Airy Wave Theory 1: Wave Length and Celerity Wave Theories Mathematical relationships to describe: (1) the wave form, (2) the water motion (throughout the fluid column) and pressure in waves, and (3) how (1) & (2) change with shoaling. We ll obtain expressions for the movement of water particles under passing waves - important to considerations of sediment transport --> coastal geomorphology. No single theory best describes the full range of conditions found in nature! 1
Linear (Airy) Wave Theory Originates from Navier Stokes --> Euler Equations Works very well in deep water, but only applicable when L >> H, so it breaks down in shallow water. Solution is eta relationship: George Biddell Airy (1801-1892) Wave Number: k = 2π/L Radian Frequency: σ = 2π/T Water Surface Displacement Equation What is the wave height? What is the wave period? 2
Dispersion Equation: Convert to a general expression for Wave Celerity Fundamental relationship in Airy Theory, which illustrates how waves segregate according to wave period: Substitute the relationships for radian frequency and wave number, respectively to get an equation for wavelength. Divide both sides by wave period to obtain an equation for wave speed (celerity). These are tough to solve, as L is on both sides of equality and contained within hyperbolic trigonometric function. Effect of the Hyperbolic Trig Functions on Wave Celerity What s the relationship for celerity in deep water? What s the relationship for celerity in shallow water? 3
So the celerity illustrated is General Expression: Airy Wave Celerity: General Expression, Deep & Shallow Approximatio 45 40 35 SWS, only depth dependent Celerity (m/s) 30 25 20 15 DWS, T=16 s DWS, T=14 s DWS, T=12 s DWS, T=10 s DWS, T=8 s Gen l Soln., T=16 s Gen l Soln., T=14 s Gen l Soln., T=12 s Gen l Soln., T=10 s Gen l Soln., T=8 s Deep-water expression: 10 5 0 0 50 100 150 200 Depth (m) Shallow-water expression: Example 1 of Shallow Water 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/s 2 *4000 m) 1/2 ~200 m/s ~450 mph! (Alaska to Hawaii in 4.7 hours) 4
Example 2 of Shallow Water Wave Speed Tow-In Surfing How fast does a Laird Hamilton surf? wave speed C = (gh) 1/2 tow-in waves: H = ~8 m C = (10 m/s 2 * 10 m) 1/2 ~ 10 m/s ~25 mph! waves surfable by mortals: C = (10 m/s 2 * 2 m) 1/2 ~ 4.4 m/s ~9 mph! Airy Wave Theory 2: Wave Orbitals and Energy 5
Compilation of Airy Equations Orbital Motion of Water Particles Airy Wave Theory also predicts water particle orbital path trajectories. Orbital path divided by wave period provides the wave orbital velocity. Show code for this: /Users/pna/Work/mFiles/pna_library/wave_pna_codes/waveOrbVelDeep.m 6
Orbital Motion of Water Particles 0.8 0.6 H=2m, T=10s, h=4000m Horizontal Vertical Tangential 0.4 velocity (m/s) 0.2 0 0.2 A B C D 0.4 0.6 0.8 0 10 20 30 40 50 60 Where is the wave crest? The trough? time (sec) Code for this: /Users/pna/Work/mFiles/pna_library/wave_pna_codes/waveOrbVelDeep.m Orbital Motion of Water Particles Deep water (h>l/2): s=d=he kz, circular orbits whose diameters decrease through water column to zero at h = L/2. At water surface, diameter of particle motion = wave height, H Intermediate water (h<l/2): elliptical orbits, whose size 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 valid in shallow water. 7
Orbital Motion at the Bed in Shallow Water The horizontal diameter at the bed simplifies to And the maximum horizontal velocity at the bed, which relates conveniently to the shear stress, is Derivation of Wave Energy Density Total Energy = Potential Energy + Kinetic Energy z E = E p + E k = 1 L L 0 ρgzdzdx + η h 1 L = 1 16 ρgh 2 + 1 16 ρgh 2 L 0 η h 1 2 ρ ( u2 + w 2 )dzdx = 1 8 ρgh 2 [dimensions] = M L L 2 ; Units = joules/m 2 or ergs/m 2 L 3 T 2 8
Wave Energy Flux Differs from energy density, as energy flux is equal to the energy density carried along by the moving waves. a.k.a. Power per unit wave crest length [dimensions] = M L L 2 L L 3 T 2 T [units] = joules/sec/m = Watts/m Deep Water n=1/2 Shallow Water n=1 P = 1 T η 0 h[ Δp(x,z,t) ]udzdt T = 1 8 ρgh 2 c 1 # 2 1+ 2kh & % ( $ sinh(2kh) ' = Ecn Wave Groups The expression Cn (sometimes written C g ) is known as the group celerity. In deep water, the first wave in a group decreases in height until it disappears and the second wave now becomes the leading wave (Figure). A new wave develops behind the last wave, thus maintaining the number of waves. 9
Individual Waves and Wave Group Velocity Group velocity approx. c g = Δσ/ Δk ~ σ/ k Deep Water: σ 2 = gk c g = σ/ k = g/2σ = 1/2 c (use implicit differentiation) Shallow Water: σ 2 = ghk 2 c g = σ/ k = (gh) 1/2 = c (use implicit differentiation) 1 & 2kh # n = $ 1 + 2! % sinh(2kh) " The effect of the dispersion process is that, in deep water, the group of waves travels at a speed equal to ½ the speed of the individual waves in the group.* This is important in forecasting wave propagation and in particular the travel time of waves generated by a distant storm (hint for a problem on Assignment 3). Stokes s 2 nd Order Wave Theory Airy (linear) wave theory which makes use of a symmetric wave form, cannot predict the mass transport phenomena which arise from asymmetry that exists in the wave form in intermediate-toshallow water. The wave form becomes distorted in shallower water. The crest narrows and the trough widens. Shoreward-directed horizontal velocity becomes higher under the wave crest than the offshore-directed velocity under the trough. Waves steepen and relative depth decreases, so that these waves are no longer considered smallamplitude. Instead they are called finite-amplitude. 10
Orbital Motion in Finite-Amplitude Wave Theory Due to the asymmetry of the wave form, orbital paths are not closed. There is a net motion of the water particle in the direction of wave advance, called Stokes drift. Stokes drift is important because it provides a mechanism of sediment transport on beaches, independent of current-driven transport. Can divide drift distance by wave period to obtain drift velocity. Shallow Water - Cnoidal and Solitary Wave Theories Wave speed in shallow-water is influenced more by wave amplitude than water depth. The water particle motion is dominated by horizontal flows - vertical accelerations are small, and Stokes's theory becomes invalid. Mathematically complex formulations have emerged that predict shallow water wave forms well Cnoidal and Solitary theory, which originates from the shallow water Boussinesq equation. 11
Limits of Application 12