View source for Exact waves at surface of liquid

'''Exact waves at surface of liquid''' refers to the solution of the equations
: $ \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ ~ \displaystyle
\frac {\mathrm d \vec v}{\mathrm d t}= - \frac{1}{\rho} \nabla p + f$

where 
: $\displaystyle \frac {\mathrm d \vec v}{\mathrm d t} = \dot{\vec v} + \vec v_{,a} v_a$

This equation neglects viscosity $\nu \nabla^2 \vec v$.
$f$ is interpreted as gravitational force.

Assume the non–compressive liquid, $\nabla \vec v =0$.

The Blue tetrad (1993) suggests  the solution

: $ \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ ~ \displaystyle X(x,y,t) = x + r(y) \cos(kx-\omega t)$
: $ \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ ~ \displaystyle Y(x,y,t) = y + r(y) \sin(kx-\omega t)$

In this notations, at $t=0$, $x=0$, the vertical speed is maximal.
Perhaps, this is not best choice; it may have sense to have the crest at $x=0$.

Notations:
: $ c=\cos(kx-\omega t)$
: $ s=\sin(kx-\omega t)$
: $ r=r(y)$ ; $~ r'=r'(y)$

Let $t=$const. Then

: $ \!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ ~ \displaystyle X=X(x,y,t) = x + r c$
: $ \!\!\!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ ~ \displaystyle Y=Y(x,y,t) = y + r s$

: $ \mathrm d X= (1-rsk) ~\mathrm d x + r' c ~ \mathrm d y$
: $ \mathrm d Y= rck ~ \mathrm d x + (1+r's) ~ \mathrm d y$

:$ (1+r's) ~\mathrm d X - r' c ~ \mathrm d Y = \Big( (1-rsk)(1+r's) - rr' c^2 k \Big)~ \mathrm d x$
:$ -rck ~ \mathrm d X + (1-rsk)~ \mathrm d Y = \Big( - r'rc^2 k (1-rsk)(1+r's) \Big)~ \mathrm d y$

Let $D=(1-rsk)(1+r'c)-r r- k c^2$

Then $D=1+ (r'-kr) s -k r r'$

:$ \frac{\partial x}{\partial X}=\frac{1}{D}(1+r's) ~ ~$, $~ ~ \frac{\partial x}{\partial Y}= \frac{-1}{D} r'c$
:$ \frac{\partial y}{\partial X}= \frac{-1}{D}r'ck ~ ~$, $~ ~\frac{\partial x}{\partial Y}= \frac{1}{D}(1-rck)$

Then 

:$ \nabla \vec v=
\frac{\partial v_x}{\partial x} \frac{ \partial x}{\partial X}+
\frac{\partial v_x}{\partial y} \frac{ \partial y}{\partial X}+
\frac{\partial v_y}{\partial x} \frac{ \partial x}{\partial Y}+
\frac{\partial v_y}{\partial y} \frac{ \partial y}{\partial Y}=$ $
(kr'-r)\omega c$
Hence, $r=r_0 \exp(kr)$

$V(X,Y,t)=v(x,y,t)$

$p=\frac{g}{\rho}(h-y)$

$P=\rho \mu$

$\mu=-gy+r_0 \omega^2(\exp(ky)-1)$


$\omega^2=gk$
==References==
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[[Journal of Computational Physics]], Volume 210, Issue 1, 20 November 2005, Pages 247–273.

А.И.Некрасов. Точная теория волн устоявшегося вида на поверхности тяжелой жидкости. Изв.АН СССР, 1951.

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Ю.З.Алешков. Теория волн на поверхности тяжелой жидкости. - Л, 1981, 196с. ФИ/I 78271

http://www.mathnet.ru/links/59058e5edc897f915d4b36c6d396ce80/vuu220.pdf
В. А. Баринов, К. Ю. Басинскии. НЕЛИНЕЙНЫЕ ВОЛНЫ СТОКСА НА ПОВЕРХНОСТИ СЛАБОВЯЗКОЙ
ЖИДКОСТИ. ВЕСТНИК УДМУРТСКОГО УНИВЕРСИТЕТА. 2011. Вып.2. p.112-122.

==Keywords==
[[Classical mechanics]]

[[Category:Waves]]
[[Category:Navier Stokes]]
[[Category:Classical mechanics]]
[[Category:Articles in English]]