2.1 Time-harmonic motion of an isolated cylinder

In this section, required information about the radiation and scattering properties of a cylinder in open water is described. All of the motions discussed in this section are time harmonic with radian frequency ${\it\omega}$ and are described by a potential of the form $\text{Re}\{{\it\phi}(\boldsymbol{x})\text{e}^{-\text{i}{\it\omega}t}\}$ , where ${\it\phi}$ is harmonic within the fluid and satisfies the linearised condition

where $S_{F}$ is the undisturbed free surface, $K={\it\omega}^{2}/g$ and $g$ is the acceleration due to gravity.

Radiation potentials ${\it\phi}_{j}$ , $j\in \{1,3\}$ , that describe the forced oscillations of the cylinder with unit velocity amplitude additionally satisfy the boundary conditions

where $S_{B}$ is the wetted surface of the cylinder cross-section, $n$ is an inward normal coordinate to $S_{B}$ , and $n_{j}$ is the component of the inward unit normal to $S_{B}$ measured parallel to the direction of mode $j$ . The sway and heave velocity potentials behave, respectively, as

and

for some complex constants $A_{1}$ and $A_{3}$ . Complex-force coefficients for the radiation problems are defined by

and each such coefficient may be separated into real and imaginary parts by writing $q_{jk}={\it\mu}_{jk}+\text{i}{\it\nu}_{jk}$ , where ${\it\mu}_{jk}$ is an added-mass coefficient and ${\it\omega}{\it\nu}_{jk}$ is a radiation damping coefficient. As $S_{B}$ is symmetric about a vertical line by assumption, $q_{jk}=0$ for $j\neq k$ .

The scattering potential ${\it\phi}_{S}$ describes the fluid motion arising when a plane wave of amplitude  $\mathscr{A}$ is incident from large positive $x$ on the fixed cylinder, and is chosen to have the behaviour

and

where $R$ and $T$ denote, respectively, the reflection and transmission coefficients. Because of the geometrical symmetry, the reflection and transmission coefficients take the same values for a wave incident from the left. For the wave incident from the right, the exciting force in the direction of mode $j$ is defined as

and by the identity

for further information about this last result see the texts by Linton & McIver (Reference Linton and McIver2001, § 1.4) and Mei, Stiassnie & Yue (Reference Mei, Stiassnie and Yue2005, § 8.6.3).

2.2 The extended wide-spacing approximation

Wide-spacing approximations to the time-harmonic radiation problems for a cylinder in the presence of a wall are given by Porter & Evans (Reference Porter and Evans2011). In these approximations only plane waves are used to determine the interactions between the cylinder and the wall, while the evanescent modes are neglected. Here, the extended wide-spacing approximation developed by McIver (Reference McIver2014) is used to account for the leading-order effects of the evanescent modes in both the radiation and scattering problems. In these approximations it is assumed that both ${\it\epsilon}=1/(2Kb)\ll 1$ and $a/b\ll 1$ , so that the distance from the wall is large compared to both the wavelength and the cylinder radius. In the case that the wall is present, the complex-force coefficients and the exciting forces are notated by adding a superscript ‘ $w$ ’ to the notation introduced for the isolated cylinder in § 2.1.

The required hydrodynamic quantities from the extended approximation can be conveniently expressed in the same form as the standard wide-spacing approximation that neglects evanescent modes by introducing the notation:

Here ${\it\kappa}=Ka$ and, from McIver (Reference McIver2014), $p_{3}=-\text{i}{\it\kappa}^{2}A_{3}^{2}/{\rm\pi}a^{2}$ and ${\it\sigma}={\it\gamma}_{13}/{\rm\pi}+(1/2){\it\kappa}q_{33}/M$ ( ${\it\gamma}_{13}$ depends on the geometry of the structure (McIver Reference McIver2014, equation (4.28)), but for the half-immersed circular cylinder ${\it\gamma}_{13}=(1/2){\rm\pi}{\it\kappa}-2$ ). Note that the results in McIver (Reference McIver2014) are given in terms of complex-force coefficients that are scaled by ${\it\rho}a^{2}$ , and exciting forces that are scaled by ${\it\rho}ga\mathscr{A}$ . However, in the present work, dimensional coefficients are retained in order to make a direct connection with the results in Porter & Evans (Reference Porter and Evans2011).

The problem of the synchronised heaving of two identical cylinders is treated in McIver (Reference McIver2014) and, by symmetry, half of the heave coefficient calculated there provides the heave force coefficient needed here for a cylinder in the presence of a wall, so that

(note that a factor of two is missing from the last term in (6.18) of McIver (Reference McIver2014)). Other required results are obtained here in appendix A. The remaining complex-force coefficients are

and

while the exciting forces are

and

Results given for the standard wide-spacing approximation are recovered by setting ${\it\epsilon}$ to zero in (2.10)–(2.11), which is equivalent to dropping the hats in (2.12)–(2.16). When this is done, (2.12)–(2.14) become identical to results given for added-mass and damping coefficients in Porter & Evans (Reference Porter and Evans2011), while, with the aid of (2.9), (2.15)–(2.16) become identical to their equation (3.8).

3.1 General observations

The first specific problem to be considered is the time-dependent motion that results when the cylinder is displaced vertically a distance $z=X_{3}(0)$ and released from rest in calm water. The subsequent time-domain displacements are given by the inverse Fourier transforms

where $v\in \mathbb{R}$ is, in general, positive to allow for the possibility that a frequency-domain displacement $x_{j}({\it\omega})$ may have singularities on the real axis (the absence of motion for $t<0$ implies that each $x_{j}({\it\omega})$ is analytic in $\text{Im}\,{\it\omega}>0$ ); see Stakgold (Reference Stakgold2000, § 5.6). In practice, the limit ${\it\nu}\rightarrow 0$ is taken on the right-hand side of (3.1). From McIver & McIver (Reference McIver and McIver2011, equation (40))

and

A matrix formulation of the initial-value problem for floating elastic or rigid structures in two dimensions is given by Meylan (Reference Meylan2014) and the last two results may also be obtained from this. Note that, because of the definition of the Fourier transform, each $x_{j}({\it\omega})$ has dimensions of length $\times$ time.

In the absence of the trapped modes discovered for this geometry by Porter & Evans (Reference Porter and Evans2009) (and discussed here in § 4), the leading contributions to the large-time asymptotics of the displacements are governed by the behaviour of each $x_{j}({\it\omega})$ as ${\it\omega}\rightarrow 0$ . In this limit, $q_{11}^{w}({\it\omega})$ and $q_{13}^{w}({\it\omega})$ have finite non-zero limits while

(Porter & Evans Reference Porter and Evans2011, table 1). It follows that

and

and hence (McIver & McIver Reference McIver and McIver2011, § 5) that

and

From (3.7), the release of the cylinder from rest results ultimately in a net horizontal translation in a direction that depends upon the sign of the initial vertical displacement (see McIver (Reference McIver2012) for a discussion of circumstances under which a net horizontal translation might occur). From the numerical results in Porter & Evans (Reference Porter and Evans2011), ${\it\mu}_{13}^{w}(0)<0$ for a half-immersed cylinder (in fact, a separate numerical solution of the zero-frequency problem gives ${\it\mu}_{13}^{w}(0)/M=-0.327$ ) so that the net translation is towards the wall when $X_{3}(0)>0$ . Equation (3.8) shows that the vertical motion tends to zero algebraically, as found by Ursell (Reference Ursell1964) for the purely vertical motion of a cylinder in open water (Ursell’s result differs from (3.8) by a factor of two due to the presence of the wall in the current work). In fact, (3.8) holds for any two-dimensional structure with a free-surface intersection that is a line of length $2a$ .

In addition to the singularities at the origin indicated in (3.5) and (3.6), each $x_{j}({\it\omega})$ has poles in the lower half of the complex ${\it\omega}$ plane that arise from the zeros of the identical factors in the denominators in (3.2) and (3.3), and hence the poles have identical positions for the horizontal and vertical displacements. Suppose that these poles are at ${\it\omega}={\it\omega}_{n}-\text{i}{\it\delta}_{n}$ , with $n\in \mathbb{N}\cup \{0\}$ and ${\it\omega}_{n},{\it\delta}_{n}\in \mathbb{R}^{+}$ , so that

where $x_{jn}$ is the residue at the $n$ th pole of $x_{j}$ , and $\{{\it\omega}_{0},{\it\omega}_{1},\ldots ,{\it\omega}_{n},\ldots \}$ is a monotonically increasing sequence. It follows from a property of the Fourier transform (namely, $f(-\overline{{\it\omega}})=\overline{f({\it\omega})}$ for a transform $f({\it\omega})$ with ${\it\omega}\in \mathbb{C}$ ) that the frequency-domain displacements also have a pole at ${\it\omega}=-{\it\omega}_{n}-\text{i}{\it\delta}_{n}$ for each $n$ , so that

The contribution of each pair of poles to the behaviour of $X_{j}(t)$ is

(see McIver & McIver Reference McIver and McIver2011, § 5) and, herein, such a motion will be termed a transient. Each transient is a decaying oscillation with an amplitude of twice the modulus of the residue at the relevant pole. In the following, asymptotic methods valid for large $n$ are used to approximate each ${\it\omega}_{n}$ , ${\it\delta}_{n}$ and $x_{jn}$ using known high-frequency results for a half-immersed circular cylinder. In fact, later results are given mostly in terms of the complex wavenumber ${\it\kappa}={\it\kappa}_{n}+\text{i}{\it\tau}_{n}\equiv ({\it\omega}_{n}-\text{i}{\it\delta}_{n})^{2}a/g$ , so that ${\it\kappa}_{n}=({\it\omega}_{n}^{2}-{\it\delta}_{n}^{2})a/g$ and ${\it\tau}_{n}=-2{\it\omega}_{n}{\it\delta}_{n}a/g$ .

Numerical results for the magnitudes of the frequency-domain displacements, $|x_{1}|$ and $|x_{3}|$ , as functions of ${\it\kappa}\in \mathbb{R}^{+}$ are shown in figure 2 (the required hydrodynamic coefficients were computed using the multipole method described in Porter (Reference Porter2008)). Both displacements show a prominent resonance just below ${\it\kappa}=1$ (indexed in this paper by $n=0$ ) that is a perturbation of that found near ${\it\kappa}=0.8$ for the cylinder when in open water (Maskell & Ursell Reference Maskell and Ursell1970). There are also resonances at higher frequencies that are associated with near-standing waves between the cylinder and the wall. For the particular geometry corresponding to figure 2, these resonances occur near ${\it\kappa}=n{\rm\pi}$ , $n\in \mathbb{N}$ , and are much less prominent than the $n=0$ resonance. More detail of the first standing-wave resonance, just above ${\it\kappa}={\rm\pi}$ , is shown in the inset to figure 2. One feature of the inset is that, in the case of heave, the perturbation to the curve caused by the resonance is smaller in magnitude than nearby values of the heave displacement itself, which is in agreement with the asymptotic results obtained later in this paper.

Figure 2.

The modulus of the scaled frequency-domain displacements $|x_{j}|/(X_{3}(0)\sqrt{a/g})$ versus  ${\it\kappa}$ for a half-immersed cylinder of radius $a$ at a distance $2a$ from a rigid vertical wall. The solid and dashed lines show, respectively, the sway ( $j=1$ ) and heave ( $j=3$ ) displacements, and the inset shows the resonance near ${\it\kappa}=3.3$ in more detail.

Figure 3.

The non-dimensional added masses ${\it\mu}_{jj}^{w}/M$ versus ${\it\kappa}$ , with $j\in \{1,3\}$ , for a half-immersed cylinder of radius $a$ at a distance $2a$ from a rigid vertical wall. The solid and dashed lines show, respectively, the sway ( $j=1$ ) and heave ( $j=3$ ) values.

There are also resonances in the complex-force coefficients, and it is of particular note that the standing-wave resonances are much more prominent than those in the displacement; one such resonance is shown in figure 3 for the sway and heave added-mass coefficients ${\it\mu}_{11}^{w}$ and ${\it\mu}_{33}^{w}$ , which may be compared with the corresponding features in figure 2. The literature contains many reported calculations of standing-wave resonances in hydrodynamic coefficients for surface-piercing structures (e.g. Newman Reference Newman1977; Linton & Evans Reference Linton and Evans1992; Yeung & Seah Reference Yeung and Seah2007), and it has been found that, as frequency increases, the resonance peaks in the added-mass and damping coefficients become higher and narrower. It should be noted that, in general, the locations of the resonances of complex-force coefficients $q_{jk}^{w}({\it\omega})$ do not coincide with those of the displacements (McIver Reference McIver2005), and a consequence of this is that highly detailed computation of the hydrodynamic coefficients around a resonance may be unnecessary when performing computations of the motion of a freely floating structure. Rather, attention should be paid to the resonances in the displacement which occur at nearby, but distinct, complex frequencies (Lewandowski Reference Lewandowski2008), although it should be noted that the differences in the locations can be quite small.

The main aim now is to use the extended wide-spacing approximations given in § 2.2 to determine approximations to the resonances. Substitution of the approximations (2.12)–(2.14) into (3.2) yields

where

and

while substitution of (2.12)–(2.14) into (3.3) gives

the second form for $x_{3}$ will be useful later.

3.2 Location of the resonances

The results of the extended wide-spacing approximation are truncated expansions in powers of the small parameter ${\it\epsilon}=1/(2Kb)$ (McIver Reference McIver2014), and each expansion coefficient depends upon the non-dimensional wavenumber ${\it\kappa}=Ka$ . In the following, each coefficient in the wide-spacing expansion is approximated using asymptotic results for the isolated cylinder valid as ${\it\kappa}\rightarrow \infty$ . For the leading term in the wide-spacing approximation a number of terms in the high-frequency expansions will be used in the development of approximations to the resonances. In addition, to give an indication of the significance of evanescent modes, the leading-order effect as ${\it\kappa}\rightarrow \infty$ on the resonances will be retained for those terms identified in (2.10) and (2.11) by the explicit appearance of ${\it\epsilon}$ . The retention of terms in the high-frequency expansions at the expense of terms in the wide-spacing expansion implicitly assumes that ${\it\epsilon}\ll {\it\kappa}^{-1}$ , that is $b\gg a$ , which is consistent with one of the wide-spacing assumptions. This section begins by reviewing the required high-frequency results for the half-immersed circular cylinder when in isolation.

The reflection coefficient

where $R_{32}$ is a real number whose value is unknown. This expression up to the term in $1/{\it\kappa}$ was obtained by Leppington (Reference Leppington1973), while the term in $1/{\it\kappa}^{2}$ is given by Robertson (Reference Robertson1986, table 1). Given the expansion up to $1/{\it\kappa}^{2}$ , the real part of the term in $1/{\it\kappa}^{3}$ is obtained by ensuring that the energy-flux identity

is consistent with the result

also given by Robertson (Reference Robertson1986), although the leading term was first obtained by Ursell (Reference Ursell1961). (It is not done in Robertson (Reference Robertson1986), and hence not in (3.18) either, but generally in this paper the order notation $O\left(1/{\it\kappa}^{n}\right)$ will be used here to include possible terms of the form $(\log {\it\kappa})^{m}/{\it\kappa}^{n}$ for integers $m\geqslant 0$ .) For the radiation problems, as ${\it\kappa}\rightarrow \infty$

The expressions for $A_{1}$ , $A_{3}$ , ${\it\mu}_{33}$ and the leading-order term for ${\it\mu}_{11}$ are given by Simon (Reference Simon1985). The numerical coefficient in the expansion of ${\it\mu}_{11}$ was calculated by Greenhow (Reference Greenhow1986, equation (27)), while the expressions for the damping coefficients ${\it\nu}_{11}$ and ${\it\nu}_{33}$ are obtained from Simon’s results with the help of a known relation between damping coefficients and radiated wave amplitudes (Linton & McIver Reference Linton and McIver2001, equation (1.52)).

The resonances of the displacements given in (3.12) and (3.15) occur at the solutions for ${\it\kappa}$ of

where $n\in \mathbb{N}$ is chosen, $S=G_{2}/G_{1}\hat{R}$ and

From this form, and the asymptotics in (3.19)–(3.22), it may be seen that at high frequencies the sway characteristics of the cylinder have a stronger influence on the locations of the resonances than the heave characteristics. This is physically reasonable because, as a result of the vertical free-surface intersections, a surging cylinder generates larger waves at high frequencies than a cylinder heaving with the same amplitude, and hence the surging cylinder interacts more strongly with the wall.

Suppose that a resonance, indexed by the integer $n$ , occurs at ${\it\kappa}={\it\kappa}_{n}+\text{i}{\it\tau}_{n}$ , where ${\it\kappa}_{n},{\it\tau}_{n}\in \mathbb{R}$ and $|{\it\tau}_{n}|\ll |{\it\kappa}_{n}|$ . The modified reflection coefficient $\hat{R}$ may be extended into the complex ${\it\kappa}$ plane within a neighbourhood of ${\it\kappa}_{n}$ (McIver Reference McIver2005), and then by Taylor series expansion

where, by a property of functions of a complex variable, the derivative $\hat{R}^{\prime }$ may be taken along the real axis (this idea is also used in Meylan & Tomic (Reference Meylan and Tomic2012, § 3) for the calculation of resonances). A similar expression holds for $S$ and hence, from (3.23),

where $\hat{R}$ , $S$ and their derivatives are now evaluated at ${\it\kappa}={\it\kappa}_{n}$ .

The real part of (3.26) yields

From (3.17) and (3.18), as ${\it\kappa}\rightarrow \infty$

and from results in appendix B

so that

The imaginary part of (3.26) yields

From (3.16), as ${\it\kappa}\rightarrow \infty$

and from results in appendix B

Hence

and reversion gives

The first approximation from (3.37) in (3.31) yields

To the order given above, the above approximations to ${\it\kappa}_{n}$ and ${\it\tau}_{n}$ contain no contributions from the evanescent modes that are included in the extended wide-spacing approximation, so that, to the order given, such modes have no influence on the locations of the displacement resonances.

The resonances of the hydrodynamic coefficients in (2.12)–(2.14) occur at the solutions for ${\it\kappa}$ of

and hence may be recovered by setting $S=0$ in the above calculations. In a similar fashion to the above, suppose that a resonance, indexed by the integer $n$ , occurs at ${\it\kappa}={\it\kappa}_{q,n}+\text{i}{\it\tau}_{q,n}$ , where ${\it\kappa}_{q,n},{\it\tau}_{q,n}\in \mathbb{R}$ and $|{\it\tau}_{q,n}|\ll |{\it\kappa}_{q,n}|$ . Equation (3.27) with $S=0$ yields

so that for each high-frequency resonance $|{\it\tau}_{q,n}|\ll |{\it\tau}_{n}|$ . The term in $a^{2}/b^{2}$ in the expression for ${\it\tau}_{q,n}$ arises from evanescent terms in the extended wide-spacing approximation. Equation (3.32) with $S=0$ gives

and then the first approximation from (3.41) in (3.40) yields

The results in (3.37) and (3.41) reveal a positive shift in the real part of a displacement resonance frequency relative to the corresponding value for the hydrodynamic coefficients (such shifts in a resonance are discussed in McIver (Reference McIver2005)).

Table 1.

Resonances: comparison of the asymptotic formulae (3.37) and (3.41) with numerical calculations for $b/a=2$ .

From the leading-order approximations ${\it\kappa}_{n}\approx {\it\kappa}_{q,n}\approx n{\rm\pi}/(b/a-1)$ it is apparent that the validity of this procedure depends upon the spacing $b/a$ as well as the mode number  $n$ . As $b/a$ increases, the real parts of the resonance locations decrease, and hence these high-frequency approximations cannot be expected to work well for small $n$ when $b/a$ is large. On the other hand, when $b/a$ takes a modest value, say less than approximately two, all standing-wave resonances can be thought of as lying in the high-frequency regime. Although the validity of the wide-spacing approximation might then be questioned, there is much published work to suggest that it performs well even when the basic assumptions are violated. This view is supported by the results in table 1 which, for $b/a=2$ , compare the asymptotic approximations to the displacement resonances given in (3.37) and (3.38), and the real parts of the resonances in the hydrodynamic coefficients (3.41) (the imaginary parts are very small), with numerical estimates obtained using the method described in Fitzgerald & McIver (Reference Fitzgerald and McIver2009). The lowest resonances ( $n=0$ ) are not revealed by the asymptotic method, as the above procedure relies on the poles being both close to the real frequency axis and within the high-frequency regime. The distance from the real axis also makes an $n=0$ resonance more difficult to locate accurately from the numerical results. However, for the geometry chosen here, for $n\geqslant 1$ the accuracy of the approximations is good even for small $n$ .

3.3 Residues

The residues at simple poles of $x_{j}$ in the complex plane are estimated as follows. Suppose that

where ${\it\kappa}({\it\omega})={\it\omega}^{2}a/g$ , and $h({\it\kappa})$ has a simple zero at ${\it\kappa}=\widetilde{{\it\kappa}}\equiv \widetilde{{\it\omega}}^{2}a/g$ so that $h(\widetilde{{\it\kappa}})=0$ , $h^{\prime }(\widetilde{{\it\kappa}})\neq 0$ and $f_{j}(\widetilde{{\it\kappa}})\neq 0$ (here $\widetilde{{\it\omega}}$ is complex and has positive real part – see § 3.1). When $x_{j}$ is regarded as a function of ${\it\kappa}$ , so that

then the residue

When $x_{j}$ is regarded as a function of ${\it\omega}$ , so that (3.9) holds, then the residue

where the positive real number ${\it\omega}_{n}=\sqrt{g{\it\kappa}_{n}/a}$ .

With

then, as ${\it\kappa}\rightarrow \infty$ ,

so that

As ${\it\kappa}\rightarrow \infty$

so that, to leading order in ${\it\kappa}$ , the contributions from the evanescent terms (identified by $a^{2}/b^{2}$ ) to $\hat{A}_{3}$ and $G_{1}$ cancel. From (3.12), for the horizontal displacement

so that from (3.45) and (3.46) the residue

From the second form for the vertical displacement in (3.15), the term that is singular at resonance gives

and hence the residue

so that the amplitude of a vertical transient is asymptotically an order of magnitude smaller than the corresponding horizontal transient, although both decay very rapidly with increasing $n$ .

For real ${\it\kappa}$ near a resonance at ${\it\kappa}_{n}+\text{i}{\it\tau}_{n}$ , the magnitude of the contribution to $x_{j}$ , when regarded as a function of ${\it\kappa}$ , from that resonance is $|x_{jn}^{({\it\kappa})}|/\sqrt{({\it\kappa}-{\it\kappa}_{n})^{2}+{\it\tau}_{n}^{2}}$ . This is greatest at ${\it\kappa}={\it\kappa}_{n}$ , and as $n\rightarrow \infty$ the maximum value is

with ${\it\kappa}_{n}\sim n{\rm\pi}/(b/a-1)$ . The powers of ${\it\kappa}_{n}$ in these expressions indicate that the contribution to the displacement from a particular resonance is smaller for heave, which confirms the behaviour seen in the numerical results given in the inset to figure 2.

3.4 High-frequency asymptotics away from resonance

Away from the resonances, so that $\text{e}^{-2\text{i}{\it\kappa}b/a}-R=\text{ord}\left(1\right)$ as ${\it\kappa}\rightarrow \infty$ through suitable values ( $\text{ord}\left(1\right)$ denotes ‘strictly order one’ (Hinch Reference Hinch1991, p. 6)), the leading-order approximation to the denominator in (3.12) arises from the term in $G_{1}$ to give

so that, to leading order, and unlike the imaginary part, $\text{Re}\,x_{1}({\it\omega})$ decays monotonically with increasing frequency. Similarly, from (3.15),

(the asymptotics of $\hat{q}_{33}$ are not known in sufficient detail to allow the imaginary part of the second term on the left-hand side to be expanded to the same order as the term on the right – more information about the real part is given below). These results confirm one of the features shown in figure 2, namely that the general trend with increasing frequency is for the sway displacement to decrease more rapidly than the heave displacement. As

the residues in (3.52) and (3.54) are recovered once account has been taken of (3.46). The above expressions confirm that the imaginary parts of the frequency-domain displacements are most strongly influenced by the resonances.

The imaginary part of the vertical displacement decays particularly slowly with increasing frequency as

(by a result on Fourier transforms (Bleistein & Handelsman Reference Bleistein and Handelsman1986, p. 80), (3.59) is effectively a restatement of the initial condition on $X_{3}(t)$ ). Further expansion in (3.57) yields

which decays with increasing frequency much more quickly than $\text{Im}\,x_{3}({\it\omega})$ . While (3.59) holds for a cylinder in open water, the limit $b/a\rightarrow \infty$ in (3.60) does not recover the open-water result for the real part, as interactions with the wall through propagating waves are present for any $b/a>1$ ; in fact, for open water,

3.5 Computation of the displacements

The time-domain displacements $X_{j}(t)$ are recovered from the frequency-domain displacements $x_{j}({\it\omega})$ through the inverse Fourier transform (3.1). In § 3.1 it is noted that $x_{1}({\it\omega})$ has a simple pole at the origin, leading to a non-zero large-time displacement $X_{1}^{\infty }$ given by (3.7). It is also possible that, for the chosen value of $a/b$ , there may be simple poles at ${\it\omega}={\it\omega}_{0}$ with ${\it\omega}_{0}\in \mathbb{R}^{+}$ corresponding to the existence of a trapped mode (equations (3.9) and (3.10) with $n=0$ and ${\it\delta}_{0}=0$ give the asymptotic behaviours at such poles); the existence of trapped modes for $a/b=0.22504,0.60333$ is established numerically in Porter & Evans (Reference Porter and Evans2009). Frequency-domain quantities with these poles extracted are

where, to unify the notation, $X_{3}^{\infty }=0$ . As each $y_{j}({\it\omega})$ is now free of singularities in $\text{Im}\,{\it\omega}\geqslant 0$ , it follows from (3.1) that

where the integration is now taken along the real ${\it\omega}$ axis and the pole contributions are obtained by closing the contour in the lower half-plane. Further, as noted by Meylan (Reference Meylan2014, § 3) (see also McIver Reference McIver2012, appendix A), by virtue of the analytic properties of $y_{j}({\it\omega})$ in the upper half-plane, the inverse Fourier transform of $y_{j}({\it\omega})$ may be rewritten either in terms of the cosine transform of the real part of $y_{j}({\it\omega})$ or the sine transform of the imaginary part of $y_{j}({\it\omega})$ . From the high-frequency results in (3.56) and (3.59)–(3.60), it is apparent that the real parts of the frequency-domain displacements are better suited to computations, and hence it is appropriate to use

For $x_{j0}\neq 0$ , the initial conditions determine each $x_{j0}$ . As both $\text{Re}\{y_{j}({\it\omega})\}$ and ${\it\omega}\,\text{Re}\{y_{j}({\it\omega})\}$ are absolutely integrable over $(0,\infty )$ , the limit $t\rightarrow 0$ may be taken under the integral sign in both the integral term in $X_{j}(t)$ and its first derivative with respect to $t$ . The velocity conditions ${\dot{X}}_{1}(0)={\dot{X}}_{3}(0)=0$ yield that the real part of each $x_{j0}$ is zero so that, in fact, $\text{Re}\{y_{j}({\it\omega})\}=\text{Re}\{x_{j}({\it\omega})\}$ . The conditions on the initial displacements then give

which determines the amplitude of the trapped mode, and hence

In the absence of a trapped mode no special treatment of the initial condition is needed.

Figure 4.

The scaled displacements $X_{j}/X_{3}(0)$ versus scaled time $t/\sqrt{a/g}$ , with $j\in \{1,3\}$ , for a half-immersed cylinder of radius $a$ at a distance $2a$ from a rigid vertical wall. The solid and dashed lines show, respectively, the horizontal ( $j=1$ ) and vertical ( $j=3$ ) displacements, and the straight lines indicate the corresponding large-time asymptotic values of these displacements. The dash-dot line is the vertical displacement for a cylinder in open water.

Numerical results for $b/a=2$ are shown in figure 4 (there is no trapped mode for this geometry and hence $x_{j0}=0$ in (3.66)). After a time equivalent to approximately two periods of the $n=0$ transient, the vertical motion settles to a slowly decaying oscillation about the mean water level, and the horizontal motion to a slowly decaying oscillation about the asymptotic value $X_{1}=X_{1}^{\infty }\approx -0.153X_{3}(0)$ . For the computations shown in figure 4, the Fourier integral was written in terms of the non-dimensional frequency $\sqrt{{\it\kappa}}$ and the range split at $\sqrt{{\it\kappa}}=2$ . Numerical integration was used for $\sqrt{{\it\kappa}}\in (0,2)$ , corresponding to the range of ${\it\kappa}$ displayed in figure 2, and explicit integration of the asymptotic forms given in § 3.4 was used for $\sqrt{{\it\kappa}}>2$ . A consequence of this is that only the first two resonances are accounted for in the calculations. However, on the scales used for the figure, only the slowly decaying oscillation arising from the lowest resonance ( $n=0$ ) is apparent; the higher-frequency oscillation due to the $n=1$ resonance is not discernible. From (3.11) and (3.52), the amplitude of the $n=1$ transient

for $b/a=2$ , which confirms the numerical observation that it does not contribute significantly to the motion (the amplitudes of other transients are even smaller). An indication of the accuracy of the results is obtained from the observation that the absolute errors in the initial displacements are less than $5\times 10^{-4}$ . For comparison, the motion of a cylinder in open water when displaced vertically a distance $X_{3}(0)$ and released from rest is also shown in figure 4. In this case, the sole transient is given in Maskell & Ursell (Reference Maskell and Ursell1970, equation (5.2)) and is due to resonances at ${\it\kappa}\approx \pm 0.8141-0.2387\text{i}$ (the approximate method in Fitzgerald & McIver (Reference Fitzgerald and McIver2009) yields ${\it\kappa}\approx \pm 0.82-0.29\text{i}$ ). For the cylinder in the presence of a wall, the method of Fitzgerald & McIver (Reference Fitzgerald and McIver2009) gives that the lowest-frequency resonances are at ${\it\kappa}\approx \pm 0.96-0.01\text{i}$ , which confirms the much slower rate of decay seen in figure 4. Computation of a trapped-mode case is discussed in the following section.