The older classic theories i.e. the equilibrium theory and the dynamic theory failed to completely solve the tidal problems, and phases and amplitudes of the actual oceanic tides could not be computed.
Therefore, W. Whewell, in 1,833, advanced an altogether new theory based on observation of the progress of tides with time. As a matter of fact, this theory known as the Progressive Wave theory aimed at solving the various complexities and anomalies of oceanic tides.
Following his recommendations tide observations were taken at 666 coastal points in June 1835 along the North Atlantic Ocean (figure 13.9). W. Whewell also prepared a map of the co-tidal lines for the semi-diurnal tide.
However, in 1848, Whewell, influenced by Airy’s channel theory developed in 1842, changed his original concept of tidal waves as pure progressive waves. Thereafter he revoked his map of the oceanic tides.
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The progressive wave means a wave in which the waveform progressively moves. Before going into the details of this theory, it is necessary to grasp the exact meaning of certain technical terms that may be used in connection with this theory.
A free standing wave is generated by the atmospheric disturbances, and its period is determined by the size and shape of the basin. A forced standing wave is generated by the tide-producing forces such as the sun and the moon.
According to C.A.M. King, a wave propagated in a channel of infinite length will have certain properties; it is known by the term ‘progressive wave’. Further, where the wave is long compared with the depth of water, the streams will extend uniformly from the top to the bottom of the water.
The distance between two adjacent crests of the wave is called the length of the progressive wave. The time the progressive wave takes to move one wave length is called its period. The velocity of travel of the wave form, not of the water in the wave, is dependent on the depth of water.
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The waves of tidal length are very long compared with the depth of the ocean. The wave velocity is equal to the square root of depth multiplied by the force of gravity. Except in very shallow coastal waters, the variation in the height of the wave is small compared with the depth of the water.
This is true over the oceans. Besides, the rate of propagation of the wave is dependant only on the depth of water and nothing else. The following formula gives the velocity of wave; c = √gd; where c = velocity, g = force of gravity and d = depth.
The essential characteristics of a progressive wave are that its velocity is dependent on depth, and that the current flows in the direction of wave propagation and is at its maximum at the crest. At the mean water level, there is no current; at the trough the speed is at its maximum, but flows in the opposite direction.