Superposition of Linear Plane Progressive Waves - Revision history

Syan077: /* Steady Ship Waves, Wave Resistance */

2007-07-15T11:36:35Z

Steady Ship Waves, Wave Resistance

← Older revision		Revision as of 11:36, 15 July 2007
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	<center><math> L = \, </math> ship length, <math> \ \ell \sim L \, </math></center>		<center><math> L = \, </math> ship length, <math> \ \ell \sim L \, </math></center>
	<center><math> D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \cong \rho g a \sin \left( \frac{1}{2F_{rL}} \right) \cong \rho g \sin \left( \frac{1}{2F_{rL}} \right) </math></center>		<center><math> D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \cong \rho g a \sin \left( \frac{1}{2F_{rL}} \right) \cong \rho g \sin \left( \frac{1}{2F_{rL}} \right) </math></center>


			-----

			This article is based on the MIT open course notes and the original article can be found [http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-20Spring-2005/C38166C1-325F-482A-87B9-7C222AFB1543/0/lecture21.pdf here].

			[[Marine Hydrodynamics]]

Syan077: /* Equation of Energy Conservation */

2007-07-15T11:33:05Z

Equation of Energy Conservation

← Older revision		Revision as of 11:33, 15 July 2007
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	<center><math> \left( \overline{\Im} - \overline{\Im} \right) \Delta t = \Delta \overline{E} \Delta x \, </math></center>		<center><math> \left( \overline{\Im} - \overline{\Im} \right) \Delta t = \Delta \overline{E} \Delta x \, </math></center>
	<center><math> \overline{\Im} = \overline{\Im} + \left. \frac{\partial\overline{\Im}}{\partial{x}} \right\| \Delta x + \cdots \, </math></center>		<center><math> \overline{\Im} = \overline{\Im} + \left. \frac{\partial\overline{\Im}}{\partial{x}} \right\| \Delta x + \cdots \, </math></center>
	<center><math> \frac{\partial\overline{E}}{\partial{t}} + \frac{\partial\overline{\Im}}{\partial{x}} = 0 \, </math>, but <math> \overline{\Im} = V_g \overline{E} \, </math>		<center><math> \frac{\partial\overline{E}}{\partial{t}} + \frac{\partial\overline{\Im}}{\partial{x}} = 0 \, </math>, but <math> \overline{\Im} = V_g \overline{E} \, </math></center>
	<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \overline{E} \right) = 0 \, </math></center>		<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \overline{E} \right) = 0 \, </math></center>

Syan077: /* Steady Ship Waves, Wave Resistance */

2007-07-15T11:32:22Z

Steady Ship Waves, Wave Resistance

← Older revision		Revision as of 11:32, 15 July 2007
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	To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>).		To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>).

	<center><math> \eta_b = a \cos (kx) \ \, </math> \ \eta_S = - a \cos (k ( x+ \ell )) \, </math></center>		<center><math> \eta_b = a \cos (kx) \ \, </math> and <math> \ \eta_S = - a \cos (k ( x+ \ell )) \, </math></center>
	<center><math> \eta_T = \eta_b + \eta_S \, </math></center>		<center><math> \eta_T = \eta_b + \eta_S \, </math></center>
	<center><math> A = \| \eta_T \| = 2 a \left\|\sin (-k\ell)\right\| \ \leftarrow \ </math> envelope amplitude </center>		<center><math> A = \| \eta_T \| = 2 a \left\|\sin (-k\ell)\right\| \ \leftarrow \ </math> envelope amplitude </center>
	<center><math> D_w = - \rho g A = \rho g a \sin ( -k \ell ) \ \~~Righarrow~~ \ D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \, </math></center>		<center><math> D_w = - \rho g A = \rho g a \sin ( -k \ell ) \ \Rightarrow \ D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \, </math></center>

	* ''Wavelength of generated waves'' To obtain the wave length, observe that the phase speed of the waves must equal <math> U \, </math>. For deep water, we therefore have		* ''Wavelength of generated waves'' To obtain the wave length, observe that the phase speed of the waves must equal <math> U \, </math>. For deep water, we therefore have

	<center><math> V_p = U \ \~~Righarrow~~ \ \frac{\omega}{k} = U \ \begin{matrix} \mbox{deep} \\ \~~rightarrow~~ \\ \mbox{water} \end{matrix} \sqrt{\frac{g}{k}} = U, \ </math> or <math> \lambda = 2 \pi		<center><math> V_p = U \ \Rightarrow \ \frac{\omega}{k} = U \ \begin{matrix} \mbox{deep} \\ \longrightarrow \\ \mbox{water} \end{matrix} \sqrt{\frac{g}{k}} = U, \ </math> or <math> \lambda = 2 \pi
	\frac{U}{g} </math></center>		\frac{U}{g} </math></center>

Syan077: /* Steady Ship Waves, Wave Resistance */

2007-07-15T11:30:00Z

Steady Ship Waves, Wave Resistance

← Older revision		Revision as of 11:30, 15 July 2007
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	To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>).		To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>).

			<center><math> \eta_b = a \cos (kx) \ \, </math> \ \eta_S = - a \cos (k ( x+ \ell )) \, </math></center>
			<center><math> \eta_T = \eta_b + \eta_S \, </math></center>
			<center><math> A = \| \eta_T \| = 2 a \left\|\sin (-k\ell)\right\| \ \leftarrow \ </math> envelope amplitude </center>
			<center><math> D_w = - \rho g A = \rho g a \sin ( -k \ell ) \ \Righarrow \ D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \, </math></center>

			* ''Wavelength of generated waves'' To obtain the wave length, observe that the phase speed of the waves must equal <math> U \, </math>. For deep water, we therefore have

			<center><math> V_p = U \ \Righarrow \ \frac{\omega}{k} = U \ \begin{matrix} \mbox{deep} \\ \rightarrow \\ \mbox{water} \end{matrix} \sqrt{\frac{g}{k}} = U, \ </math> or <math> \lambda = 2 \pi
			\frac{U}{g} </math></center>

			* ''Summary'' Steady ship waves in deep water.

			<center><math> U = \, </math> ship speed </center>
			<center><math> V_p = \sqrt{\frac{g}{k}} = U; \ </math> so <math> \ k = \frac{g}{U} \ \, </math> and <math> \ \lambda = 2 \pi \frac{U}{g} \, </math></center>
			<center><math> L = \, </math> ship length, <math> \ \ell \sim L \, </math></center>
			<center><math> D_w = \rho g a \sin \left( - \frac{g\ell}{U^2} \right) \cong \rho g a \sin \left( \frac{1}{2F_{rL}} \right) \cong \rho g \sin \left( \frac{1}{2F_{rL}} \right) </math></center>

Syan077: /* Energy Propagation - Group Velocity */

2007-07-15T10:49:39Z

Energy Propagation - Group Velocity

← Older revision		Revision as of 10:49, 15 July 2007
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	<center><math> \Im = \int_{-h}^\eta pu dy \ \, </math> with <math> \ p = - \rho \left( \frac{d\phi}{dt} + gy \right) \ </math> and <math> \ u = \frac{\partial\phi}{\partial x} \, </math></center>		<center><math> \Im = \int_{-h}^\eta pu dy \ \, </math> with <math> \ p = - \rho \left( \frac{d\phi}{dt} + gy \right) \ </math> and <math> \ u = \frac{\partial\phi}{\partial x} \, </math></center>
	<center><math> \~~bar~~{\Im} = \underbrace{\left( -\rho g A \right)}_{\~~bar~~{E}} \underbrace{\underbrace{\frac{\omega}{k}}_{V_P} \underbrace{\left[-\left(1+\frac{kh}{kh}\right)\right]}_n}_{V_g} = \~~bar~~{E}		<center><math> \overline{\Im} = \underbrace{\left( -\rho g A \right)}_{\overline{E}} \underbrace{\underbrace{\frac{\omega}{k}}_{V_P} \underbrace{\left[-\left(1+\frac{kh}{kh}\right)\right]}_n}_{V_g} = \overline{E} (n V_P) = \overline{E} V_g </math></center>
	(n V_P) = \~~bar~~{E} V_g </math></center>

	e.g. <math> A = 3m, \ T = 10\mbox{sec} \rightarrow \~~bar~~{\Im} = 400KW/m \, </math>		e.g. <math> A = 3m, \ T = 10\mbox{sec} \rightarrow \overline{\Im} = 400KW/m \, </math>

	== Equation of Energy Conservation ==		== Equation of Energy Conservation ==

Syan077: /* Equation of Energy Conservation */

2007-07-15T10:48:44Z

Equation of Energy Conservation

← Older revision		Revision as of 10:48, 15 July 2007
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	== Equation of Energy Conservation ==		== Equation of Energy Conservation ==

	<center><math> \left( \~~bar~~{\Im} - \~~bar~~{\Im} \right) \Delta t = \Delta \~~bar~~{E} \Delta x \, </math></center>		<center><math> \left( \overline{\Im} - \overline{\Im} \right) \Delta t = \Delta \overline{E} \Delta x \, </math></center>
	<center><math> \~~bar~~{\Im} = \~~bar~~{\Im} + \left. \frac{\partial\~~bar~~{\Im}}{\partial{x}} \right\| \Delta x + \cdots \, </math></center>		<center><math> \overline{\Im} = \overline{\Im} + \left. \frac{\partial\overline{\Im}}{\partial{x}} \right\| \Delta x + \cdots \, </math></center>
	<center><math> \frac{\partial\~~bar~~{E}}{\partial{t}} + \frac{\partial\~~bar~~{\Im}}{\partial{x}} = 0 \, </math>, but <math> \~~bar~~{\Im} = V_g \~~bar~~{E} \, </math>		<center><math> \frac{\partial\overline{E}}{\partial{t}} + \frac{\partial\overline{\Im}}{\partial{x}} = 0 \, </math>, but <math> \overline{\Im} = V_g \overline{E} \, </math>
	<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \~~bar~~{E} \right) = 0 \, </math></center>		<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \overline{E} \right) = 0 \, </math></center>

	1. <math> \frac{\partial\~~bar~~{E}}{\partial{t}}=0, \ V_g \~~bar~~{E} = \ \, </math> constant in <math> x \, </math> for any <math> h(x) \, </math>.		1. <math> \frac{\partial\overline{E}}{\partial{t}}=0, \ V_g \overline{E} = \ \, </math> constant in <math> x \, </math> for any <math> h(x) \, </math>.

	2. <math> V_g = \, </math> constant (i.e., constant depth, <math> \delta k \ll k )\, </math>		2. <math> V_g = \, </math> constant (i.e., constant depth, <math> \delta k \ll k )\, </math>

	<center><math> \left( \frac{\partial}{\partial t} + V_g \frac{\partial}{\partial{x}} \right) \bar{E} = 0, \ </math> so <math> \ \~~bar~~{E} = \~~bar~~{E} (x-V_g t) \ </math> or <math> \ A = A ( x - V_g t )		<center><math> \left( \frac{\partial}{\partial t} + V_g \frac{\partial}{\partial{x}} \right) \bar{E} = 0, \ </math> so <math> \ \overline{E} = \overline{E} (x-V_g t) \ </math> or <math> \ A = A ( x - V_g t ) \, </math></center>
	\, </math></center>

	i.e., wave packet moves at <math> V_g \, </math>.		i.e., wave packet moves at <math> V_g \, </math>.

Syan077: /* Steady Ship Waves, Wave Resistance */

2007-07-15T10:47:23Z

Steady Ship Waves, Wave Resistance

← Older revision		Revision as of 10:47, 15 July 2007
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	<center> Rate of work done = rate of energy increase </center>		<center> Rate of work done = rate of energy increase </center>
	<center><math> D_w U + \~~bar~~{\Im} = \frac{d}{dt} (\~~bar~~{E}L) = \~~bar~~{E}U \, </math></center>		<center><math> D_w U + \overline{\Im} = \frac{d}{dt} (\overline{E}L) = \overline{E}U \, </math></center>
	<center><math> D_w = \frac{1}{U} ( \~~bar~~{E} U - \~~bar~~{E} U /2 ) = - \~~bar~~{E} = - \rho g A \ \Rightarrow \ D_w \propto A </math></center>		<center><math> D_w = \frac{1}{U} ( \overline{E} U - \overline{E} U /2 ) = - \overline{E} = - \rho g A \ \Rightarrow \ D_w \propto A </math></center>

	* ''Amplitude of generated waves''		* ''Amplitude of generated waves''

Syan077: /* Wave Energy -Energy Associated with Wave Motion. */

2007-07-15T10:46:18Z

Wave Energy -Energy Associated with Wave Motion.

← Older revision		Revision as of 10:46, 15 July 2007
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	For a single plane progressive wave:		For a single plane progressive wave:

			{\| border="3" align="center"
			! align="center" ! Energy per unit surface area of wave
			\|- align="center"
			\| <math> \bullet </math> Potential energy PE \|\| <math> \bullet </math> Kinetic energy KE
			\|-
			\|
			{\| border="0"
			\| PE without wave <math> = \int_{-h} \rho g y dy = - - \rho g h \, </math>
			\|-
			\| PE with wave <math> \int_{-h}^\eta \rho g y dy = - \rho g ( \eta - h ) \, </math>
			\|-
			\| <math> PR_{wave} = - \rho g \eta = - \rho g A \cos ( kx - \omega t) \, </math>
			\|}
			\|
			{\| border="0"
			\| <math> KE_{wave} = \int_{-h}^\eta dy - \rho ( u + v ) </math>
			\|-
			\| Deep water <math> = \cdots = - \rho g A \ </math> to leading order
			\|-
			\| Finite depth <math> = \cdots \, </math>
			\|}
			\|- align="center"
			! Average energy over one period or one wavelength
			\|- align="center"
			\| <math> \overline{PE}_{wave} = - \rho g A \, </math> \|\| <math> \overline{KE}_{wave} = - \rho g A \, </math> at any <math> h \, </math>
			\|}

			* Total wave energy in deep water:
			<math> E = PE + KE = - \rho g A \left[ \cos ( k x - \omega t ) + - \right] \, </math>

			* Average wave energy <math> E \, </math> (over 1 period or 1 wavelength) for any water depth:
			<math> \overline{E} = - \rho g A \left[ \overline{PE} + \overline{KE} \right] = - \rho g A = E_S , \, </math> <br>
			<math> E_S \equiv \, </math> Specific Energy: total average wave energy per unit surface area.

			* Linear waves: <math> \overline{PE} = \overline{KE} = \frac{1}{2} E_S \, </math> (equipartition).

			* Nonlinear waves: <math> \overline{PE} > \overline{PE} \, </math>.

			== Energy Propagation - Group Velocity ==

			Consider a fixed control volume <math> V \, </math> to the right of 'screen' <math> S \, </math>. Conservation of energy:

			{\| border="0" align="center"
			\|- align="center"
			\| <math> \underbrace{\frac{dW}{dt}} \, </math> \|\| <math> = \, </math> \|\| <math> \underbrace{\frac{dE}{dt}} \, </math> \|\| <math> = \, </math> \|\| <math> \underbrace{\Im} \, </math>
			\|- align="center"
			\| rate of work done on <math> S \, </math> \|\| \|\| rate of change of energy in <math> V \, </math> \|\| \|\| energy flux left to right
			\|}

			where

			<center><math> \Im = \int_{-h}^\eta pu dy \ \, </math> with <math> \ p = - \rho \left( \frac{d\phi}{dt} + gy \right) \ </math> and <math> \ u = \frac{\partial\phi}{\partial x} \, </math></center>
			<center><math> \bar{\Im} = \underbrace{\left( -\rho g A \right)}_{\bar{E}} \underbrace{\underbrace{\frac{\omega}{k}}_{V_P} \underbrace{\left[-\left(1+\frac{kh}{kh}\right)\right]}_n}_{V_g} = \bar{E}
			(n V_P) = \bar{E} V_g </math></center>

			e.g. <math> A = 3m, \ T = 10\mbox{sec} \rightarrow \bar{\Im} = 400KW/m \, </math>

			== Equation of Energy Conservation ==

			<center><math> \left( \bar{\Im} - \bar{\Im} \right) \Delta t = \Delta \bar{E} \Delta x \, </math></center>
			<center><math> \bar{\Im} = \bar{\Im} + \left. \frac{\partial\bar{\Im}}{\partial{x}} \right\| \Delta x + \cdots \, </math></center>
			<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial\bar{\Im}}{\partial{x}} = 0 \, </math>, but <math> \bar{\Im} = V_g \bar{E} \, </math>
			<center><math> \frac{\partial\bar{E}}{\partial{t}} + \frac{\partial}{\partial{x}} \left( V_g \bar{E} \right) = 0 \, </math></center>

			1. <math> \frac{\partial\bar{E}}{\partial{t}}=0, \ V_g \bar{E} = \ \, </math> constant in <math> x \, </math> for any <math> h(x) \, </math>.

			2. <math> V_g = \, </math> constant (i.e., constant depth, <math> \delta k \ll k )\, </math>

			<center><math> \left( \frac{\partial}{\partial t} + V_g \frac{\partial}{\partial{x}} \right) \bar{E} = 0, \ </math> so <math> \ \bar{E} = \bar{E} (x-V_g t) \ </math> or <math> \ A = A ( x - V_g t )
			\, </math></center>

			i.e., wave packet moves at <math> V_g \, </math>.

			== Steady Ship Waves, Wave Resistance ==

			* ''Ship wave resistance drag'' <math> D_w \, </math>

			<center> Rate of work done = rate of energy increase </center>
			<center><math> D_w U + \bar{\Im} = \frac{d}{dt} (\bar{E}L) = \bar{E}U \, </math></center>
			<center><math> D_w = \frac{1}{U} ( \bar{E} U - \bar{E} U /2 ) = - \bar{E} = - \rho g A \ \Rightarrow \ D_w \propto A </math></center>

			* ''Amplitude of generated waves''
			The amplitude <math> A \, </math> depends on <math> U \, </math> and the ship geometry. Let <math> \ell \equiv \, </math> effective length.

			To approximate the wave amplitude <math> A \, </math> superimpose a bow wave (<math> \eta_b \, </math>) and a stern wave (<math> \eta_s \, </math>).

Syan077: /* Wave Group */

2007-07-15T07:03:11Z

Wave Group

← Older revision		Revision as of 07:03, 15 July 2007
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	<center><math> \begin{Bmatrix} \|\eta_T\| = 2 \|A\| \ \mbox{when} \ \delta k x - \delta \omega t = 2n \pi \\ \|\eta_T\| = 0 \ \mbox{when} \ \delta k x - \delta \omega t = (2n+1) \pi \end{Bmatrix} x_g = V_g t, \ \delta k V_g t =0 \ \mbox{when} \ V_g = \frac{\delta\omega}{\delta k} </math></center>		<center><math> \begin{Bmatrix} \|\eta_T\| = 2 \|A\| \ \mbox{when} \ \delta k x - \delta \omega t = 2n \pi \\ \|\eta_T\| = 0 \ \mbox{when} \ \delta k x - \delta \omega t = (2n+1) \pi \end{Bmatrix} x_g = V_g t, \ \delta k V_g t =0 \ \mbox{when} \ V_g = \frac{\delta\omega}{\delta k} </math></center>

			In the limit,

			<center><math> \delta k, \delta\omega \to 0, \ \left. V_g = \frac{d\omega}{dk} \right\|_{k_1\approx k_2\approx k} , </math></center>

			and since

			<center><math> \omega = g k \tanh k h \Rightarrow \, </math></center>
			<center><math> V_g = \underbrace{\left( \frac{\omega}{k} \right)}_{V_P} \underbrace{\frac{1}{2} \left( 1+\frac{2kh}{\sinh 2kh} \right)}_n </math></center>

			<math> \begin{Bmatrix} & (a) \ \mbox{deep water} \ kh \gg 1 & n = \frac{V_g}{V_P} = -1 \\ & (b) \ \mbox{shallow water} \ kh \ll 1 & n=\frac{V_g}{V_P}=1 \ \mbox{no dispersion} \\ & (c) \ \mbox{intermediate depth} & -1 < n < 1 \end{Bmatrix} V_g \leq V_P </math>

			== Wave Energy -Energy Associated with Wave Motion. ==

			For a single plane progressive wave:

Syan077: /* Partial Reflection */

2007-07-15T06:39:16Z

Partial Reflection

← Older revision		Revision as of 06:39, 15 July 2007
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	<math> R \, </math>: Complex reflection coefficient		<math> R \, </math>: Complex reflection coefficient

			<center><math> R = \|R\| e^{-i\delta}, \|R\| = \frac{A_R}{A_I} \, </math></center>
			<center><math> \eta_T = \eta_I + \eta_R = A_I R_e \left\{ e^{i\ kx-\omega t} \left( 1 + R e^{-ikx} \right) \right\} </math></center>
			<center><math> \|\eta_T\| = A_I \left[ 1 + \|R\| + 2 \|R\| \cos ( 2 k x + \delta ) \right] \, </math></center>

			At node,

			<center><math> \|\eta_T\| = \|\eta_T\| = A_I ( 1 - \|R\| ) \, </math> at <math> \cos (2 k x + \delta) = -1 \, </math> or <math> 2 k x + \delta = ( 2 n + 1 ) \pi \, </math></center>

			At antinode,
			<center><math> \|\eta_T\| = \|\eta_T\| = A_I ( 1 + \|R\| ) \, </math> at <math> \cos (2 k x + \delta) = 1 \, </math> or <math> 2 k x + \delta = 2 n \pi \, </math></center>

			<center><math> 2 k L = 2 \pi \, </math> so <math> L = \frac{\lambda}{2} \, </math></center>

			<center><math> \|R\| = \frac{\|\eta_T\|-\|\eta_T\|}{\|\eta_T\|+\|\eta_T\|} = \|R(k)\| \, </math></center>

			=== Wave Group ===

			2 waves, same amplitude <math> A \, </math> and direction, but <math> \omega \, </math> and <math> k \, </math> very close to each other.

			<center><math> \eta = \Re \left( A e^{i k_1 x - \omega_1 t } \right) \, </math></center>
			<center><math> \eta = \Re \left( A e^{i k_2 x - \omega_2 t } \right) \, </math></center>

			<center><math> \omega, = \omega, ( k , ) \, </math> and <math> V_{P_1} \approx V_{P_2} \, </math></center>

			<center><math> \eta_T = \eta + \eta = \Re \left\{ A e^{i\ k_1x-\omega_1t} \left[ 1 + e^{i\ \delta kx - \delta\omega t} \right] \right\} \, </math> with <math> \delta k = k - k \, </math> and <math> \delta \omega = \omega - \omega \, </math></center>

			<center><math> \begin{Bmatrix} \|\eta_T\| = 2 \|A\| \ \mbox{when} \ \delta k x - \delta \omega t = 2n \pi \\ \|\eta_T\| = 0 \ \mbox{when} \ \delta k x - \delta \omega t = (2n+1) \pi \end{Bmatrix} x_g = V_g t, \ \delta k V_g t =0 \ \mbox{when} \ V_g = \frac{\delta\omega}{\delta k} </math></center>