Properties of shoaling waves by theory and experiment

Properties of Shoaling Waves by Theory and Experiment

PETER S. EAGLESON

AbsfractF.xperimental results on the transformation of wave height, length and steepnc on a plane beach of 1/15 slOpe are compared with the small amplitude theory of Airy. The

effects of wave shape and beach slope on the applicability of the theory are shown through

com-parison with these data and with those of former invesUgatora. The theory is found applicable to the prediction of steepness. Additional observations on the transformation of crest height and length are presented and discussed with respect to their influence on the mass-transport

veloelties.

Introduc/jonIn laboratory studies of beach

processes it is convenient to dassify the incident

wave according to

its so-called

'deep water'

(d/L >

) characteristics which by definition

are independent of depth. Upon assumption of the

deep-water profile form this independence of

depth allows complete description of geometry and kinematics through specification of the

steep-ness, Ho/Lo, and the period T.

it is not always desirable, nor indeed possible because of limitations of equipment, to generate

the desired wave in its deep-water form in the

laboratory. It then becomes necessary to define through theory an equivalent deep-water wave on

the basis of measured shallow-water (d/L <)

characteristics.

The reverse of this problem, the prediction of

shallow-water characteristics as a function of

depth and given deep-water properties, has been the subject of considerable analytical and of some experimental investigation as summarized below.

Unfortunately, the recent theoretical results of

Eckart [1951], Stoker [1947], Biesel [1951] and others

are not readily converted for the purpose on hand while the results of various experiments by Wiegel [1950], Ivessen

[1953], and others seem

incon-dusive when correlated on the basis of the dassical small-amplitude theory.

NotalionsThe following symbols are used. a1 = wave-crest height, ft

c = subscript referring to wave crest

C = wave celerity, ft/sec

d = local water depth measured from still-water level to the bottom, ft

E = wave energy per unit of surface area, ft lb/ft'

g = acceleration due to

gravity, 32.2 ft/sec2 H = wave height, crest to trough, ft

k = mean diameter of beach roughness

L = wave length, ft

n = ratio of group

velocity to -wave velocity as specified

o = subscript referring to deep water

P = power trnnqmitted by a wave per foot of

crest width, ft.lb/sec ft

T = wave period, sec

Ti = mass transport velocity, ft/sec = specific weight, lb/ft3

o

angular displacement in wave, origin as

specified, radians

= 3.14.

Results of previous workRayleigh [18771 was

one of the first to deal theoretically with the

transformation of shallow-water waves moving

into shoaling water. He introduced the idea of

energy partition which states that the power per unit of crest width in any depth of water may be given by

P=nEC

(1)

For the progressive wave of finite amplitude it can be shown [Gerstner, 1802] that

E=c[' 2tanh24(12)]

in which the kinetic energy is slightly greater than the potential.

For waves of low steepness this reduces to

E

'yH2/8 (2b)

which, in a non-dispersive medium, is divided

equally between potential and kinetic energies.

The energy-transformation factor n accounts

for the fact that in deep water, where partide

orbits are circular and the kinetic energy along any horizontal plane through the fluid is a con-stant, only the potential energy travels with the wave As the wave shoals, more and more of the kinetic energy also travels with it, n approaching unity as the wave becomes a purely translatory phenomenon.

(6)

Iversen [1953] measured the transfonnation of height for a wide range of initial steepness on slopesof from 1/10 to 1/S0andmadeacomparison with (9). He found the experimental height to fall well below the theoretical in the region of minimum

height and then to rise to values greater than

theory at lower d/L.

In addition, Wiegel (19501 made measurements of wave length in the shoaling region. His investi-gations, using slopes .of 1/10.8 and 1/20, were

chiefly concerned with comparisons of profile with the extrapolated trochoidal theory, but in addition, induded comparisons of wave length and steepness

transformation with Airy's theory. Wave length (and thus celerity) agreed well with theory over a range of H/L from 0.009 to 0.05 regardless of

beach slope. However, the experimental H/L

values on the beach became much larger than theory for H/L > 0.0279.

During the course of a study on the sorting of beach sediments by shoaling waves (Ippen and

Eagleson, 19551, it became necessary to determine accurately the local or transformed characteristics of the shoaling wave. In view of the shortcomings of existing knowledge, it was found desirable to measure these characteristics again.

Experimenial equipment and procedures-The experiments were carried out in the 100-ft wave channel of the Hydrodynamics Laboratory at the Massachusetts Institute of Technology. The

working section of the glass-sided channel is 30 in.

wide, 36 in. deep and 90 ft long.

At one end of the channel a plane, false-bottom beach of 1/14.8 slope was installed occupying 36 ft of the working section. The surface was coated with a carefully graded angular sand of mean

diameter, k = 0.0060 ft. Shallow-water waves were generated at the other end by means of a vertical faced piston which reciprocated horizontally and was controlled by a hydraulic servo-mechanism

TABLE 1- Characieris&sof iesi waves before lransfo,mai ion

H L T d d/L H/L (theo-retacal) Ap. ft ft uc ft I 0.186 8.92 1.428 1.75 0.196 0.0209 0.0176 0.020 2 0.265 11.16 1.684 1.75 0.157 0.0237 0.0200 0.020 3 0.334 8.42 1.389 1.75 0.208 0.0420 0.0390 0.040 4 0.234 5.91 1.101 1.75 0.296 0.0396 0.0411 0.040 5 0.230 4.36 0.38 1.75 0.401 0.0528 0.0535 0.054 6 0.440 7.20 1.233 1.73 0.243 0.0611 0.0611 0.060 7 0.337 5.99 1.105 1.75 0.293 0.0598 0.0601 0.060

For the'small amplitude wave

(3)

sinh

T

Since the rate of energy transmission across any two sections must be a constant under the assump-tion of negligible dissipaassump-tion and reflecassump-tion,

LnEClo = [nEC] (4)

which, for low steepness, may be written

H0 L2

n CJ

(5)

Development of a usable expression for trans-formation of wave characteristics requires expres-sion of the wave celerity in terms of its geometrical properties. Of all the existing wave theories, Stokes' irrotational theory for waves of finite height comes dosest by experimental evidence [Mason, 19481 to representing actual wave kinematics in water of constant depth. Stokes found the speed of propaga-tion to the third approximapropaga-tion to be given by

C2

[2 (cosh ) 2+2 cosh

+5

L

8smh-r)

I

1.

2wd\4

Neglecting the effect of, height, (6) reduces to that given by Airy's [1842, p. 2891 theory

tanh (7)

2w L

which, for deep-water conditions reduces further to

Co2 ₍₈₎

Assuming the celerity of a wave at any depth on a sloping bottom to be given by (7) for the same depth on a horizontal bottom, (5) becomes

H014wd

HI2T

. 4wd

I

_h2

LT+s1Y

(9)

and (7) and (8) yield

L 2wd

of continuously variable stroke and frequency.

An expanded aluminum wave filter immediately following the generator removed secondary dis-turbances

Wave characteristics were determined by capacitance-type profile wires in conjunction with

a commercial recorder. Details of these

com-ponents, together with calibration procedures, are given by Ippen and Eagkron [1955]. Two fixed gages were used in the constant-depth portion of the channel ten feet apart, the downstream gage being 10.5 ft from the beach toe. A movable gage with two wires one foot apart was used in shoaling water. The distance between the wires was large enough to permit satisfactory travel-time measure-ments and at the same time was short enough to

keep the error due to linear, interpolation to a

minimum. Since the celerity is a non-linear, de-creasing function of distance, the assumption of linear variation between the probes gives a celerity somewhat less than actually exists. Owing probably to the presence of reflected waves, the individual measurements of wave height varied ±4 pct from

a mean curve drawn through them. Thus

while one gage will suffice for height measurement, the heght was interpolated linearly between gages.

fhe celerity of all waves whether stable or

transforming was defined as the average speed of

travel of the front and rear intersections of the

wave profile. and the still water surface.

Char-7

4

3

acteristics of the seven test waves in the constant

depth portion of the l4ulLnnel (that is, before

transformation) are given in Table 1.

Transformation characteristics of Wave 7 were also obtained on bottom roughnesses k = 0.0025 ft and k 0 ft. The only effect of roughness noted was a thange in the position and thus character-istics of the breaker.

PRESENTATION AND DiscussioN o RESULTS Transformation of wave ceierily or lengthLocal celerities of the transforming waves were

com-puted at two-foot intervals along the sloping

beach by (6) and (7) using the measured geometric characteristics.

As a check on the basic assumption of Rayleigh that waves of the same geometric characteristics have the same kinematic characteristics in water of

the same depth whether the bottom is

flat or gradually sloping, these local celerities were compared with the value determined by experi-ment. The results of this comparison may be seen in Figure 1.

Considering the fact that (7) neglects the effect of height, it is surprising to note the dose agree-ment of theory and experagree-ment over the full range of transformation. For most of the test waves the experimental celerity is

slightly greater than

theory early in the transformation, owing probably

to neglect of height, but as the transformation

4 5 6

AIRY'S ThEORETICI. CELERITY (E7) STCSCES THRETICAL CELERITY (EQ.6)

FIG. 1 Transformation of celerity 3

- II,tL Isndws cpcifte.

..

_S

I

wow HlL0(1D4

o

ii

S

2J

51

4J'

S

-:

:}

- I SLGE I 1/14.8

C.

₅ S35 - -. -:<

;

3 4 5

6__

progresses and the height increases (celerity de-creases) the expected divergence of theory and experiment does not occur.

On the other hand, examining the correspond-ence between Stokes' finite amplitude theory (6)

and experiment, a divergence is found in the

latter stages of transformation. In the early stages of transformation where the wave profile is still 'Stokian' the height correction improves agree-ment, but as the wave deforms and departs from the theoretical in shape, the correction becomes too large. The implication is that the effect of shape on the local celerity of the transforming wave is

op-1.4

Q12

I.0

0.4

posite to that of height and that at least for the experimental conditions investigated is approxi-mately equal thereto and independent of incident steepness.

Since the period of the transforming wave

remains constant, Figure 1 shows the transforma-tion of wavelength as well. In order to obtain a better picture of the position of each point in the transformatiOn process, the wavelength data are plotted in Figure 2 against d/L, L0 being deter-mined from the period through (8). Data of Wsegd [1950] (obtained by srnling from enlargements of the figures in the author's paper and hence subject

-WtGrL MIT

op. '1S Slops 'ISO s

S_ t.

S 6 o i O3S3 0325 0 lI ZJ 3j.

L.1

mi.vp q.9 : SI 9fllV ,.

a..

6.

0 01 02 0.3 0.4 d/L

to all errors inherent in such a process) for 1/10.8 and 1/20 slopes have been added to the figure and

show the same trend mentioned above, that of

slightly higher

experimental than

theoretical lengths over the entire range of transformation. However, no trend with slope is visible.

Trasfovmation of wave heightSince the incident wave was in all cases a shallow-water wave, it was necessary to define an equivalent deep water wave in order toobtain H0 . Division of (9) by(10) Yields

an expression for transformation of steepness

solely as a function of d/L

r

2irdl1

I

2cosh2

I.

HL

I L I

2d

L Ho= I

4d . 4,rd

I cotanh

T

LT+smh1TJ

Substitution of the incident values of H, d, and L in (11) defines an equivalent deep water steepness

H0/L0. Multiplication of this steepness by the

deep water wave length as dCflned by (8) yields

the equivalent deep water wave height, H0.

In Figure 2 wave-height-transformation data

from all three slopes are plotted versus d/L in

comparison with (9).

Considering for the time being only the data

for the 1/14.8 slope because of their regularity, the experiment is seen to agree fairly well with theory up to the point of minimum height,

shore-30

20

ID

ward of which it becomes divergently greater than

the theory.

As can be seen from (5), any correction to the theoretical celerity for the. effects of finite ampli-tude or to the height for the effects of viscosity merely serves to amplify the divergence. Reflec-tions have been shown to be negligible for slcpà less than 4.5° by Caidwell [19491.

Use of (2a) in place of (2b) in (4) corrects the theoretical height ratio by a maTimum of +12 pet f or the data presented. This correction is

approx-imately 50 pct of that necessary for agrenent

with experiment and thus cannot be neglected,

but it is the rmining error whici is of primary.

interest here.

ExplaMtion for this divvicuce probably lies

in the inapplicability of the expression fox nE,

the amount of energy traveling with the wave.

Assuming conservation of power, the quantity

,zE/li5 is thus too large as given by even the

finite amplitude theory. E/112 is essentiallyashape parameter and thus wifi take on different values reflecting the changing asymmetry of the wave

while n depends upon the energy partition.

Rayleigh [19111, has shown the wave energy to be

divided equally between kinetic and potential

only for waves of small amplitude in Constant

depth. Both of these criteria are violated in the

shoaling wave. Further experimental work is

d/L

Fm. 3-Transformation of wave steepness

I. I-I0.8 Slops 0208 0248 .0325 1 WlEl. I Slops 0' S' I 1/20 ,v_ 0251 0353 I I-0 0 a 0 a Slops I UIT I 5 1/14.0

2J

31

4J

no 6 6 .054

A

[spUI 1/10.8 0 C tlO

needed to define the actual energy budget in a

transforming wave.

It is again significant that the divergence corn-mences at apprnriniitely the point of minimum

height where the wave deformation becomes more

pronounced and gets worse as the asymmetry

increases.

Although the data for both the 1/20 and 1/10.8 slopes show scatter, mean curves show the same trend as that for the 1/14.8 slope, and show an effect of slope itself only for a slope of 1/10.8. This is again in agreement with the findings of

Caldwdl 119491 with respect to reflection.

The effect of increased slope (1/10.8) is to make

the heights greater early in the transformation. In this stage, portions of the kinetic energy are added, through reflection, to a potential energy which is very little reduced from its deep water value. Later in the transformation, when reflec-tions and repartition have reduced the quantity of available energy, the heights should be less than for flatter slopes at the same d/L.

Transformation of steepnessEq. (11) was used with values of the experimental wave in the con-stant depth portion of the channel, to define an equivalent deep water wave. Thus in Figure 3 a comparison of the data with this equation gives a curve of relative steepness which has been forced into agreement with theory at the beginning of the transformation. The agreement for both the 1/20

3

2

and 1/14.8 beaches continues to be excellent up to breaking, with the slope effect showing. up in slightly higher steepnesses for the 1/10.8 slope. This is due to the above-mentioned effect of slope on the height.

Asymmetry about the still water ievdBecause of the role played by asymmetry of the surface profile in determining the magnitude of the mass trans-port velocities, it is interesting to compare the experimental ratios of crest height to wave ampli-tude and crest length to half wave length with the corresponding ratios predicted by theory. Since mass transport is a 'second-order' effect the Stokes relationships for surface profile are used. The ex-pression for crest height a1 may be written

.4Td

smh-2a H L

H1+ITLI

_2id\4

while, referring to Figure 4, the trough length may be given by L 01 (13) where

I.

2id\4

2cos201 1 8 L,.5mhL)

cos01

3H

. 4.wd

smh

--Ut 0 05 24

2-I

Fic. 4- Transformation of crest height and length

30 (12)

T

sLI/I4.a

I I I

:

S _,002

0T'

P!

.o06 (14) 4 2

'2

21

Ht.

Os

a.-FIG. 5- Asymmetry with respect to still water level

Figure 4 shows the asymmetry of height to be

greater than Stokes predicts for relative

steep-nesses, (H/L)(Lo/Ho), less than 1.5. Beyond this point the theoretical wave becomes divergently more asymmetrical. The asymmetry of length is also shown in Figure 4 and indicates the theory to predict a more symmetrical wave up to a relative

steepness of about two. On the other

hand, for

(H/L)(L0/Ho) > 2 the actual wave becomes

divergently more symmetrical than the theory.

The combined effect of these two asymmetries is to

give a wave crest which is steeper

than Stokes indicates early in the transformation and flatter than theory in the breaker region.

An indication of the applicability of theStokes equation for mass transport velocity

4a(d z)

U

(H)2csuL

to the condition of a sloping bottom may possibly be obtained by examination of the asymmetry of the transforming wave about thestill water level.

It is the magnitude of these

asymmetries which controls, to a large extent, the magnitude of the

mass transport velocity. In order to show the

combined effect of these two asymmetries, Figure 5 was prepared. In this figure the ratio

(a1

H/2)/(L/2

L)

is plotted against

(H/L)(L0/Ho). The solid

lines

represent the

theoretical value of the ratio using the

experi-mental values of H, L, T and d at

the particular point. This parameter may be considered a measure of the mass transport vElocity, and thus wherever

the theory lies to the right

of the experimental curve the indication

is that the theory predicts

(15)

too large a mass transport velocity. In general, for waves of high steepness, Figure 5 indicates the mass transport velocities as given by (15) to be too large early in the transformation and near the beeaker, while at intermediate values of the relative wave steepness, (H/L)(Lo/Ho), it is too

low.

CONCLUSIONS

For slopes 1/10.8, 1/14.8 and 1/20, the small amplitude theory of Airy is satisfactory for the prediction of celerity of the transforming wave up to the point of breaking. This agreement is a result of an effect of wave asymmetry which evidently opposes the ederity-increasing

effect of finite

amplitude.

For these slopes this theory also predicts the loal wave height smaller than experimentfor

d/L <0.2. This error, which is of the order

of 25 pet at d/L = 0.07, may be halved by correcting the basic energy relation for finite amplitude.

For increasing values of slope (steeper than 1/14.8) the theory predicts incieasingiy smaller heights than experiment

The small amplitude theory of Airy is

satis-factory to within about 3 pct for d/L >

0.10 for the prediction of wave steepness transformation for slopes 1/20, 1/14.8 and 1/10.

ACKNoWLEDGMENT

The research for this paper was made pocslhle

through the sponsorship of the Beach

Erosion Board, Corps of Engineers, U. S. Army, and was carried out under the supervision

of Arthur T.

Ippen, Professor of Hydraulics at the Massichu-setts Institute of Technology.

Particular credit is due Luis A. Peralta, Research

1'

.0PI un4a iiI. tspto.) 0.02

zJ.

31₆₀₀₆

4j

61 a

--I.-

_It.o.It.

_/

/

-V

Assistant at the Hydrodynami Laboratory, MIT, for performance of much of the experimental

pro-gram, analysis of data and preparation of the

figureL

REFERENCES

Any, G. B., On tides and waves, Encydopae4ia

Mefropditana, 6, London, 241-396, 1845.

BIES., F., Study of wave propagation in water of gradually varying depth, Gravity Waves, Nat. Bur.

Standards, Circ. 521, 243-253, 1951.

CAIDWELL, J. M., Reflection of solitary waves, Beach Erosion Board Tech. Memo. 11, 19 pp., 1949.

Ecxan, C, The propagation of gravity waves from deep to shallow water, Gravity Waves, Nat. But. Standards, Circ. 521, 165-173, 1951..

GzasmEn, F. V., Theorie der Wellen, Abh. K. Be/sm.

Get. Wiss Prague, 1802; also, Gilbert's Annoien der Physik, 33, 412-445, 1809.

1mw, A. T., m P. S. EAGLESON, A study of

sedi-mont sorting by waves ahoaling on a plane beach, Beach Erosion Board Tech. Memo. 63,83 pp., 1955. IVERSEN, H. W., Waves and breakers in shoiling water, Fi-oc. Third Con!. on Coashil Engrnwing,

1-12, Cambridge, 1953.

MASON, M. A., Study of progressive oi'dilbitory waves

in water, Beach Erosion Board Tech. Rep. 1,39 pp., 1948

RAYLEIGH, LonD, On progressive waves, Proc. London Math. SOc., 9, 21-26, 1877.

RAYLEIGH, LORD, Hydrodynamiciti notes, P/il. hog., set. 6, 31, 177-193, 1911.

Soxzu, J. H., Surface waves in water of variable depth, Q. App. Math., 6, 1-54, 1947.

Wzzczi., R. L., Experimental study of surface waves

in shoaling water, Trans. Ama-. Geophys. UnIon, 31, 377-383, 1950.

Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received May 20, 1956; presented at the Thirty-Seventh Ainual Meeting, Washington, D. C.,

April 30, 1956; open for formal discussion until March 1, 1Q57.)