# When is a water surface not dispersive?

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• Daniel Petka
In summary, the article discusses the concept of water waves and how they move at different speeds depending on the frequency. The key idea is that when a raindrop hits the water surface, it creates a pulse with all frequencies, but in shallow water, the high frequencies move faster, leading to a chirp sound. However, this is contradictory to the concept of shallow water waves having no dispersion. The article also mentions how raindrops may react differently in other situations, such as hitting a lake or ocean, where the force of gravity becomes more important and all wave sizes move at the same speed. This leads to the question of whether deep water waves also disperse, with low frequencies moving faster. The article suggests that there may be a "
Daniel Petka
TL;DR Summary
Shallow water waves => high freqs faster, deep water waves => low freqs faster. When are low freqs and high freqs equally fast = no dispersion?
For my project, I need water waves of all frequencies to move at the same speed. I read this article, but struggled to grasp some concepts. The key idea of the article is that a raindrop hitting a water surface basically creates a pulse containing all the frequencies, and since the water is very shallow, the high frequencies move faster, which creates the chirp. There is something I am confusing, since according to wikipedia, shallow waves should have no dispersion, but the raindrop clearly shows otherwise.

Then, there is another part in the article which I don't get:

" Raindrops may react differently in other situations. Imagine that rain is hitting a lake or ocean – or those deep pothole puddles that require galoshes. Here, the raindrop hits the water, but the force due to gravity becomes more important. It moves waves of all sizes at the same speed which may overpower the rippling effect due to the surface force. "

But don't deep water waves disperse too? Low frequencies move faster in deep water.. The only thing that makes sense to me rn, is that there has to be some sweet spot where the effects of shallow water (high freq faster) and deep water (low freq faster) cancel out?.The article shows a perfect non-dispersive wave when the force of gravity goes up, but, of course the model is incomplete. The math here is a bit over my head, I would highly appreciate any help!

So, the ripple effect that the article is talking about are not gravitational waves, but capillary waves. They are dominated by surface tension. These waves are much faster apparently.

Gravity waves in shallow water (depth much less than wave length) indeed do not disperse, so all frequencies move indeed at the same speed.

Gravity waves in deep water do disperse, there the higher wavelength (lower frequency) waves are fastest.

## 1. When is a water surface considered to be non-dispersive?

A water surface is considered to be non-dispersive when the waves on the surface remain unchanged as they travel, meaning that they do not change direction or shape.

## 2. What factors can affect the dispersion of water waves?

The dispersion of water waves can be affected by factors such as wind speed, water depth, and the nature of the water surface (e.g. smooth or rough).

## 3. How does the depth of water affect dispersion?

The depth of water can affect dispersion by changing the speed of the waves. In shallow water, waves travel slower and are more likely to be dispersive, while in deeper water, waves travel faster and are less likely to be dispersive.

## 4. Can a water surface be both dispersive and non-dispersive?

No, a water surface can only be either dispersive or non-dispersive. It cannot exhibit both properties at the same time.

## 5. Why is it important to understand the dispersion of water waves?

Understanding the dispersion of water waves is important for various applications, such as predicting the behavior of tsunamis and designing structures that can withstand wave forces. It also helps in understanding the dynamics of ocean currents and weather patterns.

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