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SHALLOW-WATER THEORY
Waves and Forces Caused by Oscillation of a Floating Body
Determined through a Unified Nonlinear Shallow-Water Theory
Paper for Presentation at the Twenty-Third Symposium on Naval Hydrodynamics Val de Reuil, France, September 17–
22, 2000
Rupert Henn, Tao Jiang & Som Deo Sharma
(Institute of Ship Technology, Mercator University D-47048 Duisburg, Germany)
ABSTRACT
Based on a unified nonlinear shallow-water theory, waves and forces caused by vertical oscillations of floating
bodies in shallow water were numerically investigated. Whereas the classical set of Boussinesq's equations for the outer-
field flow vertically under the free surface was solved in a standard grid based on a complete Crank-Nicolson scheme, the
new set of Boussinesq-type equations for the inner-field flow vertically under the body surface was solved in a staggered
grid using partly a Crank-Nicolson scheme and partly a fully implicit scheme. For small oscillation amplitudes of the
body, a good agreement between the unified theory and the linear wave theory was achieved for amplitudes of the wave
elevation as well as for the wave force. However, there was a phase shift in the wave force. As the oscillation amplitude
increased, the unified theory showed strong nonlinear effects on the wave elevation and particularly on the wave forces.
INTRODUCTION
In the last two decades a considerable amount of research effort has been devoted to the application of Boussinesq
type equations to various shallow-water wave-related problems.
In simulation of wave propagation from deep water to shallow water, a major achievement was the extension of the
applicability of Boussinesq's equations to short waves in moderate water-depth. There exist two principal ways to obtain
this kind of modified Boussinesq's wave-models. One possibility is to use a modified depth-averaged horizontal velocity
in the classical formulation, see Witting (1984), or to add high-order terms to the classical formulation, see Madsen et al.
(1991). The parameter occurring in these treatments can be found by applying a Padé expansion for the associated
dispersion relation of modified Boussinesq's equations and of the linear wave theory. The other, a more rational, way is to
use a horizontal velocity at an arbitrary vertical level in deriving Boussinesq-type equations, see Nwogu (1993) and
Schröter (1995). The parameter required for defining the vertical level can again be specified by comparing the dispersion
relations between the resulting linearized equations and the linear wave theory. By suitable selection of the associated
parameter values the modified Boussinesq's equations are valid for a ratio of wave length to water-depth down to a value
of 2, practically to the deep-water region.
In computation of waves generated by ships, a significant contribution was the inclusion of the ship's influence on
the ambient flow. As reported by Jiang (2000b), three different approximations were applied to dealing with this ship-
generated nearfield flow. For a slender ship, the technique of matched asymptotic expansions was applied in the works of
Pedersen (1988) and Jiang (1998). The so-called technique of matched asymptotic expansions was first introduced by
Tuck (1966) in a linearized version and then extended to forced KdV equation by Mei (1986) and to a KP equation by
Mei & Choi (1987). This nonlinear version was refined by Chen & Sharma (1994) by using a modified KP equation and a
improved slender-body theory. For a wall-sided ship, a method based on the mass conservation law was derived by
Ertekin et al. (1997). For a flat ship, a pressure distribution proportional to the local ship-draft was applied by Jiang and
Sharma (1998). Due to the possibility of the direct implementation of a pressure distribution on the free surface in
Boussinesq's equations, wave generation by analytical pressure distributions was frequently studied by many authors, see
e.g. Wu & Wu (1982), Ertekin et al. (1986), and Pedersen (1988).
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SHALLOW-WATER THEORY
To remove the above restrictions on the hull form and to extend the applicability of shallow-water equations of
Boussinesq type for wave-body interactions, a unified shallow-water theory was recently derived by Jiang (2000a), which
comprises a classical set of Boussinesq's equations for the outer-field flow vertically under the free surface and a new set
of nonlinear partial-differential equations of Boussinesq type for the inner-field flow vertically under the wetted body-
surface. In comparison to the simplified approximations mentioned above, this new set of Boussinesq-type equations for
the inner-field flow satisfies the tangential-flow condition on the wetted body-surface and enables the direct calculation of
the pressure distribution on it. Thus, this unified theory yields a theoretically consistent approximation of many wave-
related three-dimensional problems in the horizontal two-dimensional plane.
In the present paper we apply this unified nonlinear theory to investigate waves and forces caused by a surface-
piercing body oscillating in shallow water. After reporting the main derivation procedure of the theory, a suitable
numerical method will be implemented for solving the initial/boundary value problems governed by the two sets of
nonlinear partial-differential equations. To meet the numerical difficulties arising from the nonlinear terms and the linear
high-order terms, we apply again an implicit Crank-Nicolson scheme in the temporal and spatial discretizations. To
overcome the difficulty caused by the decoupling of the pressure on the wetted body-surface from the depth-averaged
continuity equation, the well-proven concept of a staggered grid will be implemented in our computer code. To store the
resulting diagonal sparse matrix, a compressed diagonal storage format will be used. The corresponding large
nonsymmeric linear equation system will then be solved by means of the GMRES method given by Saad & Schultz
(1986). To verify this unified theory and to examine our numerical method, we investigate the wave generation of a
vertically oscillating circular cylinder in shallow water. For small amplitudes of body motion, the calculated wave profiles
and wave forces will be compared with those from the linear wave theory. The nonlinear effects occurring at larger
motion amplitude will also be discussed.
THEORETICAL FORMULATION
General Description
Considering wave generation by a surface-piercing body oscillating in shallow water of varying water depth h′(x′, y′)
and using prime ′ as superscript to denote all dimensional quantities, a Cartesian coordinate system Ox′y′z′ is used to
describe the velocity and pressure field, where the plane Ox′y′ is on the quiet free surface with the axis z′ positive upward.
Assume that the fluid is incompressible and inviscid, the velocity components u′, v′, w′ and pressure field p′ can be
described by the continuity equation
(1)
and Euler's equations
(2)
(3)
(4)
in the whole fluid domain. Herein, y′ denotes the acceleration due to gravity, p′ the water density, and is the
independent time variable.
The solution of the above governing equations is determined by the following boundary conditions:
• kinematic condition on the free surface
(5)
where the subscript f stands for the free surface, assumed to be differentiable in time and space,
• dynamic condition on the free surface
(6)
with the atmospheric pressure on the free surface,
• tangential-flow condition on the water bottom surface
(7)
where the subscript b stands for the bottom surface, assumed to be differentiable in space,
• tangential-flow condition on the wetted body surface
(8)
where the subscript s stands for the body (ship) surface, assumed to be differentiable in time and space,
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SHALLOW-WATER THEORY
and appropriate initial conditions.
Using the two standard nondimensional parameters in the shallow-water approximation
with wave amplitude wave length λ′ or wave number k′, and the mean water-depth the physical variables are
nondimensionally scaled as follows, now denoted without the primes ′ for distinction:
(x, y)=k′(x′, y′)
horizontal coordinates
vertical coordinate
water depth
independent time-variable
horizontal velocity components
vertical velocity component
wave elevation
local ship draft
pressure
This multiple scaling leads to a nondimensional formulation of the governing equations:
(9)
(10)
(11)
(12)
and of the corresponding boundary conditions:
• kinematic condition on the free surface
(13)
• dynamic condition on the free surface
(14)
• tangential-flow condition on the water bottom surface
(15)
• tangential-flow condition on the body surface
(16)
Approximation of the Outer-Field Flow
To describe the flow in the outer field vertically under the free surface, the continuity equation (9) is integrated from
the water bottom h(x, y) to the free surface For a single-valued function of ζ(x, y, t) and h(x, y) at each
horizontal point (x, y) it yields:
(17)
By inserting the kinematic condition on the free surface (13) and the tangential-flow condition (15) on the water
bottom into equation (17), a vertically integrated continuity equation can be derived as
(18)
with the depth-averaged horizontal velocity components
(19)
and the two-dimensional nabla-operator It is worthwhile to mention that this vertically integrated
continuity equation in the outer field satisfies the continuity equation (9), the kinematic condition (13) on the free surface,
and the tangential-flow condition (15) on the water bottom.
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By virtue of Kelvin's Theorem for inviscid fluids, wave generation can be considered to be an irrotational flow.
Following a standard derivation procedure and exploiting the absence of horizontal vorticity-components, see e.g. Jiang
(2000a), the outer velocity-field can be represented by the depth-averaged horizontal velocity ū:
(20)
(21)

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SHALLOW-WATER THEORY
To approximate the outer pressure-field the vertical momentum equation (12) is integrated over a local
submerged depth . After a rearrangement the outer pressure-field can be then represented by the depth-
averaged horizontal velocity ū and the wave elevation ζ:
(22) (23)
The relation between the depth-averaged horizontal velocity ū(x, y, t) and the wave elevation ζ(x, y, t) can be
obtained by integrating the horizontal momentum equations (10) and (11) over the water depth
Approximation of the Inner-Field Flow
To describe the flow in the inner field vertically under the wetted body surface, the continuity equation (9) is now
integrated from the water bottom h(x, y) to the body surface T(x, y, t). For a single-valued function for the water bottom h
(x, y) and for the local body draft T(x, y, t) at each horizontal point (x, y) it yields:
(24)
Inserting equations (15) and (16) into equation (24) yields the following vertically integrated continuity equation:
(25)
where the depth-averaged horizontal velocity ū is defined by
(26)
Equation (25) satisfies the continuity equation (9), the tangential-flow condition of equation (15) on the water bottom
surface, and the tangential-flow condition of equation (18) on the wetted surface.
Similar to the outer velocity-field, the inner velocity-field can also be represented by the corresponding depth-
averaged horizontal velocity ū:
The inner pressure-field
(27)
(30)
(28)
(29)
can be obtained by integrating the vertical momentum equation (12) over the submerged depth where ps
denotes the pressure on the wetted body surface.
The required relation between depth-averaged horizontal velocity ū(x, y, t) and pressure ps(x, y, t) on the wetted
surface can be derived by integrating the horizontal momentum equations (10) and (11) over the water depth (h−T):
Unified Shallow-Water Theory
According to the basic Boussinesq's assumption, the nondimensional parameters and µ , being responsible for
nonlinear and dispersive characteristics, respectively, are matched as
(31)
The leading-order approximation of equations (23) and (30) together with equations (18) and (25) leads then to a
new unified nonlinear shallow-water theory, which comprises the classical Boussinesq's equations
(32)
(33)
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SHALLOW-WATER THEORY
for the outer-field flow with depth-averaged horizontal velocity ū and wave elevation ζ as unknowns, and a new set
of nonlinear partial-differential equations of Boussinesq type
(34)
(35)
for the inner-field flow with depth-averaged horizontal velocity ū and pressure ps acting on the wetted body-surface
as unknowns.
Equation (34) would satisfy the tangential-flow condition on the wetted surface exactly if the depth-averaged
velocitiy ū were exact. Since ū is approximated to the leading order of and µ 2 in the Boussinesq theory, the new set of
equations leads to a Boussinesq's approximation in the inner field. However, it is consistent with the Boussinesq's
approximation in the outer field. The resulting unified shallow-water theory thus yields a consistent approximation of an
originally three-dimensional problem in the reduced two-dimensional horizontal plane. Moreover, it enables the direct
calculation of the pressure distribution on the wetted surface for the first time using a shallow-water wave theory.
In shallow water at constant water-depth h=1, this unified theory can be simplified to:
(36) (38)
(37)
(39)
for the outer-field flow, and to:
for the inner-field flow.
Coupling Conditions
The two sets of Boussinesq type equations are coupled by the associated interfacial conditions at the instantaneous
waterline yw=f(xw, t) as an instantaneous intersection of the body surface with the free surface.
At locations without a vertical sidewall near the waterline, i.e., ζ=−T, the interfacial coupling conditions read:
(40) (41)
where the subscripts I and O denote the flow in the inner and outer field, respectively. The condition expressed by
equation (40) means a continuous change of the depth-averaged horizontal velocity at the waterline.
At locations with a vertical sidewall near the waterline, i.e., ζ>−T, the interfacial coupling conditions read:
(42)
(43)
where and denote the depth-averaged horizontal normal velocity component at the waterline in the inner
and outer field, respectively. The condition expressed by equation (42) ensures mass conservation through the waterline.
NUMERICAL IMPLEMENTATION
In the present study we apply the unified nonlinear theory to investigate the waves generated by a surface-piercing
circular cylinder oscillating vertically in shallow water of constant water-depth h=ho and the associated forces. For a
problem having axial symmetry about the vertical axis z, the dimensional equations, now omitting the primes ′ for
simplicity, in a cylindrical coordinate system corresponding to equations (36–39) reduce to:
(44)
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(45)
for the outer-field flow r>R and to
(46)

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SHALLOW-WATER THEORY
for the inner-field flow r

WAVES AND FORCES CAUSED BY OSCILLATION OF A FLOATING BODY DETERMINED THROUGH A UNIFIED NONLINEAR 999
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SHALLOW-WATER THEORY
(58)
(57)
at i=1, 2, 3, · · ·, iR−1. It is worth mentioning that the terms associated with c are discretized using the Crank-
Nicolson scheme and the pressure term using a fully implicit scheme. Numerical test runs have shown that this treatment
suppressed the pressure oscillations caused by a complete Crank-Nicolson scheme. In the latter case an underrelaxation
procedure with a factor of approximately 0.6 had to be additionally performed.
Scheme for Coupling Conditions
At the intersection i=iR, the vertically integrated outer continuity equation (44) is one-sided discretized as:
Instead of using the vertically integrated outer momentum equation (45) at iR, the coupling equation (48) is
discretized as follows:
(59)
Since the depth-averaged continuity equation (45) can be directly integrated for the given boundary value c(0, t)=0,
the value can be determined either by the analytical solution
(60)
or by the numerical integration of equation (56) in advance.
The required value of the pressure at iR can be obtained from the coupling condition (49). Since the vertically
integrated outer momentum equation (45) is not used at the grid point iR, the high-order term
in equation (50) is neglected in the present study. Consequently, the pressure at iR is
approximated by
(61)
Solution Method
The above discretization scheme leads to a linear algebraic equation system. The corresponding system matrix is a
sparse matrix where only diagonal blocks have non-zero elements (so-called diagonal with fringes). For economy of
storage, a compressed diagonal storage format is used. Since the pressure ps appears only as a spatial derivative term in
the vertically integrated inner momentum equation, the system matrix thus has zero elements on the main diagonal. This
implies that classical iteration methods, like Jacobi or Gauss-Seidel, cannot be applied. Although the resulting linear
equation system can be directly solved by using elimination methods, such methods generally need a full storage of the
system matrix and consume a large computing time. Therefore, the more advanced GMRES (Generalized Minimal
Residual) method given by Saad & Schultz (1986) is implemented to solve the resulting nonsymmetric linear equation
system using a compressed diagonal storage format. To accelerate the convergence process, a restarted version of
GMRES is applied.
RESULTS AND DISCUSSION
In the present study we apply this unified theory to simulate the waves generated by two vertically oscillating bodies.
One corresponds to the horizontal semi-submerged 2-D circular cylinder with a raduis R=0.5 m, as investigated by Yu &
Ursell (1961) and by Keil (1974). The draft at rest reads:
(62)
The second is a vertical circular cylinder with a radius R=2.3 m and with a constant draft at rest
TS=0.4 m. These two bodies, both having a wall-sided freeboard over the still waterline, are forced to oscillate
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harmonically in shallow water of depth h=1 m. The resulting instantaneous local draft under the stillwater level is then:
(63)
where ω is the forcing frequency and a the forcing amplitude.

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SHALLOW-WATER THEORY
Note that for the vertical circular cylinder the numerical implementation formulated in a cylindrical coordinate
system can be directly applied. For the horizontal circular cylinder a numerical implementation corresponding to the
governing equations in a Cartesian coordinate system on the vertical plane can be performed. The latter case was
completely documented by Jiang (2000b).
To examine the numerical method proposed, we study first the vertical circular cylinder oscillating in shallow water
at a forcing period Fig. 1 compares the calculated depth-averaged radial velocity with the analytical solution
given by equation (60). There is no visible difference in these two predictions. Furthermore, since the bottom of the
subject vertical cylinder is flat, the dynamic pressure can be analytically described by
(64)
where ζ takes the same value both for the analytical solution and for the numerical approximation.
Fig. 2 shows the comparison of the numerically calculated pressure with the analytical solution. Again, there is no
visible difference in these two predictions.
After verifying the numerical method, we apply our computer code to compute the vertical cylinder oscillating with
a small amplitude a=0.001 m in a range of wavenumbers kho<2. In can be seen in Fig. 3 that within the range of validity
of the classical Boussinesq's equations the amplitude ratio of wave forces determined through the unified nonlinear theory
agrees well with that from WAMIT (1995) based on the linear wave theory. (The WAMIT forces were first computed in
the frequency domain in terms of added mass and damping and then synthesized in the time domain.) As expected, the
longer the waves are, the better the agreement. Looking now at the time histories in Fig. 4, we can observe that there is a
phase shift between the two different calculations. There may be two possible reasons: (i) The two calculations used
different initial conditions. WAMIT yields only the asymptotic solution and our computation assumed the motion started
from rest. (ii) The coupling conditions have not yet been implemented perfectly or they may be physically inexact. We
plan to compare further with the analytical solution derived by Yeung (1981) and with physical-model tests to clarify this
phase shift.
For a small oscillating amplitude a=0.001 m, Fig. 5 compares the calculated amplitude ratio of the wave elevation to
the body motion with those given by Yu & Ursell (1961) (solid line) and Keil (1974) (dashed line), both based on the
linear wave theory for a horizontal circular cylinder. Since the results cited agree well with those from experiments, the
good agreement between our nonlinear theory and the linear wave theory, at least within the expected range of validity of
Boussinesq's equations, namely, wavenumbers kho<1.5, can be interpreted as a practical validation.
Turning now to the specific advantage of the unified nonlinear shallow-water theory, namely, its capability to
account for nonlinear effects of shallow-water waves, Fig. 6 shows instantaneous radial wave profiles normalized by the
amplitude of motion of the vertical cylinder for systematically increasing amplitudes a=0.001 m (dotted line), a=0.01 m
(dashed line), and a=0.1 m (solid line). As expected, the wave elevation decays with increasing distance from the body
owing to dispersion in the unrestricted horizontal domain. Hence, nonlinear effects are more visible near the body. They
are better seen in the response forces since the dynamic pressure is proportional to the wave elevation directly on the body
as described by the coupling equation (50). Fig. 7 compares the three time histories of the response force also normalized
by the amplitude of motion. Furthermore, as no difference in the normalized wave elevation or response force can be
noticed for oscillation amplitudes smaller than 0.01 m, linear theory may be safely used up to a<0.01 m in the present
configuration.
CONCLUSIONS
A unified nonlinear shallow-water theory comprising two sets of Boussinesq type equations was introduced to
determine waves and forces caused by oscillations of a floating body in shallow-water. A numerical method based on the
Crank-Nicolson scheme combined with a fully implicit scheme was successfully implemented to solve the governing
equations in a staggered grid. For small motion amplitudes a good agreement between the unified nonlinear theory and
linear wave theory was observed for wave elevation and response force within the range of validity of the classical
Boussinesq's equations. However, the present unified theory did display nonlinear effects for larger motion amplitudes.
More fundamentally, we can now confidently apply the new unified nonlinear shallow-water theory to simulate wave-
body interactions.
REFERENCES
Chen, X.-N. & Sharma, S.D. 1994: Nonlinear theory of asymetric motion of a slender ship in a shallow channel. Proc. of the 20th Symp. on Naval
Hydrodynamics, CA, USA, pp. 386–407.
Ertekin, R.C., Qian, Z.M. &Wehausen, J.V. 1997: Upstream soliton generation by a slender, vertical strut and ship: Boussinesq equations. Proc. of the
7th Int.
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SHALLOW-WATER THEORY
Offshore and Polar Engrng Conf., Honolulu, USA, Vol. III, pp. 238–246.
Ertekin, R.C., Webster, W.C. & Wehausen, J.V. 1986: Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech., Vol.
169, pp. 275–292.
Jiang, T. 2000a: A unified nonlinear theory for approximations of wave-related problems in shallow water. Submitted for publication in the Journal of
Ship Research.
Jiang, T. 1998: Investigation of waves generated by ships in shallow water. Proc. of the 22nd Symp. on Naval Hydrodynamics, Washington, D.C., USA.
Jiang, T. 2000b: Ship waves in shallow water. Habilitation Thesis, Mercator University Duisburg.
Jiang, T. & Sharma, S.D. 1998: Wavemaking of flat ships at transcritical speeds. Proc. of the 19th Duisburger Colloquium, Institute of Ship Technology,
Mercator-Univ., Duisburg, Germany, pp. 171–189 (in German).
Keil, H. 1974: The hydrodynamic forces due to the perodic motion of 2-D bodies on the free surface of shallow water. Institut fuer Schiffbau, Report
No. 305 (in German).
Madsen, P.A., Murray, R. & Sorensen, O.R. 1991: A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal
Engrng., Vol. 15, pp. 371–388.
Mei, C.C. 1986: Radiation of solitons by slender bodies advancing in a shallow channel. J. Fluid Mech., Vol. 162, pp. 53–67.
Mei, C.C. & Choi, H.S. 1987: Forces on a slender ship advancing near thecritical speed in a wide canal. J. Fluid Mech., Vol. 179, pp. 59–76.
Nwogu, O. 1993: Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coast. and Ocean Engrng., Vol. 119, pp.
618–638.
Pedersen, G. 1988: Three-dimensional wave patterns generated by moving disturbances at transcritical speeds. J. of Fluid Mech., Vol. 196, pp. 39–63.
Saad, Y., & Schultz, M.H. 1986: A generalized minimal residual algorithm of solving nonsymmetric linear system. SIAM J. Sci. Stat. Comput., Vol. 7,
No. 3.
Schröter, A. 1995: Nonlinear time-discretized seaway simulation in shallow and deeper water. Institut für Strömungsmechanik und Elektronisches
Rechnen im Bauwesen der Universität Hannover, Report No. 42 (in German).
Tuck, E.O. 1966: Shallow water flows past slender bodies. J. Fluid Mechanics, Vol. 26, pp. 81–95.
WAMIT. 1995: WAMIT user manual. Department of Ocean Engineering, MIT.
Witting, J.M. 1984: A unified model for the evolution of nonlinear water waves. J. Comp. Phys., Vol. 56, pp. 203–236.
Wu, D.-M. & Wu, T.Y. 1982: Three-dimensional nonlinear long waves due to moving surface pressure. Proc. of the 14th Symp. on the Naval
Hydrodynamics, Ann Arbor, USA, pp. 103–129.
Yu, Y.S. & Ursell, F. 1961: Surface waves generated by an oscillating circular cylinder on water of finite depth: theory and experiment. J. Fluid Mech.,
Vol. 11, pp. 529–551.
Yeung, R.W. 1981: Added mass and damping of a vertical cylinder in finite-depth waters. Applied Ocean Research, Vol. 3, No. 3, pp. 119–133.
Fig. 1 Comparison of numerical and analytical solutions for the depth-averaged radial velocity generated by a vertically
oscillating vertical circular cylinder
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SHALLOW-WATER THEORY
Fig. 2 Comparison of numerical and analytical solutions for the dynamic pressure on the subject cylinder
Fig. 3 Comparison of response-force amplitudes determined from the unified nonlinear shallow-water theory with those from
the linear wave theory for the subject cylinder
Fig. 4 Comparison of time histories of the response force determined from the unified nonlinear shallow-water theory with
that from the linear wave theory for the subject cylinder
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WAVES AND FORCES CAUSED BY OSCILLATION OF A FLOATING BODY DETERMINED THROUGH A UNIFIED NONLINEAR 1003
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SHALLOW-WATER THEORY
Fig. 5 Comparison of normalized wave-amplitudes determined from the unified nonlinear shallow-water theory with those
from the linear wave theory for a vertically oscillating horizontal circular cylinder
Fig. 6 Comparison of normalized wave-profiles generated by a vertically oscillating vertical circular cylinder with a forcing
period 2.3 s
Fig. 7 Comparison of normalized response forces acting on the subject cylinder when vertically oscillating with a forcing
period 2.3 s
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SHALLOW-WATER THEORY
DISCUSSION
H.B.Bingham
University of Denmark, Denmark
I would like to congratulate the authors on the most successful demonstration I have yet seen of the direct use of a
Boussinesq-type method to solve the unsteady ship motions problem. I have a comment and a question.
Comment: Your discussion of the disagreement between your results and WAMIT for the phase of the radiation
force (Fig. 4), suggests that there is some ambiguity about the definition of this quantity. It is quite well defined and, as
long as your computations have reached a steady-state, you will converge to the same answer if you solve the problem
correctly. You do not discuss convergence, perhaps these results have not yet converged.
Question: The two examples presented here are special geometries in the sense that your discretization fit them
exactly. Do you expect a finite-difference implementation on a uniform Cartesian grid to be able to cope with a general
ship-like geometry, or will you need to develop a boundary-fitted solution in this case?
AUTHOR'S REPLY
We greatly appreciate Dr. Bingham's discussion.
Regarding his comment, Fig. 8 shows our simulated time histories of the response force along with the
corresponding WAMIT result. Graph (a) demonstrates the convergence of our numerical result with respect to space
discretization. Graph (b) compares our asymptotic time history with that of WAMIT. Again, the agreement is remarkable
for the amplitude. Unfortunately, the phase shift between the two different calculations persists and thus can not be
explained by possible lack of convergence, neither in space nor in time discretization. Moreover, our additional
simulations show that the coupling conditions between the inner and the outer flow field have some influence on the
phase shift.
We agree that a uniform Cartesian grid will not work well for a ship-like geometry. Currently, we are implementing
our computer program in a curvilinear grid and we hope to present the new results at the next Symposium on Naval
Hydrodynamics.
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SHALLOW-WATER THEORY
Fig. 8: Time histories of the response force
WAVES AND FORCES CAUSED BY OSCILLATION OF A FLOATING BODY DETERMINED THROUGH A UNIFIED NONLINEAR
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