Surface Gravity Waves-1 Knauss (1997), chapter-9, p Descriptive view (wave characteristics) Balance of forces, wave equation Dispersion relation Phase and group velocity Particle velocity and wave orbits MAST-602: Introduction to Physical Oceanography Andreas Muenchow, Sept.-30, 2008

Distribution of Energy in Surface Waves tides, tsunamiswind wavesCapillary 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 tsunamis Crossing entire Pacific Ocean in 12 hours

Definitions: Wave number  = 2  /wavelength = 2  / Wave frequency  = 2  /waveperiod = 2  /T Phase velocity c =  /  = wavelength/waveperiod = /T

Wave1 Wave2 Wave3 Superposition: Wave group = wave1 + wave2 + wave3 3 linear waves with different amplitude, phase, period, and wavelength

Wave1 Wave2 Wave3 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 bottom: w(z=-h) = w(z=  ) = ∂  /∂t Bottom z=-h is fixed Surface z=  (x,t) moves

Combine dynamics and boundary conditions to derive Wave Equation c 2 ∂ 2  /∂t 2 = ∂ 2  /∂x 2 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/waveperiod without changing its form. Pressure and velocity then vary as p(x,z,t) = p a +  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/waveperiod without changing its form. Pressure and velocity then vary as p(x,z,t) = p a +  g  cosh[  (h+z)]/cosh[  h] u(x,z,t) =   cosh[  (h+z)]/sinh[  h] if, and only if c 2 = (  /  ) 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. c 2 = (  /  ) 2 = g/  tanh[  h] Dispersion:

c 2 = (  /  ) 2 = g/  tanh[  h] c 2 = ( /T) 2 = g ( /2  ) tanh[2  / h]  h>>1  h<<1

c 2 = (  /  ) 2 = g/  tanh[  h] Dispersion means the wave phase speed varies as a function of the wavenumber (  =2  / ). Limit-1: Assume  h >> 1 (thus h >> ), then tanh(  h ) ~ 1 and c 2 = g/  deep water waves Limit-2: Assume  h << 1 (thus h << ), then tanh(  h) ~  h and c 2 = ghshallow water waves

Deep water Wave Shallow water wave Particle trajectories associated with linear waves

Deep water waves (depth >> wavelength) Dispersive, long waves propagate faster than short waves Group velocity half of the phase velocity c 2 = g/  deep water waves phase velocity red dot c g = ∂  /∂  = ∂  (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) Dispersion Relation c 2 = ( /T) 2 = g ( /2  ) tanh[2  / h] c 2 = g/  deep water waves

Blue: Phase velocity (dash is deep water approximation) Red: Group velocity (dash is deep water approximation) Dispersion Relation c 2 = ( /T) 2 = g ( /2  ) tanh[2  / h] c 2 = g/  deep water waves

Particle trajectories associated with linear waves

Wave refraction in shallow water c =  (gh)

Lituya Bay, Alaska 1958: Tsunami 1720 feet height link Next: Tides and tsunamis