## China earthquake and a word about seiches

The tally of death and devastation in China in the aftermath of the earthquake continues to grow; now the official death toll is 22,000, with 14,000 still buried under rubble. In addition, repeated aftershocks are hitting the region.

Numerous eyewitness videos have been posted online since the event. This one in particular caught my eye, which shows a group of students outdoors experiencing the quake firsthand. It is a bit chilling to see their enthusiasm, knowing the devastation that was being wrought far away, but the students clearly felt that they were experiencing a small local quake, and had no idea that they were in fact 500 miles from the epicenter.

The part of the video that caught my eye was the sloshing of the water in the small pond. I believe this could be considered a small-scale version of a relatively little-known water wave phenomenon known as a seiche.

Wave propagation very roughly can be broken into two categories: traveling waves and standing waves. Focusing on water waves, a traveling wave is a wave which propagates in an open body of water and has a definite direction of propagation. Standing waves arise in closed bodies of water and consist of counterpropagating waves interfering with each other: the waves ‘stand still’ and seemingly only oscillate up and down in place. The difference is illustrated below, for waves on a string:

In water, a tsunami is a long-wavelength traveling wave and a seiche is a long-wavelength standing wave. While a tsunami can typically only be caused by a seismic event, seiches can also be generated by wind. Conceptually, one can imagine a seiche as the ‘sloshing’ of water that occurs when one shakes a filled bowl from side to side.

Both seiches and tsunamis differ from ordinary water waves in that their long wavelength results in so-called ‘nonlinear’ effects. Because the wavelength is comparable to or greater than the depth of the water (100’s of kilometers for both seiches and tsunamis), the water depth affects the wave propagation and results in a wave which can travel long distances without dissipating. An ordinary water wave involves motion of the water only near the surface, while seiches and tsunamis in principle involve a collective motion of all the water, from surface to sea bed.

A related class of waves, known as solitons, have been known to exist since 1834, when John Scott Russell observed them on the Edinburgh-Glasgow canal. Russell’s description of the event to the British Association for the Advancement of Science in 1844 is one of my favorite scientific descriptions of all time:

I believe I shall best introduce the phaenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel.

This is perhaps the only physical phenomenon which was studied by a scientist literally chasing down his research on horseback!

There is significant debate on whether a tsunami is truly a soliton wave; it is clear that a tsunami is a nonlinear wave, but it is not easy to determine if a tsunami fits the specific prescription of a soliton, for which the natural spreading and dispersion of the wave are perfectly canceled by a nonlinear ‘clumping’ effect. In any case, the formula for the velocity of a nonlinear gravity wave seems to apply, i.e.

$c=\sqrt{g(h+a)}$,

where c is the speed of the wave, g is the acceleration of gravity, h is the depth of the water and a is the amplitude of the wave. Assuming the amplitude a is much smaller than the depth of the water, we have the formula

$c=\sqrt{gh}$.

For a standing wave such as a seiche, the speed is not as important as the period of oscillation, i.e. how often does a wave maximum crash on shore? The period of oscillation T is related to the speed of the wave and its wavelength by the formula

$c = \nu \lambda = \lambda/T$.

The period of oscillation of a seiche comes out to be

$T= \lambda/\sqrt{gh}$.

Since the longest wavelength of a standing wave which can fit in a rectangular lake is $\lambda = 2L$, where L is the width of the lake, we find a formula for the period to be:

$T = 2L/\sqrt{gh}$.

This formula is known as the Merian formula. If we look at waves traveling east-west on Lake Michigan, L = 190 km and h = 85 m, we find that

$T = 1.3 \times 10^4 \quad \mbox{seconds}$,

which is approximately 3.6 wave crests per hour!

As noted on Wikipedia, tiny, unnoticeable seiches are present all the time on large lakes such as Lake Michigan. Occasionally, these seiches can grow in size and turn deadly. Eight fishermen were swept into Lake Michigan and drowned when a 10-foot seiche hit the Chicago waterfront on June 26, 1954.

I started this post by looking at the sloshing of water in a small pool during the China earthquake, and built it into a more general discussion of nonlinear wave motion. Hopefully this emphasizes that interesting physics can be found in pretty much every small occurrence in nature, if you look closely enough!

Those who are interested in donating to help those affected by the China earthquake, as well as the Myanmar cyclone, I recommend Oxfam America as a reputable charitable organization.

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