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