surface gravity waves-1 knauss (1997), chapter-9, p. 192-217 descriptive view (wave characteristics)...

Surface Gravity Waves-1

Knauss (1997), chapter-9, p. 192-217

Descriptive view (wave characteristics)Balance of forces, wave equationDispersion relationPhase and group velocityParticle velocity and wave orbits

MAST-602: Introduction to Physical OceanographyAndreas Muenchow, Sept.-30, 2008

Distribution of Energy in Surface Waves

tides, tsunamis wind waves Capillary waves

Toenning, Germany

Wave ripples at low tide

Tautuku Bay, New Zealand

Monochromatic Swell (one regular wave)

Fully developed seas with many waves of different periods

Tsunami off OR/WA

Amplitude: Low High

Travel time in hours of 2 tsunamisCrossing entire Pacific Ocean in 12 hours

Definitions:

Wave number = 2/wavelength = 2/

Wave frequency = 2/waveperiod = 2/T

Phase velocity c = / = wavelength/waveperiod = /T

Wave1Wave2Wave3

Superposition: Wave group = wave1 + wave2 + wave3

3 linear waves with differentamplitude, phase, period, and wavelength

Wave1Wave2Wave3

Superposition: Wave group = wave1 + wave2 + wave3

Phase (red dot) and group velocity (green dots) --> more later

Linear Waves (amplitude << wavelength)

∂u/∂t = -1/ ∂p/∂x

∂w/∂t = -1/ ∂p/∂z + g

∂u/∂x + ∂w/∂z = 0

X-mom.: acceleration = p-gradient

Z-mom: acceleration = p-gradient + gravity

Continuity: inflow = outflow

Boundary conditions:

@ bottom: w(z=-h) = 0

@surface: w(z= ) = ∂ /∂t

Bottom z=-h is fixed

Surface z= (x,t) moves

Combine dynamics and boundary conditions

to derive

Wave Equation

c2 ∂2/∂t2 = ∂2/∂x2

Try solutions of the form

(x,t) = a cos(x-t)

p(x,z,t) = …

(x,t) = a cos(x-t)

u(x,z,t) = …

w(x,z,t) = …

(x,t) = a cos(x-t)

The wave moves with a “phase” speed c=wavelength/waveperiodwithout changing its form. Pressure and velocity then vary as

p(x,z,t) = pa + g cosh[(h+z)]/cosh[h]

u(x,z,t) = cosh[(h+z)]/sinh[h]

(x,t) = a cos(x-t)

The wave moves with a “phase” speed c=wavelength/waveperiodwithout changing its form. Pressure and velocity then vary as

p(x,z,t) = pa + g cosh[(h+z)]/cosh[h]

u(x,z,t) = cosh[(h+z)]/sinh[h]

if, and only if

c2 = (/)2 = g/ tanh[h]

Dispersion refers to the sorting of waves with time. If wave phase speeds c depend on the wavenumber , the wave-field is dispersive. If the wave speed does not dependent on the wavenumber, the wave-field is non-dispersive.

One result of dispersion in deep-water waves is swell. Dispersion explains why swell can be so monochromatic (possessing a single wavelength) and so sinusoidal. Smaller wavelengths are dissipated out at sea and larger wavelengths remain and segregate with distance from their source.

c2 = (/)2 = g/ tanh[h]Dispersion:

c2 = (/)2 = g/ tanh[h]

c2 = (/T)2 = g (/2) tanh[2/ h]

h>>1

h<<1

c2 = (/)2 = g/ tanh[h]

Dispersion means the wave phase speed variesas a function of the wavenumber (=2/).

Limit-1: Assume h >> 1 (thus h >> ), then tanh(h ) ~ 1 and

c2 = g/ deep water waves

Limit-2: Assume h << 1 (thus h << ), then tanh(h) ~ h and

c2 = gh shallow water waves

Deep waterWave

Shallow waterwave

Particle trajectories associated with linear waves

Particle trajectories associated with linear waves

Deep water waves (depth >> wavelength)Dispersive, long waves propagate faster than short wavesGroup velocity half of the phase velocity

c2 = g/ deep water waves phase velocityred dot

cg = ∂/∂ = ∂(g )/∂ = 0.5g/ (g ) = 0.5 (g/) = c/2

Blue: Phase velocity (dash is deep water approximation)Red: Group velocity (dash is deep water approximation)

DispersionRelation

c2 = (/T)2 = g (/2) tanh[2/ h]c2 =

g/

dee

p w

ater

wav

es

Blue: Phase velocity (dash is deep water approximation)Red: Group velocity (dash is deep water approximation)

DispersionRelation

c2 = (/T)2 = g (/2) tanh[2/ h]c2 =

g/

dee

p w

ater

wav

es

Particle trajectories associated with linear waves

Wave refraction inshallow waterc = (gh)

Lituya Bay,Alaska 1958: Tsunami1720 feet height

link

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