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#author("2021-08-18T15:04:16+09:00","default:Miyashita","Miyashita")
#author("2024-02-27T12:02:57+09:00","default:Miyashita","Miyashita")
*微小振幅波 [#j521df4f]
#contents

そのうちまとめます．~
#ref(https://main-t-miyashita.ssl-lolipop.jp/hydrocoast/image/Notebooks/smallamplitude_deep.gif)
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**基礎式 [#cfafad1f]
~
水平・鉛直の断面2次元で，一様水深の空間を考える．~
非圧縮の連続式
\[
\nabla \cdot \boldsymbol{u} = 0
\]
（\(\boldsymbol{u}\) は水粒子の速度）に対して下記のように非回転としてポテンシャル理論を適用し，
\[
\frac{\partial \phi}{\partial x} = u, \: \frac{\partial \phi}{\partial y} = v
\]
この2式からラプラスの方程式を得る．ここで　\( \phi \)　は速度ポテンシャル．
\[
\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2} = 0 \\
\nabla^2 \phi = 0
\]
これが渦なし流れの基礎方程式であるが，この微分方程式を満たす \( \phi \) は無数に存在する．~
~
底面および水面での境界条件式と，速度ポテンシャル \( \phi \) や水面 \( \eta \) が周期性を持つ波であることの仮定を用いて~
これらの変数の解を得る．

**解の導出 [#z5c0ff6a]
気が向いたらちゃんと書く．
~

**参考図 [#t884192a]
アニメーションは Julia で作成した gif．~
***深海波 [#t6e9544b]
#ref(https://main-t-miyashita.ssl-lolipop.jp/hydrocoast/image/Notebooks/smallamplitude_deep.gif)
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***浅海波 [#g7611323]
#ref(https://main-t-miyashita.ssl-lolipop.jp/hydrocoast/image/Notebooks/smallamplitude_shallow.gif)
~
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***極浅海波（長波） [#k12d238c]
#ref(https://main-t-miyashita.ssl-lolipop.jp/hydrocoast/image/Notebooks/smallamplitude_veryshallow.gif)
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~
*参考文献 [#aea2deef]
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Takuya Miyashita
