle{ n\geq N }[/math]. We prove the boundedness of [math]\displaystyle{ \left| \hat{w}\right| }[/math] by showing that [math]\displaystyle{ \left| \hat{w}\left( x+\mathrm{i}y\right) \right| }[/math] is bounded for [math]\displaystyle{ y=\pm\epsilon_{n} }[/math], [math]\displaystyle{ n=N,N+1,... }[/math], and [math]\displaystyle{ x,y\in\mathbf{R} }[/math], and then for [math]\displaystyle{ x=\pm\epsilon_{n},\,n=N,N+1,..., }[/math] [math]\displaystyle{ y\in\left[ -\epsilon_{n},\epsilon_{n}\right] }[/math].
The detailed observation on the poles of [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math] will be given in the next section, but for now we only need to know that [math]\displaystyle{ \hat{w} }[/math] has two real, four complex and an infinite countable number of imaginary poles. Let [math]\displaystyle{ K }[/math] be the set of all poles of [math]\displaystyle{ \hat{w} }[/math] and [math]\displaystyle{ K^{} }[/math] be the set of a positive real pole and poles with positive imaginary parts.
For any [math]\displaystyle{ n\gt N }[/math] we have
where [math]\displaystyle{ C }[/math] is a constant determined by [math]\displaystyle{ u }[/math]. When [math]\displaystyle{ y=\epsilon_{n} }[/math] we have
for any [math]\displaystyle{ x\in\mathbf{R} }[/math]. (We used [math]\displaystyle{ \exp\left( \mathrm{i}\left( 2n+1\right) \pi\right) =-1 }[/math] and
to show this.) For large [math]\displaystyle{ \left| \gamma\right| }[/math] we have
Since the RHS of this inequality is positive from Eqn.~((ap-1)) and Eqn.~((ap-2)),
for any [math]\displaystyle{ n\geq N }[/math]. Note that the same relationship holds for [math]\displaystyle{ y=-\epsilon_{n} }[/math].
For [math]\displaystyle{ \gamma }[/math] on the line segment [math]\displaystyle{ \epsilon_{n}-\mathrm{i}\epsilon_{n} }[/math] to [math]\displaystyle{ \epsilon_{n}+\mathrm{i}\epsilon_{n} }[/math] we use the fact that
for any [math]\displaystyle{ y }[/math], [math]\displaystyle{ n\geq N }[/math], where [math]\displaystyle{ E_{N} }[/math] is defined as
From Eqn.~((ap-1)) and the first line of Eqn.~((ap-3)), we have
for any [math]\displaystyle{ n\geq N }[/math]. The same proof can be applied for the line segment [math]\displaystyle{ -\epsilon_{n}-\mathrm{i}\epsilon_{n} }[/math] to [math]\displaystyle{ -\epsilon_{n} +\mathrm{i}\epsilon_{n}\lt math\gt . We have proved that }[/math]\left| \hat{w}\left( \gamma\right) \right| </math> is bounded on all sides of the contours [math]\displaystyle{ C_{n},\,n\geq N }[/math] where [math]\displaystyle{ N }[/math] is chosen to be large so that the contours are a certain distance away from all the poles of [math]\displaystyle{ \hat{w} }[/math].
Hence, the expansion of [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math] becomes, from [math]\displaystyle{ \hat{w}\left( 0\right) =0 }[/math],
Note that the summation on the first line is over all poles of [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math]. Note that [math]\displaystyle{ R\left( q\right) =-R\left( -q\right) }[/math], since [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math] is an even function and
Note that the term [math]\displaystyle{ \sum2R\left( q\right) /q }[/math] is zero. Indeed, expansion of the function [math]\displaystyle{ \hat{w}\left( \gamma\right) \gamma }[/math] which has the same analytic properties and poles as the function [math]\displaystyle{ \hat{w} }[/math] and residues [math]\displaystyle{ R\left( q\right) q }[/math] at [math]\displaystyle{ \gamma=q }[/math]. Hence, [math]\displaystyle{ \hat{w}\left( \gamma\right) \gamma }[/math] is expanded as,
The fact that [math]\displaystyle{ \sum2R\left( q\right) /q }[/math] is zero can also be confirmed by using the contour integration of the function [math]\displaystyle{ \hat{w}\left( \gamma\right) /\gamma }[/math] (Fig.~((fig:3-15)) in section 3.5 shows this integration). The function [math]\displaystyle{ \hat{w}\left( \gamma\right) /\gamma }[/math] is an odd function and has the same poles as the function [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math] with the residues [math]\displaystyle{ R\left( q\right) /q }[/math]. Notice that [math]\displaystyle{ \gamma=0 }[/math] is not a singular point of [math]\displaystyle{ \hat{w}\left( \gamma\right) /\gamma }[/math]. Hence, the integration over the real axis is zero and [math]\displaystyle{ \hat{w}\left( \gamma\right) /\gamma\rightarrow0 }[/math] on the semi-arc with order of [math]\displaystyle{ A^{-3} }[/math] as [math]\displaystyle{ A\rightarrow\infty }[/math].