Can someone explain why waves shoal?

So, I was talking about Tsunamis with friends and as I started looking into it, I learned about the phenomena of shoaling. As water begins moving from deep areas into shallow areas, the waves increase in height until they crash or break onto shore. Can anyone help explain why this happens physically (without equations). I understand the equations that explain why it happens- $$v = f \cdot\lambda$$ (speed = frequency x wavelength). Since the frequency doesn't change, as the waves get slower in shallower water, the wavelength must also decrease. I guess I don't understand why this translates into a physically taller wave (higher amplitude); why don't the waves just kind of pass over each other. What causes them to (compress?) and increase in size. Can anyone explain the physics behind this specifically?

• If the energy does not change (much) with time then: Wavelength shortening -> same energy in shorter distance -> more potential energy density -> increased height of water. Commented Sep 30, 2022 at 13:50
• @Peter maybe I'm asking the wrong questions. What property of water/physics allows the waves to grow in height from this phenomena. From just a basic perspective, it would seem to make more sense that the waves would just roll over each other rather than accumulate in height.. Commented Sep 30, 2022 at 14:00
• If waves slow down then the wavelength must get shorter. If waves roll over each other that would mean the front waves slowed down and the back ones didn't in the same place. The energy keeps arriving at the same rate, so it is squeezed into a smaller distance. That means the water (not the wave) is moving faster, so when it rises it can go further. Commented Sep 30, 2022 at 14:29
• @Peter Ahhh, I guess I was overlooking that part. That makes sense. Thank you! Commented Sep 30, 2022 at 14:41
• The phase speed of the wave depends on amplitude (higher amplitude = faster wave), which results in something called nonlinear wave steepening. If there is insufficient energy dissipation to stop/limit the steepening, the wave will break (since water isn't a solid, if you put water over nothing but air, it will fall). The waves increase in amplitude in shallow water since their group speed depends on the depth of the water, and group speed is tied to energy density... Commented Dec 5, 2022 at 21:57

It sounds like you're asking for a way to think about it which makes sense. You don't disagree with the math, it just doesn't make sense.

You don't have a problem with the velocity of the waves slowing down. And you see that the frequency will be the same, so the wavelength must decrease. Your question is why the amplitude increases.

One alternative would be for the amplitude to stay the same with the waves closer together. Another would be for the waves to somehow roll over each other.

Consider the first possibility. Since it's water waves, the energy in the wave is the potential energy represented by water at a distance from the average water level. A wave with double the frequency and the same amplitude, has less energy. Right?

A simple bit of calculus, and we find they -- wait a minute, they have the same energy! WTF?

Oh! We're counting the energy in two wave crests compared to one. If you squeeze one crest-and-trough into half the space, of course the crest needs to be higher to get the same volume.

OK, the second possibility. The wave moves slower, so the wave crests roll over each other? I don't see how to imagine that. Some of the waves move slower and some don't, so the fast ones move past the slow ones? The waves don't slow down after all so they keep going at the same height? I don't see how to imagine it, so I don't know how to argue against it.

But I can see it the regular way. The wave slows down so it gets squeezed into a smaller volume. The same amount of water squeezed into a smaller space goes higher and lower.

• Thank you! I'm glad you were able to understand where my disconnect was. Oftentimes with physics questions, I usually never disagree with he math but I have trouble conceptualizing and making it make sense in... I guess practical terms. This was exactly what I was looking for to help me conceptualize it. Commented Sep 30, 2022 at 15:37

The wave is a series of crests and troughs. The crest is a region where water is above the mean level.

As the wave nears shore, the leading edge of a crest slows down, allowing the trailing edge (and the crests and troughs behind) to catch up. The length of the crest shrinks, but it contains the same amount of water. The water has to pile up higher.

The trailing edge of the crest cannot pass the leading edge because it reaches shallow water and slows down too.

It is the same for the following crests and troughs. Suppose the bottom got shallow in the middle of the ocean, not near shore. A wave in the shallow area would travel more slowly. Waves behind would get closer, and slow down as the entered the shallow area.

They arrive one per wave period. They don't travel as far in a period, so the distance between is smaller.

• This is what I was looking for.. Thank you! Commented Sep 30, 2022 at 14:41