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Access provided by: anon Sign Out. Weakly dispersive spectral theory of transients, part II: Evaluation of the spectral integral Abstract: In the spectral theory of transients formulated in Part I of this paper, the transient response for weakly dispersive wave processes has been expressed in terms of canonical integrals in the complex spatial wavenumber domain. The real and complex singularities in the integrands, which dominate the behavior of the spectral integrals, have been categorized and associated with generic physical wave processes.

Maarten V. de Hoop | Free oscillations, surface waves and spectral theory

The integrals are now evaluated by Contour deformation around the singularities. This yields general expressions for the transient Green's function that are applicable to a broad class of propagation and diffraction problems. The generic results, which can be grouped into contributions from real or complex singularities; express the transient field in terms of compact and therefore physically incisive wave spectra, in contrast to alternative procedures that always constrain the spectra to be real.

These aspects, together with simplifying explicit wavefront approximations, are explored in the present paper, with the application to specific problems relegated to Part III. Article :. This is the case if a wave record is made at a single point as a function of time: the waves go past and their elevations are recorded, but no information is obtained about the direction the waves are traveling. As is to be expected, there is no uniformity of notation for this spectrum, but seems to be the most common symbol--and what is used in both Pierson and Moskowitz and Elfouhaily et al.

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Equation shows that the units of are clearly. Equations and show that integrating the omnidirectional variance spectrum over all frequencies gives the total elevation variance:. The notation in the last two equations is non-standard. The convention is to use for frequencies in the x direction above , and for frequencies in the direction above.

The of Eq. This leads to confusion in the subscripts, which can denote either frequency variables , or specific discrete values. However, the notation is standard in the literature, so that is what is used below. Its units are clearly.

This spectrum is often called the "two-dimensional wavenumber spectrum," and its arguments are often written in vector form, , where denotes the location of the frequency point in the 2-D frequency plane. Equation is the conceptual definition of. In practice, if we have discrete measurements of the two-dimensional sea surface elevation at a given time, , then the two-dimensional discrete Fourier transform of gives the two-dimensional amplitudes.

A 2-D spectrum depends on direction, i. Usually, the direction is chosen to be pointing downwind and, correspondingly, represents the spatial frequencies of the waves propagating downwind.

In this case, the angle gives the direction relative to the wind direction. As Eq. It is also common to define a directional spectrum in terms of polar coordinates given by the magnitude and direction of the vector. These are are related to by. The ECKV directional spectrum given on the next page is specified in terms of polar coordinates.

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However, on the next pages we will need a spectrum in terms of Cartesian coordinates for use in a rectangular FFT grid. The change of variables from polar to Cartesian coordinates is effected by the Jacobian. Integration of Eq. Unfortunately, making measurements of 2-D sea surfaces is extremely difficult.

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There are very few such data sets--obtained, for example, by laser reflectance measurements e. Given the paucity of empirical 2-D wave data from which to develop 2-D variance spectra, the common procedure is to start with a 1-D or omnidirectional spectrum and add an angular spreading function to distribute the wave energy over different directions relative to the downwind direction. In nature, most waves travel more or less downwind, a small amount of energy i. The spreading function must capture this behavior. Although omnidirectional wave spectra are well grounded in observations, the choice of a spreading function is still something of a black art.

Now return to Eq.

Comment: There is a subtle inconsistency in the units of mean square slopes as seen in the literature. As obtained above from the slope variance spectrum, the mss has units of radians squared. However, as defined using a finite difference of a sea surface elevation sample , the slope of the surface between two sample points and is. There is another way to view slope spectra. As we know from Eq. The corresponding relations for two dimensions are derived in the same fashion and lead to similar results.

Assuming that the wind is blowing in the direction, the mean-square slope in the along-wind direction is given by either of.

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Table summarizes the spectral quantities used on the following pages. It looks like you're using Internet Explorer 6: Features on this site are not supported by that browser version. Please upgrade to the latest version of Internet Explorer. Ocean Optics Web Book Skip to main content.