Long Waves

Standing Waves in Bays. Seiches. In bays, standing waves may develop which are similar to the oscillations in lakes. These waves are known as seiches, and were first studied by Forel (Defant, 1929; Thorade, 1931) in the Lake of Geneva. They are free oscillations of a period that depends upon the horizontal dimensions and the depth of the lake and upon the number of nodes of the standing wave. The wave length will be of the same order of magnitude as the length of the lake, and, as the length of a lake is usually great compared to the depth, the waves will have the character of long waves.

A long wave that proceeds in water of constant depth in the positive or negative x direction must satisfy the equations of motion and continuity in the form (p. 425 and p. 432)

if frictional forces and the deflecting force of the earth's rotation are neglected. Here dv_x/dt has been replaced by ∂v_x/∂t, because in the case of a long wave ∂v_x/∂x, can be considered a small quantity. The vertical displacement of the surface is called η, and ∂η/∂x is therefore the inclination of the free surface. Introducing the horizontal displacement ξ, one has v_x = ∂ξ/∂t, and the above equations take the form These equations are satisfied if where the velocity of progress of the wave, c = σ/k, is equal to [Equation].

Equations (XIV, 11) define two waves that progress in opposite directions. The equations of motion are therefore also satisfied by a superposition of two waves that proceed in opposite directions and for which the velocity of progress is c:

and For a free oscillation the antinodes of a wave in a closed basin must be located at the ends of the basin, where only vertical motion can exist.

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If the length of the basin is l and is measured from x = 0. the boundary conditions can be written x = 0, ξ = 0, and x = l, ξ = 0. The first condition is fulfilled by (XIV, 12), and the second is fulfilled if the period of oscillation is such that where n is a positive integer greater than zero. Because σ = 2π/T and c = [Equation],

It is readily seen that n is the number of nodes of the standing wave. The standing wave of the longest period is the one that has only one node and the period

This relation, derived by J. R. Merian in 1828, is known as “Merian's formula.” In fig. 128, (p. 517), is shown a standing wave with two nodes.

In applying this very simple theory to actual oscillations of the water in lakes, considerable modifications must be made, because the shape of a lake deviates very much from that of a rectangular basin of constant depth. Crystal (Defant, 1925) has developed the theory of standing waves in basins of different shapes, and more recently Defant (1925) has introduced a convenient method of determining the possible periods of oscillation in lakes by means of a numerical integration of the hydrodynamic equations. Defant's method can be directly applied to lakes or bays of any shape and permits the computation of periods, the relative magnitudes of vertical displacements, and the position of nodal lines. Defant's reasoning as presented in his book of 1929 will be briefly summarized here.

In a basin of variable width b, and variable cross-section area S, the equations of motion and of continuity can be written in the form

The vertical and horizontal displacements are supposed to vary periodically:

Replacing the differentials in equation (XIV, 16) by the small quantities Δξ and Δη, one obtains

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These equations can be used for a stepwise computation of the displacements if it is possible to determine independently an approximate value of the period of the free oscillations. If then one cross section after another is considered, and if these cross sections are placed close together, it can be assumed that the changes of the displacement from one cross section to the next are linear, in which case equation (XIV, 18) can be written

where α = (4π²/gT²)Δx and where the quantities indicated by subscript 1 and subscript 2 represent the values for two successive sections, and where v_i is the surface area of the sea between the sections i − 1 and i. The quantity q⁰ is equal to zero.

The period T for any given basin can be computed in the following manner. First, an approximate value of T is found by means of formula (XIV, 15), introducing the average depth of the basin. The result is an approximate value of α. At the end of the basin at which the computation begins (x = 0), the horizontal displacement must be zero, and for the vertical displacement an arbitrary value can be selected. In this manner the boundary condition at the one end is fulfilled. By means of equation (XIV, 19), one can now, step by step, compute the displacements for all cross sections of the basin, and, if the approximate period that was derived by means of formula (XIV, 15) is correct for the simplest seiche, the computation must give the value ξ = 0 at the other end of the lake in order to fulfill the boundary condition. The computed value will usually differ from zero, and hence it is necessary to select another value of the period and to repeat the entire computation. If this new value does not lead to a correct result, one has to select a third one, but as a rule it is possible to select the first two values of the period in such a way that the correct one lies between them and can be determined by suitable interpolation. The final result will give relative values of the displacements and the exact locations of the nodal line. In a similar manner, one can find the period of an oscillation with two nodal lines and fix their location. Figure 135 shows the computed displacement of seiches in Garda Lake, according to Defant.

So far, only lakes have been considered. In a bay that is in open communication with a large body of water, horizontal flow can take place through the opening. For a rectangular bay the simplest form of a standing wave will be that which has a nodal line across the opening and

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an antinode at the closed end of the bay. Therefore the total length of the bay is occupied by only one fourth of a wave length (fig. 128), and the corresponding period is Standing oscillations of shorter periods are also possible, but one of the nodal lines must always be located at the opening of the bay.

If the opening of the bay is very wide, it is necessary to introduce a correction that increases the period. The increase is 32 per cent, according to Rayleigh (Thorade, 1931), if the width of the bay equals the length, but is reduced to about 10 per cent if the width is one tenth of the length. Actual experimental verification of this theory has not been obtained.

In a canal that is open at both ends, standing waves can also be present, but these must be such that nodal lines are located at the two openings of the canal. The longest possible period of a standing oscillation is therefore the same as the period of a lake of similar shape, but in a lake antinodes are located at the ends of the lake, whereas in a canal nodes are found at the ends.

The period and the character of the oscillation in bays or canals can be found by Defant's method, taking the proper boundary conditions into account. In a bay the vertical displacement must be equal to zero at the opening, and in a canal it must be equal to zero at both openings.

Seiches occur commonly in bays, as is evident from records of tidal gauges in such localities. Studies of oscillations in bays along the coast of Japan have been conducted by Honda, Terada, Yoshida, and Isitani

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(Defant, 1929), who have examined records of tidal gauges and have conducted experiments on models. Such experiments have given results in good agreement with observations in cases where the computation of the possible oscillations is difficult, owing to the complicated configuration of the basins. Figure 136 shows the streamlines of a possible oscillation of the waters in San Francisco Bay according to the experiments by Honda et al. This oscillation, which is characterized by two nodal lines, one near the Golden Gate and one running north-south across the San Francisco Bay, appears to be easily produced and has a period of 38 to 48 minutes.

The causes of such oscillations are not fully understood. It should be observed, however, that only a small amount of energy is needed for producing standing oscillations in a body of water and for maintaining them. Very weak periodical variations in wind or barometric pressure may therefore produce seiches if these variations are of a period length corresponding to one of the possible free oscillations. It has also been shown by J. Proudman and A. T. Doodson (Defant, 1929) that a sudden change of wind or a rapid variation in pressure may cause oscillations that will gradually die out because of friction.

Seiches do not appear to be confined only to bays, but occur frequently on practically open coasts. As an example, Patton and Marmer (1932) mention that at Atlantic City, on the open coast of New Jersey, heavy winds will frequently bring about a seiche oscillation with a period of about 15 minutes. They assume that this seiche represents an oscillation of some part of the wide embayment of the coast between Nantucket Island and Cape Hatteras. The theory of such seiches has been developed by Hidaka (1935).

Destructive Waves. The waves that occasionally inundate low-lying coasts and cause enormous damage are considered here, although they are commonly known as “tidal waves.” However, they have nothing in common with the tides, but the name “tidal wave” has become so firmly established in the English language that the popular use will probably be continued in spite of the unfortunate confusion to which it gives rise. The destructive waves known as tidal waves are caused by

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earthquakes or by severe storms blowing against the coast, for which reason it is necessary to distinguish between earthquake waves and storm waves, the former being real waves and the latter being not even related to waves.

Waves in the sea caused by earthquakes are of two different types. In the first place a submarine earthquake may produce longitudinal oscillations that proceed at the velocity of sound waves. When reaching the surface, such longitudinal oscillations will be felt on board a ship as a shock that violently rocks the vessel. The shock may be so severe that the sailors believe their vessel has struck a rock, and several such reported “rocks” were indicated on early charts in waters where recent soundings have shown that the depth to the bottom is several thousand meters. There are many ship reports dealing with shock waves, particularly from regions in which seismological records show that submarine earthquakes are frequent. Explosion waves of this character usually occur as independent phenomena, but occasionally they are accompanied by the release of large amounts of gases that rise toward the surface and may lift the surface up like a dome, thus producing a transverse wave that behaves like any other gravitational wave. Observations of this kind of waves are rare, but it is possible that ships which have been lost at sea have been completely destroyed by such enormous disturbances. A wave of this nature spreads out from the place where it is formed and decreases in amplitude. By the time it reaches the coast, it has usually become so reduced that it does not cause much damage.

Destructive waves caused by earthquakes, dislocation waves, or “tsunamis” are in general associated with submarine landslides which directly create transverse waves. These waves may reach enormous dimensions both in the open sea and near the coasts, and they proceed as ordinary long gravitational waves. Many records exist of such waves which, near their origin, have caused enormous damage by completely inundating low-lying areas and which have subsequently traversed the entire Pacific or Atlantic Ocean. Thus, the great damage caused by the earthquake at Lisbon on November 1, 1755, was mainly due to the gigantic wave which was set up and which continued across the Atlantic Ocean, reaching the West Indies as a “tidal wave” 4 to 6 m high. In Japan, similar earthquake waves have on many occasions brought great destruction and have led to the loss of many lives. As an example, it may be mentioned that in 1703 more than 100,000 persons lost their lives when the coast of Awa was flooded. Among the most discussed waves are those that accompanied the eruption of the volcano Krakatao in the Sunda Strait on August 26 and 27, 1883. Several waves occurred after the different eruptions, and the highest ones caused great devastation on some of the East Indian Islands, where more than 36,000 persons lost their lives and where the waves in certain localities must have reached a

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height of 35 m. These waves did not enter the Pacific Ocean, but crossed the Indian Ocean and could be traced up the Atlantic Ocean, where they were recorded as far north as the English Channel, having traveled a distance corresponding to half the circumference of the earth in 32½ hours. In the English Channel their height had decreased to a few centimeters.

These waves proceed, as already stated, as long gravitational waves and their velocity of progress should therefore, over a uniform bottom, be equal to [Equation]. Where the depth to the bottom is variable, the velocity of progress will be somewhat less than [Equation] where h_m is the average depth, but it has been found that the velocity of progress is smaller than should be expected even if variations in depth are considered. In spite of this circumstance, the study of the rate of propagation of these waves served to give an idea of the average depth of the ocean prior to the time of deep-sea soundings. Thus, in 1856, A. D. Bache computed the average depth of the oceans to be about 4000 m, whereas Laplace had assumed an average depth of about 18,000 m.

The period length of tsunamis varies between 15 and 60 minutes (Krümmel, 1911, Gutenberg, 1939). Where the depth to the bottom is 200 m, the velocity of progress of a long wave ( [Equation]) is 44.2 m/sec, and with a period length of 30 minutes the wave length is 79.5 km. The corresponding maximum particle velocity is independent of the wave period and equals [Equation] η₀/h (p. 565), where η₀ is the amplitude of the wave and h is the depth to the bottom. With the above numerical values the maximum particle velocity becomes equal to 0.22η₀. The energy of the wave per unit area of the surface is ½gη₀², and, thus, equal to that of surface waves or tides of the same amplitude.

Destructive “waves” caused by wind are of an entirely different nature. In this case one has to deal, not with the effect of a wave, but instead with inundations which are caused by the ocean waters being swept up against the coast by violent storms. Abnormally high water levels caused by strong winds are frequent on many coasts, but fortunately the sea level rarely rises so much that great damage occurs. The most destructive storm “wave” known in the history of the United States is that which practically destroyed Galveston on September 8, 1900. A West Indian hurricane approached the coast of the Gulf of Mexico, where at Galveston the barometric pressure fell from 996.2 millibars, (29.42 inches) at noon to 964.4 millibars (28.48 inches) at 8:30 P.M. At the same time the wind velocity increased to 45 m/sec (100 miles per hour) at about 6:00 P.M., when the anemometer was broken to pieces. It has been estimated that the average wind velocity between 6:00 and 8:00 P.M. must have been about 55 m/sec, or 120 miles per hour. During the day of September 8 the water rose steadily but slowly until the wind had reached hurricane force, when a much more rapid rise took place. In the evening the water level was nearly 5 m (15 feet) above mean high

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water, and large districts of the city were flooded. Nearly 6000 persons were drowned, and the property damage ran into tens of millions of dollars.

The hurricane that on September 21, 1938, struck the coast of New England brought an even higher water level in many localities, but did not cause so much loss of life. At Buzzard's Bay the highest water level ranged from 4 to 5 m above mean low water, and at Fall River it was reported that “the water came up rapidly in a great surge,” rising to about 6 m above normal. More than 600 persons lost their lives in the hurricane, and the property damage was estimated at $250,000,000 to $330,000,000, although only part of this damage was due to destructive waves (Tannehill, 1938).