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, \(r=r_0 \exp(kr)\)
\(V(X,Y,t)=v(x,y,t)\)
\(p=\frac{g}{\rho}(h-y)\)
\(P=\rho \mu\)
\(\mu=-gy+r_0 \omega^2(\exp(ky)-1)\)
\(\omega^2=gk\)
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