I made this presentation for a graduate course in engineering called "Advanced Fluid Mechanics."

1. The Study of Water Waves
Breaking
December 7th 2006

2. Contents
• Introduction
• Shallow Water Waves
• Deep Water Waves
• Beach Waves
• Breaking Waves
• Solutions
• Analytical
• Experimental
• Computational
• Conclusion

3. Introduction: Water Waves
• Stokes wave train
generated by an
oscillating plunger
• 28 wavelengths
away from the
generation point, the
waves disintegrate

4. The Waves Become Unstable

5. In sight
• In the following slides the depth is 22% of the
wavelength.
• White particles are photographed over one
period demonstrating the trajectories of the water
particles
• A standing wave is created by propagating
waves at the left and then having the waves
reflected at the far right
• The radius of the circles traced by the particles
trajectories decreases exponentially with depth.

9. Shallow Water Waves
• In shallow water waves, the particle
trajectories are elliptical versus the
circular paths found in deep water
waves
Deep Water Waves
• Deep water waves are those whose
wavelength is less than the height of the
body of water

10. Beach Waves
• Biesel noted that, as ocean waves approach
the beach the water gets shallower. This
causes:
• The amplitude of the waves to increase
• The wave speed decreases
• The wavelength decreases
• The water particles tend to align there elliptical
trajectories with the bottom s slope
Circles -> Ellipses
Floor

11. Breaking Waves
• The changes in the properties of the wave
lead to structural instability in a non-linear
manner. As the wave approaches the beach
the bottom is slowed down while the top part
continues forward.
• Thusly, the wave breaks.

15. Solutions
• Analytical
• Experimental
• Computational

16. Analytical
• A parametric solution for a breaking
wave has been developed by Longuet
and Higgins.
• This solution only describes the flow up
to the moment of impact. Another
solution involving turbulence is required
to describe the aftermath.

17. Parametric representation;
branch points
• Consider the flow to be incompressible,
irrotational and in two dimensions
• x = ! + i! and z = x +iy are the particle
position coordinates
• Assume x and z are analytic functions of complex
! and or t
• x can be expresssed as a function of z if ! is
eliminated
• The following suffixed terms represent the partial
differentiation with respect to ! or t
• W* is the particle velocity, where W=X!/Z!

18. Say z! = 0 at !=!0, where z=z0 then,
near here,
z-z0~1/2(!-!0)2z!!
!-!0 !(z- z0)1/2, then,
x-x0 ~ (!-!0)(z-z0)1/2

19. Boundary Condition
From Bernoulli;
-2p = (xt-Wzt) + c.c. + WW* - g(z+z*) – 2!,
-2 Dp/Dt = (xtt-Wztt) + 2K(x!t-Wz!t) + K2(x!!-Wz!!) + c.c. – g(W+W*)– 2!,
where
K = D!/Dt = W*-zt .
z!
at the boundary condition of the free moving surface,
p = 0, Dp/Dt = 0 so,
zt = W*, K= 0
then,
-2 Dp/Dt = (xtt-Wztt) + c.c. – g(W+W*)– 2!,

20. We ve assumed that the flow is Lagrangian and ! is real at the free surface. Now,
ztt – g = irz!
where r is some function of ! and t which is real on the boundary
it was found that the particle acceleration is,
a = D/Dt zt(!*) = ztt(!*) + K*zt!*(!*),
K= [zt(!*)-zt(!)]/z! ,
Frames of Reference
ztt – g = irz!
is a non-homogeneous linear diferential equation for z(!,t) with solutions z0(!,t) and z0 + z1(!,t). then,
ztt = irz!
z1 = z0 -1/2 gt2,

21. The Stokes Corner Flow
Velocity potential
X = - 1/12 g2(!-t)3 = 2/3 ig1/2z3/2
Longuet and Higgins proved that at the tip of
the plunging wave there s an interior flow which
is the focus of a rotating hyperbolic flow.

22. i! = !
½ t (ztt-g) = z!,
Upwelling Flow
z = - ½ gt!
z! = - ½ gt,
W = zt*(-!) = ½ g!,
x! = ¼ g2t!,
x = - 1/8 g2t!2 = - z2/2t
The free surface is the y-axis
Velocity potential
! = - x2 – y2
2t
the streamlines are
!=- xy = constant,
t
For t > 0 the flow represents a
decelerated upwelling, in which
the vertical and horizontal
components of flow are given by
!x = - x/t, !y = y/t
at x = 0 the pressure is constant
-py = vt +(uvx+vvy),
where (u,v) = (!x,!y) à py = 0
-p = !t + ½ (!x2 +!y2) – gx

23. Now, we want the solution to the homogeneous boundary condition. This will describe the flows
complementary to the upwelling flow.
½ tztt = z!
when ! +!* = 0
Make z a polynomial, z = bn!n + bn-1!n-1 + … + b0; bn is a function of time
bn!n + is of the form At + B and A and B are constant
P0 = t,
P1 = t! + t2,
P2 = t!2 + 2t2! + 2/3 t3,
P3 = t!2 + 3t2!2 + 2t3! 1/3 t4.
Q0 = 1,
Q1 = ! + 2tln|t|,
Q2 = !2 + 4t!ln|t| + 4t2(ln|t| - 3/2),
Q3 = !3 + 6t!2ln|t| + 12t2!(ln|t| - 3/2) + 4t3(ln|t| - 7/3)
z = !n (AnPn + BnQn)

24. Physical Meaning
To understand these flows; consider a linear,
cubic, and quadratic flow.

25. Comparison with Observation
Only the front face of the
wave is being described

26. Experimental
• The following is a numerical approach
involving coefficients that were
experimentally determined.

28. The Setup
The wave s velocity
fields were measured
laser Doppler
velocimeter (LDV) and
particle image
velocimetry (PIV)

29. Computational
• The computation of a breaking wave
acts as a good test to see if the
numerical model accurately depicts
nature.

30. A Popular Model
In 1804 Gerstner Developed a wave model whose particle
(x,z) coordinates are mapped as so:
z
x = x0 –Rsin(Kx0-!t)
z = z0 +Rcos(Kx0-!t) x
Where,
R= R0eKz0 R0 = particle trajectory radius = 1/K
K= number of waves != angular speed
z0 = !A2/4! A= 2R != 2!/K

31. Biesel improved on this model to account for the particle trajectory s
tendency toward an elliptical shape.
Improving still on Biesel s model was Founier-Reeves
x = x0 + Rcos(!)Sxsin(!) + Rsin(!)Szcos(!)
z = z0 – Rcos(!)Szcos(!) + Rsin(!)Sxsin(!)
Sx = (1-e-kxh)-1,Sz(1 – e-Kzh)
sin(!) = sin(!)e-K0h
! = - !t + !0x0K(x)!x
K(x) = K!/(tanh(K!h))1/2

32. Here:
K0 – relates depth to to the angle of the particles elliptical trajectory
Kx – is the enlargement factor on the major axis of the ellipse
Kz – is the reduction factor of the minor axis
These variables range between 0 and 1. They are used to tune the model in order to avoid
unreal results that arise from a negatively sloping beach.
Waves
(-)
Beach
Floor

33. Conclusion
An accurate model of a wave crashing on
the beach could yield beneficial
information for coastal structures such
as boats, or break-walls.
Also,
Perhaps a more accurate wave model
could assist in the design of surf boards

37. References
• Cramer, M.s. "Water Waves Introduction." Fluidmech.Net. 2004.
Cambridge University. 6 Dec. 2006 <http://www.fluidmech.net/tutorials/
ocean/w_waves.htm>.
• Crowe, Clayton T., Donald F. Elger, and John A. Roberson.
Engineering Fluid Mechanics. 7th ed. United States: John Wiley &
Sons, Inc., 2001. 350.
• Gonzato, Jean-Christophe, and Bertrand Le Saec. "A
Phenomenological Model of Coastal Scenes Based on Physical
Considerations." Laboratoire Bordelais De Recherche En Informatique.
• Lin, Pengzhi, and Philip L. Liu. "A Numerical Study of Breaking Waves
in the Surf Zone." Journal of Fluid Mechanics 359 (1998): 239-264.
• Longuet, and Higgins. "Parametric Solutions for Breaking Waves."
Journal of Fluid Mechanics 121 (1982): 403-424.
• Richeson, David. "Water Waves." June 2001. Dickinson College. 6
Dec. 2006 <http://users.dickinson.edu/~richesod/waves/index.html>.
• Van Dyke, Milton. An Album of Fluid Mechanics. 10th ed. Stanford,
California: The Parabolic P, 1982.

44. Hydraulic Jump