浅水方程式を解く：慣性重力波：地衡流：ケルビン波

$ \frac{du}{dt}-fv=-g\frac{\partial\eta}{\partial x}, \qquad \frac{du}{dt}+fu=-g\frac{\partial\eta}{\partial y}

$ \frac{\partial \eta}{\partial t}+\frac{\partial}{\partial x}((H+\eta)u)+\frac{\partial}{\partial y}((H+\eta)v)=0

つまり、非線形項を無視した形にして

u, v, ηは十分に小さいものとしているので，u^2は無視できる，としている

$ \tag{1} \frac{\partial u}{\partial x}-fv=-g\frac{\partial\eta}{\partial x}

$ \tag{2} \frac{\partial v}{\partial t}+fu=-g\frac{\partial\eta}{\partial y}

$ \tag{3} \frac{\partial \eta}{\partial t}+\frac{\partial}{\partial x}(Hu)+\frac{\partial}{\partial y}(Hv)=0

$ \tag{4} \frac{\partial}{\partial x}(2)-\frac{\partial}{\partial y}(1) \rightarrow f\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}\right)=-\frac{\partial}{\partial t}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial x}\right)

$ \tag{5} \frac{\partial}{\partial t}\left(\frac{\partial}{\partial x}(1)+\frac{\partial}{\partial y}(2)\right) \rightarrow \frac{\partial^2}{\partial t^2}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)-f\frac{\partial}{\partial t}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+g\frac{\partial}{\partial t}\left(\frac{\partial^2 \eta}{\partial x^2}+\frac{\partial^2 \eta}{\partial y^2}\right)=0

(5)に(4)を代入し，

$ \tag{6} \left(\frac{\partial^2}{\partial t^2}+f^2\right)\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)+g\frac{\partial}{\partial t}\left(\frac{\partial^2 \eta}{\partial x^2}+\frac{\partial^2 \eta}{\partial y^2}\right)=0

$ -\frac{1}{H}\frac{\partial \eta}{\partial t}= \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}

これを6式に代入することで，次の波動方程式(7)を得る．

$ \tag{7} \frac{\partial}{\partial t}\left(\left(\frac{\partial^2}{\partial t^2}+f^2 \right) \eta+gH \nabla_H^2 \eta\right)=0

ここで，波動解を複素数波で，

$ \eta=\eta_0 \exp(i(kx+ly-\sigma t+\phi))

と置く．

ただし，

(7)式に波動解を代入することで，

$ \tag{8} \sigma(\sigma^2-gH|\mathbf K|^2-f^2)=0

この解として，次の二つの分散関係式を得る．

$ \tag{9} \sigma=0

(9)式は定常であることを示す．つまり，

$ \frac{\partial}{\partial t}=0

$ \tag{10} \sigma=\pm \sqrt{gH|\mathbf K|^2+f^2}

波数が虚数の時