# Tsunami in Bays

Once again, this post is prompted by repeated questions from friends (particularly those in the Bay Area) about what happens to tsunami as they enter bays. Two things can happen:

1. Diffusion: waves spread out in bays by refraction, reducing the energy impacting any particular location.
2. Amplification: if the wave frequency matches the natural resonance frequency, wave energy is amplified through constructive interference.

As these are pretty much polar opposites, a bit more explanation is warranted.

### Refraction

Refraction at Sitges, Barcelona. Click for image location & credit.

Waves in shallow water slow down while the wave in deeper water continues at its original speed. This results in refraction, bending of waves so that wave energy is concentrated on headlands and spread out in bays. This is why it always looks like the waves are coming straight at you when you’re standing on the beach looking at the ocean.

Application to Tsunami
Refraction happens for ocean waves, and it happens for tsunami. Most of the time, when a tsunami enters a bay, the wave energy is spread out around the whole bay, diffusing the energy impacted at any one location. For example, the 1906 San Francisco earthquake produced a tiny (10 cm) tsunami; a propagation model shows the wave energy quickly decreasing within the bay compared to the open coast outside the bay.

### Resonance

Everything has a natural frequency at which it likes to wiggle. Any time you’ve played on a swing set, you’ve found the natural frequency and constantly fed a little more energy in each swing to build up the energy into larger amplitude arcs. When waves perfectly reflect within an so that they constructively interfere, building bigger and bigger and bigger waves, standing waves are created. In enclosed water bodies, this is called a seiche, and it happens any time the wave frequency is a multiple of the natural resonance frequency.

Resonance frequencies depend on size. The natural resonance of a coffee cup is less than a second; of a bathtub is about a second (you’ve probably found that by scooting back-and-forth until the water slops onto the floor), and swimming pools are around 20-30 seconds (roughly matching earthquakes). The period of small harbors are usually around 10 to 30 minutes, Lake Tahoe an hour, and the North Sea on order of 36 hours.

Application to Tsunami
Tsunami have a period of anywhere from 10 minutes to an hour. Sometimes, that matches up with the natural resonance of a harbour. When it does, the tsunami will resonate within the harbour, amplifying the wave height and exacerbating damage.

2004 Sumatra tsunami bouy data for Winter Harbour, BC.

Hilo Bay is famous for tsunami resonating within the harbor, resonating with a half-hour period (thus amplifying tsunami with 15 minute, 30 minute, or 1 hour periods). The 2004 Sumatra tsunami had a period of 40 to 50 minutes. Although the wave height was tiny when it reached the west coast of Vancouver Island, the wave resonated within Tofino (50 minute period) and Winter Harbor (30 to 40 minute period) for over a day.

### Tsunami & the San Francisco Bay

Using the Merian formula to calculate the resonance frequency for water bodies, and a range of lengths and average depths, the resonance frequency of the San Francisco Bay is somewhere between 1 and 10 hours. Although the lack of precision on that estimate is downright painful, and using the Merian formula is not entirely justified, the reality check of “More than Lake Tahoe, less than the North Sea” makes sense. Irrelevant of the exact number, tsunami have frequencies too short to resonate within the bay, so a seiche is unlikely. Instead, future tsunami will most likely do exactly what this tsunami did, and the 1906 tsunami did: refract within the bay, diffusing wave energy. This does not mean that you’re free to ignore tsunami warnings: even with refraction spreading the energy out, tsunami are nasty and can do a lot of damage. Always pay attention to tsunami warnings, and remember that the strong currents can make the water unsafe for days.

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