Vibration of Elastic Plates Submerged in a Fluid with a Free Surface November 2009 Michael J. A. Smith Supervised by Dr. Michael H. Meylan A dissertation submitted for the degree of Bachelor of Science (Honours) in Applied Mathematics at the University of Auckland, New Zealand
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Vibration of Elastic Plates
Submerged in a Fluid with a Free
Surface
November 2009
Michael J. A. Smith
Supervised by
Dr. Michael H. Meylan
A dissertation submitted for the degree of
Bachelor of Science (Honours) in Applied Mathematics
at the University of Auckland, New Zealand
Abstract
The vibration of an elastic plate by floating and submerged bodies, and the associated
problem of wave scattering, is a problem of great interest in the fields of engineering and
mathematics, with applications in offshore platform construction and shipbuilding design,
as well as in the vibration of icebergs and sea ice. This field is generally known as hy-
droelasticity.
We investigate the response of floating elastic plates, subject to a wave forcing, using
linearised governing equations. Using Green’s functions, we formulate the problem in
terms of integral equations, which are then solved with the use of numerical methods,
including boundary element methods.
A solution is presented for the vibration of an elastic plate of variable stiffness and mass,
which floats on the surface of a fluid, with negligible submergence. The solution method
is based closely on the equivalent problem for a floating elastic plate of constant proper-
ties. We show that the solution for a plate of variable properties can be calculated from
the solution of a uniform plate, by expressing the natural modes for the variable plate in
terms of the modes for a constant plate.
A solution is also presented for the problem of a submerged thin beam. This is anal-
ogous to solving the problem of wave scattering by a submerged thin structure when the
beam is rigid. The solution method for this problem is also closely related to the prob-
lem for a floating uniform plate, but with the additional consideration of hypersingular
integral equations.
i
ii
Acknowledgements
The author would like to acknowledge the support and dedication of Dr Mike Meylan, an
excellent supervisor from whom I have learnt a great deal.
The author would also like to thank the University of Auckland for the financial sup-
port given by a University of Auckland PGDip/Honours/Masters Scholarship.
Acknowledgements also to Dr Howie Cohl for his editorial assistance.
iii
iv
Contents
Abstract i
Acknowledgements iii
Contents iii
List of figures vii
Nomenclature ix
1 Introduction 1
2 Solution for a uniform free beam floating on water 3
Φ - fluid potentialφ - potential in the frequency domainφI - incident potentialφS - scattered potentialφD - diffraction/diffracted potentialφR - radiated potential
∂jxφ - denotes ∂jφ
∂xj
∂n - denotes ∂∂n
Re - real part of an expressionχ - displacement of the beamζ - displacement in the frequency domainρ - density of mediumg - gravitational constantP - pressureD - stiffness function for a beamβ - non-dimensional stiffnessm - linear mass density function for a beamγ - non-dimensional linear mass densityH - dimensional depth of mediumh - non-dimensional depth of mediumL1 - dimensional length of beamL - non-dimensional length of beamω - frequency of incident waveα - ω2
kn - positive real solution of dispersion relation k tanh(kh) = αk0 - first imaginary solution of dispersion relation k tanh(kh) = αR - reflection coefficientT - transmission coefficientXn - eigenfunctions for uniform floating beamµn - eigenvalues for uniform floating beamǫ[v] - energy functional
ix
x
1Introduction
The problem of a floating elastic plate, subject to linear wave forcing has been studied for
a considerable time and is discussed in some detail by Stoker (1957) , where the solution
for shallow water is given for both a finite and semi-infinite plate. For non-shallow water,
the solution for a semi-infinite plate was first found by Fox & Squire (1994), and for a
finite plate by Meylan & Squire (1994); Newman (1994). Recently, the floating plate
problem has received considerable research attention, motivated largely by the construc-
tion, or proposed construction, of very large floating structures (VLFS). These are used
for industrial space, airports, storage facilities and habitation. We do not discuss in detail
the numerous methods developed to solve these problems, instead readers are referred to
the excellent review articles by Watanabe et al. (2004); Squire (2007).
The most straightforward method to solve the linear response of a variable hydroelastic
body is to generalize the method used to solve for a rigid body. In this method the body
motion is solved by an expansion in the rigid body modes, and the effect of the fluid
is found by solving an integral equation over the wetted surface of the body, using the
Green’s function for the fluid (Mei, 1989; Newman, 1977). To include the effect of the
elasticity, the elastic modes, as well as the rigid body modes, need to be included. This
method is the basis of the solution we present here.
1
2 Introduction
Current computational methods allow for the solution of very complicated hydroelastic
problems, for example, to calculate the response of a container ship. However, there still
exists a need for methods to solve simple hydroelastic problems. The case of a floating
plate in two-dimensions is exactly one such well studied problem. The solution for a
plate of variable thickness has been developed by Bennetts et al. (2007) and by Meylan
& Sturova (2009).
To consider the problem of wave scattering in general, we suspend an object in a medium,
and subject it to an incident forcing wave. This incident wave will be scattered by the
structure, and the object itself will vibrate, causing the propagation of waves into the
medium.
This work primarily focuses around the use of linearised models for the fluid flow and
beam response. Although the actual problem is non-linear, we assume that the amplitude
of the waves is much smaller than their wavelength, which admits the use of linearised
equations for fluid flow. We can also assume that the displacement of the mean is small
compared to these waves, which allows the use Euler-Bernoulli beam theory, as opposed
to more complicated theories, such as Timoshenko beam theory (Rao, 1986).
More specifically, this work applies the ideas on uniform floating beams from Stoker (1957)
to the more complicated problem of a floating elastic plate of variable stiffness and mass.
The technique for computing the response of a floating body focuses primarily on the use
of integral equations (particularly Green’s second identity). An additional extension is
made to this theory for the case of a submerged thin plate of varying stiffness and mass,
however, the integral equation formulation gives rise to added complications, including
hypersingular integrals. The solution for the problem of wave scattering by soft and hard
bodies is discussed in detail by Yang (2002).
The primary goal of this dissertation is to solve the response of a submerged variable
plate, subject to an incident wave forcing. This solution is considered in a number of
stages. Firstly, the solution method for a finite elastic plate of constant properties floating
on water is considered. Using concepts from this established theory, the solution for a
floating variable plate is then considered. The final goal is to consider the vibration of a
submerged thin beam of variable stiffness and mass. Unfortunately, while the theory has
been fully developed, we have only developed computational methods for a rigid plate
(which contains most numerical difficulties).
2Solution for a uniform free beam
floating on water
We begin with the problem of a two dimensional uniform elastic beam, which is floating
on the surface of a fluid, where the domain is of constant finite depth, H . The solution
presented here is closely related to the solution method presented by Newman (1994).
2.1 Outline of beam theory
Our beam is modelled using the Euler-Bernoulli beam equation:
D∂4xχ +m∂2
t χ = P, (2.1)
where χ is the displacement, D is the stiffness, m is the mass per unit length of the beam,
P denotes pressure, and the notation ∂jxχ denotes the partial derivative ∂jχ
∂xj . We also have
conditions at the ends of the beam, which are assumed free here (although other edge
conditions could be easily included in the current formulation). The free conditions are
∂2xχ = ∂3
xχ = 0, x = ±L1. (2.2)
3
4 Uniform floating beam
2.2 Equations in the time domain
We express our problem in terms of a set of linearised equations, assuming that the fluid
flow is irrotational and inviscid, and that the amplitude of the fluid and structure motion
is small. This gives rise to the equations for the problem as follows:
∆Φ = 0, −H < z < 0, (2.3a)
∂zΦ = 0, z = −H, (2.3b)
∂zΦ = ∂tχ, z = 0, and (2.3c)
−ρ(gχ+ ∂tΦ) = P, z = 0, (2.3d)
where Φ is the velocity potential for the fluid, ∆ is the Laplace Operator (∆Φ = ∂2xΦ +
∂2zΦ), ρ is the fluid density, and g is the acceleration due to gravity. Note that equation
(2.3a) is Laplace’s equation for an irrotational, inviscid medium, (2.3b) is the condition
that there is no flow through the seabed (at depth H), (2.3c) is the kinematic condition,
and (2.3d) is the dynamic condition for the plate.
We assume that the pressure is zero except under the plate which occupies the region
−L1 ≤ x ≤ L1. Under the plate the dynamic condition, from the Euler-Bernoulli beam
equation, is
D∂4xχ+m∂2
t χ = −ρ(gχ + ∂tΦ), x ∈ (−L1, L1), z = 0. (2.4)
A diagram outlining the problem is shown below:
∂nΦ = ∂tχ∂nΦ = αΦ∂nΦ = αΦ
∆Φ = 0
∂nΦ = 0
−L1 L1z = 0
z = −H
Figure 2.1: Floating elastic beam in a semi-infinite domain
5
The condition (2.3b) can be alternatively expressed in the form ∂zΦ∣∣z=−H
= 0, as can
expressions (2.3c) – (2.3d).
2.3 Non-dimensionalisation and transformation to the
frequency domain
We non-dimensionalise all lengths with respect to an arbitrary length parameter L∗, and
non-dimensionalise time with respect to√L∗/g (so that gravity is unity). L denotes the
non-dimensional length of the beam (L = L1/L∗), and h denotes the non-dimensional
depth (h = H/L∗). We now consider the equations in the frequency domain and assume
that all quantities are proportional to eiωt so that we can write
χ(x, t) = Reζeiωt
, (2.5a)
Φ(x, t) = Reφeiωt
, (2.5b)
where Re denotes the real part of the expression.
Under this assumption, equations (2.3) and (2.4) become
∆φ = 0, −h < z < 0, (2.6a)
∂zφ = 0, z = −h, (2.6b)
∂zφ = iωζ, x ∈ (−L,L), z = 0, (2.6c)
∂zφ = αφ, x /∈ (−L,L), z = 0, and (2.6d)
β∂4xζ − (γα− 1) ζ − αφ = 0, x ∈ (−L,L), z = 0, (2.6e)
where α = ω2, β = D/ρgL∗4, and γ = m/ρL∗. The edge conditions (2.2) also apply. Note
that expression (4.1) can be found in Tayler (1986).
2.4 Boundary conditions at x→ ±∞Equations (2.6a) to (4.1) also require boundary conditions as x → ±∞. To investigate
the wave scattering from our structure, we consider an incident wave of unit amplitude
6 Uniform floating beam
(in potential) incoming from the left. This gives rise to boundary conditions involving an
incident wave, plus corresponding reflected and transmitted waves from our plate. These
imply that
limx→−∞
φ =cosh(k(z + h))
cosh(kh)e−ikx +R
cosh(k(z + h))
cosh(kh)eikx, (2.7)
where k is the first positive, real solution of the dispersion relation: k tanh(kh) = α, and
R is the reflection coefficient. We also have
limx→∞
φ = Tcosh(k(z + h))
cosh(kh)e−ikx, (2.8)
where T is the transmission coefficient. These expressions for the potential, as |x| → ∞,
come from separation of variables. Details for this can be found in Linton & McIver
(2001).
2.5 Modes of vibration for a uniform beam
To determine the modes of vibration for our plate, we now expand the displacement of
the beam in terms of the modes of vibration, and obtain the following series:
ζ =
N∑
n=0
ζnXn, (2.9)
where the beam eigenfunctions satisfy the eigenvalue equation
∂4xXn = µ4
nXn, (2.10)
with the free edge conditions
∂2xXn = ∂3
xXn = 0, x = ±L. (2.11)
These modes can be determined analytically, and are given by
X0 =1√2L, (2.12a)
X1 =
√3
2L3x, (2.12b)
X2n(x) =1√2L
(cos(µ2nx)
cos(µ2nL)+
cosh(µ2nx)
cosh(µ2nL)
), (2.12c)
7
and
X2n+1(x) =1√2L
(sin(µ2n+1x)
sin(µ2n+1L)+
sinh(µ2n+1x)
sinh(µ2n+1L)
), (2.12d)
where µ0 = µ1 = 0 and µn for n ≥ 2 are the roots of
(−1)n tan(µnL) + tanh(µnL) = 0. (2.12e)
Note that we have defined the beam eigenfunctions so that they satisfy the following
normalising condition ∫ L
−L
Xm(x)Xn(x)dx = δmn, (2.13)
where δmn is the Kronecker delta.
2.6 Radiated, diffracted, and incident potentials
For the purposes of computing the potential φ, we make use of the fact that all equations
are linear, and split the computation of the potential into two parts:
φ = φD + φR,
where φD denotes the diffraction potential, and φR denotes the radiation potential. The
diffraction potential relates to the wave scattering problem, as this is the potential as-
suming that the structure is completely rigid, whereas the radiation potential considers
the response of the structure, and the resulting propagation of waves into the medium.
Note that φD also incorporates the incident potential, and can therefore be split into
scattered and incident potentials (φD = φI + φS).
After expanding the displacement of the plate in terms of the modes (2.9), and applying
this to equations (2.6a) – (2.6c) we obtain the following equations for the potential
∆φ = 0, −h < z < 0, (2.14a)
∂zφ = 0, z = −h, (2.14b)
∂zφ = αφ, x /∈ (−L,L), z = 0, and (2.14c)
∂zφ = iω
N∑
n=0
ζnXn, x ∈ (−L,L), z = 0. (2.14d)
8 Uniform floating beam
From linearity the potential can be further expanded as
φ = φD +N∑
n=0
ζnφRn . (2.15)
The radiation potentials satisfy the conditions:
∆φRn = 0, −h < z < 0, (2.16a)
∂zφRn = 0, z = −h, (2.16b)
∂zφRn = αφR
n , x /∈ (−L,L), z = 0, and (2.16c)
∂zφRn = iωXn, x ∈ (−L,L), z = 0, (2.16d)
plus the Sommerfeld radiation condition that:
∂φRn
∂x± ikφR
n = 0, as x→ ±∞. (2.17)
The diffracted potential satisfies
∆φD = 0, −h < z < 0, (2.18a)
∂zφD = 0, z = −h, (2.18b)
∂zφD = αφD, x /∈ (−L,L), z = 0, and (2.18c)
∂zφD = 0, x ∈ (−L,L), z = 0, (2.18d)
with an additional radiation condition that:
∂
∂x
(φD − φI
)± ik
(φD − φI
)= 0, as x→ ±∞. (2.19)
The incident potential is given to be
φI =cosh(k(z + h))
cosh(kh)e−ikx, (2.20)
which is unit amplitude with respect to potential, and travelling towards positive infinity.
9
Incidentally, the scattered potential satisfies
∆φS = 0, −h < z < 0, (2.21a)
∂zφS = 0, z = −h, (2.21b)
∂zφS = αφS, x /∈ (−L,L), z = 0, and (2.21c)
∂zφS = −∂zφ
I, x ∈ (−L,L), z = 0. (2.21d)
We aim to determine the radiation and diffraction potential through integral equations,
which require the use of Green’s functions.
2.7 Solution for radiation and diffracted potentials
using Green’s functions
For this problem, we can use a special Green’s function, which is given by the following
system of equations
∆G(x,x′) = δ(x− x′), −h < z < 0, (2.22a)
∂zG = αG, z = 0, and (2.22b)
∂zG = 0, z = −h, (2.22c)
plus the Sommerfeld radiation condition of no incoming waves. In our case, we denote our
field points by x = (x, z) and our source points by x′ = (x′, z′). We express the solution
to our system above as the following series
G(x,x′) =∞∑
n=0
an(x)fn(z), (2.23)
where fn(z) is chosen in such a way as to satisfy the boundary conditions above:
fn(z) =cos(kn(z + h))
Nn
. (2.24)
Note that the normalization coefficient Nn is:
Nn =
√cos(knh) sin(knh) + knh
2kn. (2.25)
10 Uniform floating beam
The Green’s function can then be expressed as
(∂2x + ∂2
z )G = δ(x− x′)δ(z − z′), (2.26)
and the delta function can be expanded as follows
δ(z − z′) =
∞∑
n=0
fn(z)fn(z′), (2.27)
which gives rise to the expression
∞∑
n=0
(∂2x − k2
n)an(x)fn(z) = δ(x− x′)
∞∑
n=0
fn(z)fn(z′). (2.28)
Therefore in order to find the coefficients an, we must solve
This allows us to express kmn as follows (with weights wh)
kmn =
∫ L
−L
β(x)X′′
mX′′
ndx ≈∑
β(xh)X′′
m(xh)X′′
n(xh)wh
≈ ~X′′
mH~X
′′Tn , (3.12)
where
~X′′
m = [X′′
m(x1), X′′
m(x2), ..., X′′
m(xh)],
H =
γ(x1)w1 0 ... 0
0 γ(x2)w2 ... 0...
. . ....
0 ... ... γ(xh)wh
,
and the weights are defined as: w1 = h/3, w2 = 4h/3, ..., wh = h/3.
We can extend this concept to the the full matrix K if we form
K = X′′
matHX′′Tmat, (3.13)
where
X′′
mat =
X′′
1 (x1) X′′
1 (x2) ... X′′
1 (xh)
X′′
2 (x1) X′′
2 (x2) ... X′′
2 (xh)...
......
X′′
N(x1) X′′
N(x2) ... X′′
N(xh)
. (3.14)
21
For M we use an identical approach
M = XmatJXTmat, (3.15)
where
J =
γ(x1)w1 0 ... 0
0 γ(x2)w2 ... 0...
. . ....
0 ... ... γ(xh)wh
, and (3.16)
Xmat =
X1(x1) X1(x2) ... X1(xh)
X2(x1) X2(x2) ... X2(xh)...
......
XN(x1) XN(x2) ... XN(xh)
. (3.17)
Having determined the matrices K and M, as well as solving the eigenvalue problem
(3.11), we can determine the non-uniform modes of vibration. This allows us to expand
the displacement in terms of the non-uniform modes to obtain
ζ =N∑
n=0
ζnXn, (3.18)
and as the displacement was previously defined as
Xm =
N∑
n=0
amnXn(x), (3.19)
we can obtain the following relationship
ζn = Aζn and ζn = A−1ζn. (3.20)
In other words, we can express the displacement coefficients for a non-uniform beam in
terms of the displacement coefficients for a uniform beam, allowing us to connect with
established theory in the previous chapter.
22 Non-uniform floating beam
3.5 Equations for a floating non-uniform beam
The equations for a non-uniform beam, floating on the fluid surface, are virtually the
same as those given for a uniform beam (2.6a) to (4.1). If we apply these results, the
governing equations can be expressed as
∆φ = 0, −h < z < 0, (3.21a)
∂zφ = 0, z = −h, (3.21b)
∂zφ = αφ, x /∈ (−L,L), z = 0, (3.21c)
∂zφ = iω
∞∑
n=0
ζnXn, x ∈ (−L,L), z = 0, and (3.21d)
∂2x
(β(x)
∞∑
n=0
ζn∂2xXn
)− (γ(x)α − 1)
N∑
n=0
ζnXn + iωφ = 0, x ∈ (−L,L), z = 0. (3.21e)
From linearity the potential can be written as
φ = φD +
N∑
n=0
ζnφRn , (3.22)
where
φRn =
∑
m
amnφRn . (3.23)
Substituting this expression into equation (3.21e) gives the following
N∑
n=0
(µ4
nγ(x) − γ(x)α + 1)ζnXn = −iω
(φD +
N∑
n=0
ζnφRn
), x ∈ (−L,L), z = 0. (3.24)
If we now multiply by Xm and integrate from −L to L, we obtain
N∑
n=0
(µ4
n − α) [∫ L
−L
γ(x)XnXmdx+
∫ L
−L
XnXmdx+ iω
∫ L
−L
φRn Xmdx
]ζn =
−iω∫ L
−L
φDXmdx, x ∈ (−L,L), z = 0. (3.25)
Using the series expansion for Xn at (3.7), and (3.20), we obtain the following equation
[(D − αI)ATMA + ATA + ATΛA
]~ζ = A~f , (3.26)
23
where the elements of the diagonal matrix D are given by
dmm = µ4m. (3.27)
Equation (3.26) is closely related to equation (2.45).
3.6 Numerical examples
To demonstrate the theory outlined in this chapter, we consider the β(x) and γ(x) profiles
outlined in the following graph. Using these profiles, we are able to compute the reflec-
tion and transmission coefficients for different multiples of γ and β. The reflection and
transmission coefficients provide a useful way of comparing the behaviour of the beam
from different density and mass profiles.
−1 −0.5 0 0.5 10
0.5
1
1.5
2
x
β , γ
Density and Mass profiles for a Non−Uniform beam
Body 1Body 2Body 3Body 4
Figure 3.1: Density and mass function profiles for Bodies 1 to 4
24 Non-uniform floating beam
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
β = γ = 0.01
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
β = γ = 0.01
Figure 3.2: All bodies for β = γ = 0.01
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
β = γ = 0.02
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
β = γ = 0.02
Figure 3.3: All bodies for β = γ = 0.02
25
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
β = γ = 0.05
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
β = γ = 0.05
Figure 3.4: All bodies for β = γ = 0.05
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
β = γ = 0.1
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
β = γ = 0.1
Figure 3.5: All bodies for β = γ = 0.1
26 Non-uniform floating beam
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
shape 1
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
shape 1
Figure 3.6: Body 1 for β = γ = 0.01, 0.02, 0.05, 0.1
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
shape 2
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
shape 2
Figure 3.7: Body 2 for β = γ = 0.01, 0.02, 0.05, 0.1
27
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
shape 3
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
shape 3
Figure 3.8: Body 3 for β = γ = 0.01, 0.02, 0.05, 0.1
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|R|
shape 4
0 5 10 15 200
0.2
0.4
0.6
0.8
1
α
|T|
shape 4
Figure 3.9: Body 4 for β = γ = 0.01, 0.02, 0.05, 0.1
28 Non-uniform floating beam
4Non-uniform submerged beam
The problem of determining the vibration of a submerged beam is a problem of great
theoretical and practical importance, and is the ultimate aim of the project. Our beam
under consideration is assumed to have negligible thickness (considered to be a thin body).
This simple problem is, in fact, particularly challenging, as when formulating the integral
equation form of the problem, we encounter hypersingular integral equations.
∆Φ = 0
∂nφ = αφ
∂nφ = 0
Figure 4.1: Submerged thin structure
29
30 Submerged Beam
We consider a two-dimensional beam in a semi-infinite region (−∞ < x < ∞ and −h <z < 0), completely submerged in a fluid. To the best knowledge of the author, this
problem has never been solved.
4.1 Thin structures
When submerging a beam in a semi-infinite region, we need to impose artificial boundaries
in order to utilise boundary element methods, which assume a closed domain. This
effectively splits our domain into three parts. Our equations for the fluid flow in the
central domain, Ω (immediately surrounding the structure), are as follows:
∆φ = 0, (x, z) ∈ Ω, (4.1a)
∂zφ = 0, z = −h, (4.1b)
∂zφ = αφ, z = 0, (4.1c)
φ = φ1(z), x = a1, and (4.1d)
φ = φ2(z), x = a2, (4.1e)
plus radiation conditions:
∂
∂xφ± ikφ = 0, as x→ ±∞. (4.2)
Additionally, there is a kinematic condition for the response of the beam, which is sub-
merged at constant depth
∂nφ = g(x), x ∈ (−L,L), z = −d, (4.3)
as well as a dynamic condition for the plate:
N∑
n=0
(µ4
nγ(x) − γ(x)α)ζnXn = −iω
(
[ φD ] +
N∑
n=0
ζn[ φRn ]
)
, x ∈ (−L,L), z = −d.
(4.4)
This expression is closely related to (3.21e) for the case of a non-uniform floating beam,
and for the case of a uniform floating beam (there is no restoring force for the case of a
submerged beam).
31
For a non-uniform submerged beam, we can use:
∂nφ = iωXn, x ∈ (−L,L), z = −d, (4.5)
and in the case of a rigid structure, we specify g(x) = 0.
Our problem can be illustrated by the following image:
n
∂nφ = αφ
Γ−
Γ+
φ = φ1 φ = φ2
∂nφ = 0
z = 0
z = −h
z = −d
Ω
Figure 4.2: Submerged thin structure in a finite region
As the obstacle is thin, we need to split the object (denoted by Γ) into two regions (Γ±),
as well as distinguish their normal derivatives (n into n±).
The boundary condition is now written as
∂n±φ = ±g(x), (4.6)
with the additional constraint that the normal derivatives are related in the following
manner
∂n− = −∂n+ . (4.7)
The notation ∂n and ∂n′ represents the normal derivatives on the boundary of our domain.
If we lie on the outer boundary ∂Ω, then the normal derivative is always outwards pointing
relative to Ω, whereas if we lie on the inner boundary Γ, the normal derivative is always
in the positive z direction.
32 Submerged Beam
As a consequence of the relationship (4.7), we can express everything in terms of Γ+,
and omit the notation in the future (in particular, ∂n = ∂n+ ). Therefore, Green’s second
identity for our submerged structure, integrated along both intervals Γ+ and Γ−, becomes
∫
Γ
(φ∂n′G−G∂n′φ)ds′ =
∫
Γ
[ φ ]∂n′Gds′ , (4.8)
where [ φ ] = φ(x+) − φ(x−). This function represents the difference in potential above
and below the beam, and is known as the jump in potential across Γ (Linton & McIver,
2001).
4.2 Integral equations for the submerged beam prob-
lem
In this section, we aim to formulate our problem in terms of integral equations. In doing
this, we define our Green’s function to be the fundamental solution of Laplace’s equation
in R2 (not the Green’s function in the case of a floating plate). From Greenberg (1971),
G(x,x′) = − 1
4πln((x− x′ )2 + (z − z′ )2
). (4.9)
Employing Green’s second identity, (2.33), around Ω and Γ, in connection with the well
known result from Zwillinger (1992)
∫
Ω
δ(x,x′)φ(x′)ds′ =
0 if x /∈ Ω and x /∈ ∂Ω12φ(x) if x ∈ ∂Ω
φ(x) if x ∈ Ω
, (4.10)
we can obtain the following relationship for the total boundary of our domain (where x
and x′ are the source and field points in our region, restricted to lie on the boundary,
with n and n′ their respective normals)
1
2φ(x) =
∫
Γ
(φ∂n′G−G∂n′φ)ds′ +
∫
∂Ω
(φ∂n′G−G∂n′φ)ds′ . (4.11)
Using our result from the previous section, we can express this as
1
2φ(x) =
∫
Γ
[ φ ]∂n′Gds′ +
∫
∂Ω
(φ∂n′G−G∂n′φ)ds′ , (4.12)
33
To evaluate the second integral in the above expression, we revisit boundary element
methods.
4.3 Boundary element methods
As mentioned earlier, the boundary element method is a useful tool for solving elliptic
problems such as Laplace’s equation, in a closed region. Providing we correctly discretise
our boundary into suitably small elements, we can express our integral as a sum over
these elements, evaluating φ and φn at the panel midpoints.
For example, ∫
∂Ω
(φ∂n′ nG− ∂nG∂n′φ)ds′ ,
can be expressed in the form
G(1)~φ− G
(2)~φn′ ,
where
G(1)ij =
∫ xi+l/2
xi−l/2
∂n′ nG(xi, x′
j)dh, and (4.13)
G(2)ij =
∫ xi+l/2
xi−l/2
∂nG(xi, x′
j)dh. (4.14)
Recall that we have restricted x to be on Γ, and that x′ is restricted to be on ∂Ω. As a
consequence of this, G(1) and G(2) are matrices of dimension (Γ, ∂Ω), and φ and φn′ are
vectors of dimension (∂Ω, 1). We now consider in more detail, the computation of the
fluid potential.
4.4 Radiation and diffraction potentials
Recall that the potential can be expressed as the sum
φ = φD +∑
m
ζmφRm. (4.15)
For our example of a submerged beam, we first consider computing the diffraction poten-
tial (with φ = φD to avoid notation), which is equivalent to solving the original system of
34 Submerged Beam
equations ((4.1a) – (4.69d)), assuming that the structure is completely rigid:
∂nφ = 0 on Γ. (4.16)
In order to compute [ φ ], we will restrict all x points in (4.12) to be on Γ, and then take
the normal derivative of that expression, to obtain:
×∫
Γ
[ φ ]∂n′ nGds′ +
∫
∂Ω
(φ∂n′ nG− ∂nG∂n′φ)ds′ = 0. (4.17)
where the first integral is known as a Hadamard finite part integral (discussed shortly).
To avoid any possible confusion, it is important to note that for the first integral of (4.17),
(x,x′ ) ∈ (Γ,Γ), and for the second integral, (x,x′ ) ∈ (Γ, ∂Ω).
Note that when considering the radiated potentials (φ = φRm), the kinematic condition
becomes
∂nφRm = iωXm. (4.18)
However, for the moment, we will consider φ = φD
4.5 Hypersingular integral equations
Hypersingular integral equations are integral equations whose kernel has a singularity of
order greater than one. They typically arise in the study of problems in fluid dynamics.
We can obtain hypersingular integral equations from reducing Neumann boundary value
problems for Laplace’s equation to integral equations by means of the double-layer po-
tential (Lifanov et al., 2004). In other words, they effectively arise from deriving the log
singularity of the Green’s function twice.
We define a Cauchy principle value of a singular integral to be
P
∫ b
a
f(x)dx = limǫ→0
[∫ x0−ǫ
a
f(x)dx+
∫ b
x0+ǫ
f(x)dx
], ǫ > 0,
providing the limit exists (Kwok, 2002). It can be shown that the derivative of a Cauchy
principle value integral gives rise to a Hadamard principal value integral (Aliabadi, 2002)
∂
∂xP
∫ b
a
f(x)
x− x′dx = ×
∫ b
a
f(x)
(x− x′ )2dx.
35
Using the Green’s function defined earlier:
G(x,x′) = − 1
4πln((x− x′ )2 + (z − z′ )2
), (4.19)
we can express the first term of equation (4.12) in the following form
∫
Γ
[ φ ]∂n′Gds′ =1
2π
∫
Γ
[ φ ](z − z′ )
(x− x′ )2 + (z − z′ )ds′ , (4.20)
which has an interior singularity in Γ. Consequently, this is a Cauchy principal value
integral. Therefore, when we take the derivative of this, we obtain a Hadamard integral
of the form∂
∂n
∫
Γ
[ φ ]∂n′Gds′ =1
2π×∫
Γ
[ φ ]
(x− x′ )2dx′ . (4.21)
In order to evaluate this integral, we express [ φ(x′ ) ] as a series of Chebyshev polynomials
of the second kind
[ φ(x′ ) ] = (1 − x′2)1/2
M∑
m=0
bmUm(x′ ), (4.22)
and use the result from Linton & McIver (2001)
×∫ 1
−1
(1 − v2)1/2Um(v)
(u− v)2dv = −π(m+ 1)Um(u). (4.23)
From this we can obtain
−1
2
M∑
m=0
bm(m+ 1)Um(x) +
∫
∂Ω
∂n′ nGφds′ −∫
∂Ω
∂nG∂n′φds′ = 0, (4.24)
providing that the non-dimensionalised length L is unitary.
Multiplying this expression by (1 − x2)1/2Un(x) and integrating yields
M~b+ Y[G(1)~φ− G(2)~φn′ ] = 0, (4.25)
where
Y = UTW
(1)W
(2), (4.26)
W(1) = diag((1 − x2
k)1/2), (4.27)
U = [U1(x), U2(x), . . . , Um(x)], (4.28)
36 Submerged Beam
Mmm = −π4m, and W(2) is a weights matrix associated with the choice of a numerical in-
tegration technique. The matrices G(1) and G(2) denote the integration of our appropriate
Green’s function expressions, and the matrix U has columns of Chebyshev polynomials
of the second kind.
This result arises through the use of the orthogonal relation (Mason, 2003)
∫ 1
−1
(1 − x2)1/2Um(x)Un(x)dx =π
2δmn. (4.29)
We now focus our attention to the original expression (4.12)
1
2φ(x) =
∫
Γ
[ φ ]∂n′Gds′ +
∫
∂Ω
(φ∂n′G−G∂n′φ)ds′ . (4.30)
Using the series approximation for [ φ ] as at (4.22), we obtain
1
2~φ = G
(3)W
(1)U~b+ G
(4)~φ− G(5)~φn′ , (4.31)
where
G(3)ij =
∫ xi+l/2
xi−l/2
∂n′G(xi, x′
j)dh, (4.32)
G(4)ij =
∫ xi+l/2
xi−l/2
∂n′G(xi, x′
j)dh, and (4.33)
G(5)ij =
∫ xi+l/2
xi−l/2
G(xi, x′
j)dh. (4.34)
Note that this time, G(3) has (x,x′ ) ∈ (∂Ω, Γ), whereas G(4) and G(5) are both restricted
to ∂Ω. Practically speaking, it is important to take note of line elements when integrating
over ∂Ω, and the direction of integration.
This leaves us with the following two equations
M~b+ YG(1)~φ− YG
(2)~φn′ = 0, and (4.35)
G(3)
W(1)
U~b+ G(4)~φ− G
(5)~φn′ =1
2~φ. (4.36)
In order to evaluate this, we need to find a way to express our boundary conditions in
Neumann form, so that we can remove ~φn′ from the expressions above. This is discussed
in the following section.
37
4.6 Neumann form
In order to utilise boundary element methods, we previously split our domain into three
parts, and Dirichlet boundary conditions were imposed at the artificial boundaries of Ω.
We now attempt to express these boundary conditions in Neumann form by examining
this separation in more detail.
a1 a2
∂nφ = αφ
∂nφ = 0
∆φ = 0
∂nφ = αφ
∂nφ = 0
∆φ = 0
∂nφ = αφ
∂nφ = 0
∆φ = 0
φ=φ
1(z
)
φ=φ
2(z
)
z = 0
z = −h
Ω− Ω Ω+
Figure 4.3: Semi-infinite domain split into 3 regions
In the Ω− region (−∞ < x < a1, −h < z < 0), the following equations model the
potential of the fluid
∆φ = 0, −h < z < 0, (4.37a)
∂nφ = αφ, z = 0, (4.37b)
∂nφ = 0, z = −h, and (4.37c)
φ = φ1(z), x = a1, (4.37d)
plus a radiation condition
limx→−∞
φ = φ0(z)e−k0x +Rφ0(z)e
k0x. (4.38)
Note that we use the same expression for the incident potential as (2.20)
φI = φ0(z)eikx =
cosh(k(z + h))
cosh(kh)eikx =
cos(k0(z + h))
cos(k0h)e−k0x, (4.39)
and that a1 < 0.
38 Submerged Beam
For our problem in Ω−, separation of variables can be used
φ =∞∑
m=0
CmXm(x)Zm(z), (4.40)
with the following general solutions
Zn(z) =cos(kn(z + h))
cos(knh), and (4.41)
Xn(x) = ekn(x−a1). (4.42)
A normalisation factor can be computed as follows
∫ 0
z=−h
Zn(z)Zm(z)dz = Amnδmn, (4.43)
where
An =1
2
cos(knh) sin(knh) + knh
kn cos2(knh). (4.44)
Using condition (4.37), and orthogonality, allows us to determine that
Cm =1
Am
∫ 0
−h
φ1Zm(z)dz =1
Am
⟨φ1, Zm(z)
⟩. (4.45)
Employing the radiation condition finally yields the potential in the region Ω−
φ = φ0(z)e−k0x +
∞∑
m=0
1
Am
⟨φ1, Zm(z)
⟩Xm(x)Zm(z)
= φ0(z)e−k0x +
∞∑
m=0
1
Am
⟨φ1,
cos(kn(z + h))
cos(knh)
⟩[cos(kn(z + h))
cos(knh)
]ekn(x−a1).(4.46)
From this expression, we can form a Neumann boundary condition for our artificial bound-
ary at x = a1. For convenience, we define
QLφ1(z) = −∞∑
m=0
km
Am
⟨φ1, Zm(z)
⟩Zm(z), (4.47)
and evaluating (4.46) at x = a1 yields the following
φ∣∣x=a1
= φ0(z)e−a1k0 +
∞∑
m=0
CmZm(z). (4.48)
39
Using expressions (4.37) and (4.48), it can be shown that
φ1(z) = φ0(z)e−a1k0 +
∞∑
m=0
CmZm(z). (4.49)
To determine our Dirichlet boundary condition, we first multiply this expression by Z0(z)
and integrate (then multiply the expression by Zn(z) and integrate), to obtain
⟨φ1, Z0
⟩= A0e
−a1k0 + A0C0 (for m = 0), and⟨φ1, Zn
⟩= AnCn for m ≥ 1.
Therefore we can say that
∂nφ∣∣x=a1
= −∂xφ∣∣x=a1
(4.50)
= k0φ0(z)e−k0a1 −
∞∑
m=0
kmCmZm(z)
= k0φ0(z)e−k0a1 − k0C0Z0(z) −
∞∑
m=1
kmCmZm(z)
= k0φ0(z)e−k0a1 −
[k0
A0
⟨φ1, Z0
⟩Z0(z) − k0Z0(z)e
−k0a1
]−
∞∑
m=1
kmCmZm(z)
= 2k0φ0(z)e−k0a1 −
∞∑
m=0
km
Am
⟨φ1, Zm
⟩Zm(z).
Consequently, the normal derivative at x = a1 can be expressed as follows
∂nφ(a1, z) = QLφ1(z) + 2k0φ0(z)ek0a1 . (4.51)
Observe that for our particular case, as the incident wave potential is unit amplitude in
potential, we have φ0(z) = Z0(z). Using the approach above for the region Ω+, with
equations
∆φ = 0, −h < z < 0, (4.52a)
∂nφ = αφ, z = 0, (4.52b)
∂nφ = 0, z = −h, and (4.52c)
φ = φ2(z), x = a2, (4.52d)
40 Submerged Beam
plus a radiation condition
limx→∞
φ = Tφ0(z)e−k0x, (4.53)
it is straightforward to show that
φn(a2, z) = QRφ2(z). (4.54)
However, all this still leaves us with the unknown quantities, QL and QR. Recall that we
earlier defined
QLφ1(z) = −∞∑
m=0
km
Am
⟨φ1, Zm(z)
⟩Zm(z), at x = a1. (4.55)
If we discretise the integral, assuming that φ1(z) is piecewise constant over each element,
it is clear to see that
⟨φ1, Zm(z)
⟩=
∫ 0
−h
φ1(p)Zm(p)dp =∑
i
φ1(xi)
∫xi+l/2
xi−l/2
Zm(p)dp. (4.56)
After truncating the number of modes, the following expression for QL can be obtained
QLφ1(z) = −M∑
m=0
∑
i
[1
Am
Zm
][km
∫xi+l/2
xi−l/2
Zm(p)dp
]φ1(xi), (4.57)
which can be expressed in the form
QLφ1 = SRφ1, (4.58)
where
sim = − 1
Am
cos(km(xi + h)
cos(kmh)), and (4.59)
rmi = km
∫xi+l/2
xi−l/2
cos(km(p+ h))
cos(kmh)dp =
[sin(km(p+ h))
cos(kmh)
]xi+l/2
p=xi−l/2
. (4.60)
We now have Neumann boundary conditions along our entire outer boundary. This allows
us to express the boundary conditions as
~φn′ = A~φ− ~f, (4.61)
41
where
A =
QL 0 0 0
0 0 0 0
0 0 QR 0
0 0 0 αI
, and ~f =
−2k0φ0e−k0a1
0
0
0
, (4.62)
when orientating anticlockwise around the outer boundary from (a1, 0).
After converting all the boundary conditions to Neumann form, we can express our system
(4.36) in the following form
[M Y(G(1) − G(2)A)
G(3)W(1)U (G(4) − G(5)A − 12I)
][~b~φ
]
=
[−YG(2) ~f
−G(5) ~f
]
. (4.63)
Solving this allows us to recover the potential around the outer boundary, ∂Ω, as well as
[ φ ] (or [ φD ] to be precise).
4.7 Solution for the beam
For the case of the radiated potentials ( φ = φRm) we use an identical process, which yields
the system of equations
M~b+ YG(1)~φ− YG
(2)~φn′ = iωYAXmat, and (4.64)
G(3)
W(1)
U~b+ G(4)~φ− G
(5)~φn′ =1
2~φ, (4.65)
(for all m). This system can be expressed in the following form
[M Y(G(1) − G(2)A)
G(3)W(1)U (G(4) − G(5)A − 12I)
][~b~φ
]
=
[iωYAXmat
0
]
. (4.66)
Solving this system gives us the critical [ φRm ]. Having found [ φ ] for both the radiated
and diffraction potentials, the displacement coefficients for our submerged beam can now
be determined, providing we solve the dynamic condition
N∑
n=0
(µ4
nγ(x) − γ(x)α)ζnXn = −iω
([ φD ] +
N∑
n=0
ζn[ φRn ]
), x ∈ (−L,L), z = −d.
(4.67)
42 Submerged Beam
After multiplying by Xn and integrating over the submerged plate, we obtain
N∑
n=0
(µ4
n − α) [∫ L
−L
XnXmdx+ iω
∫ L
−L
[ φRn ]Xmdx
]ζn = −iω
∫ L
−L
[ φD ]Xmdx.
This can be expressed in the following matrix form
[(D − αI) + ATA + ATΛA
]~ζ = A~f , (4.68)
where the matrices and vectors are defined analogously to the case of a submerged beam.
This system is closely related to (3.26).
Unfortunately we do not have the numerical solution for the elastic plate, so for the
purposes of this dissertation, we only consider the case of wave scattering from a rigid
beam. This is equivalent to only considering the diffraction potential for our elastic beam
(and ignoring the radiated potentials entirely).
In doing this, we can use the method of eigenfunction matching to determine whether
our results are consistent with established theory.
43
4.8 Eigenfunction matching for a submerged rigid
body
We use the established technique of eigenfunction matching, as discussed by Heins (1950),
to determine whether our integral equation formulation and use of boundary element
methods is producing consistent results with the calculation of the diffraction potential.
For the purposes of the eigenfunction matching, we model our problem in the frequency
domain, as previously
∆φ = 0, −h < z < 0, (4.69a)
∂zφ = 0, z = −h, (4.69b)
∂zφ = αφ, z = 0, and (4.69c)
∂zφ = 0, z = −d, x ∈ (−L,L), (4.69d)
plus the Sommerfeld radiation condition
∂
∂xφ± ikφ = 0, as x→ ±∞. (4.70)
We have 4 primary regions, x < −L, x > L, and x ∈ (−L,L) for z ∈ (−d, 0) and z ∈(−h,−d) (referred to as regions 1 to 4 respectively). In the first three regions, we use the
free-surface eigenfunctions (due to their contact with the free surface itself), and in the
last, we use the structure eigenfunctions.
z = 0
z = −d
z = −h−L L
Figure 4.4: Regions for eigenfunction matching method
44 Submerged Beam
In section (4.6) earlier, the free surface eigenfunctions were derived in the case of depth h
Zn(z) =cos(kh
n(z + h))
cos(khnh)
, (4.71)
where the new notation khn denotes the roots of the dispersion relation for depth h.
Separation of variables for a floating structure yields the eigenfunctions
Zn(z) = cos(κn(z + h)), (4.72)
where κn = nπh
. Adjusting for the fact the beam is submerged gives rise to κn = nπh−d
(the
eigenfunction is unchanged). To avoid confusion, we define
φhm =
cos(khn(z + h))
cos(khnh)
, (4.73)
φdm =
cos(kdn(z + d))
cos(kdnd)
, and (4.74)
ψm = cos(κm(z + h)), (4.75)
then expand the potential as follows
φ1(x, z) = e−kh0(x+L)φh
0 (z) +
∞∑
m=0
amekh
m(x+L)φhm(z), (4.76a)
φ2(x, z) =∞∑
m=0
bme−kd
m(x+L)φdm(z) +
∞∑
m=0
cmekd
m(x−L)φdm(z), (4.76b)
φ3(x, z) = d0L− x
2L+
∞∑
m=1
dme−κm(x+L)ψm(z) + e0
x+ L
2L+
∞∑
m=1
emeκm(x−L)ψm(z), and
(4.76c)
φ4(x, z) =
∞∑
m=0
fme−kh
m(x−L)φhm(z), (4.76d)
where φ1 denotes the potential in the first region, and so forth. The potential and its
derivative must be continuous between the regions, that is, the potentials and their deriva-
tives must match at x = ±L:
φ1 = φ2 + φ3, at x = −L, and (4.77)
φ4 = φ2 + φ3, at x = L, (4.78)
45
plus their respective derivatives. Multiplying these equations by φhl , and integrating from
−h to 0, allows us to find our required coefficients, when the sums are suitably truncated.
4.9 Numerical results
Using MATLAB, we are able to reconstruct [ φ ], and the potential around the boundary
for a rigid structure, for an incident wave of a given frequency. The solution obtained from
the method outlined in this dissertation is compared with the solution from eigenfunction
matching.
−1 −0.5 0 0.5 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Non−dimensional plate length
Dis
plac
emen
t
[ φD ]
E−fn MatNum. Est
0 50 100 150 200 250 300 350 400−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(x,z) coords of ∂Ω
Re(
φ)
φ
E−fn MatNum. Est.
Figure 4.5: [ φ ] and φ for α = 0.5
−1 −0.5 0 0.5 1−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Non−dimensional plate length
Dis
plac
emen
t
[ φD ]
E−fn MatNum. Est
0 50 100 150 200 250 300 350 400−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(x,z) coords of ∂Ω
Re(
φ)
φ
E−fn MatNum. Est.
Figure 4.6: [ φ ] and φ for α = 2.5
46 Submerged Beam
−1 −0.5 0 0.5 1−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Non−dimensional plate length
Dis
plac
emen
t
[ φD ]
E−fn MatNum. Est
0 50 100 150 200 250 300 350 400−1
−0.5
0
0.5
1
1.5
(x,z) coords of ∂Ω
Re(
φ)
φ
E−fn MatNum. Est.
Figure 4.7: [ φ ] and φ for α = 4.5
−1 −0.5 0 0.5 1−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
Non−dimensional plate length
Dis
plac
emen
t
[ φD ]
E−fn MatNum. Est
0 50 100 150 200 250 300 350 400−1.5
−1
−0.5
0
0.5
1
1.5
(x,z) coords of ∂Ω
Re(
φ)
φ
E−fn MatNum. Est.
Figure 4.8: [ φ ] and φ for α = 6.5
The examples shown above demonstrate good agreement for reconstructing the potential
at the boundary, and also show that our solution is well matched at the plate.
5Discussion
This dissertation examines three problems, specifically, the displacement of a uniform
floating beam, the displacement of a variable floating beam, and the response from a
submerged plate.
The first part relating to uniform beam theory was an implementation of an established
solution by Newman (1994). However, the solution method for variable floating plates
is original research. The solution method for variable plates involves an expansion of
the non-uniform modes of vibration in terms of the uniform modes using Rayleigh-Ritz
methods, so as to determine the displacement of the beam. Numerical methods have been
able to produce results consistent with the solution for uniform beams.
A solution method is also presented for the problem of a submerged plate. The solu-
tion method for this problem is also similar to the technique for floating beams, involving
the separation of the potential. This is also original work, as previous authors (Yang,
2002), have only considered wave scattering from submerged rigid structures.
The solution method for submerged beams had the additional complication of hyper-
singular integral equations. This problem was overcome by expanding an unknown in
terms of Chebyshev polynomials of the second kind.
47
48 Discussion
Theoretically, a full solution method is shown for the case of an submerged variable
beam, however, we have only been able to numerically demonstrate the theory for the
case of a rigid body. These results were compared against the established approach of
eigenfunction matching, and results were found to be consistent with this.
Details of this dissertation, and MATLAB code, can be found online at www.wikiwaves.org,