Wave Theories and Their Applicability - Essay Example

Shylesh Rajith Rasquinha 01 April 2008 Topic: Wave theories and their applicability work of coastal and river engineering module) OBJECTIVES The main purpose of this coursework is to understand the waves generated by wind and how they have been described using different approaches and wave theories lead to their applicability in deep and shallow waters. BRIEF DESCRIPTION OF THE COURSEWORK This coursework is aimed at documenting the types of waves, generation of waves, wave theories and their applicability on different environmental conditions. WATER WAVES To understand the wave theories in a better way, first I would like to present the definition of a water wave and a table represents "Beaufort Scale Of Wind And Sea State" (Kamphuis 24). In simple words a water wave is a disturbance generated by the wind. This disturbance carries the energy and travels through the water from one location to another without or temporarily transporting the water on which the wind acts. It is like transfer of power or energy from one thing to another. The wind exerts the force on water to form the wave to carry its energy. Then where the wind energy comes from It's again recursive. It's not the topic of discussion in this coursework. Table 1: Beaufort Scale Of Wind And Sea State Beaufort Wind Force Wind Speed (knots) 1 knot=0.5m/s Description of Wind Description of Sea Approx. H (m) Approx. T (sec) 0 0-1 Calm Sea is like a mirror 0 1 2 4-6 Light Breeze Small wavelets. 0.1 3 6 22-27 Strong Breeze Large waves begin to form. The white foam crests are more extensive everywhere. 4 8 7 28-33 Moderate Gale Sea heaps up and white foam from breaking waves begins to be blown in streaks along the direction of wind (spindrift). 7 10 9 41-47 Strong Gale High waves. Dense streaks of foam along the direction of the wind. Sea begins to roll. Visibility is affected. 18 16 10 48-55 Whole Gale Very high waves with long overhanging crests. The resulting foam is in great patches and is blown in dense white streaks along the direction of the wind. On the whole, the surface of the sea takes a white appearance. The rolling of the sea becomes heavy and shock like. Visibility is affected. 25 18 11 56-63 Storm Exceptionally high waves (small and medium sized ships might for a long time be lost to view behind the waves). The sea is completely covered with white patches of foam lying along the direction of the wind. Visibility is affected. 35 20 12 64-71 Hurricane Air filled with foam and spray. Sea completely white with driving spray. Visibility is very seriously affected. 40 22 WAVE THEORIES After many years of experience and deep observations with water waves lead to the development of different wave theories. And Stokes Wave Theory is the one among them. In this theory Stokes uses the common approach of generation of waves due to the wind. The three major parameters involved in the theory and they are the waves, which are subjected to certain boundary conditions, the equations of motion and the fluid (ideal or perfect or frictionless) on which the wind acts to form the wave. Nowadays there are several variations of Stokes Theory developed using the computer implementations and are called the Extension of the Stokes theory to higher orders. For example, Stokes 2nd order, 3rd order and so on. Dean's (1965) stream function theory is the theory, which uses the stream function in place of the velocity potential to develop it. Dean (1974) did a limited comparison of measured horizontal particle velocity in a wave tank with the tenth-order stream function theory and several other theories. Again the variations of the stream function theory developed using different numerical methods by Dalrymple 1974, Chaplin 1980, Reinecker and Fenton 1981. Stokes finite amplitude wave theory is a non-linear shallow-water wave theory and is applicable when the depth to wavelength ratio d/L is greater than about 1/8 or kd > 0.78 or Ur < 79. As waves move into shallow water, portions of the wave travel faster because of amplitude dispersion or waves travel faster because they are in deeper water. The frequency dispersion will be less in shallow water and waves get affected. Various shallow-water equations can be derived by assuming the pressure to be hydrostatic so that vertical water particle accelerations are small and imposing a horizontal velocity on the flow to make it steady with respect to the moving reference frame. Korteweg and de Vries (1895) developed a wave theory called Cnoidal theory. The theory is defined in terms of the Jacobi elliptic function, cn (Weisstein 1), hence the name Cnoidal. This theory can be applied when there are periodic waves progressing in water whose depth is less than about one-tenth the wavelength. For example it is applied to generate the radiation stress tensor (Musick 1). This theory is very much applicable to finite-amplitude shallow-water waves where both no linearity and dispersion effects are included. This theory is based on the Boussinesq approximation (Wikipedia), but is restricted to waves progressing in only one direction. Cnoidal waves are periodic with sharp crests separated by wide flat troughs. Validity range for Cnoidal theory is d/L < 1/8 when the Ursell number UR > 20. As wavelength becomes long and approaches infinity, cnoidal wave theory reduces to the solitary wave theory. Also, as the ratio of wave height to water depth becomes small (infinitesimal wave height), the wave profile approaches the sinusoidal profile predicted by the linear theory. Based on the foundation of analytical mechanics, the theory of a solitary wave is evolved. And it was purely mathematical. The exact solution of the wave theory, published by Gerstner in 1802 assumes that the motion of a wave and that of particle coincide. Till then it was recognized that the motion of a wave differs from that of the water particles. In 1838, Russell's experimental observations revealed the motion of transmission, i.e. the motion of the wave, as the transfer of momentum from one spatial point to its neighbour. He classified the waves into four orders. He discovered that the velocity of the solitary wave of permanent form depends on its height and which lead to the mathematical dispute. In 1845, Airy gave his most successful contributions towards a non-linear shallow water theory of waves of non-permanent form and Stokes in 1849 with a theory of deep-water waves. Stokes concluded that his theory holds whenever the amplitude of the surface elevation relative to the water depth is smaller than the ratio of the water depth to the wavelength: max/h < (h/) 2, whereas Airy's results imply max/h < (h/) 2 thus providing different physical phenomena. In 1871, Boussinesq obtained the first mathematical description of a wave of permanent form. In 1876, Lord Rayleigh rediscovered another method for the same. Rayleigh said in his solution that the pressure distribution at the free surface is not constant, thus the stationarity of solution is invalidated. McCowan in 1894 and Kortewegde Vries in 1895 gave more accurate results for the waveform and the maximum possible wave height (Mechanica 111). The applicability of different wave theories is summarized in the following figure adapted from Le Mhaut (1976) (Kamphuis 30). Figure: Applicability of various wave theories (after Le Mhaut, 1976) OBSERVATIONS After reading a lot of books and searching in the web, I found that there are no theories so far proven to be accurate so that they can be applied in general for every case. I think that, the generation of water wave itself is a great puzzle. To solve these puzzles you need to have real time experience in the subject and strong knowledge in mathematics and concourse deep observations required. The above figure illustrates clearly about the applicability of the different wave theories. Works Cited Kamphuis, J. William. Introduction to Coastal Engineering and Management, 2000, World Scientific. ISBN 9810238304 Weisstein, Eric W. "Jacobi Elliptic Functions." From MathWorld--A Wolfram Web Resource. Musick,George Meredith , III. "CNOIDAL WAVE THEORY APPLIED TO RADIATION STRESS PHENOMENA", 1970 Wikipedia, the free encyclopedia, Boussinesq approximation (water waves), < http://en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves)> Mechanica, Acta. Journal, "On the development of the theory of the solitary wave. A historical essay". 2005 , ISSN:0001-5970 (Print) 1619-6937 (Online), Read More