By Zarubin A.N.
Translated from Differentsialnye Uravneniya, Vol. forty, No. 10, 2004, pp. 1423-1425.
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Extra resources for A boundary value problem with an involutive shift in the boundary condition
For such long waves the water-particle velocities under the wave are equal to the product of the ratio of the amplitude of the wave crest to the depth and the phase speed: ⎛a ⎞ u = ⎜ c ⎟ gh. ⎝h⎠ For example, with an abnormally large wave amplitude of 61 centimeters (2 feet), a depth of 4,000 meters (13,120 feet), and a wave period of 20 minutes, the water-particle velocity on the bottom would be about 3 centimeters (about 1 inch) per second. A tsunami spreads from its source and travels great distances before running ashore.
Consider the spectrum of a sound. If the sound is one pure frequency, the spectrum that indicates the distribution of sound intensity (or energy) with frequency will be a vertical line at the frequency of the signal. However, if the sound has intensities that are the same for a wide range of frequencies, the energy spectrum will consist of a horizontal line at the energy WAVES 21 corresponding to that of these frequencies. This is sometimes called a “top hat” distribution. Generally the energy distribution of speech falls between these two extremes.
The typical length of tsunami waves is about 100 times the depth. Thus, the speed of travel of the wave in the ocean with an average depth of about 4,000 meters (13,120 feet) would be about 700 kilometers per hour (440 miles per hour)—the speed of a jet plane. Re- 44 CHAPTER 1 member that this is the velocity of the “wave shape” (that is, the phase speed), not the velocity of the water particles under the wave. For such long waves the water-particle velocities under the wave are equal to the product of the ratio of the amplitude of the wave crest to the depth and the phase speed: ⎛a ⎞ u = ⎜ c ⎟ gh.