Special Public Seminar on Earthquakes and Tsunamis Tsunami from the Perspective of Ocean Waves Professor K. W. Chow Department of Mechanical Engineering University of Hong Kong

GOAL: Try to understand how tsunamis can cause so much damage and destruction from the perspective of ocean waves: (A) water waves in the open oceans, (B) their dynamics near the seashore.

General ideas about wave motion Very often, a ‘fluid’ (liquid or gas) transmits energy and information by small disturbances or waves. Sound waves – a sequence of compressions and relaxations in a gas.

How about waves in water? Try to classify them through the periods of wave motion: (1) We jump up and return to the ground in a few seconds, due to gravity. The same principle applies to water particles. Hence ordinary gravity waves on the sea surface have a period of a few seconds. (2) Tides have periods of roughly 12 hours. (3) Anything in between?

In fact a whole spectrum of wave motions is possible: In fact a whole spectrum of wave motions is possible:

IMPLICATIONS:

What are tsunamis? Commonly asked questions: (1) How high do the waves need to be to qualify as a ‘tsunami’? (2) Will we see a 10-meter high wall of water in the open ocean? (3) Waves shown in television news reports are not particularly huge, why are they so damaging? (4) Will I see a 10-foot tsumami in Hong Kong?

These are not the most appropriate questions. The proper question to ask, perhaps, is: How can Nature transfer a huge amount of energy from the epicenter to the coast through water waves ?

The answer: Through ‘long oceanic waves’, since such long waves are: (1) moving very fast; (2) non-dispersive; and (3) can excite motion through a large depth of water. Results: most dissipation mechanisms (geometry, friction, scattering…) do not have time to attenuate this flow of energy.

Moving Fast?? (1) Generally, any disturbance or pattern in water consists of various components with different wavelengths. (2) Water waves: Long waves travel at a HIGH velocity. Short waves travel at a LOW velocity.

Long waves are thus the first wave group observed at the coast. All long wave components travel at the speed of square root (g H), where g = acceleration due to gravity, 9.8 m s –2, H = water depth.

For the Japan earthquake: (1) At a depth of around 1000 meters, the speed is around 100 m s –1. Speed of a Boeing 747 at cruising altitude is approximately 150 – 250 m s –1. (2) At the shore, the depth is zero. Let us simplify the dynamics by taking an ‘average depth’ of 500 meters, an average speed is then 70 m s –1. (3) Time to travel 100 km ≈ 1400 seconds ≈ 23 minutes!! For the Japan earthquake: (1) At a depth of around 1000 meters, the speed is around 100 m s –1. Speed of a Boeing 747 at cruising altitude is approximately 150 – 250 m s –1. (2) At the shore, the depth is zero. Let us simplify the dynamics by taking an ‘average depth’ of 500 meters, an average speed is then 70 m s –1. (3) Time to travel 100 km ≈ 1400 seconds ≈ 23 minutes!!

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Non-dispersive?? Dispersion of white light (sunlight) into components: as light rays of different colors have different refractive indices (or speeds) inside the glass prism.

DISPERSION (1) Components of different wavelengths move with different speeds. (2) Example: Many students are walking to the canteen. If they all walk at the same speed, they arrive at the same time. If each student walks at a different speed, the group will ‘disperse’.

Dispersion If dispersion is present, the pulse broadens. Animation courtesy of Dr. Dan Russell, Kettering University

Long waves are non – dispersive In the case of water waves, all long waves travel at the same speed (= square root of (g H)), and are thus non–dispersive. The various long wave components do NOT separate from each other, and pound on the coast at the same time.

Particle paths?? Particle paths?? As the wave form progresses, each particle oscillates in a circular or elliptic path, with dimensions decreasing with depth.

IMPLICATIONS: Dimensions of particle trajectories decrease exponentially as we go deeper into the fluid. Typically these paths will become very small at a water depth larger than a few wavelengths.

A water wave moving from left to right. Each particle moves in an elliptical path of decreasing dimension.

For short waves of say 1 meter, water motion is negligible in water deeper than say 5 meters. Motions in such a small region are easily scattered or dissipated.

For long waves of say 10 km (which is greater than the ocean depth), this motion will persist throughout most of the ocean!!

Waves in the open ocean explained, how can we understand the dynamics as the waves approach the shore?

Several Factors: (1) On approach to shore, the wavelength must be compressed to a smaller value, and this will ‘squeeze’ the wave to a larger amplitude. (2) Geometric configuration: an ‘enclosed harbour’ or ‘gulf’ might elevate the wave amplitude.

Effect of a gulf

(3) ‘Piling up’ or ‘Nonlinear’ effects – Faster moving waves from the back catch up with the slower moving waves in the front.

The speed of long waves is square root of (g H) g = acceleration due to gravity, 9.8 m s –2 H = water depth. As the waves approach shore, H decreases and hence the speed will decrease. Hence the waves start to pile up and the amplitude increases.

How a Tsunami Increases in Height

Large scale tsunamis: Can it happen in Victoria Harbour? My personal (emphasize ‘My’) opinion – should be unlikely (??): (1) Geography: Hong Kong, especially Victoria Harbour, is surrounded or shielded by islands and peninsulas, unless the epicenters are located very nearby and in very special positions. Waves seldom turn 90 degrees.

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(2) ‘Geometric factors / configurations’: Many disturbances or vibrations, including ‘tsunamis’ or shallow water waves, are governed by the ‘wave equation’.

● Propagation in one spatial dimension: No attenuation. ● Propagation in two spatial dimensions: Amplitude decays like 1/(square root of the distance from the epicenter).

Unless the epicenter is really close to Hong Kong, Victoria Harbour would be less vulnerable than say Banda Aceh (Indonesia, 2004) or Sendai (Japan 2011).

As a rough comparison: December 26, 2004: Banda Aceh, Indonesia, about 200,000 (??) people perished. Time for tsunamis to reach shore: roughly 30 minutes. March 11, 2011: Sendai, Japan. Time for tsunamis to reach shore: roughly 30 minutes as well.

CONCLUSIONS (1) Tsunamis propagate as small amplitude, long surface waves in the open ocean. (2) On approach to shore, the geometry of the coast line, the shape of the sea floor and other factors will distort, compress or amplify the wave amplitudes.

(3) Computer simulations take a couple of days, but the most damaging, killer waves typically arrive in 30 minutes or so. Better run to a higher ground immediately during an earthquake. (4) We have ‘oversimplified’ the situation. The ocean is a complex system. Many other factors, e.g. internal waves, thermal stratification, and the rotation of the earth, have been ignored.

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Scenes from Japan before and after the tsunami Extracted form :

Scenes from Japan before and after the tsunami Extracted form :

Scenes from Japan before and after the tsunami Extracted form :

The wave equation for ‘tsunamis’ and shallow water waves. Mathematics: ∂ 2 u/∂x 2 + ∂ 2 u/∂y 2 + ∂ 2 u/∂z 2 = (∂ 2 u/∂t 2 )/c 2 u = displacement, c = velocity