Second-Order Wave Theory - Revision history

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

2019-07-17T08:41:58Z

Second-Order Bi-Chromatic Wave Theory

← Older revision		Revision as of 08:41, 17 July 2019
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	It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.		It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.

	The solution of the above forced free surface problems together with <math> \nabla^2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:		The solution of the above forced free surface problems together with <math> \nabla^2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2^\pm \, </math> are:

	<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>		<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

2019-07-17T08:41:04Z

Second-Order Bi-Chromatic Wave Theory

← Older revision		Revision as of 08:41, 17 July 2019
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	It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.		It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.

	The solution of the above forced free surface problems together with <math> \~~nabla_2~~\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:		The solution of the above forced free surface problems together with <math> \nabla^2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:

	<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>		<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

2019-07-17T08:38:32Z

Second-Order Bi-Chromatic Wave Theory

← Older revision		Revision as of 08:38, 17 July 2019
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	They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:		They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

	<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>		<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial^2 \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

	Making use of the above identity it is easy to show that:		Making use of the above identity it is easy to show that:

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

2019-07-17T08:35:37Z

Second-Order Bi-Chromatic Wave Theory

← Older revision		Revision as of 08:35, 17 July 2019
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	They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:		They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

	<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial^\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>		<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

	Making use of the above identity it is easy to show that:		Making use of the above identity it is easy to show that:

Adi Kurniawan: /* Special Cases */

2009-12-08T16:07:40Z

Special Cases

← Older revision		Revision as of 16:07, 8 December 2009
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	==Special Cases==		==Special Cases==

	A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in ~~MEI~~.		A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in [[Mei 1983\|Mei 1983]].

	<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>		<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>

Adi Kurniawan: /* Second-Order Bi-Chromatic Wave Theory */

2009-12-08T16:05:27Z

Second-Order Bi-Chromatic Wave Theory

Adi Kurniawan: /* Second-Order Bi-Chromatic Wave Theory */

2009-12-08T16:04:50Z

Second-Order Bi-Chromatic Wave Theory

← Older revision		Revision as of 16:04, 8 December 2009
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	They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:		They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

	<center><math> \frac{d}{dZ} \left( \frac{\partial^\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial Z} \right) = 0 , \quad \mbox{any} \ Z \leq 0 \, </math></center>		<center><math> \frac{\mathrm{d}}{\mathrm{d}Z} \left( \frac{\partial^\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial Z} \right) = 0 , \quad \mbox{any} \ Z \leq 0 \, </math></center>

	Making use of the above identity it is easy to show that:		Making use of the above identity it is easy to show that:

Adi Kurniawan: /* Second-Order Free Surface Condition */

2009-12-08T16:03:41Z

Second-Order Free Surface Condition

← Older revision		Revision as of 16:03, 8 December 2009
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	Kinematic condition:		Kinematic condition:

	<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1; \quad z=0 \, </math></center>		<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1 \quad \text{at }z=0 \, </math></center>

	Dynamic condition:		Dynamic condition:

Adi Kurniawan: /* Second-Order Free Surface Condition */

2009-12-08T16:02:35Z

Second-Order Free Surface Condition

← Older revision		Revision as of 16:02, 8 December 2009
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	Kinematic condition:		Kinematic condition:

	<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial Z^2} - \nabla\phi_1 \cdot \nabla\zeta_1; \quad Z=0 \, </math></center>		<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1; \quad z=0 \, </math></center>

	Dynamic condition:		Dynamic condition:

	<center><math> \zeta_2 + \frac{1}{g} \frac{\partial\phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial Z \partial t} \right)_{Z=0} </math></center>		<center><math> \zeta_2 + \frac{1}{g} \frac{\partial\phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

	Hydrodynamic pressure:		Hydrodynamic pressure:
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	<center><math> \nabla^2\phi_2 = 0 \,</math></center>		<center><math> \nabla^2\phi_2 = 0 \,</math></center>
	<center><math> \frac{\partial^2\phi_2}{\partial t^2} + g \frac{\partial\phi_2}{\partial Z} = - \frac{\partial}{\partial t} \left( \nabla\phi_1 \cdot \nabla\phi_1 \right) + \frac{1}{g} \frac{\partial\phi_1}{\partial t} \frac{\partial}{\partial Z} \left( \frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial \phi_1}{\partial Z} \right)_{Z=0} </math></center>		<center><math> \frac{\partial^2\phi_2}{\partial t^2} + g \frac{\partial\phi_2}{\partial z} = - \frac{\partial}{\partial t} \left( \nabla\phi_1 \cdot \nabla\phi_1 \right) + \frac{1}{g} \frac{\partial\phi_1}{\partial t} \frac{\partial}{\partial z} \left( \frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial \phi_1}{\partial z} \right)_{z=0} </math></center>
	<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1\cdot\nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial Z \partial t} \right)_{Z=0} </math></center>		<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1\cdot\nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

	The solution of the above boundary-value problem is available in closed from when the linear potential <math>\phi_1\,</math> is the linear superposition of regular plane progressive waves.		The solution of the above boundary-value problem is available in closed from when the linear potential <math>\phi_1\,</math> is the linear superposition of regular plane progressive waves.

Adi Kurniawan at 16:01, 8 December 2009

2009-12-08T16:01:45Z

← Older revision		Revision as of 16:05, 8 December 2009
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	The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:		The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:

	<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k Z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>		<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>

	where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.		where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.
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	They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:		They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

	<center><math> \frac{\mathrm{d}}{\mathrm{d}Z} \left( \frac{\partial^\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial Z} \right) = 0 , \quad \mbox{any} \ Z \leq 0 \, </math></center>		<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial^\phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

	Making use of the above identity it is easy to show that:		Making use of the above identity it is easy to show that:

	<center><math> - {\Omega^+}^2 \varphi_2^+ + g \frac{\partial \varphi_2^+}{\partial z} = - i \Omega^+ \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln} \quad Z=0 </math></center>		<center><math> - {\Omega^+}^2 \varphi_2^+ + g \frac{\partial \varphi_2^+}{\partial z} = - i \Omega^+ \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln} \quad z=0 </math></center>

	and		and

	<center><math> - {\Omega^-}^2 \varphi_2^- + g \frac{\partial \varphi_2^-}{\partial z} = - i \Omega^- \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln}^* \quad Z=0 </math></center>		<center><math> - {\Omega^-}^2 \varphi_2^- + g \frac{\partial \varphi_2^-}{\partial z} = - i \Omega^- \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln}^* \quad z=0 </math></center>

	where <math> \varphi_{1km}\,</math> and <math> \varphi_{2ln}\,</math> are the complex velocity potentials of the regular plane progressive waves the ambient seastate consists of.		where <math> \varphi_{1km}\,</math> and <math> \varphi_{2ln}\,</math> are the complex velocity potentials of the regular plane progressive waves the ambient seastate consists of.
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	The solution of the above forced free surface problems together with <math> \nabla_2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:		The solution of the above forced free surface problems together with <math> \nabla_2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:

	<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{Z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>		<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

	where:		where:

← Older revision		Revision as of 16:01, 8 December 2009
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	The above conclusion is analogous to the following identity:		The above conclusion is analogous to the following identity:

	<center><math> \~~mathbb~~{~~R}\mathbf{e~~} \left\{ A_1 e^{i\omega t} \right\} \~~mathbb{R}\mathbf~~{e} \left\{ A_2 e^{i\omega t} \right\} = \frac{1}{2} \~~mathbb{R}\mathbf~~{e} \left\{ A_1 A_2 e^{i\left(\omega_1+\omega_2\right)t} + A_1 A_2^* e^{i\left(\omega_1-\omega_2\right)t}\right\} </math></center>		<center><math> \mathrm{Re} \left\{ A_1 e^{i\omega t} \right\} \mathrm{Re} \left\{ A_2 e^{i\omega t} \right\} = \frac{1}{2} \mathrm{Re} \left\{ A_1 A_2 e^{i\left(\omega_1+\omega_2\right)t} + A_1 A_2^* e^{i\left(\omega_1-\omega_2\right)t}\right\} </math></center>

	It follows that the forcing and hence the solution of the second-order problem is the linear super position of a sum and a difference-frequency components.		It follows that the forcing and hence the solution of the second-order problem is the linear super position of a sum and a difference-frequency components.
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	The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:		The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:

	<center><math> \Phi_1 = \~~mathbb~~{~~R}\mathbf{e~~} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k Z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>		<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k Z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>

	where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.		where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.
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	We therefore define the yet unknown second-order potential as follows:		We therefore define the yet unknown second-order potential as follows:

	<center><math> \Phi_2 = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ \varphi_2^+ e^{i\left(\omega_k+\omega_\rho\right)t} + \varphi_2^- e^{i\left(\omega_k-\omega_\rho\right)t} \right\} </math></center>		<center><math> \Phi_2 = \mathrm{Re} \left\{ \varphi_2^+ e^{i\left(\omega_k+\omega_\rho\right)t} + \varphi_2^- e^{i\left(\omega_k-\omega_\rho\right)t} \right\} </math></center>

	So the principal task of second-order theory in the absence of floating bodies is to determine the complex second-order potentials <math>\varphi_2^+\,</math> and <math> \varphi_2^-\,</math>.		So the principal task of second-order theory in the absence of floating bodies is to determine the complex second-order potentials <math>\varphi_2^+\,</math> and <math> \varphi_2^-\,</math>.
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	The real second-order potential follows by quadruple summation over all pairs of frequencies and wave headings and is not stated here.		The real second-order potential follows by quadruple summation over all pairs of frequencies and wave headings and is not stated here.

	~~A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials.~~		==Special Cases==

	All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in MEI.		A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in MEI.

	~~==Special Cases==~~

	<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>		<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>