Mittag-Leffler Expansion for the Floating Elastic Plate Dispersion Relation
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. This results is used to calculate the Free-Surface Green Function for a Floating Elastic Plate. The Mittag-Leffler expansion (Carrier, Krook and Pearson carrier) is a tool for expressing functions of a complex variable. We will use the Mittag-Leffler expansion to show that the function
where
is the Dispersion Relation for a Floating Elastic Plate can be expressed by a linear sum of terms like [math]\displaystyle{ 1/\left( \gamma-a\right) }[/math], [math]\displaystyle{ a }[/math] being a zero of [math]\displaystyle{ d\left( \gamma,\omega\right) }[/math]. We first remind ourselves of the Mittag-Leffler expansion that can be found in most text books on complex analysis, and then show that it can indeed be applied to [math]\displaystyle{ \hat {w}\left( \gamma\right) }[/math].
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].
The residues [math]\displaystyle{ R\left( q\right) }[/math] can be calculated using the usual formula. Since each of the poles of [math]\displaystyle{ \hat{w}\left( \gamma\right) }[/math] is simple, the residue [math]\displaystyle{ R\left( q\right) }[/math] at a pole [math]\displaystyle{ q }[/math] can be found using the expression
As each pole [math]\displaystyle{ q }[/math] is a root of the dispersion equation, we may substitute [math]\displaystyle{ \tanh qH=\omega^{2}/\left( q^{5}+uq\right) }[/math], where for brevity we have defined [math]\displaystyle{ u=\left( 1-m\omega^{2}\right) }[/math]. The residue may then be given as the rational function of the pole
This form avoids calculation of the hyperbolic tangent which becomes small at the imaginary roots. This causes numerical round-off problems and rapid variation in computing Eqn.~((3-16)) since [math]\displaystyle{ q_{n}H }[/math] tends to [math]\displaystyle{ n\pi }[/math] as [math]\displaystyle{ n }[/math] becomes large, which makes [math]\displaystyle{ \tan q_{n}H }[/math] become small. Fig.~((fig:3-31)) shows the graphs of the two expressions, Eqn.~((3-16)) and Eqn.~((3-17)) of the residue for imaginary argument. The imaginary roots [math]\displaystyle{ \mathrm{i}q_{1},\mathrm{i}q_{2},\mathrm{i}q_{3},... }[/math] are located where the two curves in Fig.~((fig:3-31)) coincide (the spiky parts of the solid curve). There are two such points at a spike and the root is the one on the left. Eqn.~((3-16)), from the direct calculation, is a rapidly varying function near the roots [math]\displaystyle{ \left\{ \mathrm{i} q_{n}\right\} _{n=1}^{\infty} }[/math], hence a small numerical error in the values of the roots will result in a large error in the residue. In contrast, Eqn.~((3-17)) gives us a smooth function, and the resulting calculation of the residue is stable.