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
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Distribution of Energy in Surface Waves
tides, tsunamis wind waves Capillary waves
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Toenning, Germany
Wave ripples at low tide
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Tautuku Bay, New Zealand
Monochromatic Swell (one regular wave)
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Fully developed seas with many waves of different periods
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Travel time in hours of 2 tsunamisCrossing entire Pacific Ocean in 12 hours
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Definitions:
Wave number = 2/wavelength = 2/
Wave frequency = 2/waveperiod = 2/T
Phase velocity c = / = wavelength/waveperiod = /T
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Wave1Wave2Wave3
Superposition: Wave group = wave1 + wave2 + wave3
3 linear waves with differentamplitude, phase, period, and wavelength
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Wave1Wave2Wave3
Superposition: Wave group = wave1 + wave2 + wave3
Phase (red dot) and group velocity (green dots) --> more later
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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
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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)
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p(x,z,t) = …
(x,t) = a cos(x-t)
u(x,z,t) = …
w(x,z,t) = …
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(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]
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(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]
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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:
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c2 = (/)2 = g/ tanh[h]
c2 = (/T)2 = g (/2) tanh[2/ h]
h>>1
h<<1
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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
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Deep waterWave
Shallow waterwave
Particle trajectories associated with linear waves
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Particle trajectories associated with linear waves
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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
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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
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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
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Particle trajectories associated with linear waves
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Wave refraction inshallow waterc = (gh)
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Lituya Bay,Alaska 1958: Tsunami1720 feet height
link
Next: Tides and tsunamis