Linear Plane Progressive Regular Waves - Revision history

Meylan: /* Plane progressive waves in general directions */

2010-05-02T10:28:17Z

Plane progressive waves in general directions

← Older revision		Revision as of 10:28, 2 May 2010
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	For a linear plane progressive regular waves in a general direction the free surface elevation is given by		For a linear plane progressive regular waves in a general direction the free surface elevation is given by
	<center><math> \zeta = A \cos \left( \omega t - k x \cos \beta -k y \sin \beta +\chi \right) =		<center><math> \zeta = \bar{A} \cos \left( \omega t - k x \cos \beta -k y \sin \beta +\chi \right) =
	\mathrm{Re} \left \{ A e^{- i k x \cos \~~beta~~ - i k y \sin \~~beta~~ + i \omega t} \right \}		\mathrm{Re} \left \{ A e^{- i k x \cos \theta - i k y \sin \theta + i \omega t} \right \}
	</math></center>		</math></center>
			where <math>A</math> is the complex amplitude which includes the phase factor <math>\chi</math>.
			<math>\theta</math> is the angle the wave makes to the <math>x</math> axis.
	and the velocity potential in finite depth is given by		and the velocity potential in finite depth is given by
	<center><math> \Phi = \mathrm{Re} \left \{ \frac{\mathrm{i}gA}{\omega} \frac{\cosh k (z+h)}{\cosh k h} \		<center><math> \Phi = \mathrm{Re} \left \{ \frac{\mathrm{i}gA}{\omega} \frac{\cosh k (z+h)}{\cosh k h} \

Meylan: /* Plane progressive waves in general directions */

2010-05-01T11:00:38Z

Plane progressive waves in general directions

← Older revision		Revision as of 11:00, 1 May 2010
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	For a linear plane progressive regular waves in a general direction the free surface elevation is given by		For a linear plane progressive regular waves in a general direction the free surface elevation is given by
	<center><math> \zeta = A \cos \left( \omega t - k x \cos \beta -k y \sin \beta \right) =		<center><math> \zeta = A \cos \left( \omega t - k x \cos \beta -k y \sin \beta +\chi \right) =
	A \mathrm{Re} \left \{ e^{- i k x \cos \beta - i k y \sin \beta + i \omega t} \right \}		\mathrm{Re} \left \{ A e^{- i k x \cos \beta - i k y \sin \beta + i \omega t} \right \}
	</math></center>		</math></center>
	and the velocity potential in finite depth is given by		and the velocity potential in finite depth is given by
	<center><math> \Phi = A \mathrm{Re} \left \{ \frac{\mathrm{i}gA}{\omega} \frac{\cosh k (z+h)}{\cosh k h} \		<center><math> \Phi = \mathrm{Re} \left \{ \frac{\mathrm{i}gA}{\omega} \frac{\cosh k (z+h)}{\cosh k h} \
	e^{ -\mathrm{i} k x \cos \beta - \mathrm{i} k y \sin \beta + \mathrm{i} \omega t} \right \} </math></center>		e^{ -\mathrm{i} k x \cos \beta - \mathrm{i} k y \sin \beta + \mathrm{i} \omega t} \right \} </math></center>
	where		where

Meylan: /* Plane progressive waves in general directions */

2010-05-01T10:59:16Z

Plane progressive waves in general directions

← Older revision		Revision as of 10:59, 1 May 2010
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	[[Image:Plane_wave.jpg\|thumb\|right\|400px\|Plane wave]]		[[Image:Plane_wave.jpg\|thumb\|right\|400px\|Plane wave]]

	~~Consider~~ a ~~[[Linear Plane Progressive Regular Waves\|Plane~~ progressive waves in a general direction~~]]. The~~ free surface elevation is given by		For a linear plane progressive regular waves in a general direction the free surface elevation is given by

	<center><math> \zeta = A \cos \left( \omega t - k x \cos \beta -k y \sin \beta \right) =		<center><math> \zeta = A \cos \left( \omega t - k x \cos \beta -k y \sin \beta \right) =
	A \mathrm{Re} \left \{ e^{- i k x \cos \beta - i k y \sin \beta + i \omega t} \right \}		A \mathrm{Re} \left \{ e^{- i k x \cos \beta - i k y \sin \beta + i \omega t} \right \}

Meylan: /* Flow Velocity and Pressure */

2010-04-29T23:04:56Z

Flow Velocity and Pressure

← Older revision		Revision as of 23:04, 29 April 2010
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	The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is		The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is
	<center><math> P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h }		<center><math> P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h }
	e^{\mathrm{i} kx - \mathrm{i} \omega t}} \right\} </math></center>		e^{\mathrm{i} kx - \mathrm{i} \omega t} \right\} </math></center>

	== Plane progressive waves in general directions ==		== Plane progressive waves in general directions ==

Meylan: /* Flow Velocity and Pressure */

2010-04-29T23:01:20Z

Flow Velocity and Pressure

← Older revision		Revision as of 23:01, 29 April 2010
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	The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is		The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is
	<center><math> P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{-~~ikx +~~ i\omega t} \right\} </math></center>		<center><math> P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h }
			e^{\mathrm{i} kx - \mathrm{i} \omega t}} \right\} </math></center>

	== Plane progressive waves in general directions ==		== Plane progressive waves in general directions ==

Meylan at 23:00, 29 April 2010

2010-04-29T23:00:41Z

← Older revision		Revision as of 23:00, 29 April 2010
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	}}		}}

	{{~~complete~~ pages}}		{{incomplete pages}}

	== Introduction ==		== Introduction ==

Meylan: /* Propagating Wave */

2010-04-29T23:00:09Z

Propagating Wave

← Older revision		Revision as of 23:00, 29 April 2010
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	The free surface elevation is		The free surface elevation is
	<center><math> \zeta (x,t) = A \mathrm{Re} \{ e^{-~~ikx+~~i\omega t} \} \,</math></center>		<center><math> \zeta (x,t) = A \mathrm{Re} \{ e^{\mathrm{i} kx - \mathrm{i} \omega t} \} \,</math></center>
	where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) \,</math>		where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) \,</math>
	pair have the same definitions as in all wave propagation problems, namely:		pair have the same definitions as in all wave propagation problems, namely:

Meylan: /* Expression in terms of velocity potential */

2010-04-29T22:59:37Z

Expression in terms of velocity potential

← Older revision		Revision as of 22:59, 29 April 2010
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	The resulting plane progressive wave velocity potential takes the form:		The resulting plane progressive wave velocity potential takes the form:
	<center><math> \phi(x,z,t) = A \ \mathrm{Re} \left\{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h} e^{ - i k x + i \omega t} \right\} </math></center>		<center><math> \phi(x,z,t) = A \ \mathrm{Re} \left\{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
			e^{ \mathrm{i} k x - \mathrm{i} \omega t} \right\} </math></center>
	where <math>A</math> is the amplitude in displacement.		where <math>A</math> is the amplitude in displacement.

Meylan: /* Plane progressive waves in general directions */

2010-03-28T09:32:34Z

Plane progressive waves in general directions

← Older revision		Revision as of 09:32, 28 March 2010
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	where		where
	<center><math> \omega^2 = g k \tanh k h \,</math></center>		<center><math> \omega^2 = g k \tanh k h \,</math></center>
	from the [[Dispersion Relation for a Free Surface]].		from the [[Dispersion Relation for a Free Surface]] as before


	== [[Dispersion Relation for a Free Surface]] in [[Infinite Depth\| Deep]] and [[:Category:Shallow Depth\| Shallow Waters]] ==		== [[Dispersion Relation for a Free Surface]] in [[Infinite Depth\| Deep]] and [[:Category:Shallow Depth\| Shallow Waters]] ==

Meylan at 09:31, 28 March 2010

2010-03-28T09:31:43Z

@@ Line 68: / Line 68: @@
 The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is
 <center><math> P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{-ikx + i\omega t} \right\} </math></center>
 == [[Dispersion Relation for a Free Surface]] in [[Infinite Depth| Deep]] and [[:Category:Shallow Depth| Shallow Waters]] ==