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On the Dispersion Relation of Sea Waves

3.1 Introduction The multi-channel optical systems installed on modern satellites allow receiving images of the sea surface with high spatial resolution, which allows restoring its relief on scales smaller than the dominant waves [19]. Small temporal lag image sequences make it possible to determine the phase velocity of surface waves and, based on these data, to estimate the current velocity and bathymetry of the coastal zone [20, 47]. Navigational and specialized ship and shore-based radars are used to provide operational information on the current field [9, 13]. Signals from the ridges of long surface waves are clearly visible on radar images, which makes it possible to use series of sequential images to determine current velocity [14]. When interpreting remote sensing data, the question of spatial-temporal relationships in the wave field, nonlinear effects and dispersion relation disturbances becomes very important. The remote methods of determining the velocity of the current are based on the assumption that in the absence of a current for gravity waves in deep water the dispersion relation is fair [7] ω2 = gk

(3.1)

where ω is an angular frequency; g is gravitational acceleration; k is wave number. For the first time, the dispersion relation disorder in field conditions for gravity waves in deep water was experimentally shown [44]. This result was obtained on the basis of measurements of an array of spaced wave sensors mounted on fixed supports. The dispersion relation disturbances were later confirmed in a series of laboratory and field experiments [26, 27, 38, 39, 50]. A number of physical mechanisms lead

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Zapevalov et al., Simulation of the Sea Surface for Remote Sensing, Springer Oceanography, https://doi.org/10.1007/978-3-030-58752-9_3

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3 On the Dispersion Relation of Sea Waves

to dispersion relation disorders. These include: kinematic nonlinearity, presence of bound components, wave-wave interactions, impact of longer waves on short waves, etc. [11, 18, 22, 34, 35]. One of the important characteristics of the wave field, describing its space-time structure, is the spreading function that describes angular distribution of wave energy. The bulk of information on the frequency-angle characteristics of surface waves was obtained with the help of wave buoys of “heavy-pitch-roll” type [4, 32, 40, 42, 45]. Significantly fewer studies have been carried out using a high resolution directional array consisting of string sensors [2, 10, 52]. Multi-point measurements provide more detailed estimates of frequency and angle characteristics, but are technically more complex and require significantly higher financial costs. This chapter analyses the spatio-temporal structure of the sea surface wave from in situ measurements and builds models to describe it.

3.2 Wave Measurement Equipment and Measurement Conditions The results of field studies presented