Difference between revisions of "Exact waves at surface of liquid"

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'''Exact waves at surface of liquid''' refers to the solution of the equations
  
'''Exact waves at surface of liquid''' refers to the solution of the equations
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ ~ \displaystyle
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ ~ \displaystyle
−
\frac {\mathrm d \vec v}{\mathrm d t}= - \frac{1}{\rho} \nabla p + f$
 +
\frac {\mathrm d \vec v}{\mathrm d t}= - \frac{1}{\rho} \nabla p + f\)
   
 
where
 
where
−
: $\displaystyle \frac {\mathrm d \vec v}{\mathrm d t} = \dot{\vec v} + \vec v_{,a} v_a$
 +
: \(\displaystyle \frac {\mathrm d \vec v}{\mathrm d t} = \dot{\vec v} + \vec v_{,a} v_a\)
   
−
This equation neglects viscosity $\nu \nabla^2 \vec v$.
 +
This equation neglects viscosity \(\nu \nabla^2 \vec v\).
−
$f$ is interpreted as gravitational force.
 +
\(f\) is interpreted as gravitational force.
   
−
Assume the non–compressive liquid, $\nabla \vec v =0$.
 +
Assume the non–compressive liquid, \(\nabla \vec v =0\).
   
 
The Blue tetrad (1993) suggests the solution
  
The Blue tetrad (1993) suggests the solution
   
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ ~ \displaystyle X(x,y,t) = x + r(y) \cos(kx-\omega t)$
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ ~ \displaystyle X(x,y,t) = x + r(y) \cos(kx-\omega t)\)
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ ~ \displaystyle Y(x,y,t) = y + r(y) \sin(kx-\omega t)$
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ ~ \displaystyle Y(x,y,t) = y + r(y) \sin(kx-\omega t)\)
   
−
In this notations, at $t=0$, $x=0$, the vertical speed is maximal.
 +
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 $x=0$.
 +
Perhaps, this is not best choice; it may have sense to have the crest at \(x=0\).
   
 
Notations:
  
Notations:
−
: $ c=\cos(kx-\omega t)$
 +
: \( c=\cos(kx-\omega t)\)
−
: $ s=\sin(kx-\omega t)$
 +
: \( s=\sin(kx-\omega t)\)
−
: $ r=r(y)$ ; $~ r'=r'(y)$
 +
: \( r=r(y)\) ; \(~ r'=r'(y)\)
   
−
Let $t=$const. Then
 +
Let \(t=\)const. Then
   
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ ~ \displaystyle X=X(x,y,t) = x + r c$
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ ~ \displaystyle X=X(x,y,t) = x + r c\)
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ ~ \displaystyle Y=Y(x,y,t) = y + r s$
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ ~ \displaystyle Y=Y(x,y,t) = y + r s\)
   
−
: $ \mathrm d X= (1-rsk) ~\mathrm d x + r' c ~ \mathrm d y$
 +
: \( \mathrm d X= (1-rsk) ~\mathrm d x + r' c ~ \mathrm d y\)
−
: $ \mathrm d Y= rck ~ \mathrm d x + (1+r's) ~ \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$
 +
:\( (1+r's) ~\mathrm d X - r' c ~ \mathrm d Y = \Big( (1-rsk)(1+r's) - rr' c^2 k \Big)~ \mathrm d x\)
−
:$ -rck ~ \mathrm d X + (1-rsk)~ \mathrm d Y = \Big( - r'rc^2 k (1-rsk)(1+r's) \Big)~ \mathrm d y$
 +
:\( -rck ~ \mathrm d X + (1-rsk)~ \mathrm d Y = \Big( - r'rc^2 k (1-rsk)(1+r's) \Big)~ \mathrm d y\)
   
−
Let $D=(1-rsk)(1+r'c)-r r- k c^2$
 +
Let \(D=(1-rsk)(1+r'c)-r r- k c^2\)
   
−
Then $D=1+ (r'-kr) s -k r r'$
 +
Then \(D=1+ (r'-kr) s -k r r'\)
   
−
:$ \frac{\partial x}{\partial X}=\frac{1}{D}(1+r's) ~ ~$, $~ ~ \frac{\partial x}{\partial Y}= \frac{-1}{D} r'c$
 +
:\( \frac{\partial x}{\partial X}=\frac{1}{D}(1+r's) ~ ~\), \(~ ~ \frac{\partial x}{\partial Y}= \frac{-1}{D} r'c\)
−
:$ \frac{\partial y}{\partial X}= \frac{-1}{D}r'ck ~ ~$, $~ ~\frac{\partial x}{\partial Y}= \frac{1}{D}(1-rck)$
 +
:\( \frac{\partial y}{\partial X}= \frac{-1}{D}r'ck ~ ~\), \(~ ~\frac{\partial x}{\partial Y}= \frac{1}{D}(1-rck)\)
   
 
Then
 
Then
   
−
:$ \nabla \vec v=
 +
:\( \nabla \vec v=
 
\frac{\partial v_x}{\partial x} \frac{ \partial x}{\partial X}+
  
\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_x}{\partial y} \frac{ \partial y}{\partial X}+
 
\frac{\partial v_y}{\partial x} \frac{ \partial x}{\partial Y}+
  
\frac{\partial v_y}{\partial x} \frac{ \partial x}{\partial Y}+
−
\frac{\partial v_y}{\partial y} \frac{ \partial y}{\partial Y}=$ $
 +
\frac{\partial v_y}{\partial y} \frac{ \partial y}{\partial Y}=\) \(
−
(kr'-r)\omega c$
 +
(kr'-r)\omega c\)
−
Hence, $r=r_0 \exp(kr)$
 +
Hence, \(r=r_0 \exp(kr)\)
   
−
$V(X,Y,t)=v(x,y,t)$
 +
\(V(X,Y,t)=v(x,y,t)\)
   
−
$p=\frac{g}{\rho}(h-y)$
 +
\(p=\frac{g}{\rho}(h-y)\)
   
−
$P=\rho \mu$
 +
\(P=\rho \mu\)
   
−
$\mu=-gy+r_0 \omega^2(\exp(ky)-1)$
 +
\(\mu=-gy+r_0 \omega^2(\exp(ky)-1)\)
   
   
−
$\omega^2=gk$
 +
\(\omega^2=gk\)
 
==References==
  
==References==
 
<references/>
  
<references/>

Latest revision as of 18:26, 30 July 2019

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


http://shipdesign.ru/Chizhiumov/SeaKeeping.pdf С.Д. Чижиумов. ОСНОВЫ ДИНАМИКИ СУДОВ НА ВОЛНЕНИИ. Учебное пособие. Комсомольск на Амуре 2010

http://surflibrary.org/wavephysics.pdf Michael Twardos mtwardos(Гав!)uci.edu The Physics of Ocean Waves (for physicists and surfers). August 31, 2004

http://www.scribd.com/doc/45280909/18/Potential-Flow I.R.Young. Wave generated ocean waves. 1999.

http://www.sciencedirect.com/science/article/pii/S0021999105002196 Dmitry Chalikova, Dmitry Sheinin. Modeling extreme waves based on equations of potential flow with a free surface. 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.

Л.И.Седов. Методы подобия и размерности в механике. Гл.11. §13 (1951)

Н.Е.Кочин, И.А.Кисель, Н.В.РОся. Теоретическая гидромеханика часть 1, гл VIII, §9,10, rostexizdat, 1948.

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

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