GLY 4734 - Coastal Geomorphology Notes

Dr. Peter N. Adams

February-March 2010

3 Waves

In this section we’ll cover waves from their generation to dispersion and travel toshoaling transformation to breaking in the nearshore zone, where they release theirenergy to do geomorphic work on the coast. Our introduction will cover basic def-initions, measurement, analysis, data sources, and wave climate. We can organizeour thoughts by loosely categorizing waves into three personas, according to theirenvironment and maturity:

1. Sea: Arising in the localized region of wave generation, where water motionsare irregular and appear quite disorganized.

2. Swell: Covering the broad region of wave propagation, wherein the wave watermotions transport energy, and the waves themselves organize according to theirperiodicity.

3. Surf: Occurring in the very narrow band adjacent to a shoreline where wavesshoal rapidly, transform, and release energy in breaking and run-up.

3.1 Periodic Waves

• We can think of waves with respect to a spatial framework, or with respect toa temporal framework. Examine the anatomy of a wave - period, frequency,length, phase velocity, height, displacement, depth, particle orbital diameter,particle orbital velocity.

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• Numerous relationships exist among the wave variables listed above, like forexample: There exists a direct relationship between wave length and waveperiod in deep water, and thus between celerity and wave period. The mostbasic relationship is:

C =L

T(1)

• It is important to pay attention to the details of water motion in waves:

1. Individual particles move in circular patterns with the size of the circlesdecreasing (exponentially) with water depth.

2. The wave form, however, propagates through the medium.

3. So, a cork floating on the surface of the water makes no net advance inthe direction of wave motion - it simply returns back to its starting pointafter a wave passes.

• Dispersion: Wave have a tendency to sort themselves out by wave period. So,from a region of wave generation (open ocean storm), the longer period wavestravel faster across the ocean, and hence arrive at a point on the coast earlierthan short period waves.

• Superposition: Addition of two or more wave forms leads to constructive anddestructive interference as dictated by the heights and periods of the con-stituent wave forms. See Animation shown in class.

• Spectral Energy of Water Level Fluctuations - The classification of wave mo-tions is based on the restoring force (e.g. gravity, surface tension, coriolis, etc.).Wind waves occupy the 10−2 − 102 Hz band.

3.2 Measurement of Waves

We now understand that a wave field may arise from multiple superimposed waveforms from different sources with different heights, periods, and directions. So ac-curate measurement of waves, for the purposes of understanding individual con-stituents, requires more than a meter stick and stopwatch. In addition, the equip-ment must tolerate the corrosive, oceanic environment. Here we describe some ofthe techniques employed.

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3.2.1 In-situ Measurement Devices

1. Surface Piercing. Example: graduated staff with electrodes attached to a thevertical support of a pier or drilling platform. As water surface rises and falls,a specific level of electrodes are recorded as closed circuits providing a timeseries of water surface levels.

2. Pressure Sensing. Measures time series of height of column of water above theinstrument.

3. Surface Following. Example: Waverider buoy with an accelerometer.

3.2.2 Analytical Techniques

• First, a time series (then a populationdistribution) of waves must be computedfrom the time series of water level variations. One common technique is thezero upcrossing method (see lecture slide).

• Statistical Analysis - time domain analysis which analyzes the population dis-tribution of wave heights to determine a series of characteristic heights (Hsig,Hrms, H1/10, Hmax). For example, the root-mean-square wave height is deter-mined thusly:

Hrms =

√√√√ 1

N

N∑i=1

H2i (2)

Significant wave height (Hs) is commonly reported, and is determined by tak-ing the average of the highest one-third of waves over a specific interval ofobservation.

• Spectral Analysis - which is carried out in the frequency domain, and is afairly standard technique today. This method uses Fourier decomposition.Fourier Analysis is based on the concept that any complex time series canbe represented by a combination of various sine and cosine functions. Byperforming a Fourier Transform of the time domain data, we obtain a functionin the frequency domain which describes which frequencies are present in theoriginal function.

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• Several in-class examples of Fourier deconvolution illustrated with MATLABcodes.

• Sources of wave data are easily accessible (nowadays) online. For example,NOAA maintains a network of buoys (http://www.ndbc.noaa.gov/). Hindcastsare also available (WIS).

3.2.3 Wave Record Analyses

• Continuous records, if of sufficient length, can be examined for trends in chang-ing wave conditions. Recently, numerous studies have investigated various datasets indicating that changes in ocean wave storminess patterns are indeed oc-curring (Allan and Komar , 2006; Adams et al., 2008; Komar and Allan, 2008).

• Example of probability of occurrence analysis.

3.3 Wave Generation

The process of wave generation is frequently observed, but the physics and specificmechanisms are still poorly understood. In general, the longer and harder the windblows, the larger and longer period the waves generated (up to a point).

3.3.1 Processes of Wave Generation

• Generally accepted theory to account for growth of waves is the Miles-Phillipsmechanism, which incorporates 2 processes of energy transfer from wind towaves, covered on pp. 151-153 of Komar:

– Initial growth stage where there is a linear increase in wave energy withtime due to resonance between atmospheric pressure fluctuations and de-veloping, wave-covered, water surface.

– 2nd mechanism causes exponential growth of already developed waves.Under a logarithmic wind velocity profile, airflow over the sinusoidally-shaped water surface creates a distribution of high pressure over troughsand low pressure over crests, which drives flow separation at the boundarylayer, further amplifying wave height.

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• As wave generation continues, increasing energy is pumped toward the longperiod portion of the spectrum. Long period waves travel more quickly throughthe medium, and overtake short period waves, sweeping up their momentumand energy. However, short period waves are continuously being generated bythe Miles-Phillips mechanism, so they are always present.

3.3.2 Wave Predictions

Semi-empirical/semi-theoretical methods have been developed for wave prediction -theory is involved in their formulation, but data are required for the evaluation ofvarious coefficients. Heights and periods of a wave field are dependent on the windvelocity (U), the duration of time that the wind blows (D), and the distance overwhich the wind blows (F ), also known as the fetch.

1. Significant Wave Approach - developed by Sverdrup, Munk, and Bretschneider,the S-M-B methods relate Hs,Ts as f(U,D, F ), codified in nomograms still usedtoday.

2. Wave Spectrum Approach - in line with the characterization of waves by theirspectra, which provide a measure of wave energy at each period/frequency.Produce wave energy spectrum from given wind speed, duration, and fetch.Two ranges - equilibrium range and growth range. Energy first fills the highfrequency bins to the equilibrium range, then spills into the longer period por-tion of the spectrum, which has increasingly greater accommodation for energyin the equilibrium range. Hence, there is a progressive shift in peak period.

3. JONSWAP spectrum - most often used today.

3.4 Wave Theories

This section focuses on the motion and energy of waves. Wave movement acrossthe ocean from the source area to coastal locations, where breaking occurs, can bequantitatively explained via several wave theories. In addition, these theories provideinformation on the motion of water particles as waves pass through the medium. Thismaterial is extensively covered on pp. 160-176 of the Komar text.

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3.4.1 Airy Wave Theory

All fluid motions are governed by the mathematical relationships (differential equa-tions) that balance mass and momentum in a control volume. Waves are no different.These relationships boil down to:

1. The equations of motion, which, when applied to an incompressible fluid, arereferred to as the Navier-Stokes equations, and when operating under certainassumptions (low viscosity), are known as the Euler equations.

2. Continuity equation, which ensures that mass is conserved.

Simultaneous solution of these equations with the right boundary conditions andassumptions provides us with the basis of Airy Wave Theory (AWT), also referredto as Linear Wave Theory (LWT). One of the main assumptions in LWT isthat wave height is much smaller than wave length and water depth, so LWT isapplicable over much of the open ocean, but breaks down somewhat in shallow,coastal areas.

• Water Surface Elevation Fluctuation

The solution for water surface elevation η as a function of space and time (xand t) is:

η(x, t) =H

2cos(kx− σt) (3)

where H is wave height, k = 2πL

is wave number, and σ = 2πT

is radian frequency,with T being wave period, of course. The Airy wave is sinusoidal in profile withrespect to time and space.

• Dispersion Equation

Another fundamental equation arising from AWT is the dispersion equation:

σ2 = gk tanh(kh) (4)

which can be manipulated with the identities for radian frequency and wavenumber to yield

L =g

2πT 2 tanh(

2πh

L) (5)

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which is quite useful, as it relates wavelength, period, and water depth, but isproblematic because wavelength appears on both sides of the equation and isimprisoned within a hyperbolic trigonometric function for good measure. Thiswill have to be solved iteratively, perhaps by the Newton-Raphson method.

• Celerity Behavior in Deep and Shallow Water

Let’s divide both sides of Eqn. 5 by the wave period and see where that getsus.

C =gT

2πtanh(

2πh

L) (6)

Now we have a relationship for wave celerity, C, alternatively referred to as thephase velocity or wave speed. Let’s explore this function in deep and shallowwater. We must first recall the behavior of the hyperbolic trig functions. Justas the sin, cos, tan, etc. describe ratios between triangle legs on the unit circle,x2 + y2 = 1, the hyperbolic trigonometric functions describe ratios between”triangle legs” on the unit hyperbola, x2 − y2 = 1. The behavior of thesefunctions is shown in a plot on p. 163 of Komar.

Note that the hyperbolic tangent of a large value approaches 1, and the hy-perbolic tangent of a small value approaches that small value. So the ef-fect on the general solution for celerity is as follows. In deep water, wheretanh(anything) ≈ 1:

C =gT

2π(7)

and in shallow water, where tanh(something) ≈ thatsomething:

C =gT

2π

2πh

L(8)

after rearranging and canceling,

C2 = gh, or, C =√gh (9)

illustrating that in shallow water, all waves should be traveling at the samespeed.

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We can view the behavior of the celerity as a function of water depth, ascomputed by the deep-water, shallow-water, and general expressions - see figureof ”Celerity Illustrated”, shown in class.

As an application, we can examine two examples of shallow water waves:

1. Tsunamis

2. Tow-In, big wave surfing

The behavior of wave length, wave celerity, and wave height are nicely summa-rized in a figure that shows dimensionless variables.

• Water Particle Movement

Water particle orbital path trajectories are also predicted by Airy Wave Theory.The orbital paths divided by wave period provide the wave orbital velocities.The general expressions for horizontal and vertical diameters are

d = Hcosh[k(zo + h)]

sinh(kh)(10)

s = Hsinh[k(zo + h)]

sinh(kh)(11)

Deep water: s = d = Hekzo = He2πzoL , circular orbits, whose diameters

decrease exponentially with depth from the water surface; at water surface thediameter of particle motion is obviously the wave height, H.

Intermediate water: ellipse sizes decrease downward through water column

Shallow water: s = 0, d = Hkh

; ellipses flatten to horizontal motions; orbitaldiameter is constant from surface to bottom.

Water particle orbital velocities, denoted u for the horizontal and v for thevertical velocity components, respectively, are simply the paths divided by theperiods, with a sine or cosine dependence on (kx−σt), since they vary spatiallyat fixed time, and temporally at fixed position. The general forms for horizontaland vertical orbital velocities are:

u =πH

T

cosh[k(zo + h)]

sinh(kh)cos(kx− σt) (12)

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w =πH

T

sinh[k(zo + h)]

sinh(kh)sin(kx− σt) (13)

• Wave Energy Density

Although no net water movement occurs during the passage of a wave (inAiry Theory), there is still gross movement which constitutes a transfer ofenergy over the sea surface. Water surface displacement from the flat, still-water conditions provides a potential energy component, and the movement ofthe water particles in their orbital paths provides a kinetic energy component.Summing the integrals of both components over one wavelength yields the totalenergy density (so named because it represents the wave energy over an areaof sea surface) of the wavy, sea surface:

E = Ep + Ek =1

8ρgH2 (14)

• Wave Energy Flux

E in equation14 is not conserved during shoaling, because it is not truly anenergy, but a distribution of energy per area of sea surface. What is conservedis the wave energy flux

P = ECg = ECn =1

8ρgH2Cn (15)

where Cg is wave group speed, and the ratio of group to individual wave speedis given by

n =1

2[1 +

2kh

sinh(2kh)] (16)

Examine how n varies as a function of water depth. In deep water, n = 12,

implying that the group velocity is 1/2 the individual velocity, whereas inshallow water , n = 1 meaning that the wave group travels at the same speedas an individual wave.

• Wave Groupiness

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• Radiation Stress

This concept is of great power in the study of wave related phenomena. TheRadiation Stress is a tensor which represents the ”excess flow of momentumdue to the presence of waves” (Longuet-Higgins and Stewart , 1964). Wave ad-vance has an associated momentum flux. The ”excess” momentum is calculatedby using the dynamic component of the pressure, i.e. the difference betweenabsolute pressure and the hydrostatic pressure, ensuring that the ”excess” mo-mentum represents the momentum due solely to the presence of waves.

Consider that there exists both x-directed and y-directed components of x- andy-oriented momentum, such that

Sxx = E

[2kh

sinh(2kh)+

1

2

]= E(2n− 1

2) (17)

Syy = E

[kh

sinh(2kh)

]= E(n− 1

2) (18)

In deep water, when n = 12, Sxx = E

2, and Syy = 0; and in shallow water,

when n = 1, Sxx = 3E2

, and Syy = E2

. We’ll return to radiation stresses in ourdiscussion of wave set-up/set-down, and longshore sediment transport.

3.4.2 Stokes Wave Theory

Airy wave theory begins to break down in shallow water and, as such, we turn toa wave theory derived by Sir George Stokes in the mid-1800’s. Stokes theory doesnot make the ”small wave height” assumption. (Show MATLAB code – airyequa-tions.m)

3.4.3 Limits of Application of Various Theories

Figure 5-29 on p. 175 of Komar text shows appropriate depth-height-wavelengthfields for each of the various wave theories.

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3.5 Wave Transformation

This section focuses on the movement and natural alteration of the characteristicsof waves as they travel from the source region toward the shore – the material isextensively covered on pp. 176-196 of the Komar text.

3.5.1 Propagation

At this point, we will redefine the term ”wave group” to mean a bundle of waveenergy that travels from source region to the shore.

Wave groups travel with speed Cg = 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 (Komar’s Fig. 5-30), so thegreater the travel distance, the narrower the strong frequency band in the energydensity spectrum (PNA Slide).

Individual waves arise at the rear of a group, move through the group, and die out atthe front. Hence, they cannot be traced across the ocean - but the groups themselvescan!

• Wave Group Rectangle

Here, we can introduce the wave group rectangle concept. Imagine a storm areaon the open ocean (deep water), whose wind blows for a duration D, and whosefetch (length) is F . This group rectangle will have one period associated withit, and each wave period generated by the storm will have its own conceptualgroup rectangle. It is truly a continuum, but we’ll conceptualize it as a discreteperiod associated with a discreet rectangle. The width of the group rectangleis the width of the storm front, and the length would be W = DCg. What’sCg again?

Cg = Cn =1

2

gT

2π(19)

The edges of rectangle will not be sharp, as there will be some lateral energyloss along wave crests, and rounded front and back edges due to this patternof individual wave movement through the group. Importantly, however, therewill not be major radial spreading, as experienced by waves generated by a

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pebble thrown into a pond. Note this clarification, which may conflict a bitwith your notion from prior lectures. A wave group rectangle retains most ofits original energy during propagation from source to shore.

• Wave Dispersion

The time from when the storm starts to when the waves first reach a shoreline,some distance R away from the storm front is

tob =R

Cg=

4πR

gT(20)

Examples of dispersion:

1. Barber and Ursell (1948) - Cornwall coast; 2-hourly spectra for 2.5 daysdocumented a tropical cyclone source of the east coast of the U.S. (KomarFigs. 5-32 and 5-33)

2. Wiegel and Kimberly (1950) - Oceanside, CA coast; southerly swell sourcedfrom (see Google Earth Polygon).

3. Munk and Snodgrass (1957) - Guadalupe Island, Baja CA coast: wit-nessed swell from Indian Ocean, 15000 km away!

These were the studies that verified the group velocity concept - then later,observations were made regarding this through-group movement of individualwaves in a group rectangle.

• Wave Energy Losses

Amazingly, very little energy is lost as waves traverse these great distances.See Komar Fig. 5-34 to illustrate spectra of a storm at various distances fromthe front. Four sources of wave energy loss have been suggested.

Viscous Damping - Mathematical relationships have been worked out tocompute the amount of viscous damping of waves during propagation, and theresult is wave height decays exponentially as a function of time and period tothe -4 power.

H = Hi exp

(−32π4νt

g2T 4

)(21)

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This may help explain why short period waves have a tendency to disappear,while long period waves persist. But the result is still negligible, as illustratedby the wave height half life diagram (Komar Fig. 5-35) and equation below.

t1/2 =0.0088

4π2

g2T 2

ν(22)

This shows that it would take a 1 sec period wave about 4.5 hours (12 km) oftravel to decrease its height by 1/2, and a 5 sec period wave about 2600 hours(37,000 km) of travel to decrease its height by 1/2!

Other Sources of wave energy loss: angular spreading, contrary winds, wavewave interactions/breaking.

A nice summary of deep-water wave propagation is provided in a succinct paragraphon p. 183 of the Komar text.

3.5.2 Shoaling

Waves moving from deep to intermediate/shallow water change their shape andcharacteristics significantly. Wave velocity and wavelength decrease, while heightincreases to conserve wave energy flux. Period remains the same, thank goodness.Just offshore of the breaker zone, the waves have peaked crests and broad troughs; avery different appearance to their deep-water sinusoidal form. Here, we will explorethe details of shoaling transformation.

• Wave Length and Celerity: A commonly accepted relationship, in Airywave theory, for wavelength in intermediate water (1

2> h

L∞> 1

20) was provided

by Eckart (1952):

L = L∞

[tanh

(2πh

L∞

)] 12

(23)

Since wave period remains constant, and recalling the general and deep-waterexpressions for wave celerity (Eqns. 6 and 7 in this set of notes), we can safelysay that

C

C∞=

L

L∞= tanh

(2πh

L

)=

[tanh

(2πh

L∞

)] 12

(24)

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which helps explain the behavior of the non-dimensional quantities graphed inKomar’s Figure 5-21. Wave length and velocity systematically decrease withdecreasing water depth. The shoaling parameter, n, changes smoothly from1/2 to 1. Since Cg/C∞ is the product of n and C/C∞, the group speed, whenentering intermediate water, initially increases, then decreases to coincide withthe individual wave speed in shallow water.

• Wave Height: Figure 5-21 in the Komar text also illustrates the variation inheights of shoaling waves, which can be understood by noting that wave energyflux is constant during the shoaling process:

P = ECn = (ECn)∞ = constant (25)

A simple algebraic rearrangement of the equated flux relationships, knowingthe relationship for wave energy density provided in Eqn. 14, yields

H

H∞=

(1

2n

C∞C

)1/2

(26)

which explains why orthogonally directed waves tend to increase their heightduring shoaling – it is a direct compensation for the slowing of individualwaves and the need to keep a constant wave energy flux. The waves converta significant fraction of their kinetic energy to potential energy. It is noted,however, that when waves initial enter water of intermediate depth, the heightdecreases initially, then increases to the break point. This temporary reductionin wave height is associated with the temporary increase in wave group speed,discussed above. As the wave group velocity increases (kinetic energy), thewave energy density necessarily decreases, which is manifested in a decrease inpotential energy (wave height).

• Wave Steepness: This is a fairly straightforward consequence of the com-bined shoaling behavior of wave height and wave length. Steepness initiallydecreases upon entry to intermediate water depth, then rapidly increases untilthe instability condition associated with wave breaking.

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3.5.3 Refraction

• At this point it is important to make a distinction between wave crests andwave rays:

1. Wave crests are the line segments that connect the peaks (or troughs)of a wave field. The crests are visible to the observer.

2. Wave rays are the lines orthogonal (perpendicular) to the wave crests,which represent the direction of wave propagation.

Waves that approach a coast with their wave crest oriented at an angle tothe shoreline orientation will refract when entering intermediate and shallowwater. This is visible in the curved appearance of the wave crests and waverays outlined on the photos shown in class. Straight coasts with shore parallelbathymetric contours will cause obliquely approaching waves to become moreshore parallel in shallow water, producing a fan-like appearance in map view.

• Refraction occurs because of the dependence of shallow water wave speed onwater depth:

1. Waves travel more swiftly in deep water.

2. If waves are approaching at an angle to a straight coast, the deeper partof the wave crest will propagate faster than the shallow part of the wavecrest.

3. The result is a bending of the wave ray, as the deeper part of the crestsweeps through a greater arc length than the shallow part of the crest,per unit time.

• The refraction process is analogous to the bending of light rays and governedby the same physical principle – Snell’s Law

sinα1

C1

=sinα2

C2

= constant (27)

In Equation 27, it is obvious that wave speed relates to direction accountingfor the refractive behavior visible in the photos shown in class.

• It should be noted that this process has a profound effect on the wave energyflux distribution on the coast.

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1. Energy flux per unit length of wave crest is not necessarily conserved. Theenergy between adjacent wave rays, separated by a distance s∞, in deepwater must is maintained as those rays converge or diverge as a result ofrefraction to a separation distance of s in shallow water.

2. This can lead to a decrease in wave height during the shoaling and refrac-tion process.

3. This process is sometimes referred to as wave crest stretching.

• We can summarize the effects of shoaling and refraction with the appropriatelynamed ”shoaling coefficient”, Ks, and ”refraction coefficient”, Kr, which actto modify the wave height via the following relationship

H

H∞= KsKr =

[1

2n

C∞C

]1/2 [s∞s

]1/2(28)

Simple geometric considerations allow for the simplification of Kr below

s∞s

=cosα∞cosα

(29)

Examples shown in class demonstrate the effects of shoaling and refraction ofa 2m 10s wave over linearly shallowing bathymetry.

3.5.4 Diffraction

• Lateral translation of energy along a wave crest.

• Most noticeable where a barrier interrupts a wave train creating a ”shadowzone”. Energy leaks along wave crests into the shadow zone.

• Also by analogy to light, Huygen’s Principle explains the physics of diffractionthrough a superposition of point sources along the wave crest.

3.5.5 Numerical Models of Wave Propagation, Shoaling, Refraction, andDiffraction

• Monochromatic wave models – Ray tracing models, grid models; e.g. REFDIF(Kirby and Dalrymple, 1986)

• Spectral wave models – e.g. SWAN (Booij et al., 1999)

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3.6 Breaking

The process of wave breaking can be thought of as the release of energy, derived fromthe wind, along the narrow coastal zone. It leads to geomorphic work done by wind,really, which is translated through medium of water. Wave breaking is responsiblefor the processes which control beach morphology: (1) nearshore current generation,and (2) sediment transport.

3.6.1 Nearshore Wave Breaking

As waves shoal into shallow water, the wave height H increases and the wave lengthL decreases, dramatically increasing the steepness H/L. This cannot continue indef-initely – something has to give.

Common misconception: breaking is a result of waves dragging on the bottom,then trip forward due to friction NO!

In reality, friction plays a very small role in the dissipation of wave energy. Computersimulations that completely neglect friction can still produce breaking waves.

A wave breaks when it becomes overly steep, because the velocity of wa-ter particles in the wave crest exceed the velocity of the wave form!

• Breaker Types The three (or four) main breaker types are:

1. Spilling Breakers – display a cascading face of bubbles and foam afterpeaking and initiating the breaking process.

2. Plunging Breakers – abrupt pattern of peaking to a vertical face, over-curling, and plunging downward and forward to unload energy in a veryconcentrated portion of beach

3. Surging Breakers – during the peaking process, the base of the wave desta-bilizes and surges forward, causing the wave top to implode/collapse

4. Collapsing Breakers – not often witnessed and difficult to identify, thiswave breaking type is thought to be intermediate between Plunging andSurging.

• Iribarren Number / Surf Similarity Parameter

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In reality, breaker types transition from one to the next through a continuum,but in general the type of breaking style correlates well with the ratio of beachslope to wave steepness. This concept was explored by Battjes (1974), and inso doing, he introduced deep-water and shallow-water forms of the IribarrenNumber (ξ∞ and ξb, respectively), which has/have since been referred to as theSurf Similarity Parameter

ξ∞ =S

(H∞/L∞)1/2(30)

ξb =S

(Hb/L∞)1/2(31)

Spilling breakers tend to occur on gently sloped beaches with waves of highsteepness (ξ∞ < 0.5, ξb < 0.4)

Plunging breakers tend to occur on intermediate beaches with waves of inter-mediate steepness (0.5 < ξ∞ < 3.3, 0.4 < ξb < 2.0)

Surging/Collapsing breakers tend to occur on high gradient beaches withwaves of low steepness (3.3 < ξ∞, 2.0 < ξb)

• Breaking Wave Condition

It is convenient to identify a critical condition at which waves break – attemptsat this have resulted in the following ratio which relates breaking wave heightHb to the breaking wave depth hb.

γb =Hb

hb(32)

Laboratory experiments have revealed that this value is not a constant, butvaries considerably with wave steepness, Hb/gT

2, and beach slope, S. Thisbehavior is illustrated in Komar’s Fig. 6-8. For a given wave steepness, higherbeach slopes yield higher γb values. Logically, this observation has led to anattempt to link γb to the deep-water Iribarren number

γb = 1.2ξ0.27∞ (33)

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Several breaker height prediction relationships have been generated based ondeep-water wave conditions including:

Hb

H∞=

1

3.3(H∞/L∞)1/3(34)

Hb

H∞=

0.563

(H∞/L∞)1/5(35)

Hb

H∞=

0.46

(H∞/L∞)0.28(36)

These various forms of the breaker height prediction relationship are plottedin Komar’s Fig. 6-9.

Rearranging Eqn. 35, derived by Komar and Gaughan (1972), we can obtaina relationship for breaking wave height as a function of deep-water height andperiod

Hb = 0.39g1/5(TH2∞)2/5 (37)

This relationship is plotted and compared to 3 data sets in Komar’s Fig. 6-10.The data span 3 orders of magnitude of breaker heights and are remarkablywell-behaved.

• Plunge Distance

As shown in Komar’s Fig. 6-11, the ratio of plunge distance to breaking waveheight tends to decrease with increasing beach slope.

3.6.2 Surf Zone Wave Decay

Within the surf zone, the wave energy dissipation pattern depends on morphologyof the beach.

• General Characterization of Wave Energy Dissipation in Surf Zone

On steep, reflective beaches, wave breaking (and energy dissipation) isconcentrated through plunging breakers. The broken wave surges up the beach

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as runup. An example of this setting, provided in class, was the Vilano beachsite, just north of St. Augustine Inlet in North Florida.

On low-slope, dissipative beaches, there is an extensive, wide surf zone overwhich spilling breakers dissipate energy. In this setting, at any time, severalbroken wave bores, and smaller unbroken waves, are visible. An example ofthis setting, provided in class, was the Anastasia Island site, just south of St.Augustine Inlet in North Florida.

In general, we note that where there is a smooth, continuous beach profile, thepattern of wave energy dissipation (breaking) is fairly uniform. In contrast,where there are alongshore bars and troughs, wave dissipation (breaking) isconcentrated on the bars and relatively absent over the troughs.

Understanding the patterns of wave decay in the surf zone is important for twosignificant reasons:

1. Wave energy dissipation is inversely related to the alongshore pattern ofwave energy delivery – so it can help identify relative vulnerability ofcoastal property.

2. Wave energy expenditure is partially transformed into nearshore currents,which are responsible for sediment transport and beach morphologic mod-ification.

In general, natural beaches are subjected to a wide spectrum of wave heights– resulting in a wide range of wave breaking conditions – larger waves break indeeper water, and smaller waves break in shallower water. This is one of theparameters that is often reported by spectral wave models such as SWAN.

• Breaking Wave Distributions

A now-famous study by Thornton and Guza (1983), used Torrey Pines Beach,a fine sand beach with minimal bars and troughs to study distributions ofbreaking wave heights within the surf zone on a natural beach. Wave staffsand current meters were emplaced to make measurements from 10 m waterdepth to inner surf zone. This study documented the following:

1. The histograms of breaking wave height distributions illustrate a greaterfraction of broken waves at shallower depths.

2. Histograms of all waves in nearshore (broken and unbroken) show skeweddistributions – many small waves and few large waves – comparable to

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that of a Rayleigh distribution, which also describe the distribution ofdeep water wave heights (Longuet-Higgins , 1975).

3. Histograms of broken waves only (cross hatched pattern in Komar’s Fig.6-12) show a more uniform distribution.

• Energy Saturation

Furthermore, in their examination of root-mean-square wave height at variousdepths in the surf zone, over four different days, Thornton and Guza (1983)found that independent of deep water wave height, waves in the surfzone decay in the same manner, i.e. following the Hrms = 0.42h line.Waves are described to be saturated with respect to their energy content withinthe inner surf zone, where local water depth h controls the wave height.

After initial breaking at Hb = 0.78hb, surf wave heights appear to decay to theγ = 0.42 ratio, as shown in Komar’s Fig.6-14.

• Models of Wave Height Decay

This material is covered on pp. 222-232 of the Komar text. The basis formost predictive models aimed at understanding wave height decay and energydissipation in the surf zone targets the energy flux relationship

∂(ECn)

∂x= −ε(x) (38)

This relationship states that the cross shore spatial rate of change of waveenergy flux is a function of cross shore position. Regarding the mechanisms ofenergy dissipation, only a small fraction is expended in frictional losses – thebulk of the dissipation occurs as a result of wave breaking and the associatedgeneration of turbulence.

Some of the models used in wave height decay/energy dissipation are listedbelow:

1. Dally et al. (1985) proposed a model where the loss in wave energy perunit area per unit time, ε, is proportional to the difference between thelocal wave energy flux and the ”stable” wave energy flux. This modelhas been calibrated with laboratory measurements and since has beenincorporated into the USACE model known as RCPWAVE.

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2. Following the work of Battjes and Janssen (1978), Thornton and Guza(1983) proposed that the energy dissipation term, ε, is proportional tothe wave frequency and the cube of the broken wave bore height, whilebeing inversely proportional to the local depth.

• Undertow

3.6.3 Set-up and Set-down

During the breaking process, within the surf zone shoreward of initial breaking, thereis a rise in mean water level which slopes up landward. This phenomenon is knownas Set-Up. There is a corresponding depression in mean water level oceanward ofthe break point, known as Set-Down.

This process was brought to public attention during the 1938 Hurricane along theNortheast Atlantic coast. Shoreline elevations in exposed regions were 1m higherthan in sheltered regions – a fact that could not be explained by storm surge, whichwould elevate shorelines uniformly.

Radiation stresses, or the momentum flux of waves, are responsible for this phe-nomenon. (Longuet-Higgins and Stewart , 1964) In a nutshell, the onshore compo-nent of the radiation stress, Sxx (the cross-shore component of shoreward directedmomentum flux), is balanced by a seaward slope of the water surface providing apressure gradient or force, in the region of set-up.

Recall from Eqn. 40,

Sxx = E

[2kh

sinh(2kh)+

1

2

]= E(2n− 1

2) (39)

which, in shallow water, becomes

Sxx =3

2E =

3

16ρgH2 (40)

The cross-shore momentum balance is then

∂Sxx∂x

+ ρg(η + h)∂h

∂x= 0 (41)

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where η is difference in water surface elevation between still water and with wavespresent, i.e. set-up or set-down.

In summary, and paraphrasing from Komar’s p. 234, both Set-Down and Set-Up areproduced by the shoaling and decay of waves. The momentum flux is conserved bybalancing the gradient in the onshore component of the radiation stress, Sxx, withthe pressure field of the sloping sea surface.

This has been verified in laboratory measurements. Notably, the width of the surfzone increases by 20% due to wave set-up. The maximum set-up is at the shorelineand is 29% of the breaking wave height. Set-down at breaker is 5% of breaking waveheight. Neglect setup much further than the breaker line.

Although there is a lack of dependence on H in set-up, for larger waves, setup be-gins further seaward and slope is constant so the result is a higher set-up on thebeach.

3.6.4 Wave Runup

Wave swash represents cutting edge of the ocean’s geomorphic buzzsaw, and we wantto know how the wave conditions and beach morphology affect the run-up.

• Methods

Several techniques have been employed to measure run-up.

1. Guza and Thornton (1981, 1982) used an 80-m long resistance wire stretchedacross the beach profile held up at 3 cm height by non-conducting supports– same general idea as electronic wave staffs discussed earlier.

2. Holman and Sallenger (1985) used video measurements to document theswash excursion. The run-up edge can be examined frame by frame.

3. Time stack video method by Holland and Holman (1993), on which theslopes of the linear features represent the speed of the run-up bores.

• Components of Run-Up

1. Wave set-up.

2. Swash of incident waves.

3. Infragravity component (> 20s) of swash oscillations.

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• Previous Estimates of Run-Up

– Hunt’s (1959) estimate: R2% = 8HsS, from the examination of run-up onrock rubble structures in The Netherlands

– Battjes’s (1971) estimate: R2% = HsCξp, related run-up to Iribarren num-

ber, ξp = S/√Hs/L∞, and an experimentally calibrated coefficient, C,

which ranges in value from 1 to 4. This relationship was also used toexplore the effect of substrate on run-up, presented in Komar’s Fig. 6-31. Run-up was found to be approximately twice as great on a smoothsurface, as compared to a rock-covered surface.

– Guza and Thornton’s (1982) estimate of avg. of highest one-third ofswashes: Rs = 0.7H∞. This study revealed some unexpected results –most of the swash excursion occurred at periods much greater than theincident wave period range. By spectral analysis of the run-up records,they were able to separate into 2 components:

1. An incident wave bore direct swash component, which displayed nodependence on deep-water wave height

2. An infragravity wave component at periods greater than 20 seconds,which DID display a dependence on deep-water wave height

3.6.5 Infragravity Water Motions

Infragravity energy is derived from incident wave energy, but how? This material iswell covered in Komar pp. 249-269.

• Evidence from Surf Zone Currents

• Edge Waves

References

Adams, P., D. Inman, and N. Graham (2008), Southern california deep-water waveclimate: Characterization and application to coastal processes, Journal of CoastalResearch, 24 (4), 1022–1035.

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Allan, J., and P. Komar (2006), Climate controls on us west coast erosion processes,Journal of coastal research, 22 (3), 511–529.

Battjes, J. (1974), Surf similarity, Proc 14th International Conference on Coastal. . . .

Battjes, J., and J. Janssen (1978), Energy loss and set-up due to breaking of randomwaves, Proceedings of the 16th international conference on Coastal Engineering,pp. 569–587.

Booij, N., R. Ris, and L. Holthuijsen (1999), A third-generation wave model forcoastal regions. i- model description and validation, Journal of Geophysical Re-search, 104 (C4), 7649–7666.

Dally, W., R. Dean, and R. Dalrymple (1985), Wave height variation across beachesof arbitrary profile, Journal of Geophysical Research - Oceans, 90 (C6), 11,917–11,927.

Eckart, C. (1952), The propagation of gravity waves from deep to shallow water,Gravity waves, National Bureau of Standards(Circular No. 521), 165–173.

Kirby, J., and R. Dalrymple (1986), An approximate model for nonlinear dispersionin monochromatic wave propagation models, Coastal Engineering, 9, 545–561.

Komar, P., and J. Allan (2008), Increasing hurricane-generated wave heights alongthe us east coast and their climate controls, Journal of coastal research, 24 (2),479–488.

Komar, P., and M. Gaughan (1972), Airy wave theory and breaker height prediction,Proceedings of the 13th Coastal Engineering Conference, ASCE, pp. 405–418.

Longuet-Higgins, M. (1975), On the joint distribution of the periods and amplitudesof sea waves, Journal of Geophysical Research.

Longuet-Higgins, M., and R. Stewart (1964), Radiation stresses in water waves; aphysical discussion, with applications, Deep-Sea Research, 11, 529–562.

Thornton, E., and R. Guza (1983), Transformation of wave height distribution, J.Geophys. Res, 88 (10), 5925–5938.

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