Waves in Collections of Circular Shoals and Bathymetric Depressions

Steward, David R.

doi:10.1061/(ASCE)WW.1943-5460.0000570

Open access

Technical Papers

Apr 28, 2020

Waves in Collections of Circular Shoals and Bathymetric Depressions

Author: David R. Steward, F.ASCE [email protected]Author Affiliations

Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering

Volume 146, Issue 4

https://doi.org/10.1061/(ASCE)WW.1943-5460.0000570

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Abstract

New approximate analytic solutions are developed to study wave propagation through collections of coastal features. Solutions are developed for circular shoals formed by a submerged cylinder with water depth shallower than the surrounding sea, and for bathymetric depressions formed by a circular pit deeper than the surrounding sea. Classic solutions for monochromatic wave propagation through a single isolated coastal feature are extended using the analytic element method to achieve analytic solutions for a number of coastal features that collectively shape the wave field. Each element is formulated as a Riemann–Hilbert interface problem, where the wave amplitude and phase are continuous between the element and its surroundings; however, a discontinuity in the normal derivative of the wave field occurs as a result of the change in water depth across the interface. Interface conditions are satisfied nearly exactly for typical problems, as demonstrated by normalized root mean square errors of the order of 10⁻¹⁶. This article contributes new mathematical and computational methods that shed insight into wave amplification and dissipation generated within collections of coastal features, with potential applications including the study of trapped waves and tsunamis near coastal features, and of waves traveling through aquatic vegetation.

Introduction

The propagation of water waves is studied for plane waves that encounter a collection of coastal features. Waves in water of uniform depth may be formulated in terms of a complex function

φ

that is a solution to the two-dimensional Helmholtz equation (Courant and Hilbert 1962)

\begin{aligned} \frac{\partial^{2} φ}{\partial x^{2}} + \frac{\partial^{2} φ}{\partial y^{2}} & = - k^{2} φ, \\ Φ (x, y, z, t) & = \frac{\cos h k (z + h)}{\cos h k h} ℜ [φ (x, y) e^{- i ω t}] \end{aligned}

(1)

where

k

= the wave number;

ω

= the angular frequency; and

h

= the water depth (Dean and Dalrymple 1984). This potential function provides periodic solutions to the wave equation (Lamb 1879), with the velocity of water particles equal to minus its gradient

\frac{\partial^{2} Φ}{\partial x^{2}} + \frac{\partial^{2} Φ}{\partial y^{2}} + \frac{\partial^{2} Φ}{\partial z^{2}} = \frac{1}{c^{2}} \frac{\partial^{2} Φ}{\partial t^{2}}, v = - \nabla Φ

(2)

Coastal features will be placed within background plane waves of amplitude

A_{0}

with complex wave function (Berkhoff 1976)

φ = φ_{0} e^{i k x}, φ_{0} = \frac{g A_{0}}{ω}

(3)

Its wavelength,

L

, is related to the wave number, and the wave period,

T

, is related to the angular frequency

L = \frac{2 π}{k} \leftrightarrow k = \frac{2 π}{L}, T = \frac{2 π}{ω} \leftrightarrow ω = \frac{2 π}{T}

(4)

Problems are solved for waves of specified wave period

T

and water depth

h

, and the wave number

k

is obtained from a dispersion relation (Airy 1845, p. 290)

ω^{2} = k g \tan h k h

(5)

which fixes the wave number

k

for a specific angular frequency and water depth.

Analytic solutions using this mathematical foundation have been widely employed in coastal engineering studies of waves near a shoal or bathymetric depression. MacCamy and Fuchs (1954) developed the foundational study of wave reflection by a fully reflective circular island using the Helmholtz equation [Eq. (1)]. This model was extended by Liu et al. (2012) to study waves near a shoal of circular geometry with the long-wave equation, similar to the experimental setup of Vincent and Briggs (1989). Analytic solutions have also been employed to study conical islands and parabolic shoals by Zhang and Zhu (1994), using the long-wave equation, and extended to circular shoals by Liu et al. (2017), who used Frobenius series in the modified mild-slope equation, and Linton (2011), who used multipole expansions in the Laplace equation. Waves near a rectangular pit (Williams 1990) and the interactions of waves near collections of rectangular pits (Lee et al. 2002) have been studied using the boundary element method. The analytic solution was also extended to a circular scour pit with the mild-slope equation by Liu et al. (2013), similar to the experiment of Sumer et al. (1992). A similar analytic solution was developed by Liu et al. (2015) to study water waves near a cylindrical patch of aquatic vegetation, utilizing the theoretical framework established by Mei et al. (2014) and Liu et al. (2015). This method was extended to problems with several patches of vegetation by Chang et al. (2017), using the boundary integral element method. While these models each develop mathematical methods to analyze water waves near an isolated coastal feature (such as a shoal or bathymetric depression) and numerical methods to analyze a number of coastal features, to date, approximate analytic solutions have not been applied to problems with collections of such elements.

Recent developments by Steward (2018) provide a computational platform to study problems with wave reflection and adsorption by collections of cylindrical features. Specifically, Steward (2018) adapted the analytic element method, which had previously been applied to solve problems in groundwater flow, to solve wave problems in coastal engineering, and the boundary condition from MacCamy and Fuchs (1954) was extended to solve the partially reflecting boundary condition of Berkhoff (1976). While Steward (2018) developed solutions to study collections of partially reflecting vertical cylinders, this study extends these methods to study water waves in collections of shoals and bathymetric depressions. These methods are easily implemented in computer codes, using open-source Scilab, and provide quick insight into how elements collectively shape a wave field. Examples provide understanding into the impacts of coastal features on wave fields, and provide insight into the range of wavelengths and water depths important to wave interactions. They also provide a nearly exact set of benchmarks for the validation of numerical methods.

Analytic Element Method Formulation

The specific problem to be studied is illustrated in Fig. 1. An open sea with uniform water depth

h^{+}

is infinite in extent and contains a collection of

I

circular elements, each of which have water depth

h^{-}

, where

h^{-} < h^{+}

for shoals with the geometry of a submerged circular cylinder, and

h^{-} > h^{+}

for bathymetric depressions with the geometry of a circular pit. This depiction of bathymetry is consistent with and builds on previous analytic element method studies of heterogeneity in groundwater (Barnes and Janković 1999; Steward 2015) and soil inclusions in a vadose zone (Steward 2016). Since water waves have the same period

T

throughout the problem domain, each element contains an interface across which the wave number has a value

k^{+}

outside an element and

k^{-}

inside the element, obtained from the dispersion relation (5)

ω = \frac{2 π}{T}, \begin{matrix} ω^{2} = k^{+} g \tan h k^{+} h^{+} \to L^{+} = \frac{2 π}{k^{+}} \\ ω^{2} = k^{-} g \tan h k^{-} h^{-} \to L^{-} = \frac{2 π}{k^{-}} \end{matrix}

(6)

This gives the wavelengths

L^{+}

and

L^{-}

that plane waves would assume in a uniform water depth.

Fig. 1. Circular elements are inserted into a background field of monochromatic waves.

This problem may be solved using the expansion for Hankel functions, outside the element, and an expansion for Bessel functions

J_{n}

inside a circle, giving (Sommerfeld 1972, p. 108)

φ (r, θ) = {\begin{matrix} \sum_{n = 0}^{N} {\overset{H \cos}{c}}_{n} H_{n}^{(1)} (k^{+} r) \cos n θ + \sum_{n = 1}^{N} {\overset{H \sin}{c}}_{n} H_{n}^{(1)} (k^{+} r) \sin n θ + \overset{add}{φ} & (r \geq r_{0}) \\ \sum_{n = 0}^{N} {\overset{J \cos}{c}}_{n} J_{n} (k^{-} r) \cos n θ + \sum_{n = 1}^{N} {\overset{J \sin}{c}}_{n} J_{n} (k^{-} r) \sin n θ & (r < r_{0}) \end{matrix}

(7)

This inner expansion uses only the Bessel

J

functions that are non-singular within a circle multiplied by complex coefficients, similar to formulations for a circular shoal with the mild-slope equation (Lin and Liu 2007) and with the long-wave equation (Liu and Li 2007). The additional function

\overset{add}{φ}

for element

i

of Steward (2018) contains contributions from the regional wave field and all other elements

φ_{i}^{add} = φ_{0} e^{i k^{+} x} + \sum_{j \neq i}^{I} \sum_{n = 0}^{N} c_{j}^{\cos}_{n} H_{n}^{(1)} (k^{+} \underset{j}{r}) \cos n \underset{j}{θ} + \sum_{n = 1}^{N} c_{j}^{\sin}_{n} H_{n}^{(1)} (k^{+} \underset{j}{r}) \sin n \underset{j}{θ}

(8)

Water waves a large distance from the circle elements will take on the far-field conditions in the infinite domain of plane waves [Eq. (3)]. A key tenet of this formulation using the analytic element method is separation of the coefficients for an element from those for all other coastal features using this additional function. This will be used in the solution methods developed in the Appendix to solve interface conditions at each circular element independently of all others, and removes the need to extrapolate interactions amongst elements using Graf’s addition theorem (Linton and Evans 1990; Linton 2011).

Two conditions, which are applied at the

M

control points in Fig. 1, are satisfied at the interface of each element. Conservation of energy requires that a wave’s amplitude and phase be continuous across the interface

φ^{+} = φ^{-}

(9)

This gives the first interface condition for control point

m

, which may be written as

f_{m}^{(1)} = φ_{m}^{+} - φ_{m}^{-}

(10)

This condition is implemented by evaluating the complex wave function (7) at these points:

\begin{aligned} φ_{m}^{+} & = \sum_{n = 0}^{N} {\overset{H \cos}{c}}_{n} H_{n}^{(1)} (k^{+} r_{0}) \cos n θ_{m} + \sum_{n = 1}^{N} {\overset{H \sin}{c}}_{n} H_{n}^{(1)} (k^{+} r_{0}) \sin n θ_{m} + {\overset{add}{φ}}_{m} \\ φ_{m}^{-} & = \sum_{n = 0}^{N} {\overset{J \cos}{c}}_{n} J_{n} (k^{-} r_{0}) \cos n θ_{m} + \sum_{n = 1}^{N} {\overset{J \sin}{c}}_{n} J_{n} (k^{-} r_{0}) \sin n θ_{m} \end{aligned}

(11)

A second interface condition is developed from the component of the vector for water particle movement normal to the interface, which is given by the partial derivative of the potential

Φ

in Eq. (1)

v_{r} = - \frac{\partial Φ}{\partial r} = \frac{\cos h k (z + h)}{\cos h k h} \frac{\partial}{\partial r} ℜ (φ e^{- i ω t})

(12)

This is vertically integrated over the depth of the water column to give

V_{r} = \int_{- h}^{0} v_{r} d z = \int_{- h}^{0} \frac{\cos h k (z + h)}{\cos h k h} d z \frac{\partial}{\partial r} ℜ (φ e^{- i ω t}) = \frac{\sin h k h}{k \cos h k h} \frac{\partial}{\partial r} ℜ (φ e^{- i ω t})

(13)

Conservation of mass is satisfied when this is continuous across the interface

\begin{aligned} {V_{r}}^{+} & = {V_{r}}^{-} \to β^{+} \frac{\partial φ^{+}}{\partial r} = β^{-} \frac{\partial φ^{-}}{\partial r}, β^{+} = \frac{\sin h k^{+} h^{+}}{k^{+} \cos h k^{+} h^{+}}, \\ β^{-} & = \frac{\sin h k^{-} h^{-}}{k^{-} \cos h k^{-} h^{-}} \end{aligned}

(14)

and the water depth (

h^{+}

and

h^{-}

) and wave number (

k^{+}

and

k^{-}

) take on values for the corresponding side of the interface. This gives a second interface condition for control point

m

f_{m}^{(2)} = \frac{\partial φ_{m}^{+}}{\partial r} - β \frac{\partial φ_{m}^{-}}{\partial r}, β = \frac{β^{-}}{β^{+}}

(15)

which reproduces Williams (1990, Eq. (4)) for shallow water waves in the limit as

\sin h k h \to k h

and as

\cos h k h \to 1

. This condition is implemented by evaluating partial derivatives of the complex wave function (7) at these points:

\begin{aligned} \frac{\partial φ_{m}^{+}}{\partial r} & = \sum_{n = 0}^{N} {\overset{H \cos}{c}}_{n} H_{n}^{' (1)} (k^{+} r_{0}) \cos n θ_{m} + \sum_{n = 1}^{N} {\overset{H \sin}{c}}_{n} H_{n}^{' (1)} (k^{+} r_{0}) \sin n θ_{m} + \frac{\partial {\overset{add}{φ}}_{m}}{\partial r} \\ \frac{\partial φ_{m}^{-}}{\partial r} & = \sum_{n = 0}^{N} {\overset{J \cos}{c}}_{n} J_{n}^{'} (k^{-} r_{0}) \cos n θ_{m} + \sum_{n = 1}^{N} {\overset{J \sin}{c}}_{n} J_{n}^{'} (k^{-} r_{0}) \sin n θ_{m} \end{aligned}

(16)

where the partial derivatives of the Bessel functions may be obtained recursively using

\begin{aligned} H_{n}^{' (1)} (k r) & = \frac{n}{r} H_{n}^{(1)} (k r) - k H_{n + 1}^{(1)} (k r), \\ J_{n}^{'} (k r) & = \frac{n}{r} J_{n} (k r) - k J_{n + 1} (k r) \end{aligned}

(17)

which was adapted from Abramowitz and Stegun (1972, p. 361). The partial derivatives of the additional function are given by

\begin{aligned} \frac{\partial \overset{add}{φ}}{\partial r} & = \cos θ \frac{\partial \overset{add}{φ}}{\partial x} + \sin θ \frac{\partial \overset{add}{φ}}{\partial y} \\ \frac{\partial φ}{\partial x} & = \cos θ \frac{\partial φ}{\partial r} - \frac{\sin θ}{r} \frac{\partial φ}{\partial θ} \end{aligned}

(18)

\frac{\partial φ}{\partial y} = \sin θ \frac{\partial φ}{\partial r} + \frac{\cos θ}{r} \frac{\partial φ}{\partial θ}

(19)

with

\begin{aligned} \frac{\partial φ}{\partial r} & = \sum_{n = 0}^{N} {\overset{\cos}{c}}_{n} H_{n}^{^{'} (1)} (k^{+} r) \cos n θ + \sum_{n = 1}^{N} {\overset{\sin}{c}}_{n} H_{n}^{^{'} (1)} (k^{+} r) \sin n θ \\ \frac{\partial φ}{\partial θ} & = - \sum_{n = 0}^{N} {\overset{\cos}{c}}_{n} H_{n}^{(1)} (k^{+} r) n \sin n θ + \sum_{n = 1}^{N} {\overset{\sin}{c}}_{n} H_{n}^{(1)} (k^{+} r) n \cos n θ \end{aligned}

(20)

The mathematical and numerical methods for solving the two interface conditions are deferred to the Appendix, and the capacity of these methods to achieve nearly exact solutions is quantified and discussed later using Tables 1 and 2.

Table 1. Parameters of wave fields for shoals and bathymetric depressions

Figure	Row	Column	(m)	(m)	(m)	(s)	(m)	(m)	$f_{m}^{(1)} / \| φ_{0} \|$	$f_{m}^{(2)} / (k_{0} \| φ_{0} \|)$	$f_{m}^{(1)} / \| φ_{0} \|$	$f_{m}^{(2)} / (k_{0} \| φ_{0} \|)$
			$r_{0}$	$L^{+}$	$L^{-}$	$T$	$h^{+}$	$h^{-}$	Residual ( $N$ = 20, $M$ = 185)		Residual ( $N$ = 40, $M$ = 365)
2	Top	Left panel	30	30	15	10	0.93	0.23	$3.60 \times 10^{- 11}$	$1.15 \times 10^{- 10}$	$3.60 \times 10^{- 16}$	$2.24 \times 10^{- 16}$
2	Top	Center panel	30	30	60	10	0.93	3.87	$3.60 \times 10^{- 11}$	$1.15 \times 10^{- 10}$	$3.82 \times 10^{- 16}$	$2.26 \times 10^{- 16}$
2	Top	Right panel	30	30	120	10	0.93	19.42	$3.60 \times 10^{- 11}$	$1.15 \times 10^{- 10}$	$4.40 \times 10^{- 16}$	$6.66 \times 10^{- 16}$
2	Bottom	Left panel	30	60	15	10	3.87	0.23	$1.29 \times 10^{- 16}$	$2.11 \times 10^{- 16}$	$4.42 \times 10^{- 16}$	$2.88 \times 10^{- 16}$
2	Bottom	Center panel	30	60	30	10	3.87	0.93	$1.14 \times 10^{- 16}$	$2.25 \times 10^{- 16}$	$3.73 \times 10^{- 16}$	$2.98 \times 10^{- 16}$
2	Bottom	Right panel	30	60	120	10	3.87	19.42	$1.42 \times 10^{- 16}$	$2.63 \times 10^{- 16}$	$6.08 \times 10^{- 16}$	$4.01 \times 10^{- 16}$
3	Top	Left panel	30	30	15	10	0.93	0.23	$1.47 \times 10^{- 11}$	$4.70 \times 10^{- 11}$	$1.03 \times 10^{- 16}$	$7.29 \times 10^{- 17}$
3	Top	Center panel	30	30	60	10	0.93	3.87	$1.47 \times 10^{- 11}$	$4.71 \times 10^{- 11}$	$1.34 \times 10^{- 16}$	$9.65 \times 10^{- 17}$
3	Top	Right panel	30	30	120	10	0.93	19.42	$1.01 \times 10^{- 11}$	$3.23 \times 10^{- 11}$	$1.20 \times 10^{- 16}$	$1.97 \times 10^{- 16}$
3	Bottom	Left panel	30	60	15	10	3.87	0.23	$5.18 \times 10^{- 14}$	$3.44 \times 10^{- 13}$	$1.83 \times 10^{- 16}$	$1.37 \times 10^{- 16}$
3	Bottom	Center panel	30	60	30	10	3.87	0.93	$5.96 \times 10^{- 14}$	$3.97 \times 10^{- 13}$	$1.73 \times 10^{- 16}$	$1.23 \times 10^{- 16}$
3	Bottom	Right panel	30	60	120	10	3.87	19.42	$3.01 \times 10^{- 14}$	$2.00 \times 10^{- 13}$	$2.24 \times 10^{- 16}$	$1.36 \times 10^{- 16}$
4		Left panel	125–500	20,000	10,000	300	450	110	$5.65 \times 10^{- 15}$	$8.17 \times 10^{- 13}$	$9.26 \times 10^{- 17}$	$6.34 \times 10^{- 16}$
4		Center panel	125–500	20,000	10,000	300	450	110	$2.38 \times 10^{- 11}$	$3.35 \times 10^{- 9}$	$9.73 \times 10^{- 16}$	$2.28 \times 10^{- 13}$
4		Right panel	125–500	20,000	10,000	300	450	110	$1.94 \times 10^{- 11}$	$2.73 \times 10^{- 9}$	$9.87 \times 10^{- 16}$	$2.06 \times 10^{- 13}$

Note: The computational residual error is reported as the root mean square error, which is normalized by dividing

f_{m}^{(1)}

[Eq. (10)] by the amplitude of the background waves,

| φ_{0} |

, and dividing

f_{m}^{(2)}

[Eq. (15)] by the open sea wave number multiplied by its amplitude,

k^{+} | φ_{0} |

.

Table 2. Numerical accuracy increases with higher-order elements

Figure	Row	Column	(m)	(m)	(m)	(s)	(m)	(m)	$f_{m}^{(1)} / \| φ_{0} \|$	$f_{m}^{(2)} / (k_{0} \| φ_{0} \|)$	$f_{m}^{(1)} / \| φ_{0} \|$	$f_{m}^{(2)} / (k_{0} \| φ_{0} \|)$
			$r_{0}$	$L^{+}$	$L^{-}$	$T$	$h^{+}$	$h^{-}$	Residual ( $N = 6$ , $M = 59$ )		Residual ( $N = 40$ , $M = 365$ )
5	Top	Left panel	30	30	15	10	0.93	0.23	$3.25 \times 10^{- 2}$	$2.31 \times 10^{- 2}$	—	—
5	Top	Center panel	30	30	60	10	0.93	3.87	$3.25 \times 10^{- 2}$	$2.31 \times 10^{- 2}$	—	—
5	Top	Right panel	30	30	120	10	0.93	19.42	$3.25 \times 10^{- 2}$	$2.31 \times 10^{- 2}$	—	—
5	Bottom	Left panel	30	30	15	10	0.93	0.23	—	—	$3.60 \times 10^{- 16}$	$2.24 \times 10^{- 16}$
5	Bottom	Center panel	30	30	60	10	0.93	3.87	—	—	$3.82 \times 10^{- 16}$	$2.26 \times 10^{- 16}$
5	Bottom	Right panel	30	30	120	10	0.93	19.42	—	—	$4.40 \times 10^{- 16}$	$6.66 \times 10^{- 16}$

Computational Results and Discussion

Waves through Shoals and Bathymetric Depressions

The wave field is analyzed first for a single, isolated element with uniform water depth

h^{-}

placed in a background with uniform water depth

h^{+}

. The results in Fig. 2 illustrate waves traveling over a shoal (with

h^{-} < h^{+}

) and waves traveling over a bathymetric depression (with

h^{-} > h^{+}

). Each example shows the amplitude normalized by

| φ_{0} |

, the magnitude of the background plane waves, which are traveling from left to right. The amplitude is contoured at intervals of

0.1 | φ_{0} |

, and a contour line is placed at

| φ | = | φ_{0} |

, the amplitude that exists a large distance from the element. Note that the amplitude of surface water waves is obtained by multiplying the normalized magnitudes in this figure by the amplitude of the background plane waves,

A_{0}

, since

φ_{0} = g A_{0} / ω

in Eq. (3). These solutions are founded in the theory of linear water waves (Berkhoff 1976), and neglect nonlinear effects such as the wave breaking that occurs when the ratio of wave height to wavelength exceeds one-to-seven (Bascom 1980, p. 34). Each example also shows the cosine of the phase,

\cos (\arg φ)

, which illustrates wave position; for example, when wave crests occur at the darkest shade, then the wave troughs are at the lightest shade.

Fig. 2. Wave fields for a single circular shoal or bathymetric depression across a range of exterior and interior wavelengths, where background monochromatic waves travel from left to right. Amplitude is contoured at intervals of $0.1 | φ_{0} |$ , and contour lines identify where $| φ | = | φ_{0} |$ equals the background.

The specific configurations for these wave fields are given in Table 1. These examples have period

T = 10

s and the shoals and depressions have radius

r_{0} = 30

m. The particular water depths (

h^{+}

and

h^{-}

) were chosen so that the wavelengths (

L^{+}

and

L^{-}

) would scale to multiples of

r_{0}

, to aid in visualizing results. As specified in Table 1, the waves that would exist in the absence of the circular feature would be separated by wavelength

L^{+}

equal to

r_{0}

for the top panels and

2 r_{0}

for the bottom panels. The occurrence of partially standing waves, and the patterns of wave resonance and dissipation, are consistent with studies of a shoal by Yu and Zhang (2003) and Niu and Yu (2011a): wave amplitude increases at locations in the shallower water above the shoal, areas with calmer water emanate radially from the leeward side of the shoal, and partially standing waves form to the windward side. Likewise, wave patterns near a bathymetric depression are similar to those obtained by Niu and Yu (2011b): wave amplitude decreases over the circular pit compared with the background, which extends to decreased amplitude on the leeward side, and partially standing waves form to the windward side. The phase of the wave illustrates that waves become spaced more closely above shoals, and a larger distance occurs between waves above a circular pit.

Next, results are extended to wave fields moving through collections of elements, as shown in Fig. 3. The specific configurations for these wave fields is given in Table 1, using the same period, water depth, and wave numbers as the single elements shown in Fig. 2. Clearly, these elements are collectively interacting to shape the wave field. Regions of high wave application occur on the windward side to the left and wave dissipation has reduced amplitudes on the leeward side to the right. For the leftmost elements, where background waves first encounter coastal features, the wave amplification above shoals and wave reduction above depressions are consistent with the results for a single element shown in Fig. 2. At locations above elements, the three sets of shoals contain highly amplified waves in their shallower water, and the three sets of bathymetric depressions contain lower amplitude waves in their deeper water, similar to the isolated features of Fig. 2. For these particular configurations, bathymetric features provide shelter to their downwind neighbors, and the wave amplitude for these elements is greatly reduced from the wave field of individual coastal features shown in Fig. 2. While the wave interactions amongst features collectively shape the phase of the wave field, the spacing of waves is consistent with what is observed for individual elements: waves become spaced more closely above shoals, and spaced at larger distances between waves above depressions.

Fig. 3. Wave fields for a collection of circular shoals or bathymetric depressions across a range of exterior and interior wavelengths, equally spaced at $δ = 4 r_{0}$ .

Wave fields are analyzed next for longer wavelengths traveling through deeper water shoals with a milder shift in wave number, as shown in Fig. 4. The wave field parameters in Table 1 are typical for tsunamis (Stefanakis et al. 2014):

ω \in (0.01 - 0.1)

/s and

h \in (100, 1, 000)

m. Circular shoals are located with random placement and radius, and contain an increasing number of elements

I

in the model domain. The results with

I = 10

contain a wave field similar to the plane waves that exist at large distances from the elements. As

I

increases to 50 and 100 elements, the deep water shoals form a region with increased wave amplification to the leeward side of the shoals. This amplification is observed with the contour intervals representing

0.1 A_{0}

, the amplitude of the background wave field; the maximum wave amplitudes for 10, 50, and 100 elements are approximately

1.1 A_{0}

,

1.4 A_{0}

, and

1.5 A_{0}

for these configurations. These results are consistent with Stefanakis et al. (2014), who found that circular islands may serve to amplify tsunami waves rather than provide leeward protection to coastlines. Since the spacing between waves is smaller above circular shoals than the background, the net impact of the collection of shoals is a shorter wavelength with more closely spaced contours in the phase diagrams within the collection of elements; the shortening of wavelengths becomes more pronounced as the number of elements increases. Wave amplification to the leeward of the collection of these elements occurs as waves converge around the collection of elements, similar to observations of diffracted waves near vegetation made by Liu et al. (2015).

Fig. 4. Wave fields for randomly placed collections of circular shoals.

Numerical Validation and Considerations

The analytic element method achieves nearly exact solutions for water waves near circular shoals and bathymetric depressions. This is quantified by computing the root mean square error of the residuals

f_{m}^{(1)}

in Eq. (10) and

f_{m}^{(2)}

in Eq. (15). This is reported in Table 1 for each example studied in Figs. 2–4. Results are reported with

N = 20

coefficients in the complex function

φ

[Eq. (7)] and

M = 185

control points, consistent with those of Steward (2018). While this accuracy was used to visualize findings in Figs. 2–4, the residual error was also computed with more terms,

N = 40

and

M = 365

. The results given in Table 1 illustrate that residual error decreases with more terms, and may achieve normalized errors with 16 significant digits.

The impact of the number of coefficients

N

on the wave fields is quantified in Fig. 5. These results provide close-up views of the waves depicted in Fig. 2; contour lines are included to illustrate the capacity of the solution to satisfy interface conditions. Results in the top panels illustrate results for a low-order solution with

N = 6

coefficients [consistent with Steward (2018, Fig. 11)], and the bottom panels illustrate a high-order solution with

N = 40

coefficients (consistent with Table 1). The capacity of these solutions to match interface conditions

f_{m}^{(1)}

[Eq. (10)] and

f_{m}^{(2)}

[Eq. (15)] is quantified in Table 2 for the root mean square error (RMSE) of these residuals. This normalized error is of the order of

10^{- 2}

for

N = 6

and

10^{- 16}

for

N = 40

. While the low-order solution reproduces the general features of wave amplitude and phase near a shoal and a depression, the RMSE of approximately 0.01 results in contour lines that are not exactly continuous across the interface, as they are for the higher-order solution. Note that the results with

N = 40

shown in Fig. 5 and those with

N = 20

shown in Fig. 2 have no discernible differences in contours of amplitude and phase. While the results with a RMSE of approximately

10^{- 10}

for

N = 20

given in Table 1 have nearly exact solutions, the results with

N = 40

with a RMSE of

10^{- 16}

approach the computational accuracy of double-precision computations.

Fig. 5. Wave field solutions in the top row for $N = 6$ and in the bottom row for $N = 40$ coefficients, with the same bathymetry and background waves as the top row in Fig. 2. Amplitude is contoured at intervals of $0.1 | φ_{0} |$ , and amplitude and phase diagrams both contain contour lines at every contour interval, except for amplitude in the left column, where lines are shown at every other contour interval.

While the results are illustrated for wavelengths with water depths chosen for ease of visualization, these methods are broadly applicable to other configurations. In the limiting case where

β^{+} = β^{-}

, the coefficients approach the Jacobi–Anger expansion for plane waves

\begin{aligned} φ^{(i)} & = φ_{0} e^{i k^{+} r \cos (θ - θ_{i})} = φ_{0} [J_{0} (k^{+} r) + 2 \sum_{n = 1}^{\infty} i^{n} J_{n} (k^{+} r) \cos n (θ - θ_{i})], \\ (φ_{0} = \frac{i g}{ω} A_{i}) \end{aligned}

(21)

which gives coefficients in Eq. (7) of

{\overset{J \cos}{c}}_{0} = ϕ_{0}, {\overset{J \cos}{c}}_{n} = 2 i^{n} ϕ_{0}, {\overset{J \sin}{c}}_{n} = 0

(22)

Note that these solutions satisfy the interface conditions very well; however, coefficients for large

N

become large, as a result of solving the inner coefficients with terms that divide by

J_{n}

, even though the outer expansion coefficients are numerically zero. This is an artifact of the solution, and only becomes evident for certain values of wave number; where inner coefficients become large but the solution remains very good.

Many of the configurations studied exhibit regions with highly amplified waves. While the depicted solutions achieve accurate solutions, conditions exist where isolated zones of amplification and standing waves become intensified at the boundaries of neighboring elements. Cases exist where small perturbations of element size and spacing would result in very different patterns of amplification and standing waves. In such cases, numerical solutions are difficult to achieve, since successive iterations may bounce across solutions. Such problems may be resolved by changing the element spacing to avoid standing wave formation within interfaces of neighboring elements.

Future Applications

Features with circular geometry are known to impact the resonance of long waves (Summerfield 1972). These methods build on a well-established foundation of analytic methods to study water waves near a single isolated shoal or bathymetric depression in water of uniform depth. New methodology advances solutions with the same level of precision to collections of interacting coastal features. This provides new approaches to studying the impacts of variations in bathymetry on the formation of trapped waves and shore protection for tsunamis (Stefanakis et al. 2014). It aids in identifying locations with wave breaking, and the impact of the wave period on these locations (Lie and Torum 1991). It also provides a numerical laboratory to gain an understanding of how coastal features interact, and promotes hypothesis testing to discover how changes in wave parameters and bathymetric configurations impact wave generation. Such insight is key to the deployment of sensors to measure waves in applications such as tsunami detectors.

This solution is easily adapted to study wave transmission through aquatic vegetation. The theoretical framework put forth by Mei et al. (2014) for a vegetative forest and by Liu et al. (2015) for a single patch of aquatic vegetation contains outer and inner expansions, which are easily extensible to multiple patches using the analytic element methodology developed here. The small change to the wave number inside an aquatic forest [Liu et al. 2015, Eq. (4.28)] would change the coefficient

β

in this formulation [Liu et al. 2015, Eq. (4.30)]. The problem formulated by Kobayashi et al. (1993) and Liu et al. (2015) for waves over submerged and emergent vegetation used wave numbers and water depth similar to the range of variables in Table 1.

Summary and Conclusions

Methods are developed to study waves in collections of shoals formed by submerged circular cylinder and bathymetric depressions formed by circular pits. Classic methods of analysis for a single element using Bessel functions and Fourier series are extended to achieve analytic solutions for several coastal features. The Riemann–Hilbert interface problem is formulated using

f_{m}^{(1)}

in Eq. (10) and

f_{m}^{(2)}

in Eq. (15), and solution methods using least squares with Gauss–Seidel iteration are formulated in the Appendix to achieve nearly exact solutions, as quantified in Table 1. These new approximate analytic solutions represent the first analytic element method applications for waves moving through elements.

Wave interactions occurring in coastal features with circular geometry are studied across typical settings. The waves near isolated elements shown in Fig. 2 illustrate wave amplification on shoals and wave dissipation by bathymetric depressions. The collections of elements shown in Fig. 3 illustrate the impact of interactions amongst elements on shaping the wave field, with partially standing waves on the windward side and calmer water to the leeward side. Deeper water shoals are studied for tsunami conditions (Fig. 4), and illustrate amplification of long-period waves occurring to their leeward side, with increased amplification for larger numbers of shoals. Increased accuracy of the solution for higher-order elements is illustrated in Fig. 5 and quantified in Table 2.

This study provides an insight into how coastal features collectively shape a wave field. Computational models may easily implement these methods to provide deeper understanding and insight into the influences of wave periods and water depths on wave interactions. Such knowledge provides a fuller understanding of wave mechanics than existing analytic solutions of a single coastal feature, and the methodology also provides an accurate benchmark for the verification and validation of numerical methods. Related applications exist in the study of water waves through aquatic vegetation, and both acoustic and electromagnetic wave propagation through heterogeneous media. The paradigm used to achieve solutions using the analytic element method for wave fields has the potential to be extended to elements with more general geometries than the circle elements used in this formulation.

Appendix: Solving Coefficients for Coastal Features

Solutions are obtained for coastal features by fixing the center (

x_{c}

,

y_{c}

) and radius

r_{0}

of the

I

elements, and specifying the important wave properties (period

T

, amplitude

| φ_{0} |

, and water depth

h^{+}

and

h^{-}

). The number of coefficients is set to

N = 20

here, but this number may be increased to provide smaller residual errors in matching the interface conditions, or it may be decreased to provide shorter computation times. The number of control points where interface conditions are applied is set to

M = 1.5 \times (6 N + 3) = 185

(a factor of 1.5 times the number of real coefficients to be determined), following the overspecification principle (Janković and Barnes 1999). The rest of this section develops methods to compute coefficients for one circular element by satisfying the interface conditions using the method of least squares. The coefficients for a collection of elements are solved using Gauss–Seidel iteration, by sequentially solving for the coefficients of element

i

while holding coefficients for all other elements fixed in the additional function [Eq. (8)]. Each element is solved independently of the others, and this cycle is repeated until convergence occurs, with a criterion used here that the maximum change in

| φ_{m} |

between successive iterations is

10^{- 12}

across all control points of all elements.

The value of

φ

at the control points [Eq. (11)] may be separated into real and imaginary parts of the complex wave function to give

\begin{aligned} φ_{m}^{+} & = \sum_{n = 0}^{N} [J_{n} (k^{+} r_{0}) ℜ {\overset{H \cos}{c}}_{n} - Y_{n} (k^{+} r_{0}) ℑ {\overset{H \cos}{c}}_{n}] \cos n θ_{m} \\ + i [Y_{n} (k^{+} r_{0}) ℜ {\overset{H \cos}{c}}_{n} + J_{n} (k^{+} r_{0}) ℑ {\overset{H \cos}{c}}_{n}] \cos n θ_{m} \\ + \sum_{n = 1}^{N} [J_{n} (k^{+} r_{0}) ℜ {\overset{H \sin}{c}}_{n} - Y_{n} (k^{+} r_{0}) ℑ {\overset{H \sin}{c}}_{n}] \sin n θ_{m} \\ + i [Y_{n} (k^{+} r_{0}) ℜ {\overset{H \sin}{c}}_{n} + J_{n} (k^{+} r_{0}) ℑ {\overset{H \sin}{c}}_{n}] \sin n θ_{m} + {\overset{add}{φ}}_{m} \end{aligned}

(23a)

and

\begin{aligned} φ_{m}^{-} & = \sum_{n = 0}^{N} [J_{n} (k^{-} r_{0}) ℜ {\overset{J \cos}{c}}_{n}] \cos n θ_{m} + i [J_{n} (k^{-} r_{0}) ℑ {\overset{J \cos}{c}}_{n}] \cos n θ_{m} \\ + \sum_{n = 1}^{N} [J_{n} (k^{-} r_{0}) ℜ {\overset{J \sin}{c}}_{n}] \sin n θ_{m} + i [J_{n} (k^{-} r_{0}) ℑ {\overset{J \sin}{c}}_{n}] \sin n θ_{m} \end{aligned}

(23b)

Likewise, the real and imaginary parts of the partial derivatives [Eq. (16)] may also be separated into real and imaginary parts. The real and imaginary parts of the first and second interface conditions may be organized as matrix multiplications with Fourier coefficients, as (Steward 2018)

\begin{matrix} A \overset{ℜ}{c} {_{}}^{(1)} = \overset{ℜ}{b} {_{}}^{(1)} \\ A \overset{ℑ}{c} {_{}}^{(1)} = \overset{ℑ}{b} {_{}}^{(1)} \end{matrix}, \begin{matrix} A \overset{ℜ}{c} {_{}}^{(2)} = \overset{ℜ}{b} {_{}}^{(2)} \\ A \overset{ℑ}{c} {_{}}^{(2)} = \overset{ℑ}{b} {_{}}^{(2)} \end{matrix}, A = [\begin{matrix} 1 & \cos θ_{1} & \sin θ_{1} & \dots & \cos N θ_{1} & \sin N θ_{1} \\ 1 & \cos θ_{2} & \sin θ_{2} & \dots & \cos N θ_{2} & \sin N θ_{2} \\ 1 & \cos θ_{M} & \sin θ_{M} & \dots & \cos N θ_{M} & \sin N θ_{M} \end{matrix}]

(24)

The coefficient and known vectors for the first condition,

f_{m}^{(1)}

, contain the following values for the real part

\overset{ℜ}{c} {_{}}^{(1)} = [\begin{matrix} J_{0} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} - Y_{0} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - J_{0} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{0} \\ J_{1} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{1} - Y_{1} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{1} - J_{1} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{1} \\ J_{1} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{1} - Y_{1} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{1} - J_{1} (k^{-} r_{0}) ℜ \overset{J \sin}{c}_{1} \\ ⋮ \\ J_{N} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{N} - Y_{N} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{N} - J_{N} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{N} \\ J_{N} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{N} - Y_{N} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{N} - J_{N} (k^{-} r_{0}) ℜ \overset{J \sin}{c}_{N} \end{matrix}], \overset{ℜ}{b} {_{}}^{(1)} = [\begin{matrix} - ℜ \overset{add}{φ}_{1} \\ - ℜ \overset{add}{φ}_{2} \\ ⋮ \\ - ℜ \overset{add}{φ}_{M} \end{matrix}]

(25a)

and the imaginary parts

\overset{ℑ}{c} {_{}}^{(1)} = [\begin{matrix} Y_{0} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} + J_{0} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - J_{0} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{0} \\ Y_{1} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{1} + J_{1} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{1} - J_{1} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{1} \\ Y_{1} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{1} + J_{1} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{1} - J_{1} (k^{-} r_{0}) ℑ \overset{J \sin}{c}_{1} \\ ⋮ \\ Y_{N} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{N} + J_{N} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{N} - J_{N} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{N} \\ Y_{N} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{N} + J_{N} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{N} - J_{N} (k^{-} r_{0}) ℑ \overset{J \sin}{c}_{N} \end{matrix}], \overset{ℑ}{b} {_{}}^{(1)} = [\begin{matrix} - ℑ \overset{add}{φ}_{1} \\ - ℑ \overset{add}{φ}_{2} \\ ⋮ \\ - ℑ \overset{add}{φ}_{M} \end{matrix}]

(25b)

Likewise, the real part of the second condition,

f_{m}^{(2)}

, contains the real parts

\overset{ℜ}{c} {_{}}^{(2)} = [\begin{matrix} J_{0}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} - Y_{0}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - β J_{0}^{'} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{0} \\ J_{1}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{1} - Y_{1}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{1} - β J_{1}^{'} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{1} \\ J_{1}^{'} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{1} - Y_{1}^{'} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{1} - β J_{1}^{'} (k^{-} r_{0}) ℜ \overset{J \sin}{c}_{1} \\ ⋮ \\ J_{N}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{N} - Y_{N}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{N} - β J_{N}^{'} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{N} \\ J_{N}^{'} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{N} - Y_{N}^{'} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{N} - β J_{N}^{'} (k^{-} r_{0}) ℜ \overset{J \sin}{c}_{N} \end{matrix}], \overset{ℜ}{b} {_{}}^{(2)} = [\begin{matrix} - ℜ \frac{\partial \overset{add}{φ}_{1}}{\partial r} \\ - ℜ \frac{\partial \overset{add}{φ}_{2}}{\partial r} \\ ⋮ \\ - ℜ \frac{\partial \overset{add}{φ}_{M}}{\partial r} \end{matrix}]

(25c)

and the imaginary parts

\overset{ℑ}{c} {_{}}^{(2)} = [\begin{matrix} Y_{0}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} + J_{0}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - β J_{0}^{'} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{0} \\ Y_{1}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{1} + J_{1}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{1} - β J_{1}^{'} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{1} \\ Y_{1}^{'} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{1} + J_{1}^{'} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{1} - β J_{1}^{'} (k^{-} r_{0}) ℑ \overset{J \sin}{c}_{1} \\ ⋮ \\ Y_{N}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{N} + J_{N}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{N} - β J_{N}^{'} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{N} \\ Y_{N}^{'} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{N} + J_{N}^{'} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{N} - β J_{N}^{'} (k^{-} r_{0}) ℑ \overset{J \sin}{c}_{N} \end{matrix}], \overset{ℑ}{b} {_{}}^{(2)} = [\begin{matrix} - ℑ \frac{\partial \overset{add}{φ}_{1}}{\partial r} \\ - ℑ \frac{\partial \overset{add}{φ}_{2}}{\partial r} \\ ⋮ \\ - ℑ \frac{\partial \overset{add}{φ}_{M}}{\partial r} \end{matrix}]

(25d)

The least squares solutions to these four systems of equations for equally spaced control points, as in Steward (2015), gives, for the first condition

\begin{aligned} J_{0} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} - Y_{0} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - J_{0} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{0} & = - \frac{1}{M} \sum_{m = 1}^{M} ℜ (\overset{add}{φ}_{m}) \\ J_{n} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{n} - Y_{n} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{n} - J_{n} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℜ (\overset{add}{φ}_{m}) \cos n θ_{m} \\ J_{n} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{n} - Y_{n} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{n} - J_{n} (k^{-} r_{0}) ℜ \overset{J \sin}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℜ (\overset{add}{φ}_{m}) \sin n θ_{m} \end{aligned}

(26a)

and

\begin{aligned} Y_{0} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} + J_{0} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - J_{0} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{0} & = - \frac{1}{M} \sum_{m = 1}^{M} ℑ (\overset{add}{φ}_{m}) \\ Y_{n} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{n} + J_{n} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{n} - J_{n} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℑ (\overset{add}{φ}_{m}) \cos n θ_{m} \\ Y_{n} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{n} + J_{n} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{n} - J_{n} (k^{-} r_{0}) ℑ \overset{J \sin}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℑ (\overset{add}{φ}_{m}) \sin n θ_{m} \end{aligned}

(26b)

and, for the second condition

\begin{aligned} J_{0}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} - Y_{0}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - β J_{0}^{'} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{0} & = - \frac{1}{M} \sum_{m = 1}^{M} ℜ (\frac{\partial \overset{add}{φ}_{m}}{\partial r}) \\ J_{n}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{n} - Y_{n}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{n} - β J_{n}^{'} (k^{-} r_{0}) ℜ \overset{J \cos}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℜ (\frac{\partial \overset{add}{φ}_{m}}{\partial r}) \cos n θ_{m} \\ J_{n}^{'} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{n} - Y_{n}^{'} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{n} - β J_{n}^{'} (k^{-} r_{0}) ℜ \overset{J \sin}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℜ (\frac{\partial \overset{add}{φ}_{m}}{\partial r}) \sin n θ_{m} \end{aligned}

(26c)

and

\begin{aligned} Y_{0}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{0} + J_{0}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{0} - β J_{0}^{'} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{0} & = - \frac{1}{M} \sum_{m = 1}^{M} ℑ (\frac{\partial \overset{add}{φ}_{m}}{\partial r}) \\ Y_{n}^{'} (k^{+} r_{0}) ℜ \overset{H \cos}{c}_{n} + J_{n}^{'} (k^{+} r_{0}) ℑ \overset{H \cos}{c}_{n} - β J_{n}^{'} (k^{-} r_{0}) ℑ \overset{J \cos}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℑ (\frac{\partial \overset{add}{φ}_{m}}{\partial r}) \cos n θ_{m} \\ Y_{n}^{'} (k^{+} r_{0}) ℜ \overset{H \sin}{c}_{n} + J_{n}^{'} (k^{+} r_{0}) ℑ \overset{H \sin}{c}_{n} - β J_{n}^{'} (k^{-} r_{0}) ℑ \overset{J \sin}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℑ (\frac{\partial \overset{add}{φ}_{m}}{\partial r}) \sin n θ_{m} \end{aligned}

(26d)

These systems of equations may be rearranged to remove the Bessel

J

terms, giving the following equations for those with a right-hand sides containing real terms:

\begin{aligned} J_{0} ℜ \overset{H \cos}{c}_{0} - Y_{0} ℑ \overset{H \cos}{c}_{0} & = - \frac{1}{M} \sum_{m = 1}^{M} ℜ [J_{0} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{0}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \\ J_{n} ℜ \overset{H \cos}{c}_{n} - Y_{n} ℑ \overset{H \cos}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℜ [J_{n} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{n}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \cos n θ_{m} \\ J_{n} ℜ \overset{H \sin}{c}_{n} - Y_{n} ℑ \overset{H \sin}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℜ [J_{n} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{n}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \sin n θ_{m} \end{aligned}

(27a)

and for those containing imaginary terms:

\begin{aligned} Y_{0} ℜ \overset{H \cos}{c}_{0} + J_{0} ℑ \overset{H \cos}{c}_{0} & = - \frac{1}{M} \sum_{m = 1}^{M} ℑ [J_{0} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{0}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \\ Y_{n} ℜ \overset{H \cos}{c}_{n} + J_{n} ℑ \overset{H \cos}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℑ [J_{n} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{n}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \cos n θ_{m} \\ Y_{n} ℜ \overset{H \sin}{c}_{n} + J_{n} ℑ \overset{H \sin}{c}_{n} & = - \frac{2}{M} \sum_{m = 1}^{M} ℑ [J_{n} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{n}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \sin n θ_{m} \end{aligned}

(27b)

using the condensed notation

\begin{aligned} J_{n} & = J_{n}^{'} (k^{+} r_{0}) J_{n} (k^{-} r_{0}) - β J_{n} (k^{+} r_{0}) J_{n}^{'} (k^{-} r_{0}) \\ Y_{n} & = Y_{n}^{'} (k^{+} r_{0}) J_{n} (k^{-} r_{0}) - β Y_{n} (k^{+} r_{0}) J_{n}^{'} (k^{-} r_{0}) \end{aligned}

(28)

Alternatively, the condensed notation may be expressed in terms of the complex Hankel function

\begin{aligned} J_{n} & = ℜ [H_{n}^{^{'} (1)} (k^{+} r_{0}) J_{n} (k^{-} r_{0}) - β H_{n}^{(1)} (k^{+} r_{0}) J_{n}^{'} (k^{-} r_{0})] \\ Y_{n} & = ℑ [H_{n}^{^{'} (1)} (k^{+} r_{0}) J_{n} (k^{-} r_{0}) - β H_{n}^{(1)} (k^{+} r_{0}) J_{n}^{'} (k^{-} r_{0})] \end{aligned}

(29)

This system of two equations with two unknowns for each

n

may be solved for the real and imaginary components, which are then combined to give the complex coefficients for the exterior expansions

\begin{aligned} \overset{H \cos}{c}_{0} & = \frac{1}{M} \sum_{m = 1}^{M} \frac{- J_{0} + i Y_{0}}{{J_{0}}^{2} + {Y_{0}}^{2}} [J_{0} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{0}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \\ \overset{H \cos}{c}_{n} & = \frac{2}{M} \sum_{m = 1}^{M} \frac{- J_{n} + i Y_{n}}{{J_{n}}^{2} + {Y_{n}}^{2}} [J_{n} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{n}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \cos n θ_{m} \\ \overset{H \sin}{c}_{n} & = \frac{2}{M} \sum_{m = 1}^{M} \frac{- J_{n} + i Y_{n}}{{J_{n}}^{2} + {Y_{n}}^{2}} [J_{n} (k^{-} r_{0}) \frac{\partial \overset{add}{φ}_{m}}{\partial r} - β J_{n}^{'} (k^{-} r_{0}) \overset{add}{φ}_{m}] \sin n θ_{m} \end{aligned}

(30)

A relation between the coefficients for the inner Bessel

J

expansions and the exterior Hankel coefficients is given by combining the real part with

i

times the imaginary parts of the

f^{(1)}

conditions:

\begin{aligned} J_{0} (k^{-} r_{0}) \overset{J \cos}{c}_{0} & = [J_{0} (k^{+} r_{0}) + i Y_{0} (k^{+} r_{0})] \overset{H \cos}{c}_{0} + \frac{1}{M} \sum_{m = 1}^{M} \overset{add}{φ}_{m} \\ J_{n} (k^{-} r_{0}) \overset{J \cos}{c}_{n} & = [J_{n} (k^{+} r_{0}) + i Y_{n} (k^{+} r_{0})] \overset{H \cos}{c}_{n} + \frac{2}{M} \sum_{m = 1}^{M} \overset{add}{φ}_{m} \cos n θ_{m} \\ J_{n} (k^{-} r_{0}) \overset{J \sin}{c}_{n} & = [J_{n} (k^{+} r_{0}) + i Y_{n} (k^{+} r_{0})] \overset{H \sin}{c}_{n} + \frac{2}{M} \sum_{m = 1}^{M} \overset{add}{φ}_{m} \sin n θ_{m} \end{aligned}

(31)

or doing the same for the second condition

\begin{aligned} β J_{0}^{'} (k^{-} r_{0}) \overset{J \cos}{c}_{0} & = [J_{0}^{'} (k^{+} r_{0}) + i Y_{0}^{'} (k^{+} r_{0})] \overset{H \cos}{c}_{0} + \frac{1}{M} \sum_{m = 1}^{M} \frac{\partial \overset{add}{φ}_{m}}{\partial r} \\ β J_{n}^{'} (k^{-} r_{0}) \overset{J \cos}{c}_{n} & = [J_{n}^{'} (k^{+} r_{0}) + i Y_{n}^{'} (k^{+} r_{0})] \overset{H \cos}{c}_{n} + \frac{2}{M} \sum_{m = 1}^{M} \frac{\partial \overset{add}{φ}_{m}}{\partial r} \cos n θ_{m} \\ β J_{n}^{'} (k^{-} r_{0}) \overset{J \sin}{c}_{n} & = [J_{n}^{'} (k^{+} r_{0}) + i Y_{n}^{'} (k^{+} r_{0})] \overset{H \sin}{c}_{n} + \frac{2}{M} \sum_{m = 1}^{M} \frac{\partial \overset{add}{φ}_{m}}{\partial r} \sin n θ_{m} \end{aligned}

(32)

Data Availability Statement

All data, models, and code generated or used during the study appear in the published article.

Acknowledgments

This study contributes toward research on electromagnetic waves in wheat for the USDA/NIFA (award 2017-67007-25943), and electrical resistivity imaging in soils to support groundwater analysis for the USDA/AFRI (award 2017-67023-26276).

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Journal of Waterway, Port, Coastal, and Ocean Engineering

Volume 146 • Issue 4 • July 2020

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

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Received: May 31, 2019

Accepted: Nov 6, 2019

Published online: Apr 28, 2020

Published in print: Jul 1, 2020

Discussion open until: Sep 28, 2020

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David R. Steward, F.ASCE [email protected]

Professor, Dept. of Civil and Environmental Engineering, North Dakota State Univ., CIE Bldg, Rm. 201, Dept. 2470, PO Box 6050, Fargo, ND 58108-6050. Email: [email protected]

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Waves in Collections of Circular Shoals and Bathymetric Depressions

Abstract

Introduction

Analytic Element Method Formulation

Computational Results and Discussion

Waves through Shoals and Bathymetric Depressions

Numerical Validation and Considerations

Future Applications

Summary and Conclusions

Appendix: Solving Coefficients for Coastal Features

Data Availability Statement

Acknowledgments

References

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Abstract

Introduction

Analytic Element Method Formulation

Computational Results and Discussion

Waves through Shoals and Bathymetric Depressions

Numerical Validation and Considerations

Future Applications

Summary and Conclusions

Appendix: Solving Coefficients for Coastal Features

Data Availability Statement

Acknowledgments

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