海岸动力学英文PPT课件Coastal Hydrodynamics_2.2

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.
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
A field of propagating waves consisting of many frequencies would separate due to the different celerities of the various frequency components.
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Chapter 2
of small amplitude waves 4. Standing waves
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Chapter 2
1845, Airy developed a theory for irrotational
waves traveling over a horizontal bottom in any
on z=η
x ,z ,t x c,zt
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Chapter 2
§2.3 Small Amplitude Wave Theory
1. Linearization of basic equations 2. Solution of the linearized equations 3. Dynamic & kinetic characteristics
the velocity potential be changed?
Chapter 2
Homework
A wave with the period of 5s travels in water of 5m, what is its celerity and what is its length?
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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.
Take the pressure on the free surface to be zero.
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Chapter 2
What is the Taylor series ?
If a continuous function f(x,y) of two independent variables x and y is known at the position where x is equal to x0, then it can be approximated at another location on the x axis, by using the Taylor series.
Case study of “wave dispersion”
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Chapter 2
MINI-EXAMINATION (1)
Had we used the surface elevation
asikn x(t) instead of
acoks x(t), how would
H 2 cg c o ko h s k s h z h h sikn x (t)
While the elevation of the water surface is
Hcosk(xt)
2
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Chapter 2
Dispersion relationship
Substituting the velocity potential and the surface elevation into the K.F.S.B.C yields the dispersion relationship.
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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.
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,
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Chapter 2
What is a small 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.

t
z
z0
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Chapter 2
Summary
G.D.E. B.B.C. D.F.S.B.C. K.F.S.B.C. L.B.C.
2 0
h z, x

z
zh
0
1
g t
z0

t z
z0
x ,z ,t x c,zt
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
on t1 2[ x2 z2]p 0g z0 z=η
0
t xx z
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Chapter 2
2. Solution
Coordinates System
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Chapter 2
How to obtain the solution ? L.B.C G.D.E B.B.C
D.F.S.B.C
Velocity Potential
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Chapter 2
Assuming that the wave slope is small (H/L<<1) and that the water depth is much greater than the wave height (h/H>>1), the solution of velocity potential is:
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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.
f x 0 x , y f x 0 , y f x x 0 , y x 2 f x x 2 0 , y 2 x ! 2 n f x x n 0 , y n x ! n
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Chapter 2