Transcript Wellen - ETH Z

Numerical Hydraulics
Waves
W. Kinzelbach
M Holzner
C. Beffa
Wave equation in 1D
2
2
 u
2 u
--------2- = v --------2t
x
u amplitude, v phase velocity
u =   kx – t 
k wave number,  angular frequency  = 2pf, f frequency
Insertion yields

v = ---k
Wave equation 2D
Analogous in 2D: 1D-wave front arbitrarily orientied, amplitude u
2
2
2
 u
2   u  u
--------2- = v  --------2- + --------2-
 x
t
y 
Solution
u =   k x x + k yy – t 
Wave velocity by insertion

v = ----------------------------2
2 12
 kx + ky 
Wave equation 2D
Position of wave front:
y
a kx
y

x
ky ky
x
y
Wave front
k
Wave vector
k  (k x , k y )
Gradient: -k /k
y
k
x
x
Abb: 7-1:
Ausbreitung einer ebenen W elle mit
W ellenvektor k = (k x ,k y ). Die W ellenfront
bezeichnet die Kurven gleicher Amplitude
Harmonic wave
Wave vector in x-direction
uk  x t  = a k sin  kx – t  + b k cos  kx – t 
More economic way of writing
i  kx – t 
u k  x t  = Ak e
Decomposition of an arbitrary wave into harmonic waves (Fourier integral)

  x t  =

–

u k  x t  dk =

Ak e
i  kx –  t 
dk
–
If domain has finite length L: Only integer k (Fourier analysis)
p
k n  = --- n
L
  x t  =
 u k  n   x t 
Group velocity
Superposition of 2 waves with slightly different ki and i:
u 1 = a  sin  k 1 x – 1 t 
u 2 = a  sin  k 2 x – 2 t 
with
1
k = ---  k1 + k2 
2
k = k 1 – k 2
1
 = ---  1 + 2 
2
 =  1 –  2
k

u tot = u 1 + u 2 = 2a  cos  ------- x – -------- t  sin  kx – t 
 2
2 
Modulated wave
Velocity of propogation of modulation = group velocity
2 p /k

vg = -------k
In the limit of small , k
x
d
v g = ------dk
2p/k
Abb: 7-2:
Überlagerung von 2 Wellen mit leicht
verschiedenen Wellenzahlen
Dispersion
If v is constant (independent of k) we get
d
vg = ------- = v
dk
If group and phase velocity are different the wave packet is smoothed out, as the
components move with different velocities. This phenomenon is called dispersion.
Waves in water aee dispersive.
e.g. deep water waves
g
v =  ------
 2p
12
g
=  ---
 k
1 2
d
1  g 1  2
1
v g = ------- = --- --= --- v
dk
2 k
2
Wave equations which lead to dispersion, have an addtional term:
2
2

 u
2 u
--------2- = v  --------2- + cu
 x

t
2 
2
d

2  
resp. with solution F(kx-t) ---------  k – ------ + c = 0
2
2
d 
v 
Damped wave
Wave equations with an addtional time derivative term lead to damped waves
2
2
 u
u
2 u
--------2- + a ------ = v --------2t
x
t
yields
with
u  kx –  t  = Ae
i kx – t 

1
2 2
2



ia

4
v
k

a
  ia  v k  0 resp.
2
With a<2k one obtains
eit  ei0t  e1t for t  
2
2
2

Non linear wave equations
Non linear wave equations lead to a coupling of harmonic components. There is
no more undisturbed superposition but rather interaction (enery exchange)
between waves with different k.
Types of waves
• Gravity waves
– are caused by gravity
• Capillary waves :
– important force is surface tension
• Shallow water wave
– Gravity wave, but at small water depth (compared to
wave length)
• Solitons (Surge waves)
– Waves with a constant wave profile
• Internal waves, seiches
Gravity waves in deep water
2p
 = -----T
Path lines of water particles: Circles
c,
ellenges
chwindigkeit
cW
wave
velocity
w1  c  r
w
1
r
r
Bernoulli along
water surface
w
/4
Abb: 7-3:
2
w1
2
w2  c  r
Schw e re w elle n
2
Decrease of amplitude with depth
z 0 – z
exp  – 2p ------------
 
w2
------ + 2r = -----2g
2g
Phase velocity c and group velocity c* of the wave
gT
c = ------2p
c
g

2p
Gravity waves are dispersive
g
k
c 
d 1 g 1

 c
dk 2 k 2
Capillary waves
In addition to pressure force the surface tension is acting as restoring force
p
p0



g g Rg 
2
2
w1
w2


Bernoulli along pathline ------ + 2r + ----------- = ------ – ----------2g
Rg
2g Rg
Radius of curvature R of water surface
c
2
R
4p 2 r
Wave length, at which capillary
and gravity contributions are equal
g 2p

2p


 1 = 2p -----g
for  << 1 c 
For water:  = 1000
 = 0.073 N/m
1 = 1.71 cm c1 = 23.1 cm/s
kg/m3
for  >> 1 c =
2p

g
-----2p
Waves at finite water depth
2


g

2
p

4
p
2
c 
tanh(
h)  1 
2 
2p


g



h/ 
2p
tanh  ------h  1
 
c
g 2p

2p

Shallow water equations
InIm
deep
water
tiefen
Wa sser
In
water
Imshallow
flachen
Wa sser
hydrostatische
hydrostatic
Druckverteilung
pressure distribution
h <  /2
Abb: 7-4:
2
Schwerewellen
2p
2p
tanh  ------h  ------h
 

c = gh
for h << /2
c =
gh
Solitons
• Dispersion (small kh)
A
k 2h2
c(k )  gh (1 
)
6
c 1 =

h
2
h
gh ------2

• Front steepening
c' =
g h + A  c =
c2  c'c 
g A
h 2
Soliton:
c 1 = c2
Abb: 7-5:
gh
Solitonen
2h 3
A 2

Equilibrium between steepening and dispersion
A
y = h  ---------------------2 x
cosh  ---

mit
Wave form does not change
2
4 3
A = --- h
3
Seiches
Base
Grundschwingung
Erste Oberschwingung
Schwingungsknoten
Schwingungsknoten
Gleichgewichtslage
der Seeoberfläche
z
Abb: 7-6:
Stehende Oberflächenwelle, genannt Seiches
Assumption: Water movement horizontally
Linearised equation:

h
u
x
0
Abb: 7-7:
Oberflächenseiche in
einem Rechtecksee
u
1 p
------ = – --- ----- x
t
h+
p z  = p0 + g

z
 dz = p0 + g  h +  – z 
Surface seiches
p

------ = g-----x
x
Assumption: Velocity u constant over depth z

u

------   h +  u  h ------ = – -----x
x
t
u

------ = – g -----x
t
2
From those:
2
 
 
--------2- = gh --------2x
t
2
2
2
2
Derivative
with resp. to t
 u
 
h ----------- = – --------2xt
t
Derivative
with resp. to x
 u
 
----------- = – g --------2x t
x
v =
gh
Standing wave (n-th Oberschwingung)
2L
2p
 = ------ = -----n
k
 n = vk n
2p
2L 1
Tn = ------ = -----------  --n
gh n
Internal waves
z
Pressure in Epilimnion/equ. of motion:

h H +h E
Epilimnion  E
p = p 0 + g E  h E +  – z 
u
hH
Hypolimnion  H
v
u

------ = – g -----x
t
p

------ = gE -----x
x

x
Abb:in
7-8:
Interne Seiche im
Pressure
Hypolimnion/equ.
of Rechtecksee
motion p = p 0 + g E h E +  –   + g H h H +  – z 
p


------ = g   ------ +  ------
 x
x
E x 
mit   H-E
v

h H ------ = – -----x
t
Continuity:
u
 
hE ------ = – ------ – -----x
t t
2
   2  2 
 
- --------2 + --------2-
And finally: --------2 = ghH  -----

t
x
x 
2
v
1 p
 

------ = –------  ------ = – g -------  ------ – g-----H x
 x
x
t
2
2
2
2
 
1     
--------2- = – ----------  -------2- – --------2-
gh E  t
x
t 
2
 
 h E h H  
 
--------2 = g -------  -------------------  --------2 = g'h' --------2
 h E + h H x
t
x
with

g' = ------- g

1
1
1
---- = ------ + -----h'
hE hH
Numerical example
• Example for basic period of surface seiches und
internal seiches
– Length of lake: L = 20 km
– Depth: Average h = 50 m, epilimnion hE = 10 m, hypolimnion hH
= 40 m
– Density difference H/E / = 10-3
• Surface-Seiche
v = (gh)1/2 = 22.2 m/s
T1 = 1800 s = 0.5 hours
• Internal Seiche
g‘ = g/ = 9.81 10-3ms-2
h‘ = 8m
v = (g‘h‘)1/2 = 0.28 m/s
T1 = 1.43 105 s = 39.7 hours