Eigenfunction Matching Method for Floating Elastic Plates
We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties This work is based on Kohout et. al. 2006. This is a generalisation of the Eigenfunction Matching Method for a Semi-Infinite Floating Elastic Plate. We assume that the first and last plate are semi-infinite. The presentation here does not allow open water (it could be included but makes the formulation more complicated). In any case open water can be considered by taking the limit as the plate thickness tends to zero. The solution is derived using an extended eigenfunction matching method, in which the plate boundary conditions are satisfied as auxiliary equations.
We consider the problem of small-amplitude waves which are incident on a set of floating elastic plates occupying the entire water surface. The submergence of the plates is considered negligible. We assume that the problem is invariant in the direction, although we allow the waves to be incident from an angle. The set of plates consists of two semi-infinite plates, separated by a region which consists of a finite number of plates with variable properties. We also assume that the plate edges are free to move at each boundary, although other boundary conditions could easily be considered using the methods of solution presented here. We begin with the Frequency Domain Problem for multiple Floating Elastic Plates in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate)
where , and and the stiffness and mass constant for the th plate. The conditions at the edges of the plates are
where and represent the left and right edge of the th plate as shown in Figure~35.
Method of solution
We will solve the system of equations using an Eigenfunction Matching Method. The method was developed by Fox and Squire 1994 for the case of a single plate as the research is described in Two-Dimensional Floating Elastic Plate. We show here how this method can be extended to the case of an arbitrary number of plates. One of the key features in the eigenfunction expansion method for elastic plates is that extra modes are required in order to solve the higher order boundary conditions at the plate edges.
The potential velocity of the first plate can be expressed as the summation of an incident wave and of reflected waves, one of which is propagating but the rest of which are evanescent and they decay as tends to . Similarly the potential under the final plate can be expressed as a sum of transmitting waves, one of which is propagating and the rest of which are evanescent and decay towards . The potential under the middle plates can be expressed as the sum of transmitting waves and reflected waves, each of which consists of a propagating wave plus evanescent waves which decay as decreases or increases respectively. We could combine these waves in the formulation, but because of the exponential growth (or decay) in the direction the solution becomes numerically unstable in some cases if the transmission and reflection are not expanded at opposite ends of the plate.
Separation of variables
The potential velocity can be written in terms of an infinite series of separated eigenfunctions under each elastic plate, of the form If we apply the boundary conditions given we obtain the Dispersion Relation for a Floating Elastic Plate
Solving for gives a pure imaginary root with positive imaginary part, two complex roots (two complex conjugate paired roots with positive imaginary part in all physical situations), an infinite number of positive real roots which approach as approaches infinity, and also the negative of all these roots (Dispersion Relation for a Floating Elastic Plate) . We denote the two complex roots with positive imaginary part by and , the purely imaginary root with positive imaginary part by and the real roots with positive imaginary part by for a positive integer. The imaginary root with positive imaginary part corresponds to a reflected travelling mode propagating along the axis. The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes. In a similar manner, the negative of these correspond to the transmitted travelling, damped and evanescent modes respectively. The coefficient is
where the root with positive real part is chosen or if the real part is negative with negative imaginary part. Note that the solutions of the dispersion equation will be different under plates of different properties, and that the expansion is only valid under a single plate. We will solve for the coefficients in the expansion by matching the potential and its derivative at each boundary and by applying the boundary conditions at the edge of each plate.
Expressions for the potential velocity
We now expand the potential under each plate using the separation of variables solution. We always include the two complex and one imaginary root, and truncate the expansion at real roots of the dispersion equation. The potential can now be expressed as the following sum of eigenfunctions:
where is the non-dimensional incident wave amplitude in potential, is the plate, is the last plate, represents the -coordinate of the right edge of the plate, () represents the -coordinate of the left edge of the plate, represents the reflected potential coefficient of the mode under the plate, and represents the transmitted potential coefficient of the mode under the plate. Note that we have divided by , so that the coefficients are normalised by the potential at the free surface rather than at the bottom surface.
Expressions for displacement
The displacement is given by
Solving via eigenfunction matching
To solve for the coefficients, we require as many equations as we have unknowns. We derive the equations from the free edge conditions and from imposing conditions of continuity of the potential and its derivative in the -direction at each plate boundary. We impose the latter condition by taking inner products with respect to the orthogonal functions , where is a natural number. These functions are chosen for the following reasons. The vertical eigenfunctions are not orthogonal (they are not even a basis) and could therefore lead to an ill-conditioned system of equations. Furthermore, by choosing we can use the same functions to take the inner products under every plate. Finally, and most importantly, the plate eigenfunctions approach for large , so that as we increase the number of modes the matrices become almost diagonal, leading to a very well-conditioned system of equations.
Taking inner products leads to the following equations
where and denotes the potential under the th plate, i.e. the expression for valid for . The remaining equations to be solved are given by the two edge conditions satisfied at both edges of each plate
We will show the explicit form of the linear system of equations which arise when we solve these equations. Let be a column vector given by and be a column vector given by .
The equations which arise from matching at the boundary between the first and second plate are
The equations which arise from matching at the boundary of the th and ()th plate boundary () are
The equations which arise from matching at the ()th and th boundary are
where , , , and are by matrices given by
, , , and are given by
is a vector which is given by
The integrals in the above equation are each solved analytically. Now, for all but the first and th plate, the edge equation becomes
The first and last plates only require two equations, because each has only one plate edge. The equation for the first plate must be modified to include the effect of the incident wave. This gives us
and for the th plate we have no reflection so
, , and are 2 by M+3 matrices given by
Now, the matching matrix is a by matrix given by
the edge matrix is a by matrix given by
and finally the complete system to be solved is given by
The final system of equations has size by .