This section deals with the development of a tsunami wave or wave train (series of waves generated in the same event), once the perturbation of the sea surface, by the processes described in the previous section, has taken place. It considers the defining characteristics and equations of motion of tsunami waves, which lie at the root of the most distinctive feature of the tsunami hazard for insurance purposes, the ability of these waves to propagate over transoceanic distances; the behaviour of tsunami waves in the shallow water of the continental shelf, which accounts for the complexity of tsunami damage patterns; and the impact of tsunami waves on different types of coastline.

Tsunami change significantly as they propagate, principally due to water depth variations as discussed below, but they can be described in terms of the following properties:

Distance between the beginning and end of a wave (or more conveniently, in the case of waves in a wave train, the distance between crests of successive waves).

The time between passage of one crest (or trough) and the next past a fixed point. In general, the waves in a tsunami wave train will in fact consist of a series of superimposed waves with different periods, although one or two will generally dominate leading to the perception in the impact area of a series of discrete waves. This is often expressed in terms of a frequency spectrum ( (frequency) = 1 / (period) ).

Refers to the motion of the wave or waves, NOT the water within it which (except when the wave is breaking or has formed a bore or surge) moves more slowly and mainly in a circular or osscilatory path with dimensions comparable to the wave amplitude. A further complication arises when a regular train or group of waves forms, in which case the phase velocity of individual waves has to be distinguished from the group velocity of the wave train as a whole. In deep water the former is twice the latter, and the propagation of individual waves forwards through the group is an important mechanism by which the energy of the waves is preserved rather than dispersed.

The vertical distance between crests and troughs. This is sometimes referred to as wave height, but in other situations, especially eyewitness descriptions, wave height refers to the height of crests above normal sea level. In water that is deep compared to the wave amplitude, the crests are the same height above normal sea level as the troughs are below: the waves are said to be symmetrical.

All of these properties are related to each other and also to the total energy of the waves. The fundamental property is in fact the period of the wave (or the frequency spectrum of a complex wave train), which is set at the source as the tsunami forms, and then varies only slowly as energy loss and spreading out (or dispersion) of the wave train preferentially eliminates the high – frequency components. In general terms, larger sources produce mainly longer – period waves, the dominant period being approximately equal to the length of time that a wave would take to cross the smallest dimension of the source in the depth of water above the source. Relatively small sources, such as landslides or volcanic explosions, tend to produce tsunami with dominant periods of the order of minutes; large sources, such as the largest earthquakes, produce tsunami with dominant periods of up to an hour or more. For comparison, even the very largest and longest surface waves produced by storms have periods of only 20 to 25 seconds.

The other parameters are related to wave period and energy through the equations in the box below.

INSERT EQUATIONS AND DISCUSSION INTO A BOX

A key feature of tsunami indicated by these equations is that, except for the period (or frequency spectrum) and energy of the wave (or wave train), all the other wave properties vary with water depth. In deep ocean water several kilometres deep, wave velocities are high (up to around 800 km / hour), wavelengths are long (up to hundreds of kilometres), and because the energy of the wave is spread through a very large volume of water both vertically and in the direction of motion, the amplitude is very low, of the order of a few centimetres for typical damaging earthquake – generated tsunami. It is for this reason that tsunami waves are never perceived by ships in the open ocean (in fact, detecting them requires extremely sensitive pressure guages or other instruments on the ocean floor, and this has only recently been achieved CROSS – REFERENCE TO SECTION ON TSUNAMI DETECTION AND WARNING). Another consequence of the very small amplitude is that the wave loses very little energy by turbulence in the water or through scouring of the sea bed: it is for this reason that tsunami can propagate very efficiently over vast distances, provided that water depths are of the order of kilometres all along the propagation path.

Another aspect of tsunami propagation revealed by these equations is that the longest – period waves (with periods of tens of minutes or longer) all travel with much the same velocity. Therefore, a wave train composed of such waves will remain together as a packet, whatever their individual wavelengths. In contrast, a wave train composed of shorter – period waves, each with its own slightly different velocity, will tend to disperse. The wave energy will then be dissipated more quickly than if the waves remain together as a train.

As a general rule, then, long – period tsunami waves generated at large sources (such as giant earthquakes with ruptures several hundreds of kilometres long) will propagate and cause damage at greater distances from source, than initially equivalent (in terms of total energy) short – period tsunami waves from smaller sources such as landslides and meteorite impacts. This does not mean, however, that the latter are of only local significance as is sometimes assumed in the literature. The 1st April 1946 Aleutian tsunami was generated by a relatively small earthquake, with a rupture area only about 70 km across, and consisted of waves with a dominant period of the order of several minutes only, yet it propagated very efficiently across the Pacific causing major damage and around 150 deaths in Hawaii and significant damage as far afield as the Marquesas islands and Chile. If, as some models for this event imply, the tsunami was primarily generated by a landslide then the source region would probably have been even smaller.