Wave Propagation and Shoaling Focus on movement and natural alteration of the characteristics of waves as they travel from the source region toward shore Waves moving from deep to intermediate/shallow water change their shape and characteristics significantly. Wave velocity and wavelength decrease, while height increases to conserve wave energy flux. Period remains the same. Just offshore of the breaker zone, the waves have peaked crests and broad troughs; a very different appearance to their deepwater sinusoidal form. Wave Groups and Dispersion Define the term "wave group" to mean a bundle of wave energy that travels from source region to the shore. Wave groups travel with speed C g = Cn, a.k.a. the "group velocity". Longer period waves outrun and leave behind the shorter period waves. Dispersion produces a narrowing of the energy spectrum, so the greater the travel distance, the narrower the strong frequency band in the energy density spectrum. Individual waves arise at rear of group, move through group, and die out at front. Hence, they cannot be traced across the ocean - but the groups themselves can! https://www.meted.ucar.edu/training_module.php?id=188 1
Arrival Times for Swells of Different Periods Wave Energy Losses Waves lose very little energy as they traverse great distances, but four sources of wave energy loss have been suggested and investigated: 1. Viscous Damping 2. Angular Spreading 3. Contrary Winds 4. Wave-wave Interactions https://www.meted.ucar.edu/training_module.php?id=188 2
Wave Shoaling Schematic Changes that occur when a wave shoals (moves into shallow water): In deep water = profile of swell is nearly sinusoidal. Enter shallow water, waves undergo a systematic transformation. Wave velocity and wave length decrease while the wave height increases. Only wave period remains constant. Wave Shoaling L & C Deep water - L, C depend only on period Shallow water - L, C depend only on the water depth Summarize regions of applications of approximations Behavior of normalized variables. 3
Wave Shoaling Effects on H the shoaling coefficient, K s Outside the surf zone, wave energy flux is conserved: P in = P out Ecn = constant Ecn = E c n The shoaling coefficient, K s, is given as the ration of shallow water to deep water wave height: P out P in Explains why orthogonally directed waves increase height during shoaling Direct compensation for slowing of individual waves and need to maintain constant wave energy flux Waves convert a significant fraction of their kinetic energy to potential energy Effect of Shoaling H = 2 m T = 10 s othogonal angle of incidence 4
Wave Shoaling Steepness (H/L) Straightforward consequence of combined shoaling behavior of H & L. Steepness initially decreases upon entry to intermediate water depth, then rapidly increases until instability condition associated with wave breaking. Wave Refraction 5
Refraction Analogous to the bending of light rays as they pass through different media glass of water, for example. Wave Refraction: Distinction - Wave Crests vs. Wave Rays Wave crests are the line segments that connect the peaks (or troughs) of a wave field. The crests are visible to the observer. Wave rays are the lines orthogonal (perpendicular) to the wave crests, which represent the direction of wave propagation 6
Wave Refraction - Snell s Law c = (gh) 1/2 sin( α1) sin( α2) = c c 1 2 sin( α) = c c sin( α ) Wave Refraction: Energy flux per unit length of wave crest Energy flux per unit length of wave crest is not necessarily conserved Can lead to a decrease in wave height during the shoaling and refraction process. 7
Wave Refraction Over Canyons and Headlands Computing the Refraction Coefficient, K r ( EnC ) = s s ( EnC) 1 1 2 2 H H & 1 = $ % 2n c c #! " 1/ 2 & $ % s s #! " 1/ 2 # K r = s & $ % s ' ( 1/ 2 $ = cos(α ) ' & ) % cos(α) ( 1/ 2 8
Effect of Shoaling and Refraction H = 2 m T = 10 s compare: orthogonal wave vs. refracting wave H H & = $ % 1 2n c # c! " 1/ 2 & s # $ % s! " 1/ 2 Modeling refraction - wave rays H = 2 m T = 10 s α = 270 9
Modeling refraction - wave rays - double period H = 2 m T = 20 s α = 270 Jaws Surfing Reef, Maui Model simulations of individual waves - not time averaged. bathymetry (in feet) focusing waves 10
Wave Diffraction Lateral translation of energy along a wave crest. Most noticeable where a barrier interrupts a wave train creating a "shadow zone". Energy leaks along wave crests into the shadow zone. Also by analogy to light, Huygen's Principle explains the physics of diffraction through a superposition of point sources along the wave crest. Wave Diffraction- Barcelona 11
Modeling with SWAN Ref/Diff numerical simulation of shoaling and refraction - monochromatic (not spectral) SWAN - used here at UF (spectral) 12
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