One of the promising approaches to the extraction of ocean wave energy utilizes a dome shaped artificial atoll to capture incident waves by refraction. The efficiency of such a device will depend on its absorption cross section. A solution to the wave equation is obtained which permits calculation of both scattering and absorption cross sections for an important class of atoll shapes at their design wavelength. A method for approximating the absorption cross section at other wavelengths is also presented.
The dome shape considered is believed to be the optimum shape for ocean wave energy concentration. At the suggestion of John Isaacs, this shape has been designated as "Arthur's Point Island" in honor of R.S. Arthur, his colleague for many years at Scripps Institute. It was Arthur who first explored the refraction around circular atolls by a geometrical optics method.
Section  Title  Page 

ABSTRACT  i  
CONTENTS  ii  
SYMBOLS  iv  
FIGURES  ix  
1.0  INTRODUCTION  1 
1.1  Historical Note  1 
1.2  Arthur's Point Island  4 
2.0  WAVE EQUATION SOLUTIONS, INNER REGION  12 
2.1  Arthur's Geometrical Optics Solution  12 
2.2  The Potential Function  16 
2.3  Modal Impedance  21 
2.4  Relation to Arthur's Solution  25 
3.0  WAVE EQUATION SOLUTIONS, OUTER REGION  28 
3.1  The Potential Function  28 
3.2  Differential Scattering  28 
3.3  Scattering Cross Section  29 
3.4  Absorption Cross Section  29 
3.5  Total Cross Section  31 
3.6  Comparison of Spheres and Cylinders  31 
3.7  Some Predictions  34 
4.0  NUMERICAL RESULTS  38 
4.1  Cross Sections  38 
4.2  Differential Scattering  42 
5.0  ARTHUR'S ISLANDS WITH CLIPPED SKIRTS  55 
5.1  Case 1; Constant r_{o}, Variable r_{c}  55 
5.2  Case 2; Constant r_{c}, Variable r_{o}  57 
6.0  RESPONSE OF ARTHUR'S ISLAND TO OFFDESIGN WAVELENGTHS  64 
6.1  The Phantom Island Method  64 
6.2  The Shape of Arthur's Island  64 
6.3  Best Fit In The Significant Region  65 
6.4  Relation of A to B  68 
7.0  ABSORPTION CROSS SECTION SPECTRA  71 
7.1  The Cross Sections of Phantom Islands  71 
7.2  Sample Spectra  71 
REFERENCES  74 
Figure No.  Title  Page 

1  The dimensions assume a design wave period of about seven seconds.  2 
2  The dynamometer is a small D.C. motor which applies a controllable buckling torque to the shaft. 
3 
3  Note that Arthur measured θ^{'} and θ^{'}_{o} from the negative x axis. 
5 
4  Any singlevalued curve in the allowed region represents a physically attainable dome shape. 
7 
5  The quantity dC ⁄ dR should be regarded as the "slope" of the bottom rather than the geometrical angle of inclination. 
9 
6  The rays are shown as terminating in a surf zone, which is the actual case. All rays traverse a logarithmic spiral and the capture appears complete. 
13 
7  The outer rays escape if the local depth is greater than one half of the incident wavelength. Beyond this depth the incident wave is oblivious of the bottom contour. 
14 
8  The celerity C is discontinuous at the perimeter. This causes the abrupt change of direction in the outer rays. This is an energy reflective situation. 
15 
9 
The radius r is normalized by the outer radius r_{0} and the polar angle is measured from the positive x axis. The inner and outer regions are characterized only by the prevailing local normalized celerity C. 
17 
10 
Each region of the wave front will be best coupled to a particular mode as indicated in the figure. 
26 
11  Arthur's Island exhibits an unexpected transparency to all higher order (nonpropagating) modes. Thus they are not reflected. 
36 
12  Arthur's Island scatters the least due partly to its transparency to higher order modes. The scattering peaks occur at integral values of K_{o}. 
39 
13  The absorptive peaks occur for K_{o} a little larger than an integer as each new mode becomes well "cuton." 
40 
14 
These curves were computed as shown in Table 1 but they may also be obtained by adding Figures 12 and 13. 
41 
15  K_{o} = 1, Q = 0 
43 
16  K_{o} = 20, Q = 0 
44 
17 
K_{o} = 1, Q =  ∞ 
46 
18  K_{o} = 20, Q =  ∞ 
29 
19  K_{o} = 1, Q = 1 
48 
20 
K_{o} = 3.6, Q = 1 
49 
21 
K_{o} = 20, Q = 1 
50 
22  K_{o} = 0.9, Q_{m} =  √(K^{2}_{o}  m^{2}) 
51 
23  K_{o} = 1, Q_{m} =  √(K_{o}^{2}  m^{2}) 
52 
24  K_{o}; = 3.6, Q_{m} =  √(K_{o}^{2}  m^{2}) 
53 
25  K_{o} = 20, Q_{m} =  √(K_{o}^{2}  m^{2} 
54 
26  Much construction material can be saved by optimum skirt clipping. 
56 
27  The response curve is shifted to the right by a factor 1 ⁄ R_{c} but very little change in shape occurs. 
59 
28  The response curve is shifted to the right by a factor 1 ⁄ R_{c} = 1.25 but little change in shape occurs.  CLIPPED AT R_{c} = 0.8
60 
29  The peaks are now sparse due to the shift to the right and the valleys have become very deep. 
61 
30  The radius r_{o} may always be chosen to provide a chosen outer radius and desired degree of clipping. 
62 
31  For R_{c} >= 0.8 the curves are nearly congruent. 
63 
32  The points represent A as a function of B that provide the best curve fit between Arthur's Island and a phantom island for two selections of "significant region." The solid curve corresponds to A = √B. 
67 
33  These phantom island shapes are calculated using A = √B. 
70 
34  The rolloff for f^{'}_{o} ⁄ f_{o} < 1 result from mode sparsity for too small an island. 
73 
The name DAMATOLL, as used in the captions of Figures 1 and 2, was coined to describe an atollshaped wave energy extraction device wherein the wave induced flow spills over a top lip.
"DAMATOLL" is pronounced in exactly the same way as the mildly profane expletive phrase ('damnitall'), frequently heard during petroleum shortages.(next)
The fact that waves spiral inward around small atolls, and thereby create surf on their lee side, was well known to the most ancient Polynesian navigators, but heavy surf on the lee side was probably an unpleasant surprise to landing parties early in the Pacific campaign of World War II. An investigation of surface wave refraction around circular atolls was undertaken at Scripps Insitute of Oceanography by R.S. Arthur. His results were not declassified until 1946.
In 1975 Wirt applied Arthur's analysis to the design of small artificial atolls intended to capture wave energy and convert it to mechanical power. This was accomplished by collecting the surf all around the atoll with guide vanes which directed the inward flow tangentially into a central vertical chamber. The result was a large vortex in the central vessel which served as a "fluid flywheel." Finally the water passed through a turbine wheel before returning to the open sea. The concept is shown schematically in Figure 1.
A crude 1/100scale model was constructed. This model operated in a very lively manner and was extensively tested in 19781979. An energy conversion efficiency of about 20% was attained. Figure 2 shows the model being subjected to dynametric tests.
During this era the model was demonstrated to John Isaacs, Robert Wiegel, Michael McCormick and others. Their encouragement provided, for a time, considerable impetus toward optimization of the design, and the concept was patented by Lockheed in twentyfour countries throughout the world, Wirt (1979).
In 1980, Professor C. C. Mei, of the Massachusetts Institute of Technology pointed out that a scattering analysis of the concept would be very useful to the optimization process. At this point in time, adverse changes in both National and Lockheed Corporate energy policy put the concept into a long hiatus. This paper may help to revive interest in this type of renewable energy source now that patent obstructions have expired.
The term, "Arthur's Point Island," as used in this paper was vigorously suggested by John Isaacs. Isaacs was aware of the shape optimization problem and proposed that, once the optimum shape was established, it be designated as "Arthur's Island." The author was happy to adopt this suggestion.
Arthur (1946) solved the general problem of wave refraction around islands with circular bottom contours by geometrical optics. His approach was to apply Fermat's principle in the polar coordinate system of Figure 3. The principle states that, given a source point and a receiver point in a medium with position dependent propagation velocity, the ray path will be that which provides the least transit time.
$$ t \quad = \quad \overset {P_2} {\underset {P_1} \int } \qquad \displaystyle \frac {ds} {c} \quad (minimum) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)$$
Application of the calculus of variations led to a solution for plane incident waves.
$$ \frac {dR}{R \sqrt {{\left(\cfrac{{{c}\,_o}\; R}{c \: \sin \: \theta\,^{'}_o }\right)}^2  1 } } \quad \quad = \quad\pm \quad {d {\theta}^{'}}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \; (2)$$
If \( \: \frac {c} {{c}\;_o} \: \) is known as a function of R, equation 2 can always be integrated, often in closed form but at least numerically, to obtain R as a function of \( \theta\,^{'}, \) i.e., the ray path.
Equation 2 may be written
$$ \cot \; \phi \quad = \frac {dR} {Rd\theta\,^{'}} \quad = \quad\pm \sqrt {{\left(\cfrac{R}{C \: \sin \: \theta\,^{'}_o }\right)}^2  1 } \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \; (3)$$
where \( \phi \) is the angle between the ray path and an intersecting radius. This illustrates the dominant role of the square root in determining the ray path. The square root is dominated by the term \( \cfrac{R}{C \: \sin \: \theta\,^{'}_o } \) and its role may be illustrated by plotting \( C \: \sin \: \theta\,^{'}_o \) vs. R as is done in Figure 4.
The entire upper left corner of Figure 4 is an "excluded" region for if \( \cfrac{R}{C \: \sin \: \theta\,^{'}_o } \; < \; 1 \) then \( \phi \) is imaginary. These rays veer out again and escape the island.
In the allowed region any singlevalued locus \( \cfrac{R}{C \: \sin \: \theta\,^{'}_o } \), R is permissable. An island could be designed to provide the chosen relation between C and R for the ray path incident at \( \theta\,^{'}_o \) at any single deep water wavelength \( \lambda\,_o \) because,
$$ h \: = \: \cfrac{\lambda\,_o}{ 2\pi } \quad C \: \, \text{arctanh} \: \: C \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (4) $$
and h(R) defines the physical shape of the atoll.
Thus the problem of selecting an optimum dome shape reduces to choosing the relation between the R and C most conducive to energy transmission inward over the atoll and least reflective to the energy at the boundary.
Arthur worked out several special cases in detail. One of these is a "point island" (case 1A) which is defined by the relation C = R. This island has a number of noteworthy features. First of all, the angle \( \phi \) between any ray path and the successive radii that it crosses remains constant and in fact,
$$ \phi \quad = \quad \theta\,^{'}_o \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (5) $$
The ray path is the locus of,
$$ R \quad = \quad \text {exp} \;  \; ( \theta\,^{'}  \theta\,^{'}_o ) \; \cot \; \theta\,^{'}_o \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6)$$
This is a logarithmic spiral.
Note that the expression
$$ \cot \; \theta\,^{'}_o \quad = \quad \sqrt {{\cfrac{1}{ \sin^{2} \: \theta\,^{'}_o }}  1 } \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (7) $$
is a trigonometric identity.
The loci of \( \, C \sin \theta\,^{'}_o \) vs. \( R \, \) for the point island \( R = C \) for the various values of \( \theta\,^{'}_o \) are shown in figure 5. For each value of \( \sin \theta\,^{'}_o \), indicated on the right hand vertical scale, the plot is a straight line originating at the origin. The limiting case, \( \text{sine} \; \theta\,^{'}_o \; = \; 1 \), is the boundary of the excluded region. The other limit is \( \sin \; \theta\,^{'}_o \; = \; 0 \) which is the abscissa.
Starting in 1975, when the use of an artificial atoll for wave energy collection was first proposed, the question of what atoll shape would best serve the purpose received considerable attention. It was generally known that "gentle" bottom slopes could transmit wave energy freely whereas "steep" slopes or discontinuities were somewhat reflective. These effects were not quantitatively known except perhaps in shoaling depths. It seems clear however, that what is meant by gentle or steep is not the angle of inclination of the bottom but rather the rate of change of wave celerity that the slope induces. Thus a fairly steep slope in deeper water probably has the same effect on energy flux as some more gentle slope in shallower water so long as \( \frac{dC}{dR} \) is the same in both cases. Then to maximize transmission and minimize reflection the contour should minumize the slope \( \frac{dC}{dR} \) everywhere (see Figure 5).
As viewed by acousticians, the problem formulation is very different but the outcome is the same. A propagating surface wave may be assigned a characteristic impedance by forming the ratio of the hydrodynamic pressure to the horizontal component of the particle velocity. the result is
$$ z \quad = \quad \rho_{o} \; c \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (8) $$
where \( \rho_{o} \) is the (constant) density of the liquid and c is the celerity of the surface wave. This ratio remains constant at any depth since both the pressure and the particle velocity undergo the same exponential decrease with depth. The celerity however is a function of local depth. Thus the characteristic impedance z is a function of local depth.
It follows that any surface wave field may be dealt with as a twodimensional disturbance in a medium of spatially varying characteristic impedance, \( \, \rho_{o} \; c \, \). Any discontinuity in characteristic impedance is a reflective situation with a reflection coefficient of:
$$ \tau \quad = \quad \frac {z_2 \;  \; z_1 } {z_2 \; + \; z_1 } \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad (9) $$
For a step from a deep water free field to a shallow water free field this becomes:
$$ \tau \quad = \quad \frac { \frac{c}{c_o} \;  \; 1 } { \frac{c}{c_o} \; + \; 1 } \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad (10) $$
The step is the limiting case of a celerity gradient. In acoustics, it is axiomatic that minimum reflection and maximum transmission require minimal gradients of characteristic impedance of the medium, i.e., \( \frac{dz}{dx} \) minimal.
The foregoing paragraphs presented two of the various lines of reasoning which converge to the conclusion that Arthur's Point Island (Case 1A) is, in all probability, the most desirable shape for an energycapturing atoll. Although this conclusion has not been quite proven, it is widely accepted. For example, McCormick (1981) identifies the point island with \( \, C \, = \, R \, \) as the optimum shape, without further discussion.
The atoll for which \( \, C \, = \, R \, \) will hereafter be designated as Arthur's Point Island, or simply as Arthur's Island, in accordance with the wish of John Isaacs.
The emergence of Arthur's Point Island as optimum for wave energy capture is a flagrant violation of Murphy's Law; for it is, by far, the most tractable mathematically. In geometrical optics analysis, only if \( R \) is proportional to \( C \) does the square root of equation 2 become a constant so that the integration is trivial. In the scattering analysis which follows, it will be shown that, if \( \, R \, = \, C \), then Bessel's Equation will collapse into an elemental differential equation, solvable by inspection in terms of elemental functions.^{1}
It is to be clearly understood that Arthur's Point Island was selected as optimum because of its physical merits and was then found to be mathematically tractable. Any allegation that it was declared optimum because it is tractable, is vigorously denied.
In a very real sense, Arthur solved the wave equation for the surface area above any domelike structure. He provided a general method for tracing the ray paths from the outer perimeter to zero depth for infinitesimal waves or to breaking depth for small finite waves. Local wave energy concentrations may be deduced from the lateral spacing of the ray paths.
Figure 6 presents the ray paths for Arthur's Point Island for wavelength \( \lambda_{\, o} \), where \( \lambda_{\, o} \) is the design wavelength of the dome shape. Note that the capture of incident waves appears total. Figure 7 presents the ray paths for shorter waves \( ( \lambda{ ^{\, '}_{\, o} } \; = \; 0.5 \; \lambda_{ \, o } ) \) incident upon the same dome shape. Now the outer rays escape outside of the depth for which \( \, h \; = \; 0.5 \; \lambda{ ^{\, '}_{\, o} } \). Figure 8 presents the ray paths generated by waves with wavelength \( \, \lambda{ ^{\, '}_{\, o} } \; = \; 2.0 \; \lambda_{\, o} \), again incident on the original dome. The capture appears total but the ray path is discontinuous across the outer boundary due to the abrupt change in refractive index at the perimeter. This is an intrinsically reflective situation.
To illustrate this, Arthur's analysis is general enough to apply to an obstacle comprising a simple solid cylinder submerged a little beneath the surface. Now all refraction occurs at the perimeter i n a discontinuous manner and the straight ray paths inside can be constructed (approximately) by applying Snell's law. The degree of convergence appears attractive but it is obvious that the reflection at the boundary must be large.
Referring again to Figure 6, one peculiarity of Arthur's solution is that this display of ray paths applies equally well to all diameters of atoll. \( \, r_{o} \) may be assigned any value large or small. In his cylindrical coordinates, \( \theta^{'} \) is "ordinary," \( r \) is normalized by (any) \( r_{o} \), \( h \) and \( \lambda_{o} \) do not appear at all except in the calculation of the local depth necessary to provide the desired relationship between \( C \) and \( R \). It taxes credulity to think that a very small Arthur's Island and a very large one could behave the same. Thus the geometrical optics analysis provides no help in actually sizing a wave collecting atoll or predicting its energy absorption capabilities. Professor C. C. Mei in 1980 pointed out the need for a scattering and absorption analysis to cope with this problem. The writer approached this task with considerable trepidation.
The polar coordinates to be used are shown in Figure 9. Plane, linear, monochromatic waves propagate in the positive \( x \,\)direction. These coordinates differ from those of Arthur in that \( \theta \) is now measured from the positive \( x \) axis. A circular boundary at \( r = r_{o}\) divides the plane into an inner region \( r \geq r_{o} \)[did the author mean \( r \leq r_{o} \)?] and an outer region \( r > r_{o} \). For the outer region, \( c = c_{o} \), the deep water celerity, whereas, in the inner region, the celerity is some function of \( r \).
The linear wave equation applies to both regions:
$$ \nabla^{2} \phi \quad = \quad \frac{1}{c^{2}} \quad \frac{a^{2} \phi } {a \, t^{2}} \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11) $$
In polar coordinates:
$$ \frac{a^2\,\phi}{ar^2} \; + \; \frac{1}{r}\; \frac{ a\phi}{ar} \; + \; \frac{1}{r^2} \frac{a^2\phi}{a\phi^2} \quad = \quad \frac{1}{c^2} \frac{a^2\phi}{at^2} \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \: \; (12) $$
At this point it is convenient to introduce Arthur's normalized radius \( R \) and celerity \( C \):
$$ R \quad = \quad \frac{r}{r_o} \quad (R_o \quad = \quad 1 )\quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \: \: \, (13) $$
$$ C \quad = \quad \frac{c}{c_o} \quad (C_o \quad = \quad 1) \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \; \: (14) $$
We choose the time dependence of \( \phi \) to be a factor \( T \):
$$ T \quad = \quad exp \;  \; i \; \, \omega_o \; \, t \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \; \; (15) $$
also introduce:
$$ K_o \quad = \quad k_o \; \, r_o \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \; \; \; (16) $$
where \( k_o \) is the deep water wave number:
$$ k_o \quad = \quad \frac{\omega_o}{c_o} \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \; \; \; \, (17) $$
Then:
$$ R^2 \; \; \frac{a^2 \; \phi}{\alpha R^2} \quad + \quad R \: \frac{\alpha \phi}{\alpha R } \quad + \quad \frac{{K^2_o}}{C^2} \; \; R^2 \; \; \phi \quad + \quad \frac{a^2 \phi}{\alpha \theta^2} \quad = \quad 0 \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \; \; \;(18) $$
The variables may be separated by setting:
$$ \phi \quad = \quad F(R) \; \, G \; \, ( \theta ) \; \, T\, (t) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \; \; \; \: (19) $$
and dividing through by \( \phi \) to obtain:
$$ \frac{R^2}{F} \; \; \frac{d^2F}{dR^2} \quad + \quad \frac{R}{F} \; \; \frac{dF}{dR} \quad + \quad \frac{{K^2_o}}{C^2} \; \; R^2 \quad + \quad \frac{1}{G} \; \; \frac{d^2G}{d\theta^2} \quad = \quad 0 \qquad \qquad \qquad \qquad \quad \quad \quad \; \; \; \: (20) $$
This is identically true if:
$$ R^2 \; \; \frac{d^2F}{dR^2} \quad + \quad R \; \; \frac{dF}{dR} \quad + \quad \frac{{K^2_o}}{C^2} \; \; R^2F \;  \; m^2F \quad = \quad 0 \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \; \; \; \; \: (21) $$
and:
$$ \frac{d^2G}{d\theta^2} \quad = \quad m^2G \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \; \; \; \; \: (22) $$
It is clear that \( G \) is harmonic in \( \theta \)and a distinct solution exists for each value of \( m \) which is allowed by the boundary conditions. By requiring continuity in the \( \theta \) direction, \( m \) is restricted to positive or negative integers or zero. Each represents a separate and distinct allowed mode. \( \, G \) is then any allowed \( \, G_m \, \) for \( \, m \: = \: \infty \) to \( \, \, m \: = \: \infty \); where,
$$ G_m \quad = \quad g_m \; e^{im\theta} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \; \; \; \; \; \: (23) $$
and \( g_m \) is allowed to be complex.
The need for symmetry about the \( x \) axis requires that each \( \phi_m \) be an even function. As a result of this boundary condition, \( g_{m} \: = \: g_m \).
It is now important to emphasize that \( \, \, g_m \; \, exp \; \, im\theta \, \) and \( \, \, g_m \; \, exp \; \, im\theta \, \, \)are two separate and distinct modes which are equal in magnitude but when multiplied by \( T \: = \: exp \:  \: i \, \omega_o \, t \) become counter rotating to form the standing wave pattern most commonly written \( \cos \: m\theta \). To consider modal impedance the modes must be kept separate.
After modal impedances have been calculated the exponentials may be combined in pairs into:
$$ G_m \quad = \quad g^{'}_{m} \; \cos \; m\theta \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \: \, \, (24) $$
in the more usual way.
The equation for \( F \) may be written:
$$ R^2 \; \frac{d^2F}{dR^2} \; + \; R \frac{dF}{dR} \; + \; K^2_o \; \frac{R^2}{C^2} \; F \; \;  \; \; m^2F \quad = \quad 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \: \, \, (25) $$
which is a standard form for Bessel's equation. Its solution is very different for the inner and outer regions and these will be considered separately.
For the particular case designated as Arthur's Island, C = R. As a result of this substitution Bessel's equation collapses to:
$$ R^2 \; \frac{d^2F}{dR^2} \; + \; R \frac{dF}{dR} \; + \; ( K^2_o \; \;  \; \; m^2 ) \; F \quad = \quad 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \; \: \, \, \, (26) $$
This reduces to a linear equation by a change of variable, e.g., see Dwight (1961).
$$ Set \quad R \quad = \quad exp \; \; u \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \; \: \, \, \, (27) $$
$$ \frac{d^2F}{du^2} \; + \; ( K^2_o \; \;  \; \; m^2 ) \; F \quad = \quad 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \; \; \: \, \, \, (28) $$
Thus F is any of an infinite number of solutions:
[Equation 29]
So long as [Mathematical Expression], the square root is real and [Mathematical Expression] represents propagating waves. Since [Mathematical Expression], the first term of [Mathematical Expression] represents outward propagation and the second term inward propagation.
It can be shown that the radially inward velocity of propagation decreases rapidly enough that the transit time from [Mathematical Expression] to R = 0 is infinite. Thus the point R = 0 is an "infinite termination." There can be no reflected wave, so [Mathematical Expression].
[Mathematical Equation 30]
The situation is reminiscent of Zeno's Paradox which states fallaciously that an arrow can never strike its target. This is because it must first traverse half the distance to the target, then half the remaining distance, etc, ad infinitum.
If however, the statement is modified to require that at each halving of the distance the velocity is also cut in half, then all finite increments of time are equal and infinite in number. Indeed the arrow will never each its target. Similarly if R = C by definition, then the wave can never arrive at R = 0 and is constrained to its logarithmic spiral forever.
This argument should be satisfactory to a mathematician. It is less likely to satisfy engineers  at least as it applies to wave concentration devices. For them, it is pointed out that in the actual application to the wave energy capture device, the real wave breaks at some point, typically at R = .3 or .4. The resulting bore is guided into a very efficient energy sink such that even the roll back of water is negligible. This energy and fluid sink assures negligible reflection.
Thus for the entire inner region over any Arthur's Island:
[Equation 31]
This solution for the inner region R < 1 bears a striking resemblance to the "Spinning Modes" of Tyler and Sofrin (1961). The wave pattern over Arthur's Island is comprised of paired, counterrotating, sinusoidal m lobed patterns. There is an abrupt cutoff of the modes. For all [Mathematical Expression] the square root is purely imaginary.
[Equation 32]
In the inner region 1n R is always negative and increases in absolute value as R decreases. Thus the nonharmonic wave amplitude decreases very rapidly towards the center. All of the higherorder modes are nonpropagating and exhibit an exponentially decaying amplitude radially inward, i.e., are cutoff.
For the first few modes [Mathematical Expression] so [Mathematical Expression] is real.
This represents a harmonic propagating wave moving radially inward with decreasing velocity. Note that the m = 0 mode always propagates. Fortunately for hte purpose of waveenergy concentration, most of the energy in an incident plane wave resides in the first few modes.
In acoustics a common approach to scattering problems is to assign a local or "point" impedance to the surface of the circle R = 1. Often this surface point impedance is assumed to be independent of mode number.
If, as is true in the case being considered, wave propagation occurs within the boundary, the surface impedance is called a distributed impedance and is not independent of mode number. If the modal impedances can be determined, then very convenient solution methods, e.g., George (1979) become available. An exploration of modal impedance also provides additional insight into the nature of the wave field over Arthur's Island.
For each distinct mode the following relations apply (the +mth mode is distinct from the mth mode), Skudzyk (1971).
Radial particle velocity:
[Equation 33]
Tangential particle velocity:
[Equation 34]
Total particle velocity in the direction of propagation, [Mathematical Expression]:
[Equation 35]
[Equation 36]
[Equation 37]
[Equation 38]
Pressure:
[Equation 39]
and thus the various modal impedances are:
[Equation 40]
[Equation 41]
[Equation 42]
These may be normalized by the local characteristic impedance [Mathematical Expression].
[Equation 43]
[Equation 44]
[Equation 45]
[Equation 46]
Note that:
[Equation 46]
All three modal impedances warrant careful consideration. [Mathematical Expression] provides the boundary condition for each mode necessary to use the impedance type of scattering analysis such as that set forth by George (1979). The coordinate system used by George, and used in this paper, considers the radius as positive in the outward direction.
The acoustical impedance of a surface is commonly taken to be that looking into the surface. Therefore the acoustical modal impedance at the perimeter of Arthur's Island is:
[Equation 47]
or:
[Equation 48]
This radial impedance looking into the perimeter of Arthur's Island is real for the first few modes for which [Mathematical Expression]. As a result, energy may cross the boundary. For the rest of the modes, [Mathematical Expression], the impedance becomes purely imaginary and no energy will cross the boundary. The behavior of these higherorder modes bounded by the imaginary [Mathematical Expression] will be discussed in detail later. It will be shown then that they are not reflected.
The tangential impedance [Mathematical Expression] is always real. This is consistent with the view that the individual circumferential modes are rotating pairs to form m lobed standing wave patterns. Because the waves are monochromatic with angular frequency [Mathematical Expression], the angular velocity of rotation is [Mathematical Expression]. The tangential phasevelocity at radius [Mathematical Expression] is:
[Equation 49]
The tangential impedance at any radius r is:
[Equation 50]
So long as [Mathematical Expression] the modes are "fast," since at the perimeter, [Mathematical Expression].
The corresponding radial modes are "cuton" and propagate inward. If [Mathematical Expression] then the modes are "slow" and the radial modes are "cutoff." They now exhibit a nonharmonic nature and an exponential type of amplitude decay. Note that the peripheral phase velocity of the zeroorder mode is infinite so it is always "cuton."
The wave impedance looking in the direction of propagation is always [Mathematical Expression] for any propagating mode at any point over the island. This is exactly the characteristic impedance at that point. Thus a perfect impedance match exists over Arthur's Island for all propagating modes.
With the aid of the modalimpedance calculations above it is possible to make an instructive comparison between the present analysis and some results of Arthur's optical approach.
Arthur (1946), case 1A, showed that a plane wave incident on the boundary at any point [Mathematical Expression] the ray continues on a logarithmic spiral. The initial angle between the ray path and the radius at the perimeter is [Mathematical Expression]. This angle remains constant at all successive crossings of radii.
From the modal analysis it is evident that the angle between the direction of propagation and any radius is a constant [Mathematical Expression]. Clearly:
[Equation 51]
For Arthur's Island:
[Equation 52]
From the modal analysis results:
[Equation 53]
[Equation 54]
Figure 10 shows the case of Arthur's Island for which:
[Equation 55]
Note that the incident vectors are rather evenly spaced. Each local region of wave front would predominantly drive the m lobed node as marked. Each ray path shown sees the impedance [Mathematical Expression] at the perimeter and continues to see the local characteristics impedance [Mathematical Expression] over its entire logarithmic spiral trajectory towards R = 0 (which in theory it never reaches).
For the outer region, R > 1, the normalized celerity C = C = 1 everywhere. As a result, equation 21 becomes,
[Equation 56]
which is Bessel's equation in standard from. F_{m} is any Bessel Function. G as presented in equation 23, and T in equation 15 remain the same, and
^{1}The author recommends "inspecting" compilations of solved differential equations, e.g., Dwight, H.B., 1961, TABLES OF INTEGRALS AND OTHER MATHEMATICAL DATA, Macmillan, N.Y.
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