So the celerity illustrated is… General Expression: SWS, only depth dependent DWS, T=16 s Gen’l Soln., T=16 s DWS, T=14 s Gen’l Soln., T=14 s Deep-water expression: DWS, T=12 s Gen’l Soln., T=12 s DWS, T=10 s Gen’l Soln., T=10 s DWS, T=8 s Gen’l Soln., T=8 s Shallow-water expression:

Wave Speed - Tsunami How fast does a tsunami travel across the ocean? What classification is this wave? Deep water? Intermediate? Shallow water? In Shallow Water wave speed C = (gh)1/2 Deep Ocean Tsunami C = (10m/s2*4000 m)1/2 ~200 m/s ~450 mph! (Alaska to Hawaii in 4.7 hours)

Wave Speed - Nearshore How fast does a Laird Hamilton surf? wave speed C = (gh)1/2 tow-in waves: H = ~8 m C = (10 m/s2 * 10 m)1/2 ~ 10 m/s ~25 mph! waves “surfable” by mortals: C = (10 m/s2 * 2 m)1/2 ~ 4.4 m/s ~9 mph!

Derivation of Deep & Shallow water Equations 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.

Airy Wave Theory Continued

Orbital Motion in Waves

Orbital Motion in Waves Deep water: s=d=Hekz, circular orbits whose diameters decrease EXPONENTIALLY (truly) through the water surface – at water surface the diameter of particle motion is obviously the wave height, H. Intermediate water: ellipse sizes decrease downward through water column Shallow water: s=0, d=H/kh; ellipses flatten to horizontal motions; orbital diameter is constant from surface to bottom. Airy assumptions not really valid in shallow water.

Derivation of Wave Energy Density z Total Energy = Potential Energy + Kinetic Energy Although there is no net movement of water during wave propagation, since the water particle orbits are closed (in this theory), the motion of the wave itself constitutes an energy transfer over the sea surface. Displacement of water away from flat still water is the potential energy and the orbital motions constitute a kinetic energy. Integrating both along the full length of the wave gives. Total energy per wave per unit width; therefore called the energy density Only proportional to the wave height but is not conserved since we know that wave height changes across the surf zone [units] = M L L2 = joules/m2 or ergs/m2 L3 T2

Wave Energy Flux [dimensions] = M L L2 L L3 T2 T Energy density carried along by the moving waves. a.k.a. “Power per unit wave crest length” Rate at which energy is transferred Is the energy flux. Energy flux is approximately constant. Energy flux is the rate at which work is being done by the fluid on the left hand side of the vertical section on the fluid on the right hand side. Dell P is the pressure deviation from hydrostatic. In deep water, energy is transmitted at only ½ the speed of the wave profile while in shallow water the wave and energy travel at the same speed. The rate of advance of the wave energy is known as the group velocity which in deep water is half the phase speed of the waves. Also called P because it is the power per unit wave-crest length. [dimensions] = M L L2 L L3 T2 T Deep Water n=1/2 Shallow Water n=1 = joules/sec/m = Watts/m

Groupiness / Group velocity Consider two waves with the same height beating together wave 1 wave 2 h = h1 + h2 h = H/2 cos(k1x - s1t) + H/2 cos(k2x - s2t) = H cos[(k1+k2)/2x - (s1+s2)/2t] * cos[(k1-k2)/2x - (s1-s2)/2t] = H cos(kx-s t)*cos[1/2Dk(x-Ds/Dk*t)] average wave group envelope Wave energy cannot propagate past the node of the group envelope since H is zero and dynamic pressure is zero. Therefore energy must travel with the group of waves. Trig identity: cos(a) +cos(b) = 2 cos(1/2(a+b))cos(1/2(a-b)) With group velocity: cg = Ds/ Dk Groupiness looks like a wave: hg = cos(Dk/2 x - Ds/2 t)

Group Velocity and n Group velocity approx. cg = Ds/ Dk ~ ∂s/∂k Deep Water s2 = gk cg = ∂s/∂k = g/2s = 1/2 c Shallow Water s2 = ghk2 cg = ∂s/∂k = (gh)1/2 = c

Radiation Stress - introduced “the excess flow of momentum due to the presence of the waves”

Nonlinear Waves - Stokes Theory Komar, 1998

Cnoidal and Solitary Wave Theory

Limits of Application