海岸动力学课件（一） 1.ppt

Bilingual Course”,ZHENG Jinhai March 2008,Coastal Hydrodynamics海 岸 动 力 学,HOHAI UNIVERSITY,Chapter 2 WAVE THEORY,Stating description of wave motion,,,Stating basic equations of wave motion,,Stating the small amplitude wave theory,,Stating the finite amplitude wave theory,,Stating wave theory limits of applicability,1/21,Chapter 2,Water is treated as a uniform,Assumptions,,continuity equation,The fluid viscosity is normally,,velocity potential,The surface tension and,,gravity only,The bottom is impermeable.,,zero velocity,ignored.,Coriolis force are ignored.,and incompressible fluid.,Waves travel in the x-z plane.,2/21,Chapter 2,G.D.E.,B.B.C.,on z= -h,L.B.C.,D.F.S.B.C.,K.F.S.B.C.,on z=η,on z=η,Boundary Value Problem of Wave Motion,3/21,1. Linearization of basic equations,Chapter 2,§2.3 Small Amplitude Wave Theory,2. Solution of the linearized equations,3. Dynamic & kinetic characteristics of small amplitude waves,4. Standing waves,4/21,Chapter 2,1. Linearization,In 1845, Airy developed a theory for irrotational waves traveling over a horizontal bottom in any depth of water. In the derivation of this theory the equation is linearized, and for this reason the theory is often referred to as the linear wave theory. In order to develop the linearization, the waves should be infinitesimally small, in this manner, this theory is also called small amplitude wave theory.,5/21,Chapter 2,Sir George Biddell Airy(1801-1892) was an English astronomer who worked in a variety of areas of science. His major work with respect to this course is his development of small amplitude water wave theory. His research encompassed magnetism, tides, geography, gravitation, partial differential equations, and sound. In 1826 he was appointed the Chair of Mathematics at Cambridge. He became the Astronomer Royal in 1835.,6/21,Chapter 2,What is linearization ?,For infinitesimally small waves, the displacement of the free surface is small, and therefore it is assumed that velocities and pressures are small; thus any products of these variables are small enough to be ignored. This process is called linearization. Linear in the sense that variables are only raised to the first power.,,,7/21,Chapter 2,A small amplitude wave is also called a linear wave. It is a wave which travels very slowly, the wave height is far smaller than the wave length and the water depth is much greater than its wave height.,What is a small amplitude wave ?,8/21,Chapter 2,Suppose that the wave is a small amplitude wave, namely H<>1), the solution of velocity potential is:,While the elevation of the water surface is,16/21,Chapter 2,Substituting the velocity potential and the surface elevation into the K.F.S.B.C yields the dispersion relationship.,Dispersion relationship,17/21,Chapter 2,This relationship shows that the wave length continually decreases with decreasing depth for a constant wave period. That is to say, waves of constant period slow down as they enter shallow water.,Dispersion relationship,A field of propagating waves consisting of many frequencies would separate due to the different celerities of the various frequency components.,18/21,Chapter 2,Case study of “wave dispersion”,19/21,Chapter 2,Had we used the surface elevation instead of , how would the velocity potential be changed?,MINI-EXAMINATION (1),Homework,Chapter 2,A wave with the period of 5s travels in water of 5m, what is its celerity and what is its length? You are on a ship( 100m in length) on the deep ocean traveling north. The regular waves are propagating north also and you note two items of information:(1)when the ship bow is positioned at a crest, the stern is at a trough, and (2) a different crest is positioned at the bow every 20s.Do you have enough information to determine the ship speed? If “no”, what additional information is needed? If “yes”, what is the ship speed?,。