Breaking wave speed given Hs and Tp

Waves break not because of their speed, which as was pointed out, tsunami waves travel at speeds of hundreds of kilometers per hour in deep water without breaking, but because the steepness of the wave (steepness = wave height / wave length) exceeds a certain value. For most waves breaking on shore, the wave steepness is usually equal to or greater than 1/7.

In deep water, tsunami wavelengths can reach 100 km, where "regular" deep water waves have lengths of a few hundred meters, at most.

As waves move shoreward from deep water, their height increases while their wavelength decreases, so that a wave which in deep water might have a steepness of 1/20 will see this value increase as the water depth decreases.

 

Is this true? It is the common explanation for why a wave slows down in shallow water, but it's never made sense to me.

Friction in fluid mechanics is typically confined to a comparatively thin boundary layer along the solid surface. The shear (friction) effects are negligible outside of this boundary layer. The shallow water equations are derived for inviscid flow, so no friction is involved. Figure 4 shows the forces acting on a volume of shallow water are pressure forces, not friction forces.

The figures shown in daiquiri's post also don't support the friction explanation. The direction of the orbital flow at the bottom is toward the sea, not the shore. So the shear stress is a force in the direction of the shore. If anything, this would accelerate the wave.

Finally, what we think of as the speed of a wave is the phase velocity of the pattern of the wave. It's not a bulk movement of the water. Friction doesn't apply to the phase velocity as it would to a stream of water flowing over a surface at the same speed. So the whole friction hypothesis doesn't hang together for me.

This paper, www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA098483, is one of the few I've seen that does talk about the contribution of friction to wave shoaling. As you'd expect there's a roughness effect and a Reynolds number that's involved. But if you set the friction coefficient to zero, the wave still shoals. Friction is a contributor, but it's not the whole explanation.

The mathematics of wave shoaling say it is actually the group velocity which slows down in shallow water, and it's the group velocity that carries the energy of the wave.

In the reports I've read, either there's a very simple "Just-So Stories" explanation (it's friction slowing the wave down) or a mathematical derivation, but not a convincing description of the actual physical mechanisms at work in a shoaling wave.

 

Valid question indeed.
However, this part is IMO not necessarily so straightforward:

What happens if you have a volume of fluid with bounded particles describing an orbital motion (hence with an angular momentum), and you slow down the bottom part of the fluid volume? IMO, the conservation of the angular momentum requires the upper part of the volume to accelerate. Hence (IMO), the observed volume changes shape, skews and forms a forward-bulging head (because the upper half now travels faster), which eventually becomes a breaking crest due to gravity.
If this consideration is true, then friction plays a central role in the formation of breaking waves. But it should be demonstrated mathematically, and I sincerely don't have time for that.
Cheers

 

Actually, it can be demonstrated mathematically, with a very simple model. I call it "two-bricks wave model" since I don't know if it has an official name...

Check the attached pic. The reference frame is lagrangean.



Hope it is clear enough.

Cheers

 

I suspect the cause of wave shoaling is not that friction slows down the orbital motion at the bottom, but that the bottom inhibits the vertical component of the orbital motion.

As I understand it, mathematically, the dispersion relationship from linear wave theory is:

This equation is derived without any reference to viscous effects (friction), but is dependent on the depth. As the wave gets into water shallower than half a wavelength, the frequency (period, T) stays the same, but the wavelength gets shorter. The group velocity (red), which was half the phase velocity in deep water becomes equal to the phase velocity (blue). The phase velocity is what we normally think of as the wave speed, but the group velocity is what determines the energy of the wave.



The fact that the group velocity equals the phase velocity means waves of all wavelengths are traveling at the same speed in shallow water (depth less than 5% of the wavelength). This keeps a wave packet concentrated, as the long wavelengths can't leave the shorter wavelengths behind like they do in deep water.

The energy flux of the wave is the product of the wave energy density with the group velocity. The average energy density of the wave is 1/2*density*g*Amplitude^2. The energy flux is conserved, so if the group velocity slows down, the energy density has to increase, which means the amplitude increases. Hence the wave gets higher in shallow water.

All this comes from linear wave theory, and friction is not involved at all. So the mathematics are clear about that. It's not primarily friction that makes a wave shoal.

But it still leaves me without a good qualitative understanding of the physical mechanisms at work.

 

Not sure if this helps, but I haven't seen it posted here yet.

https://en.wikipedia.org/wiki/Iribarren_number

 

Thanks for the link.

 

I get the point and I stand corrected. Friction is not necessary for wave distorsion and breaking. It is an inviscid phenomenon, at least in the first stages.

However, I don't think it is possible to get a qualitative understanding of the underlying physics if mathematical concepts like group velocity, phase velocity etc. are used as starting points. They are expressions of a final cumulative solution of a complex set of equations, and are not easy to break down into elementary contributions. IMO a much simpler model is necessary, capable of isolating single basic contributions - just like it is done for linearized 2D airfoil analysis. I also think that a moving reference frame might be the appropriate one for that purpose. My two-element approach was a try in that direction. Evidently an incomplete one, though it apparently gives a first hint about how momentum conservation can play a role in skewing the wave shape. Perhaps it can be improved, by integrating it with the energy equation and with appropriate boundary conditions which rappresent the varying water depth. I might try doing it, time (and my math capabilities) permitting.

Cheers

 

I think you might be able to calculate the horizontal velocity at the top (and bottom) of the wave before it starts to break, as a function of wave length and wave height, and then assume the same velocity is maintained when it breaks.