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Internal Gravity Waves
Knauss (1997), chapter-2, p. 24-34Knauss (1997), chapter-10, p. 229-234
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