Water Waves Over Obstacles: Higher Frequencies Grow, Not Decay

[person123]

TL;DR

While higher frequencies seem to generally dissipate more than lower frequencies (e.g. sound waves, vibrating strings), the opposite is the case for shallow water waves: why might this be the case?

In general, it seems that higher frequencies of a wave dissipate more than lower frequencies.

For sound waves, it explains why you can hear lower pitches from farther away. For a vibrating string or plate, the higher frequencies also dissipate first, with the fundamental fading last. For water sloshing in a container (say a cup), the fundamental, main sloshing, is also what remains. In the case of light, the greater scatter of higher frequencies explain why the sky is blue (maybe this doesn't count because it's not really dissipation though).

There is an exception though -- traveling shallow water waves. It is well established (from experimental, field, and numerical work), that when a wave passes an obstacle like an underwater reef (and so energy is dissipated) the lower frequencies die out but the higher frequencies actually increase in amplitude. It's also sort of intuitive -- it seems reasonable that a sinusoidal water wave would get morphed into messier, higher frequency waves on passing an obstacle.

Why does this break the rule? Is there anything to get out of it being reversed?
For literature on this from the linked paper:
"Generation of higher harmonics in waves propagating over submerged obstacles has long been known. Johnson et al. (1951 ) noted that over natural reefs the energy was transmitted as a multiple crest system. Jolas (1960) carried out experiments with a submerged obstacle of rectangular cross section and observed that the transmitted waves were noticeably shorter than the incident waves. In an experimental study investigating the performance characteristics of submerged breakwaters, Dattatri et al. ( 1978 ) pointed out rather complex forms of the transmitted waves, which indicated the presence of higher harmonics. Drouin and Ouellet ( 1988 ) and Kojima et al. (1990) reported their experimental results with immersed plates, the latter emphasizing the phenomenon of wave decomposition and associated harmonic generation past the obstacle. Quite recently, Rey et al. ( 1992 ) have reported similar results for laboratory waves passing over a bar."

 

[sophiecentaur]

Summary:: While higher frequencies seem to generally dissipate more than lower frequencies (e.g. sound waves, vibrating strings), the opposite is the case for shallow water waves: why might this be the case?

Why does this break the rule?

The rule refers to the Energy in the wave. What changes with a surface wave in shallow water is the shape of the wave and that doesn't necessarily involve more total energy. The total energy in the wave consists of the height of the peaks and the motion in the troughs.

You can often see the sea waves regaining their shape after passing over a shallow reef but the height, once the original shape is established, will be less.

 

[person123]

The rule refers to the Energy in the wave. What changes with a surface wave in shallow water is the shape of the wave and that doesn't necessarily involve more total energy. The total energy in the wave consists of the height of the peaks and the motion in the troughs.

I agree the total energy will decrease. However, the energy also shifts from lower frequencies to higher frequencies. If you look at the PSD before and after passing, the intensity of the fundamental frequency decreases, while the intensity of the harmonics actually increase.

You can often see the sea waves regaining their shape after passing over a shallow reef

I wasn't aware of that (although that does seem right) -- that's good to know. If the shape is restored, I guess that means the energy must have shifted back to the fundamental frequency.

 

[sophiecentaur]

The fundamental frequency cannot change*; the change in wave speed gives a change (shortening) of wavelength. The transmission medium (water of finite depth) is not linear so (if you choose to analyse the wave shape in terms of its frequency domain) then you could say that harmonics will be generated. There is nothing to forbid those harmonics combining (interfere with each other) in a way that re- generates a regular, nearly-sinusoidal. wave on the way out.

It's not easy to understand this because the speed of surface waves depends on depth and wavelength. The basic waves are only approximately sinusoidal and so the speeds of the harmonics will change, relatively. So, once the depth changes significantly, the resulting wave shape will change as the fundamental component catches up with and overtakes the higher frequency products. (That suggests that ocean wave shapes will change as they progress; I could believe that.)

*Yes; the waves become more distorted but it’s not correct that there is a change in frequency. The time for a cycle has to be unaltered because there has to be continuity of phase across a transition.

If the shape is restored, I guess that means the energy must have shifted back to the fundamental frequency.

That's how I have been looking at it. However, using the term "energy" could be misleading because what we really see is vertical displacement and the Impedance of the wave depends on the depth so it may involve no significant change in energy of the components because Energy relates to Displacement and Impedance.