# 奥尔－索末菲方程

## 公式

${\displaystyle \mathbf {u} =\left(U(z)+u'(x,z,t),0,w'(x,z,t)\right)}$ ,

${\displaystyle {\frac {\mu }{i\alpha \rho }}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi }$ ,

${\displaystyle {1 \over i\alpha \,Re}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi }$ ,

${\displaystyle \alpha \varphi ={d\varphi \over dz}=0}$ ${\displaystyle \varphi }$ 为势函数）

${\displaystyle \alpha \varphi ={d\varphi \over dx}=0}$ ${\displaystyle \varphi }$ 为流函数）。

## 参考文献

• Orr, W. M'F. The stability or instability of the steady motions of a liquid. Part I. Proceedings of the Royal Irish Academy. A. 1907, 27: 9–68.
• Orr, W. M'F. The stability or instability of the steady motions of a liquid. Part II. Proceedings of the Royal Irish Academy. A. 1907, 27: 69–138.
• Sommerfeld, A. Ein Beitrag zur hydrodynamische Erklärung der turbulenten Flüssigkeitsbewegungen. Proceedings of the 4th International Congress of Mathematicians III. Rome. 1908: 116–124.