Mild-slope Equation - Derivation of The Mild-slope Equation - Vertical Shape Function From Airy Wave Theory

Vertical Shape Function From Airy Wave Theory

Since the objective is the description of waves over mildly sloping beds, the shape function is chosen according to Airy wave theory. This is the linear theory of waves propagating in constant depth The form of the shape function is:

with now in general not a constant, but chosen to vary with and according to the local depth and the linear dispersion relation:

Here a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals and become:

$\begin{align} F &= \int_h^0 f^2\; \text{d}z = \frac{1}{g}\, c_p\, c_g \quad \text{and} \\ G &= \int_h^0 \left( \frac{\partial{f}}{\partial{z}} \right)^2\; \text{d}z = \frac{1}{g} \left( \omega_0^2\, -\, k^2\, c_p\, c_g \right). \end{align}$

The following time-dependent equations give the evolution of the free-surface elevation and free-surface potential

$\begin{align} g\, \frac{\partial\zeta}{\partial{t}} &+ \nabla\cdot\left( c_p\, c_g\, \nabla \varphi \right) + \left( k^2\, c_p\, c_g\, -\, \omega_0^2 \right)\, \varphi = 0, \\ \frac{\partial\varphi}{\partial{t}} &+ g \zeta = 0, \quad \text{with} \quad \omega_0^2\, =\, g\, k\, \tanh\, (kh). \end{align}$

From the two evolution equations, one of the variables or can be eliminated, to obtain the time-dependent form of the mild-slope equation:

$-\frac{\partial^2\zeta}{\partial{t^2}} + \nabla\cdot\left( c_p\, c_g\, \nabla \zeta \right) + \left( k^2\, c_p\, c_g\, -\, \omega_0^2 \right)\, \zeta = 0,$

and the corresponding equation for the free-surface potential is identical, with replaced by The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around

Read more about this topic: Mild-slope Equation, Derivation of The Mild-slope Equation

Famous quotes containing the words vertical, shape, function, airy, wave and/or theory:

“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)

“I will soon be going out to shape all the singing tomorrows.”
—Gabriel Péri, French Communist leader. Letter, July 1942, written shortly before his execution by the Germans. Quoted in New York Times (April 11, 1943)

“Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screwdriver, a rule, a glue-pot, nails and screws.—The function of words are as diverse as the functions of these objects.”
—Ludwig Wittgenstein (1889–1951)

“He may have seen with his mechanic eyes
A world without a meaning, and had room,
Alone amid magnificence and doom,
To build himself an airy monument”
—Edwin Arlington Robinson (1869–1935)

“The history of any nation follows an undulatory course. In the trough of the wave we find more or less complete anarchy; but the crest is not more or less complete Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.”
—Aldous Huxley (1894–1963)

“In the theory of gender I began from zero. There is no masculine power or privilege I did not covet. But slowly, step by step, decade by decade, I was forced to acknowledge that even a woman of abnormal will cannot escape her hormonal identity.”
—Camille Paglia (b. 1947)