arma-thesis

git clone https://git.igankevich.com/arma-thesis.git
Log | Files | Refs | LICENSE

commit b4929875a3d3b4db1a035c1827ba571de44c27eb
parent b7d967928bdb0edcc37ce77c4a6f16f8e9a63411
Author: Ivan Gankevich <igankevich@ya.ru>
Date:   Mon, 12 Mar 2018 14:23:55 +0300

Merge github.com:igankevich/arma-thesis

Diffstat:
LICENSE | 396+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
README.org | 6++++++
arma-thesis-ru.org | 38+++++++++++++++++++++-----------------
arma-thesis.org | 38+++++++++++++++++++++-----------------
preamble.tex | 6++++++
5 files changed, 450 insertions(+), 34 deletions(-)

diff --git a/LICENSE b/LICENSE @@ -0,0 +1,396 @@ +Attribution 4.0 International + +======================================================================= + +Creative Commons Corporation ("Creative Commons") is not a law firm and +does not provide legal services or legal advice. Distribution of +Creative Commons public licenses does not create a lawyer-client or +other relationship. Creative Commons makes its licenses and related +information available on an "as-is" basis. Creative Commons gives no +warranties regarding its licenses, any material licensed under their +terms and conditions, or any related information. Creative Commons +disclaims all liability for damages resulting from their use to the +fullest extent possible. + +Using Creative Commons Public Licenses + +Creative Commons public licenses provide a standard set of terms and +conditions that creators and other rights holders may use to share +original works of authorship and other material subject to copyright +and certain other rights specified in the public license below. The +following considerations are for informational purposes only, are not +exhaustive, and do not form part of our licenses. + + Considerations for licensors: Our public licenses are + intended for use by those authorized to give the public + permission to use material in ways otherwise restricted by + copyright and certain other rights. Our licenses are + irrevocable. Licensors should read and understand the terms + and conditions of the license they choose before applying it. + Licensors should also secure all rights necessary before + applying our licenses so that the public can reuse the + material as expected. Licensors should clearly mark any + material not subject to the license. This includes other CC- + licensed material, or material used under an exception or + limitation to copyright. More considerations for licensors: + wiki.creativecommons.org/Considerations_for_licensors + + Considerations for the public: By using one of our public + licenses, a licensor grants the public permission to use the + licensed material under specified terms and conditions. If + the licensor's permission is not necessary for any reason--for + example, because of any applicable exception or limitation to + copyright--then that use is not regulated by the license. Our + licenses grant only permissions under copyright and certain + other rights that a licensor has authority to grant. Use of + the licensed material may still be restricted for other + reasons, including because others have copyright or other + rights in the material. A licensor may make special requests, + such as asking that all changes be marked or described. + Although not required by our licenses, you are encouraged to + respect those requests where reasonable. More_considerations + for the public: + wiki.creativecommons.org/Considerations_for_licensees + +======================================================================= + +Creative Commons Attribution 4.0 International Public License + +By exercising the Licensed Rights (defined below), You accept and agree +to be bound by the terms and conditions of this Creative Commons +Attribution 4.0 International Public License ("Public License"). To the +extent this Public License may be interpreted as a contract, You are +granted the Licensed Rights in consideration of Your acceptance of +these terms and conditions, and the Licensor grants You such rights in +consideration of benefits the Licensor receives from making the +Licensed Material available under these terms and conditions. + + +Section 1 -- Definitions. + + a. Adapted Material means material subject to Copyright and Similar + Rights that is derived from or based upon the Licensed Material + and in which the Licensed Material is translated, altered, + arranged, transformed, or otherwise modified in a manner requiring + permission under the Copyright and Similar Rights held by the + Licensor. For purposes of this Public License, where the Licensed + Material is a musical work, performance, or sound recording, + Adapted Material is always produced where the Licensed Material is + synched in timed relation with a moving image. + + b. Adapter's License means the license You apply to Your Copyright + and Similar Rights in Your contributions to Adapted Material in + accordance with the terms and conditions of this Public License. + + c. Copyright and Similar Rights means copyright and/or similar rights + closely related to copyright including, without limitation, + performance, broadcast, sound recording, and Sui Generis Database + Rights, without regard to how the rights are labeled or + categorized. For purposes of this Public License, the rights + specified in Section 2(b)(1)-(2) are not Copyright and Similar + Rights. + + d. Effective Technological Measures means those measures that, in the + absence of proper authority, may not be circumvented under laws + fulfilling obligations under Article 11 of the WIPO Copyright + Treaty adopted on December 20, 1996, and/or similar international + agreements. + + e. Exceptions and Limitations means fair use, fair dealing, and/or + any other exception or limitation to Copyright and Similar Rights + that applies to Your use of the Licensed Material. + + f. Licensed Material means the artistic or literary work, database, + or other material to which the Licensor applied this Public + License. + + g. Licensed Rights means the rights granted to You subject to the + terms and conditions of this Public License, which are limited to + all Copyright and Similar Rights that apply to Your use of the + Licensed Material and that the Licensor has authority to license. + + h. Licensor means the individual(s) or entity(ies) granting rights + under this Public License. + + i. Share means to provide material to the public by any means or + process that requires permission under the Licensed Rights, such + as reproduction, public display, public performance, distribution, + dissemination, communication, or importation, and to make material + available to the public including in ways that members of the + public may access the material from a place and at a time + individually chosen by them. + + j. Sui Generis Database Rights means rights other than copyright + resulting from Directive 96/9/EC of the European Parliament and of + the Council of 11 March 1996 on the legal protection of databases, + as amended and/or succeeded, as well as other essentially + equivalent rights anywhere in the world. + + k. You means the individual or entity exercising the Licensed Rights + under this Public License. Your has a corresponding meaning. + + +Section 2 -- Scope. + + a. License grant. + + 1. Subject to the terms and conditions of this Public License, + the Licensor hereby grants You a worldwide, royalty-free, + non-sublicensable, non-exclusive, irrevocable license to + exercise the Licensed Rights in the Licensed Material to: + + a. reproduce and Share the Licensed Material, in whole or + in part; and + + b. produce, reproduce, and Share Adapted Material. + + 2. Exceptions and Limitations. For the avoidance of doubt, where + Exceptions and Limitations apply to Your use, this Public + License does not apply, and You do not need to comply with + its terms and conditions. + + 3. Term. The term of this Public License is specified in Section + 6(a). + + 4. Media and formats; technical modifications allowed. The + Licensor authorizes You to exercise the Licensed Rights in + all media and formats whether now known or hereafter created, + and to make technical modifications necessary to do so. The + Licensor waives and/or agrees not to assert any right or + authority to forbid You from making technical modifications + necessary to exercise the Licensed Rights, including + technical modifications necessary to circumvent Effective + Technological Measures. For purposes of this Public License, + simply making modifications authorized by this Section 2(a) + (4) never produces Adapted Material. + + 5. Downstream recipients. + + a. Offer from the Licensor -- Licensed Material. Every + recipient of the Licensed Material automatically + receives an offer from the Licensor to exercise the + Licensed Rights under the terms and conditions of this + Public License. + + b. No downstream restrictions. You may not offer or impose + any additional or different terms or conditions on, or + apply any Effective Technological Measures to, the + Licensed Material if doing so restricts exercise of the + Licensed Rights by any recipient of the Licensed + Material. + + 6. No endorsement. Nothing in this Public License constitutes or + may be construed as permission to assert or imply that You + are, or that Your use of the Licensed Material is, connected + with, or sponsored, endorsed, or granted official status by, + the Licensor or others designated to receive attribution as + provided in Section 3(a)(1)(A)(i). + + b. Other rights. + + 1. Moral rights, such as the right of integrity, are not + licensed under this Public License, nor are publicity, + privacy, and/or other similar personality rights; however, to + the extent possible, the Licensor waives and/or agrees not to + assert any such rights held by the Licensor to the limited + extent necessary to allow You to exercise the Licensed + Rights, but not otherwise. + + 2. Patent and trademark rights are not licensed under this + Public License. + + 3. To the extent possible, the Licensor waives any right to + collect royalties from You for the exercise of the Licensed + Rights, whether directly or through a collecting society + under any voluntary or waivable statutory or compulsory + licensing scheme. In all other cases the Licensor expressly + reserves any right to collect such royalties. + + +Section 3 -- License Conditions. + +Your exercise of the Licensed Rights is expressly made subject to the +following conditions. + + a. Attribution. + + 1. If You Share the Licensed Material (including in modified + form), You must: + + a. retain the following if it is supplied by the Licensor + with the Licensed Material: + + i. identification of the creator(s) of the Licensed + Material and any others designated to receive + attribution, in any reasonable manner requested by + the Licensor (including by pseudonym if + designated); + + ii. a copyright notice; + + iii. a notice that refers to this Public License; + + iv. a notice that refers to the disclaimer of + warranties; + + v. a URI or hyperlink to the Licensed Material to the + extent reasonably practicable; + + b. indicate if You modified the Licensed Material and + retain an indication of any previous modifications; and + + c. indicate the Licensed Material is licensed under this + Public License, and include the text of, or the URI or + hyperlink to, this Public License. + + 2. You may satisfy the conditions in Section 3(a)(1) in any + reasonable manner based on the medium, means, and context in + which You Share the Licensed Material. For example, it may be + reasonable to satisfy the conditions by providing a URI or + hyperlink to a resource that includes the required + information. + + 3. If requested by the Licensor, You must remove any of the + information required by Section 3(a)(1)(A) to the extent + reasonably practicable. + + 4. If You Share Adapted Material You produce, the Adapter's + License You apply must not prevent recipients of the Adapted + Material from complying with this Public License. + + +Section 4 -- Sui Generis Database Rights. + +Where the Licensed Rights include Sui Generis Database Rights that +apply to Your use of the Licensed Material: + + a. for the avoidance of doubt, Section 2(a)(1) grants You the right + to extract, reuse, reproduce, and Share all or a substantial + portion of the contents of the database; + + b. if You include all or a substantial portion of the database + contents in a database in which You have Sui Generis Database + Rights, then the database in which You have Sui Generis Database + Rights (but not its individual contents) is Adapted Material; and + + c. You must comply with the conditions in Section 3(a) if You Share + all or a substantial portion of the contents of the database. + +For the avoidance of doubt, this Section 4 supplements and does not +replace Your obligations under this Public License where the Licensed +Rights include other Copyright and Similar Rights. + + +Section 5 -- Disclaimer of Warranties and Limitation of Liability. + + a. UNLESS OTHERWISE SEPARATELY UNDERTAKEN BY THE LICENSOR, TO THE + EXTENT POSSIBLE, THE LICENSOR OFFERS THE LICENSED MATERIAL AS-IS + AND AS-AVAILABLE, AND MAKES NO REPRESENTATIONS OR WARRANTIES OF + ANY KIND CONCERNING THE LICENSED MATERIAL, WHETHER EXPRESS, + IMPLIED, STATUTORY, OR OTHER. THIS INCLUDES, WITHOUT LIMITATION, + WARRANTIES OF TITLE, MERCHANTABILITY, FITNESS FOR A PARTICULAR + PURPOSE, NON-INFRINGEMENT, ABSENCE OF LATENT OR OTHER DEFECTS, + ACCURACY, OR THE PRESENCE OR ABSENCE OF ERRORS, WHETHER OR NOT + KNOWN OR DISCOVERABLE. WHERE DISCLAIMERS OF WARRANTIES ARE NOT + ALLOWED IN FULL OR IN PART, THIS DISCLAIMER MAY NOT APPLY TO YOU. + + b. TO THE EXTENT POSSIBLE, IN NO EVENT WILL THE LICENSOR BE LIABLE + TO YOU ON ANY LEGAL THEORY (INCLUDING, WITHOUT LIMITATION, + NEGLIGENCE) OR OTHERWISE FOR ANY DIRECT, SPECIAL, INDIRECT, + INCIDENTAL, CONSEQUENTIAL, PUNITIVE, EXEMPLARY, OR OTHER LOSSES, + COSTS, EXPENSES, OR DAMAGES ARISING OUT OF THIS PUBLIC LICENSE OR + USE OF THE LICENSED MATERIAL, EVEN IF THE LICENSOR HAS BEEN + ADVISED OF THE POSSIBILITY OF SUCH LOSSES, COSTS, EXPENSES, OR + DAMAGES. WHERE A LIMITATION OF LIABILITY IS NOT ALLOWED IN FULL OR + IN PART, THIS LIMITATION MAY NOT APPLY TO YOU. + + c. The disclaimer of warranties and limitation of liability provided + above shall be interpreted in a manner that, to the extent + possible, most closely approximates an absolute disclaimer and + waiver of all liability. + + +Section 6 -- Term and Termination. + + a. This Public License applies for the term of the Copyright and + Similar Rights licensed here. However, if You fail to comply with + this Public License, then Your rights under this Public License + terminate automatically. + + b. Where Your right to use the Licensed Material has terminated under + Section 6(a), it reinstates: + + 1. automatically as of the date the violation is cured, provided + it is cured within 30 days of Your discovery of the + violation; or + + 2. upon express reinstatement by the Licensor. + + For the avoidance of doubt, this Section 6(b) does not affect any + right the Licensor may have to seek remedies for Your violations + of this Public License. + + c. For the avoidance of doubt, the Licensor may also offer the + Licensed Material under separate terms or conditions or stop + distributing the Licensed Material at any time; however, doing so + will not terminate this Public License. + + d. Sections 1, 5, 6, 7, and 8 survive termination of this Public + License. + + +Section 7 -- Other Terms and Conditions. + + a. The Licensor shall not be bound by any additional or different + terms or conditions communicated by You unless expressly agreed. + + b. Any arrangements, understandings, or agreements regarding the + Licensed Material not stated herein are separate from and + independent of the terms and conditions of this Public License. + + +Section 8 -- Interpretation. + + a. For the avoidance of doubt, this Public License does not, and + shall not be interpreted to, reduce, limit, restrict, or impose + conditions on any use of the Licensed Material that could lawfully + be made without permission under this Public License. + + b. To the extent possible, if any provision of this Public License is + deemed unenforceable, it shall be automatically reformed to the + minimum extent necessary to make it enforceable. If the provision + cannot be reformed, it shall be severed from this Public License + without affecting the enforceability of the remaining terms and + conditions. + + c. No term or condition of this Public License will be waived and no + failure to comply consented to unless expressly agreed to by the + Licensor. + + d. Nothing in this Public License constitutes or may be interpreted + as a limitation upon, or waiver of, any privileges and immunities + that apply to the Licensor or You, including from the legal + processes of any jurisdiction or authority. + + +======================================================================= + +Creative Commons is not a party to its public +licenses. Notwithstanding, Creative Commons may elect to apply one of +its public licenses to material it publishes and in those instances +will be considered the “Licensor.” The text of the Creative Commons +public licenses is dedicated to the public domain under the CC0 Public +Domain Dedication. Except for the limited purpose of indicating that +material is shared under a Creative Commons public license or as +otherwise permitted by the Creative Commons policies published at +creativecommons.org/policies, Creative Commons does not authorize the +use of the trademark "Creative Commons" or any other trademark or logo +of Creative Commons without its prior written consent including, +without limitation, in connection with any unauthorized modifications +to any of its public licenses or any other arrangements, +understandings, or agreements concerning use of licensed material. For +the avoidance of doubt, this paragraph does not form part of the +public licenses. + +Creative Commons may be contacted at creativecommons.org. + diff --git a/README.org b/README.org @@ -16,3 +16,9 @@ All the *code* is contained in another set of repositories: In order to *build* PDF you need to execute every code block in ~setup.org~, export the main file to LaTeX and compile it using ~make~. + +ARMA thesis is *licensed* under a Creative Commons Attribution 4.0 International +License. + +You should have received a copy of the license along with this work. If not, see +<http://creativecommons.org/licenses/by/4.0/>. diff --git a/arma-thesis-ru.org b/arma-thesis-ru.org @@ -1270,7 +1270,10 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args) \begin{align} \label{eq-problem-2d} & \phi_{xx}+\phi_{zz}=0,\\ - & \zeta_t + \zeta_x\phi_x = \frac{\zeta_x}{\sqrt{1 + \zeta_x^2}} \phi_x - \phi_z, & \text{на }z=\zeta(x,t).\nonumber + & \zeta_t + \zeta_x\phi_x + = \FracSqrtZetaX{\zeta_x} \phi_x + - \FracSqrtZetaX{1} \phi_z, + & \text{на }z=\zeta(x,t).\nonumber \end{align} Для ее решения воспользуемся методом Фурье. Возьмем преобразование Фурье от обоих частей уравнений Лапласа и получим @@ -1313,7 +1316,7 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args) \begin{equation*} \zeta_t = - \left( i f(x) - 1 \right) + \left( i f(x) - 1/\SqrtZetaX \right) \left[ \Fun{z} \ast @@ -1326,7 +1329,7 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args) E(u) = \frac{1}{2\pi u} \frac{ - \FourierY{\zeta_t / \left(i f(x) - 1\right)}{u} + \FourierY{\zeta_t / \left(i f(x) - 1/\SqrtZetaX\right)}{u} }{ \FourierY{\Fun{z}}{u} } @@ -1342,7 +1345,7 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args) \InverseFourierY{ \frac{e^{2\pi u z}}{2\pi u} \frac{ - \FourierY{ \zeta_t / \left(i f(x) - 1\right) }{u} + \FourierY{ \zeta_t / \left(i f(x) - 1/\SqrtZetaX\right) }{u} }{ \FourierY{ \Fun{\zeta(x,t)} }{u} } @@ -1387,10 +1390,11 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args) \phi(x,z) = \InverseFourierY{ \Sinh{2\pi u (z+h)} E(u) }{x}. \end{equation*} Подставляя \(\phi\) в граничное условие на свободной поверхности, получаем -\begin{equation*} - \zeta_t = f(x) \InverseFourierY{ 2\pi i u \Sinh{2\pi u (z+h)} E(u) }{x} - - \InverseFourierY{ 2\pi u \SinhX{2\pi u (z+h)} E(u) }{x}. -\end{equation*} +\begin{align*} + \zeta_t & = f(x) \InverseFourierY{ 2\pi i u \Sinh{2\pi u (z+h)} E(u) }{x} \\ + & - \frac{1}{\SqrtZetaX} + \InverseFourierY{ 2\pi u \SinhX{2\pi u (z+h)} E(u) }{x}. +\end{align*} Здесь \(\sinh\) и \(\cosh\) дают схожие результаты вблизи свободной поверхности, и, поскольку эта область является наиболее интересной с точки зрения практического применения, положим \(\Sinh{2\pi{u}(z+h)}\approx\SinhX{2\pi{u}(z+h)}\). Выполняя @@ -1403,7 +1407,7 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args) \InverseFourierY{ \frac{\Sinh{2\pi u (z+h)}}{2\pi u} \frac{ - \FourierY{ \zeta_t / \left(i f(x) - 1\right) }{u} + \FourierY{ \zeta_t / \left(i f(x) - 1/\SqrtZetaX\right) }{u} }{ \FourierY{ \FunSecond{\zeta(x,t)} }{u} } @@ -1492,10 +1496,10 @@ Mathematica\nbsp{}cite:mathematica10. В линейной теории широ \label{eq-problem-3d} & \phi_{xx} + \phi_{yy} + \phi_{zz} = 0,\\ & \zeta_t + \zeta_x\phi_x + \zeta_y\phi_y - = - \frac{\zeta_x}{\SqrtZeta{1 + \zeta_x^2 + \zeta_y^2}} \phi_x - +\frac{\zeta_y}{\SqrtZeta{1 + \zeta_x^2 + \zeta_y^2}} \phi_y - - \phi_z, & \text{на }z=\zeta(x,y,t).\nonumber + = \FracSqrtZetaY{\zeta_x} \phi_x + + \FracSqrtZetaY{\zeta_y} \phi_y + - \FracSqrtZetaY{1} \phi_z, \nonumber\\ + & \text{на }z=\zeta(x,y,t).\nonumber \end{align} Для ее решения также воспользуемся методом Фурье. Возьмем преобразование Фурье от обоих частей уравнений Лапласа и получим @@ -1530,11 +1534,11 @@ Mathematica\nbsp{}cite:mathematica10. В линейной теории широ \begin{array}{rl} \zeta_t = & i f_1(x,y) \InverseFourierY{2 \pi u \Sinh{2\pi \Kveclen (z+h)}E(u,v)}{x,y} \\ + & i f_2(x,y) \InverseFourierY{2 \pi v \Sinh{2\pi \Kveclen (z+h)}E(u,v)}{x,y} \\ - - & \InverseFourierY{2 \pi \sqrt{u^2+v^2} \Sinh{2\pi \Kveclen (z+h)}E(u,v)}{x,y} + - & f_3(x,y) \InverseFourierY{2 \pi \sqrt{u^2+v^2} \Sinh{2\pi \Kveclen (z+h)}E(u,v)}{x,y} \end{array} \end{equation*} -где \(f_1(x,y)={\zeta_x}/{\SqrtZeta{1+\zeta_x^2+\zeta_y^2}}-\zeta_x\) и -\(f_2(x,y)={\zeta_y}/{\SqrtZeta{1+\zeta_x^2+\zeta_y^2}}-\zeta_y\). +где \(f_1(x,y)=\zeta_x/{\SqrtZetaY}-\zeta_x\), +\(f_2(x,y)=\zeta_y/{\SqrtZetaY}-\zeta_y\) и \(f_3(x,y)=1/\SqrtZetaY\). Также как и в разделе\nbsp{}[[#sec-pressure-2d]] мы предполагаем, что \(\Sinh{2\pi{u}(z+h)}\approx\SinhX{2\pi{u}(z+h)}\) вблизи свободной поверхности, @@ -1556,7 +1560,7 @@ Mathematica\nbsp{}cite:mathematica10. В линейной теории широ \label{eq-phi-3d} \phi(x,y,z,t) = \InverseFourierY{ \frac{ \Sinh{\smash{2\pi \Kveclen (z+h)}} }{ 2\pi\Kveclen } - \frac{ \FourierY{ \zeta_t / \left( i f_1(x,y) + i f_2(x,y) - 1 \right)}{u,v} } + \frac{ \FourierY{ \zeta_t / \left( i f_1(x,y) + i f_2(x,y) - f_3(x,y) \right)}{u,v} } { \FourierY{\mathcal{D}_3\left( x,y,\zeta\left(x,y\right) \right)}{u,v} } }{x,y}, \end{equation*} diff --git a/arma-thesis.org b/arma-thesis.org @@ -1245,7 +1245,10 @@ Two-dimensional Laplace equation with Robin boundary condition is written as \begin{align} \label{eq-problem-2d} & \phi_{xx}+\phi_{zz}=0,\\ - & \zeta_t + \zeta_x\phi_x = \frac{\zeta_x}{\sqrt{1 + \zeta_x^2}} \phi_x - \phi_z, & \text{at }z=\zeta(x,t).\nonumber + & \zeta_t + \zeta_x\phi_x + = \FracSqrtZetaX{\zeta_x}\phi_x + - \FracSqrtZetaX{1}\phi_z, + & \text{at }z=\zeta(x,t).\nonumber \end{align} Use Fourier method to solve this problem. Applying Fourier transform to both sides of the equation yields @@ -1286,7 +1289,7 @@ boundary condition yields \begin{equation*} \zeta_t = - \left( i f(x) - 1 \right) + \left( i f(x) - 1/\SqrtZetaX \right) \left[ \Fun{z} \ast @@ -1299,7 +1302,7 @@ to both sides of this equation yields formula for coefficients \(E\): E(u) = \frac{1}{2\pi u} \frac{ - \FourierY{\zeta_t / \left(i f(x) - 1\right)}{u} + \FourierY{\zeta_t / \left(i f(x) - 1/\SqrtZetaX\right)}{u} }{ \FourierY{\Fun{z}}{u} } @@ -1314,7 +1317,7 @@ into\nbsp{}eqref:eq-guessed-sol-2d yields formula for \(\phi(x,z)\): \InverseFourierY{ \frac{e^{2\pi u z}}{2\pi u} \frac{ - \FourierY{ \zeta_t / \left(i f(x) - 1\right) }{u} + \FourierY{ \zeta_t / \left(i f(x) - 1/\SqrtZetaX\right) }{u} }{ \FourierY{ \Fun{\zeta(x,t)} }{u} } @@ -1358,10 +1361,11 @@ into\nbsp{}eqref:eq-guessed-sol-2d-full yields \phi(x,z) = \InverseFourierY{ \Sinh{2\pi u (z+h)} E(u) }{x}. \end{equation*} Plugging \(\phi\) into the boundary condition on the free surface yields -\begin{equation*} - \zeta_t = f(x) \InverseFourierY{ 2\pi i u \Sinh{2\pi u (z+h)} E(u) }{x} - - \InverseFourierY{ 2\pi u \SinhX{2\pi u (z+h)} E(u) }{x}. -\end{equation*} +\begin{align*} + \zeta_t & = f(x) \InverseFourierY{ 2\pi i u \Sinh{2\pi u (z+h)} E(u) }{x} \\ + & - \frac{1}{\SqrtZetaX} + \InverseFourierY{ 2\pi u \SinhX{2\pi u (z+h)} E(u) }{x}. +\end{align*} Here \(\sinh\) and \(\cosh\) give similar results near free surface, and since this is the main area of interest in practical applications, we assume that \(\Sinh{2\pi{u}(z+h)}\approx\SinhX{2\pi{u}(z+h)}\). Performing analogous to the @@ -1373,7 +1377,7 @@ previous section transformations yields final formula for \(\phi(x,z)\): \InverseFourierY{ \frac{\Sinh{2\pi u (z+h)}}{2\pi u} \frac{ - \FourierY{ \zeta_t / \left(i f(x) - 1\right) }{u} + \FourierY{ \zeta_t / \left(i f(x) - 1/\SqrtZetaX\right) }{u} }{ \FourierY{ \FunSecond{\zeta(x,t)} }{u} } @@ -1459,10 +1463,10 @@ Three-dimensional version of\nbsp{}eqref:eq-problem is written as \label{eq-problem-3d} & \phi_{xx} + \phi_{yy} + \phi_{zz} = 0,\\ & \zeta_t + \zeta_x\phi_x + \zeta_y\phi_y - = - \frac{\zeta_x}{\SqrtZeta{1 + \zeta_x^2 + \zeta_y^2}} \phi_x - +\frac{\zeta_y}{\SqrtZeta{1 + \zeta_x^2 + \zeta_y^2}} \phi_y - - \phi_z, & \text{at }z=\zeta(x,y,t).\nonumber + = \FracSqrtZetaY{\zeta_x} \phi_x + + \FracSqrtZetaY{\zeta_y} \phi_y + - \FracSqrtZetaY{1} \phi_z, \nonumber\\ + & \text{at }z=\zeta(x,y,t).\nonumber \end{align} Again, use Fourier method to solve it. Applying Fourier transform to both sides of Laplace equation yields @@ -1496,11 +1500,11 @@ Plugging \(\phi\) into the boundary condition on the free surface yields \begin{array}{rl} \zeta_t = & i f_1(x,y) \InverseFourierY{2 \pi u \Sinh{2\pi \Kveclen (z+h)}E(u,v)}{x,y} \\ + & i f_2(x,y) \InverseFourierY{2 \pi v \Sinh{2\pi \Kveclen (z+h)}E(u,v)}{x,y} \\ - - & \InverseFourierY{2 \pi \Kveclen \SinhX{2\pi \Kveclen (z+h)}E(u,v)}{x,y} + - & f_3(x,y) \InverseFourierY{2 \pi \Kveclen \SinhX{2\pi \Kveclen (z+h)}E(u,v)}{x,y} \end{array} \end{equation*} -where \(f_1(x,y)={\zeta_x}/{\SqrtZeta{1+\zeta_x^2+\zeta_y^2}}-\zeta_x\) and -\(f_2(x,y)={\zeta_y}/{\SqrtZeta{1+\zeta_x^2+\zeta_y^2}}-\zeta_y\). +where \(f_1(x,y)={\zeta_x}/{\SqrtZetaY}-\zeta_x\), +\(f_2(x,y)={\zeta_y}/{\SqrtZetaY}-\zeta_y\) and \(f_3(x,y)=1/\SqrtZetaY\). Like in section\nbsp{}[[#sec-pressure-2d]] we assume that \(\Sinh{2\pi{u}(z+h)}\approx\SinhX{2\pi{u}(z+h)}\) near free surface, but in @@ -1522,7 +1526,7 @@ and plugging the result into\nbsp{}eqref:eq-guessed-sol-3d yields formula for \label{eq-phi-3d} \phi(x,y,z,t) = \InverseFourierY{ \frac{ \Sinh{\smash{2\pi \Kveclen (z+h)}} }{ 2\pi\Kveclen } - \frac{ \FourierY{ \zeta_t / \left( i f_1(x,y) + i f_2(x,y) - 1 \right)}{u,v} } + \frac{ \FourierY{ \zeta_t / \left( i f_1(x,y) + i f_2(x,y) - f_3(x,y) \right)}{u,v} } { \FourierY{\mathcal{D}_3\left( x,y,\zeta\left(x,y\right) \right)}{u,v} } }{x,y}, \end{equation} diff --git a/preamble.tex b/preamble.tex @@ -63,6 +63,12 @@ % properly aligned version of sqrt for \zeta_y^2 \newcommand{\SqrtZeta}[1]{\sqrt{\vphantom{\zeta_x^2}\smash[b]{#1}}} +% normalisation +\newcommand{\SqrtZetaX}{\sqrt{1 + \zeta_x^2}} +\newcommand{\SqrtZetaY}{\SqrtZeta{1 + \zeta_x^2 + \zeta_y^2}} +\newcommand{\FracSqrtZetaX}[1]{\frac{#1}{\SqrtZetaX{}}} +\newcommand{\FracSqrtZetaY}[1]{\frac{#1}{\SqrtZetaY{}}} + % wave vector \newcommand{\Kvec}{\vec{k}} \newcommand{\Kveclen}{\lvert\smash[b]{\Kvec}\rvert}