海岸动力学英文PPT课件Coastal Hydrodynam(12)

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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
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Chapter 2
What is aFra Baidu biblioteksmall amplitude wave ?
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.
1. Linearization of basic equations 2. Solution of the linearized equations 3. Dynamic & kinetic characteristics
of small amplitude waves 4. Standing waves
The bottom is impermeable. Waves travel in the x-z plane.
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continuity equation
velocity potential
gravity only
zero velocity
Chapter 2
Boundary Value Problem of Wave Motion
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Assumptions Water is treated as a uniform and incompressible fluid.
The fluid viscosity is normally ignored.
The surface tension and Coriolis force are ignored.
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Chapter 2
How to linearize DFSBC & KFSBC ?
➢ Suppose that the wave is a small amplitude wave, namely H<<L or H<<h.
➢ Use the Taylor series expansion to relate the boundary conditions at the unknown elevation to the still water level.
G.D.E. B.B.C. D.F.S.B.C. K.F.S.B.C. L.B.C.
2 0
hz, x
0
z
on z= -h
η on z= t1 2[ x2 z2]p 0g z0
0
t xx z
on z=η
x ,z ,t x c,zt
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Chapter 2
§2.3 Small Amplitude Wave 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.
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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.
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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.
Coastal Hydrodynamics 海岸动力学
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