Many of us imagine tsunamis as tall, surf-like waves, but in the deep ocean, their amplitude is actually quite small. Tsunamis get much taller as they approach the continental shelf and coastline. This process is known as shoaling, and the devastation caused by tsunamis is linked to how high they shoal. By looking at the fundamental wave characteristics of tsunamis in deep and shallow water, we can understand why shoaling happens.

What is shoaling?

Shoaling is an increase in wave amplitude that happens when water waves (not just tsunamis) go from deep to shallow water – particularly at the coast. Tsunamis have a small amplitude in deep water (often much less than a metre), but they can shoal up to many metres high in shallow waters. For New Zealand, a tall tsunami could run up to 10 metres higher than normal sea level – that’s about the height of a 3-storey building. In the 2004 Boxing Day tsunami, there were waves 15–30m high.

Surf waves also undergo shoaling – they have a greater amplitude than the ocean swells they form from. They shoal much closer to the coastline than tsunamis, and the proportional increase in height of their wave crest height is much less.

Why shoaling happens: waves get slower, shorter and higher

Shoaling happens because waves experience force from the seabed as the water gets shallower. This slows down the wave – the shallower the water, the slower the wave.

As waves slow down, they start to bunch together, so they have a shorter wavelength than before. This can also be explained by the wave equation v = f x λ (speed = frequency x wavelength), which shows that, when a wave’s speed decreases, it must have a shorter wavelength than before – slowing down won’t change the wave’s frequency.

Having a shorter wavelength means that the waves get higher. You could think of the shortened wave as being ‘squashed’ sideways – the water in the wave has to get higher because there’s not as much room for it within the shorter wavelength. It’s a bit like squeezing a toothpaste tube – all the toothpaste is forced upwards.

Shoaling and energy conversion

Shoaling can also be thought of as conversion of a wave’s energy between different forms. When a tsunami wave propagates across the ocean, its energy is mostly in the form of kinetic (movement) energy, but as it gets near the coast, it slows down and gets higher – this means that a lot of its kinetic energy is converted into potential energy (water particles that are much higher than the normal water level will have a lot of potential energy due to gravity

So when you have a very large wave at the coast, like a 5–10 metre high tsunami wave, there is a lot of energy stored up that’s ready to be unleashed when it runs up onto and over the land.

Shoaling causes inundation by tsunamis

Shoaling is one reason why tsunamis cause so much damage to coastal areas. Tsunamis have very long wavelengths in the deep ocean and involve large volumes of water in each wavelength. This is why they can shoal as high as they do as the water gets shallower – the wavelength can decrease by a large proportion, so the amplitude can increase by a lot.

Even when tsunamis have shoaled, they still have a relatively long wavelength, which is why they seem to keep flowing inland over many minutes.

The amount of shoaling by a tsunami is described by the term ‘run-up’, which is the height of the wave crest above normal sea level at its highest point inland before it dissipates. Run-up, wave speed and slope of land at the coast all contribute towards the damage caused.

Published 2 May 2011