Forward-Speed Ship Wave Flows - Revision history

Meylan: /* Linear free surface conditions */

2011-02-23T11:28:14Z

Linear free surface conditions

← Older revision		Revision as of 11:28, 23 February 2011
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	\frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} = \frac{\partial^2 \bar{\Phi}}{\partial t^2}		\frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} = \frac{\partial^2 \bar{\Phi}}{\partial t^2}
	- 2 U \frac{\partial^2\bar{\Phi}}{\partial x \partial t}		- 2 U \frac{\partial^2\bar{\Phi}}{\partial x \partial t}
	+ U^2 \frac{\partial^ \bar{\Phi}}{\partial x^2}		+ U^2 \frac{\partial \bar{\Phi}}{\partial x^2}
	</math></center>		</math></center>

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	Hence		Hence
	<center><math> \zeta = \frac{U}{g} \frac{\partial\bar{\Phi}}{\partial x}, \qquad z=0 </math></center>		<center><math> \zeta = \frac{U}{g} \frac{\partial\bar{\Phi}}{\partial x}, \qquad z=0 </math></center>


	== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==		== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

F. Javier Arcos Zamora: /* Boundary-value problem for time-Harmonic velocity potential */

2010-09-09T16:33:35Z

Boundary-value problem for time-Harmonic velocity potential

← Older revision		Revision as of 16:33, 9 September 2010
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	Let:		Let:

	<center><math> X_0 = x+ Ut \qquad Y_0 = y \qquad ~~Z+0~~ = z \,</math></center>		<center><math> X_0 = x+ Ut \qquad Y_0 = y \qquad Z_0 = z \,</math></center>

	It follows that:		It follows that:

Meylan: /* Neumann-Kelvin linearization of U>0 ship seakeeping problem */

2010-04-06T08:50:57Z

Neumann-Kelvin linearization of U>0 ship seakeeping problem

← Older revision		Revision as of 08:50, 6 April 2010
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	{{incomplete pages}}		{{incomplete pages}}

			== Linear free surface conditions ==

			The linear free surface conditions are as follows,

			<center><math> \frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} + g \frac{\mathrm{d}\Phi}{\mathrm{d}z} = 0, \quad z=0 </math></center>

			We can calculate the second derivative above as
			<center><math>
			\frac{\mathrm{d}^2\Phi}{\mathrm{d}t^2} = \frac{\partial^2 \bar{\Phi}}{\partial t^2}
			- 2 U \frac{\partial^2\bar{\Phi}}{\partial x \partial t}
			+ U^2 \frac{\partial^ \bar{\Phi}}{\partial x^2}
			</math></center>

			When no ambient waves are present <math> \frac{\partial\bar{\Phi}}{\partial t} = 0 \, </math> and we obtain the free surface condition for the steady Kelvin ship wave problem
			<center><math> U^2 \frac{\partial^2 \bar{\Phi}}{\partial x^2}
			+ g \frac{\partial\bar{\Phi}}{\partial z} = 0, \qquad z=0 </math></center>
			This is the famous Neumann-Kelvin free-surface condition governing the linear steady wave pattern generated by a translating ship.

			Bernoulli equation:

			<center><math> P = - \rho \left( \frac{\mathrm{d}\Phi}{\mathrm{d}t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi + gz \right) = - \rho \left[ \left( \frac{\partial\bar{\Phi}}{\partial t} - U \frac{\partial\bar{\Phi}}{\partial x} \right) + \frac{1}{2} \nabla \phi \cdot \nabla \phi + g z \right] </math></center>

			Free surface elevation:

			<center><math> \zeta = - \left. \frac{1}{g} \frac{\mathrm{d}\Phi}{\mathrm{d}t} \right\|_{z=0} = -\frac{1}{g} \left. \left( \frac{\partial\bar{\Phi}}{\partial t} - U \frac{\partial\bar{\Phi}}{\partial x} \right)\right\|_{z=0} </math></center>

			In the Kelvin ship wave problem
			<center><math> \frac{\partial\phi}{\partial t} = 0 </math></center>

			Hence
			<center><math> \zeta = \frac{U}{g} \frac{\partial\bar{\Phi}}{\partial x}, \qquad z=0 </math></center>


	== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==		== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

Adi Kurniawan: /* Boundary-value problem for \bar{\Phi} */

2009-12-02T10:36:17Z

Boundary-value problem for \bar{\Phi}

Adi Kurniawan at 10:35, 2 December 2009

2009-12-02T10:35:28Z

Adi Kurniawan: /* Neumann-Kelvin linearization of U>0 ship seakeeping problem */

2009-12-02T10:33:34Z

Neumann-Kelvin linearization of U>0 ship seakeeping problem

Meylan at 09:05, 16 October 2009

2009-10-16T09:05:03Z

← Older revision		Revision as of 09:05, 16 October 2009
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	~~{{incomplete pages}}~~

	{{Ocean Wave Interaction with Ships and Offshore Structures		{{Ocean Wave Interaction with Ships and Offshore Structures
	\| chapter title = Forward-Speed Ship Wave Flows		\| chapter title = Forward-Speed Ship Wave Flows
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	}}		}}

			{{incomplete pages}}

	== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==		== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

Meylan at 09:04, 16 October 2009

2009-10-16T09:04:40Z

← Older revision		Revision as of 09:04, 16 October 2009
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			{{incomplete pages}}

			{{Ocean Wave Interaction with Ships and Offshore Structures
			\| chapter title = Forward-Speed Ship Wave Flows
			\| next chapter = [[Strip Theory Of Ship Motions. Heave & Pitch]]
			\| previous chapter = [[Wave Scattering By A Vertical Circular Cylinder]]
			}}


	== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==		== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

Adi Kurniawan at 07:10, 24 February 2009

2009-02-24T07:10:04Z

← Older revision		Revision as of 07:10, 24 February 2009
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	~~<u>~~Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem~~</u>~~		== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

	* Let <math> \Phi(X_0,Y_0,Z_0,t) \,</math>be the total potential relative to the inertial coordinate system		* Let <math> \Phi(X_0,Y_0,Z_0,t) \,</math>be the total potential relative to the inertial coordinate system
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	with <math> \phi_j, \ j=1,\cdots,6\,</math> being the radiation and <math> \phi_7\,</math> being the diffraction potentials.		with <math> \phi_j, \ j=1,\cdots,6\,</math> being the radiation and <math> \phi_7\,</math> being the diffraction potentials.

	~~<u>~~Boundary-value problem for <math> \bar{\Phi}</math>~~</u>~~		== Boundary-value problem for <math> \bar{\Phi}</math> ==

	* Free surface condition:		* Free surface condition:
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	* The ship slenderness justifies the use of the linear free-surface condition in the forward-speed problem for a broad range of speeds and hull shapes.		* The ship slenderness justifies the use of the linear free-surface condition in the forward-speed problem for a broad range of speeds and hull shapes.

	~~<u>~~Boundary-value problem for time-Harmonic velocity potential~~</u>~~		== Boundary-value problem for time-Harmonic velocity potential ==

	From the Galilean transformation:		From the Galilean transformation:
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	Be defined to be the encounter frequency which accounts for the Doppler effect included in the second term in the RHS.		Be defined to be the encounter frequency which accounts for the Doppler effect included in the second term in the RHS.

	~~<u>~~Comments on N-K formulation~~</u>:~~		== Comments on N-K formulation ==

	* The ship is assumed to be streamlined in order to justify the decomposition of the steady & time harmonic components.		* The ship is assumed to be streamlined in order to justify the decomposition of the steady & time harmonic components.

Syan077 at 08:19, 7 March 2007

2007-03-07T08:19:08Z

← Older revision		Revision as of 10:36, 2 December 2009
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	== Boundary-value problem for <math> \bar{\Phi}</math> ==		== Boundary-value problem for <math> \bar{\Phi}</math> ==

	* Free surface condition:		Free surface condition:

	<center><math> U^2 \bar{\Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 </math></center>		<center><math> U^2 \bar{\Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 </math></center>

	* Ship-hull condition		Ship-hull condition

	<center><math> \vec{n} \cdot \nabla\vec{\Phi} = \vec{n} \cdot \vec{U} = U n_1 \, </math></center>		<center><math> \vec{n} \cdot \nabla\vec{\Phi} = \vec{n} \cdot \vec{U} = U n_1 \, </math></center>
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	where <math> \vec{n} = ( n_1, n_2, n_3 ) \, </math> is the unit vector pointing inside the ship hull.		where <math> \vec{n} = ( n_1, n_2, n_3 ) \, </math> is the unit vector pointing inside the ship hull.

	* Far from the ship <math> \bar{\Phi}\,</math> represents outgoing waves which are known as the Kelvin ship wake studied earlier		Far from the ship <math> \bar{\Phi}\,</math> represents outgoing waves which are known as the Kelvin ship wake studied earlier.

	* The solution for <math> \bar{\Phi}\,</math> by the above formulation known as the Neumann-Kelvin problem and its generalizations discussed in the literature is carried out by panel methods.		The solution for <math> \bar{\Phi}\,</math> by the above formulation known as the Neumann-Kelvin problem and its generalizations discussed in the literature is carried out by panel methods.

	* The principal output quantities of interest in practice are:		The principal output quantities of interest in practice are:

	** Free-surface elevation		*Free-surface elevation

	<center><math> \zeta = - \frac{1}{g} \left. \frac{\mathrm{d}\bar{\Phi}}{\mathrm{d}t} \right\|_{\mbox{Inertial frame}} = \frac{U}{g} \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}}, \quad Z=0 </math></center>		<center><math> \zeta = - \frac{1}{g} \left. \frac{\mathrm{d}\bar{\Phi}}{\mathrm{d}t} \right\|_{\mbox{Inertial frame}} = \frac{U}{g} \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}}, \quad Z=0 </math></center>

	** Hydrodynamic pressure (linear)		*Hydrodynamic pressure (linear)

	<center><math> P = - \rho \left. \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} \right\|_{\mbox{Inertial frame}} = \rho U \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}} </math></center>		<center><math> P = - \rho \left. \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} \right\|_{\mbox{Inertial frame}} = \rho U \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}} </math></center>

	** Hydrodynamic pressure (total)		*Hydrodynamic pressure (total)

	<center><math> P_T = \rho \left( \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} = \frac{1}{2} \nabla \bar{\mathbf{\Phi}} \cdot \nabla \bar{\mathbf{\Phi}} + g Z \right)_{\mbox{Inertial frame}} </math></center>		<center><math> P_T = \rho \left( \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} = \frac{1}{2} \nabla \bar{\mathbf{\Phi}} \cdot \nabla \bar{\mathbf{\Phi}} + g Z \right)_{\mbox{Inertial frame}} </math></center>
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	The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.		The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.

	* We will derive boundary value problems for the potentials <math> \bar{\Phi}\,</math> and <math> \Phi\,</math> relative to the ship fixed frame.		We will derive boundary value problems for the potentials <math> \bar{\Phi}\,</math> and <math> \Phi\,</math> relative to the ship fixed frame.

	* The principal assumption underlying the ensuing derivation is that the ship is slender, thin or flat or in general streamlined in the longitudinal direction. More explicitly, if <math> B\,</math> is the ship beam, <math> T\,</math> its draft and <math> L\,</math> its length we will assume that:		The principal assumption underlying the ensuing derivation is that the ship is slender, thin or flat or in general streamlined in the longitudinal direction. More explicitly, if <math> B\,</math> is the ship beam, <math> T\,</math> its draft and <math> L\,</math> its length we will assume that:

	<center><math> \frac{B}{L}, \quad \frac{T}{L} = O (\varepsilon), \quad \varepsilon \ll 1 \,</math></center>		<center><math> \frac{B}{L}, \quad \frac{T}{L} = O (\varepsilon), \quad \varepsilon \ll 1 \,</math></center>
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	where <math> \varepsilon\,</math> is the slenderness parameter assumed to be small compared to <math> 1 \,</math>.		where <math> \varepsilon\,</math> is the slenderness parameter assumed to be small compared to <math> 1 \,</math>.

	* The ship slenderness justifies the use of the linear free-surface condition in the forward-speed problem for a broad range of speeds and hull shapes.		The ship slenderness justifies the use of the linear free-surface condition in the forward-speed problem for a broad range of speeds and hull shapes.

	== Boundary-value problem for time-Harmonic velocity potential ==		== Boundary-value problem for time-Harmonic velocity potential ==

← Older revision		Revision as of 10:35, 2 December 2009
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	Let <math> \Phi( X,Y,Z,t) \, </math> be the same potential expressed relative to the translating frame. It was shown before that		Let <math> \Phi( X,Y,Z,t) \, </math> be the same potential expressed relative to the translating frame. It was shown before that

	<center><math> \frac{d\Phi}{dt} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, </math></center>		<center><math> \frac{\mathrm{d}\Phi}{\mathrm{d}t} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, </math></center>

	where now the time dependence of <math>\phi\,</math> w.r.t. time will be the <math> e^{i\omega t} \,</math> form in the ship seakeeping problem.		where now the time dependence of <math>\phi\,</math> w.r.t. time will be the <math> e^{i\omega t} \,</math> form in the ship seakeeping problem.
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	** Free-surface elevation		** Free-surface elevation

	<center><math> \zeta = - \frac{1}{g} \left. \frac{d\bar{\Phi}}{dt} \right\|_{\mbox{Inertial frame}} = \frac{U}{g} \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}}, \quad Z=0 </math></center>		<center><math> \zeta = - \frac{1}{g} \left. \frac{\mathrm{d}\bar{\Phi}}{\mathrm{d}t} \right\|_{\mbox{Inertial frame}} = \frac{U}{g} \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}}, \quad Z=0 </math></center>

	** Hydrodynamic pressure (linear)		** Hydrodynamic pressure (linear)

	<center><math> P = - \rho \left. \frac{d\bar{\mathbf{\Phi}}}{dt} \right\|_{\mbox{Inertial frame}} = \rho U \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}} </math></center>		<center><math> P = - \rho \left. \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} \right\|_{\mbox{Inertial frame}} = \rho U \left. \bar{\Phi}_X \right\|_{\mbox{Ship frame}} </math></center>

	** Hydrodynamic pressure (total)		** Hydrodynamic pressure (total)

	<center><math> P_T = \rho \left( \frac{d\bar{\mathbf{\Phi}}}{dt} = \frac{1}{2} \nabla \bar{\mathbf{\Phi}} \cdot \nabla \bar{\mathbf{\Phi}} + g Z \right)_{\mbox{Inertial frame}} </math></center>		<center><math> P_T = \rho \left( \frac{\mathrm{d}\bar{\mathbf{\Phi}}}{\mathrm{d}t} = \frac{1}{2} \nabla \bar{\mathbf{\Phi}} \cdot \nabla \bar{\mathbf{\Phi}} + g Z \right)_{\mbox{Inertial frame}} </math></center>

	<center><math> = - \rho \left( - U \bar{\Phi_X} + \frac{1}{2} \nabla \bar{\Phi} \cdot \nabla \bar{\Phi} + g Z \right)_{\mbox{Ship frame}} \,</math></center>		<center><math> = - \rho \left( - U \bar{\Phi_X} + \frac{1}{2} \nabla \bar{\Phi} \cdot \nabla \bar{\Phi} + g Z \right)_{\mbox{Ship frame}} \,</math></center>
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	If <math>\bar{S_w}\,</math> is the ship wetted surface due to its steady forward translation on a free surface and <math>\bar{n}\,</math> is the unit normal vector pointing out of the fluid domain the ship ideal-fluid force is given by		If <math>\bar{S_w}\,</math> is the ship wetted surface due to its steady forward translation on a free surface and <math>\bar{n}\,</math> is the unit normal vector pointing out of the fluid domain the ship ideal-fluid force is given by

	<center><math> \overrightarrow{F} = \iint_{\overline{S_W}} P_T \vec{n} dS \, </math></center>		<center><math> \overrightarrow{F} = \iint_{\overline{S_W}} P_T \vec{n} \mathrm{d}S \, </math></center>

	The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.		The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.

← Older revision		Revision as of 10:33, 2 December 2009
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	== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==		== Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem ==

	* Let <math> \Phi(X_0,Y_0,Z_0,t) \,</math>be the total potential relative to the inertial coordinate system		Let <math> \Phi(X_0,Y_0,Z_0,t) \,</math>be the total potential relative to the inertial coordinate system

	<center><math> X_0 = x + U t \, </math></center>		<center><math> X_0 = x + U t \, </math></center>

	* Let <math> \Phi( X,Y,Z,t) \, </math> be the same potential expressed relative to the translating frame. It was shown before that		Let <math> \Phi( X,Y,Z,t) \, </math> be the same potential expressed relative to the translating frame. It was shown before that

	<center><math> \frac{d\Phi}{dt} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, </math></center>		<center><math> \frac{d\Phi}{dt} = \left( \frac{\partial}{\partial t} - U \frac{\partial}{\partial X} \right) \phi (\vec{X}, t) \, </math></center>
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	where now the time dependence of <math>\phi\,</math> w.r.t. time will be the <math> e^{i\omega t} \,</math> form in the ship seakeeping problem.		where now the time dependence of <math>\phi\,</math> w.r.t. time will be the <math> e^{i\omega t} \,</math> form in the ship seakeeping problem.

	* The total potential <math> \Phi\,</math> consists of the sum of two components in a linearized setting		The total potential <math> \Phi\,</math> consists of the sum of two components in a linearized setting

	<center><math> \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \,</math></center>		<center><math> \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \,</math></center>
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	where <math> \mathbf{\bar{\Phi}}\,</math> is the velocity potential due to the vessel forward translation with constant speed <math> U\,</math> and <math> \Phi\,</math> is the seakeeping component due to vessel motions in waves.		where <math> \mathbf{\bar{\Phi}}\,</math> is the velocity potential due to the vessel forward translation with constant speed <math> U\,</math> and <math> \Phi\,</math> is the seakeeping component due to vessel motions in waves.

	* Relative to the ship frame:		Relative to the ship frame:

	<center><math> \mathbf{\bar{\Phi}} = \bar{\Phi} (X,Y,Z) \, </math></center>		<center><math> \mathbf{\bar{\Phi}} = \bar{\Phi} (X,Y,Z) \, </math></center>

← Older revision		Revision as of 08:19, 7 March 2007
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	<u>Neumann-Kelvin linearization of <math>U>0\,</math> ship seakeeping problem</u>		<u>Neumann-Kelvin linearization of <math>U>0</math> ship seakeeping problem</u>

	* Let <math> \Phi(X_0,Y_0,Z_0,t) be the total potential relative to the inertial coordinate system		* Let <math> \Phi(X_0,Y_0,Z_0,t) \,</math>be the total potential relative to the inertial coordinate system

	<center><math> X_0 = x + U t \, </math></center>		<center><math> X_0 = x + U t \, </math></center>
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	where now the time dependence of <math>\phi\,</math> w.r.t. time will be the <math> e^{i\omega t} \,</math> form in the ship seakeeping problem.		where now the time dependence of <math>\phi\,</math> w.r.t. time will be the <math> e^{i\omega t} \,</math> form in the ship seakeeping problem.

	* The total potential <math> \Phi\,</math consists of the sum of two components in a linearized setting		* The total potential <math> \Phi\,</math> consists of the sum of two components in a linearized setting

	<center><math> \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \,</math></center>		<center><math> \mathbf{\Phi}_{TOT} = \mathbf{\bar{\Phi}} + \mathbf{\Phi} \,</math></center>
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	with <math> \phi_j, \ j=1,\cdots,6\,</math> being the radiation and <math> \phi_7\,</math> being the diffraction potentials.		with <math> \phi_j, \ j=1,\cdots,6\,</math> being the radiation and <math> \phi_7\,</math> being the diffraction potentials.

	<u>Boundary-value problem for <math> \bar{\Phi\,</math></u>		<u>Boundary-value problem for <math> \bar{\Phi}</math></u>

	* Free surface condition:		* Free surface condition:

	<center><math> U^2 \bar{Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 </math></center>		<center><math> U^2 \bar{\Phi}_{XX} + g \bar{\Phi}_Z = 0, \quad Z=0 </math></center>

	* Ship-hull condition		* Ship-hull condition
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	If <math>\bar{S_w}\,</math> is the ship wetted surface due to its steady forward translation on a free surface and <math>\bar{n}\,</math> is the unit normal vector pointing out of the fluid domain the ship ideal-fluid force is given by		If <math>\bar{S_w}\,</math> is the ship wetted surface due to its steady forward translation on a free surface and <math>\bar{n}\,</math> is the unit normal vector pointing out of the fluid domain the ship ideal-fluid force is given by

	<center><math> \~~bar~~{F} = \iint_{\~~bar~~{S_W}} P_T \~~bar~~{n} dS \, </math></center>		<center><math> \overrightarrow{F} = \iint_{\overline{S_W}} P_T \vec{n} dS \, </math></center>

	The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.		The wave resistance is: <math> R_W = \vec{i} \cdot \vec{F} \, </math>.
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	* The solution for <math> \bar{\Phi}\,</math> is far from simple numerically. A lot of research has been devoted to this effort, in particular towards the evaluation of the ship Kelvin wake and the ship wave resistance.		* The solution for <math> \bar{\Phi}\,</math> is far from simple numerically. A lot of research has been devoted to this effort, in particular towards the evaluation of the ship Kelvin wake and the ship wave resistance.

	* The linearization of the pressure and vessel wetted surface <math>\~~bar~~{S_W}\,</math> about its static value in calm water must be carried out carefully! Nonlinear effects are known to contribute appreciably to the wave resistance.		* The linearization of the pressure and vessel wetted surface <math>\overline{S_W}\,</math> about its static value in calm water must be carried out carefully! Nonlinear effects are known to contribute appreciably to the wave resistance.

	* If available, a fully nonlinear solution of the forward-speed steady ship wave problem is preferable. Numerical issues must be carefully treated and are the subject of state-of-the-art research.		* If available, a fully nonlinear solution of the forward-speed steady ship wave problem is preferable. Numerical issues must be carefully treated and are the subject of state-of-the-art research.
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	The ship equations of motion follow as in the <math> U=0\,</math> case using linear system theory:		The ship equations of motion follow as in the <math> U=0\,</math> case using linear system theory:

	<center><math> \sum_{j=1}^6 \left[ - \omega^2 \left(M_{ij} + A_{ij} \right) + i\omega B_{ij} + C_{ij} \right] \Pi_j (\~~oemga~~) = \mathbf{X}_i(\omega),\quad i=1,\cdots,6 </math></center>		<center><math> \sum_{j=1}^6 \left[ - \omega^2 \left(M_{ij} + A_{ij} \right) + i\omega B_{ij} + C_{ij} \right] \Pi_j (\omega) = \mathbf{X}_i(\omega),\quad i=1,\cdots,6 </math></center>

	where the hydrodynamic coefficients <math>A_{ij}(\omega), \ B_{ij}(\omega) \, </math> and exciting forces are now functions of the encounter frequency <math>\omega\,</math> and other forward-speed effects.		where the hydrodynamic coefficients <math>A_{ij}(\omega), \ B_{ij}(\omega) \, </math> and exciting forces are now functions of the encounter frequency <math>\omega\,</math> and other forward-speed effects.
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	It follows that:		It follows that:

	<center><math> \phi_I = \frac{i g A}{\omega_0} e^{ Kz - iKx\cos\beta - iKy\sin\beta + i (\omega_0 -UK \cos\beta) t} </math><center>		<center><math> \phi_I = \frac{i g A}{\omega_0} e^{ Kz - iKx\cos\beta - iKy\sin\beta + i (\omega_0 -UK \cos\beta) t} </math></center>

	Let:		Let:
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	* The vessel motions are assumed small and of the same order as the ambient wave amplitude. Terms omitted are of <math>O\left(A^2\right)\,</math>.		* The vessel motions are assumed small and of the same order as the ambient wave amplitude. Terms omitted are of <math>O\left(A^2\right)\,</math>.

	* When Taylor expanding the free-surface and body-boundary condition about <math>Z=0\,</math> and <math>\~~longrightarrow~~{S_B}\,</math> respectively, the steady flow potential <math> \bar{\Phi}\simeq 0 \,</math>.		* When Taylor expanding the free-surface and body-boundary condition about <math>Z=0\,</math> and <math>\overrightarrow{S_B}\,</math> respectively, the steady flow potential <math> \bar{\Phi}\simeq 0 \,</math>.

	* For ships with appreciable thickness a better approximation for <math>\bar\Phi\,</math> is that of the double-body flow disturbance such that <math> \~~longrightarrow~~{\Phi_Z} = 0\,</math> on <math> Z=0\,</math> and <math> \~~longrightarrow~~{\Phi_n} = U n\,</math>, on <math>\~~longrightarrow~~{S_B}\,</math>, This leads to the state-of-the-art linear 3D steady flow and seakeeping formulation discussed later in connection with panel methods.		* For ships with appreciable thickness a better approximation for <math>\bar\Phi\,</math> is that of the double-body flow disturbance such that <math> \overrightarrow{\Phi_Z} = 0\,</math> on <math> Z=0\,</math> and <math> \overrightarrow{\Phi_n} = U n\,</math>, on <math>\overrightarrow{S_B}\,</math>, This leads to the state-of-the-art linear 3D steady flow and seakeeping formulation discussed later in connection with panel methods.

	* The N-K formulation is the staring point of strip theory.		* The N-K formulation is the staring point of strip theory.


			[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]