Waves

W. Kinzelbach

M. Wolf

C. Beffa

Numerical Hydraulics

Wave equation in 1D

2u

t2

--------- v22u

x2

---------=

u kx t– =

u amplitude, v phase velocity

k wave number, angular frequency = 2f, f frequency

Insertion yields

v k----=

Wave equation 2D

2u

t2

--------- v2 2u

x2

--------- 2u

y2

---------+

=

u kxx kyy t–+ =

v

kx2 ky

2+ 1 2-----------------------------=

Analogous in 2D: 1D-wave front arbitrarily orientied, amplitude u

Solution

Wave velocity by insertion

Abb: 7-1: Ausbreitung einer ebenen W elle mit W ellenvektor k = (k x ,k y ). Die W ellenfront bezeichnet die Kurven gleicher Amplitude

y

k y

k x x

Wave front

Gradient: -k x /k y

Wave equation 2D

xk

k

k

ay

y

x

y

Position of wave front:

Wave vector

( , )x yk k k

Harmonic waveWave vector in x-direction

uk x t ak kx t– bk kx t– cos+sin=

uk x t Akei kx t– =

x t uk x t kd

–

Akei kx t–

kd

–

= =

k n L---n= x t uk n x t =

More economic way of writing

Decomposition of an arbitrary wave into harmonic waves (Fourier integral)

If domain has finite length L: Only integer k (Fourier analysis)

Group velocity

u1 a k1x 1t–

u2

sin

a k2x 2t– sin

=

=

k k1 k2–=k 12--- k1 k2+ =

1 2–= 12--- 1 2+ =

uto t u1 u2+ 2a k2

-------x 2

--------t– kx t– sincos= =

Abb: 7-2: Überlagerung von 2 W ellen mit leicht verschiedenen W ellenzahlen

2 / k

2 /k

x

vgk--------=

vgddk-------=

Superposition of 2 waves with slightly different ki and i:

Modulated wave

Velocity of propogation of modulation = group velocity

In the limit of small , k

with

Dispersionvg

ddk------- v= =

v g2------ 1 2 g

k---

1 2= =

vgddk------- 1

2--- g

k---

1 2 12---v= = =

2u

t2

--------- v2 2u

x2

--------- cu+

=d2

d2--------- k2 2

v2

------–

c+ 0=

If v is constant (independent of k) we get

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.

Wave equations which lead to dispersion, have an addtional term:

Waves in water aee dispersive.e.g. deep water waves

resp. with solution (kx-t)

Damped wave

2u

t2

--------- aut------+ v22u

x2---------=

0v 222 kia 222v42

1akia

u kx t– Ae i kx t– =

0 1i t ti te e e for t

Non linear wave equations

Wave equations with an addtional time derivative term lead to damped waves

with

yields

resp.

With a<2k one obtains

Non linear wave equations lead to a coupling of harmonic components. There isno 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 

Abb: 7-3: Schwerewellen

c, Wellengeschwindigkeit

r

r

w2

w1

Gravity waves in deep water 2

T------=

rcw 1

rcw 2

w12

2g------ 2r+

w22

2g------=

c gT2-------=

k

ggc

2c

k

g

dk

dc

2

1

2

1

2z0 z–

--------------– exp

Bernoulli along water surface

Decrease of amplitude with depth

Phase velocity c and group velocity c* of the wave

Gravity waves are dispersive

/4

Path lines of water particles: Circlesc wave velocity

Capillary waves

Rgg

p

g

p 0

w12

2g------ 2r

Rg-----------+ +

w22

2g------

Rg-----------–=

rR

2

2

4

2

2

gc

c g2------=

2

c

1 2 g------=

For water: = 1000 kg/m3 = 0.073 N/m1 = 1.71 cm c1 = 23.1 cm/s

In addition to pressure force the surface tension is acting as restoring force

Bernoulli along pathline

Radius of curvature R of water surface

for <<1

for >> 1

Wave length, at which capillary and gravity contributions are equal

Waves at finite water depth

2

22 4

1)2

tanh(2

gh

gc

2

------h 1tanh

2

2

gc/h

Abb: 7-4: Schwerewellen

Im tiefen W a sser Im flachen W a sser

hydrostatische Druckverteilung

h < /2

Shallow water equations

c2 gh= c gh=

2

------h 2

------htanh for h << 2

In deep water In shallow water

hydrostaticpressure distribution

Abb: 7-5: Solitonen

h

A

Solitons

• Dispersion (small kh)

• Front steepening

)6

1()(22hk

ghkc

c1 ghh2

2------=

c' g h A+ c gh= =

2'2

A

h

gccc

c1 c2= 2

32

h

A

y hA

x--- 2

cosh-----------------------= A2 4

3---h

3=

Soliton:

Equilibrium between steepening and dispersion

Wave form does not change

mit

Seiches

Abb: 7-7: Oberflächenseiche in einem Rechtecksee

u

x

h

z

0

ut------

1---px------–=

p z p0 g zd

z

h +

+ p0 g h z–+ += =

Abb: 7-6: Stehende Oberflächenwelle, genannt Seiches

Grundschwingung

Schwingungsknoten

Erste Oberschwingung

Schwingungsknoten Gleichgewichtslage der Seeoberfläche

Assumption: Water movement horizontallyLinearised equation:

Base

Surface seiches

px------ gx

------=

x------ h + u h

ux------

t------–=

2uxt----------- g

2x2---------–=

h 2uxt----------- 2

t2

---------–=

2

t2

--------- gh2

x2---------= v gh=

2Ln------ 2

k------= = Tn

2n------

2L

gh----------- 1

n---= =n vkn=

Assumption: Velocity u constant over depth z

Derivative with resp. to t

Derivative with resp. to x

From those:

Standing wave (n-th Oberschwingung)

ut------ g

x------–=

Abb: 7-8: Interne Seiche im Rechtecksee

u

x

h H

z

h H +h E

v

Epilimnion E

Hypolimnion H

Internal waves

p p0 gE hE z–+ +=

px------ gE

x------=

p p0 gE hE –+ gH hH z–+ + +=

px------ g

x------ +

E

x------

= mit H-E vt------

1H------ p

x------– g

------- x------ g

x------––= =

hHvx------

t------–= hE

ux------

t------–

t------–=

2

t2-------- ghH

-------2

x2-------- 2

x2---------+

=2

x2

--------- 1ghE---------- 2

t2

-------- 2

t2

---------–

–=

2

t2-------- g

-------

hEhH

hE hH+------------------- 2

x2-------- g'h'

2

x2--------= =

g'

-------g=

1h'---- 1

hE------ 1

hH------+=

Pressure in Epilimnion/equ. of motion:

Pressure in Hypolimnion/equ. of motion

Continuity:

And finally:

with

ut------ g

x------–=

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