Wave attenuation in a channel with roughened sides

CHAPTER 18

WAVE ATTENUATION IN A CHANNEL lvITH ROUGHENED SIDES J. A. Battjes

Department of Coastal Engineering University of Florida, Gainesville, Florida

SUMMARY

A semiempirical study l~as made of the attenuation of water l~aves in a rectangular channel, the side l~alls of which had been provided with regularly spaced roughness strips. Such strips were found to be highly effective wave dampers. A dimensionless resistance coefficient could be deduced from the attenuation data; th:i,s coefficient was mainly a function of the ratio of horizontal orbit diameter to roughness height.

The processes by which energy is transferred laterally to the side l~alls were inves tiga ted. These investigations fell short of their aim, however, mainly due to the disturbing effects of lateral secondary waves.

I. INTRODUCTION

A study was initiated with the aim of investigating the processes of transversal flux of wave energy in channels. Knowledge of such proc-esses could be helpful in the design of harbor entrances and inlets. It was known from theory and experiments that the wall friction in a smooth channel would be too small to allow direct measurements of trans-versal transport of energy. Therefore, the laboratory channel was pro-vided with large-scale roughness elements consisting of regularly spaced wood strips attached vertically to the side walls and extending as flat plates into the channel (Fig. 1). Such roughness causes an intense energy dissipation at the side walls resulting in an uneven lateral energy dis tribution. The energy deficiency at the side IQalls would in-crease l~ith distance of travel if there l~ere no processes in action which would tend to distribute the energy uniformly across the channel. The

study of such processes IQaS the main aim of the study, for which measure-ments of the transversal wave height distribution and of transversal sec-ondary currents were planned. The magnitude of the transversal flux of l~ave energy could be measured indirectly via the wave attenuation with

distance of travel, assuming that only the energy losses at the side walls were significant.

As it turned out, however, the processes of transversal flux of l~ave energy could not be studied in as great detail as was hoped. This l.as mainly due to secondary waves, which caused too much disturbance

of the transversal wave height distribution. More emphasis was therefore placed on the measurements of amplitude attenuation with distance of travel, the results of l~hich are believed to be of sufficient general interest to warrant publication. The attenuation data were analyzed and interpreted as a dimensionless resistance coefficient. Thus, the results can be applied to channels of different size and shape provided with similar roughness elements.

Figure 1. Plan View of Roughness Strips

WAVE ATTENUATION

II. AMPLITUDE ATTENUATION IN SMOOTH CHANNELS

The laminar attenuation of water waves by viscous action has been studied by various authors. The rate of attenuation is given by Lamb (1932) for two-dimensional waves in water of infinite depth, by Biesel (1949) for t~~o-dimensional waves in ~'later of finite depth, and by Hunt (1952) for waves in water of finite depth and width. In the first case

427

the internal friction only is taken into account, ~~hereas in the latter cases the boundary friction enters into the computation also. The deriva-tions are based on the linearized Navier-Stokes equations. The resulting amplitude attenuation is exponential. This can be seen from the expression equating the energy loss per unit length to the gradient of the wave power:

where

dP + F = 0

dx (1)

x horizontal dis tance in the direc tion of ~~ave propagation

F energy loss by friction per unit time per unit length of channel

IV channel width Cg group velocity

E wave energy per unit area p density of water

g acceleration of gravity h wave amplitude

Equation (1) is valid if the amplitude is not a function of time.

In a prismatic channel

(2)

where h

=

wave amplitude. The laminar shear stresses are proportional to the velocity gradients, which are proportional to h. The rate of ~qQrk done by the shear stresses is proportional to the product of the shear stresses and the velocities, or

or

From (1), (2) and (3), i t fol101~S that

dh dx -ah

l~hich upon integration becomes

h (4)

Hunt (1952), neglecting internal friction, arrived at the following

expression for

a

for a rectangular channel:

where

a

= 2m \

hI

.

mW

+

sinh 2md

\~

y ;-

2md

+

sinh 2md m 211./L

=

the wave number (L wave length)

v

=

kinematic viscosity

T wave period

d mean water depth

(5)

Internal friction can usually be neglected: it is proportional to

v and therefore, for water waves, very much smaller than the friction

along the solid boundaries, which is proportional to -fv (see equation

(5) •

Experimental values of the attenuation coefficient

a

due to boundary friction are considerably in excess of the theoretical value as given by Biesel (1949) and Hunt (1962). Eagleson (1962) reports

a value which is about 3.5 times the theoretical value for waves damped

by laminar bottom friction. Grosch and Lukasik (1961) report an excess of 80 per cent. Grosch, Ward and Lukasik (1960) suggest that this may be due to nonlinearities in the Navier-Stokes equations, boundary layer

separation during part of the l~ave cycle, or boundary layer turbulence: none of these have been considered in the derivation of

a.

An analysis

by Grosch (1962), which includes nonlinear terms, gives a clear

indica-tion that the shear stresses along the bed may attain much higher values

WAVE ATTENUATION ₄₂₉

III. AMPLITUDE ATTENUATION IN ROUGH CHANNELS

In rough channels, the boundary flO1Q is turbulent and the stresses

exerted on the boundaries are proportional to the square of the velocities;

the rate of energy dissipation at the boundaries is then proportional to the cube of the velocities:

From (1), (2) and (6), it follows that

or

dh dx

which upon integration becomes

1. =

L

+

t3x h hO assuming t3 to be independent of h. (6) (7) (8)

In lwrds: the turbulent wave attenuation follows a hyperbolic law.

It is entirely conceivable that the dissipation takes place partly

in laminar flO1Q and partly in turbulent flow. In the present study, for

instance, the bottom friction may be laminar even though the side-wall

friction is turbulent. Possibilities for similar cases are mentioned by

Biesel and Carry (1956). The resulting law of amplitude attenuation,

which the author tried in vain to find in the literature, may be determined as follows:

Instead of by (3) and (6), the rate of energy loss per unit length

of channel, F, is now given by

l~hich gives, in viel~ of (1) and (2)

dh

It should be noted that

a

in (9) need not be the same as in (5). Integration of (9) gives:

a

ln

1

+

I3h

=

a

1

+

13ho

ax

Thus the following lm~s of amplitude attenuation are found:

Damping Laminar Laminar

+

Turbulent Turbulent Differential E<J.uation dh

= -ah

dx Solution

in

1

=k

L

+

ax

h ho

1

=

L

+

I3x h ho (10) (4) (10) (8)

It should be noted that the quantity a/l3h, which appears in (10), is the ratio of the rate of laminar dissipation over the rate of turbu-lent dissipation. For very large a/l3h, solution (10) tends to solution (4), I~hereas for very small a/l3h, solution (10) tends to solution (8).

An interesting case of applicability of solution (10) might be the case of laboratory investigations of sediment transport due to I~ave action. If one uses Bagnold's approach (Bagno1d, 1963; Inman and Bowen, 1963), the I~ave pOl~er spent on the mobile bed should be knolffi, and is usually found by subtracting the power spent on the (smooth) side I.alls of the laboratory I~ave tank from the total I'lave power being spent, assuming both to give exponential attenuation. Using equation (10) hOl.ever, one can allow for the fact that the flow near the bed need not be laminar, as it lYOu1d certainly not be in case of a rippled bed, and the effective roughness coefficient of the bed may be computed.

In the present study the ratio a/l3h is very small for all the I.aves tested in the channel provided with roughness elements, and solution (8) is adequate. The coefficient 13 will be determined in terms of the wave and channel characteristics in the following section.

IV. COEFFICIENT OF TURBULENT AMPLITUDE ATTENUATION The only forces to be considered for the energy dissipation in the rough channel are, by approximation, the normal pressure forces

WAVE ATTENUATION

on the broadsides of the wood strips caused by the horizontal components of the orbital velocities. Thus internal friction, bottom friction, vertical and horizontal shear stresses on side walls and wood strips, air resistance and surface wave resistance are neglected.

The normal pressure force on the roughness strips consists of two components: a drag force due to the horizontal orbital velocities, and an inertia force due to the horizontal accelerations.

For the present problem only the work done by the normal pressures on the slats is relevant, I~hich is proportional to the product of these pressures and the horizontal velocity component of the surrounding fluid, integrated with time. It appears that the inertia force acting on the slats can perform no Iwrk inasmuch as it is 90 degrees out of phase I~ith the horizontal velocity. Therefore, the drag force only will be con-sidered.

The normal pressure on the strips due to drag may be IHitten

t;

= CD

1

pi?

1

u \

2 ( 11)

431

I~here CD is the drag coefficient of the strips, p is the density of the water, and U is the horizontal component of the orbital velocity correspond-ing to the averaged I~ave height H(x) (refer to Section IX). More gener-ally, the boundary resistance would be given by an apparent shear stress T acting uniformly on the side I~all:

--_T

_{= Cf}

_"2

1 _pU ~

I ,

_{U (12)}

where Cf is the coefficient of local boundary resistance.

Equations (11) and (12) are equivalent when the total force acting on a unit length of side wall, as given by the two equations, is the same, or when

kCD sCf (13)

where k is roughness height and s is roughness spacing. Reference is made to Figure 1. One might say that the boundary resistance is repre-sented in a microscopic way by (11) and in a macroscopic way by (12). Equation (13) may be used to convert CD into Cf and vice versa.

The rate of work done by the shear stress T is ->

Using the small amplitude theory, with x constant:

U wh cosh m (d+y) sin wt

sinh md (15)

,.here w = 2n/T and y is vertical dis tance above mean wa ter level.

The error involved in using the first order theory instead of a higher

order theory will be evaluated in Section IX).

Substituting (15) into (14) and integrating with respect to y and t

gives the loss of wave energy per unit length of channel at one side wall per wave period T, so that the total average energy loss per unit time per unit length of channel is

F 2 •

1

T

4w3

(1

+

-2 3

pCf 3nm 3 sinh md)' h

assuming Cf to be independent of y.

Substituting this value of F in (1) and remembering that

dP

=

CgWpgh dh

=

n ~ Wpgh dh

dx dx m dx (17)

where n is the ratio of group velocity over phase velocity, gives

dh

=

-Cf dx

Elimination of h 2 and dh between (7) and (18) yields dx

~

=

~

_3nWgn

(1

₃

+

_sinh -2 d)C _{m f}

(18)

(19) (16)

which is the desired relationship between ~ and the wave and channel

characteristics. For a certain wave, ~ is proportional to Cf (or to Cn). In the next section the dependency of Cn on other factors 'viII be discussed.

WAVE ATTENUATION

V. DRAG COEFFICIENT OF ROUGHNESS ELEMENTS DIMENSIONAL ANALYSIS

In the preceding section, f3 _{(thus, Cf} or CD) Ivas assumed to be independent of h. This is not necessarily true. A dimensional analysis will be made of the phenomenon of drag in unsteady flow, with special

reference to the test conditions in this study, in order to determine the dimensionless parameters on which the local CD would depend.

The quantities considered relevant are: the dependent variable

Dp, i.e. the differential pressure due to drag, and the independent

variables: roughness height k and spacing s, period T and maximum hori-zontal orbital velocity Urn of the oscillatory flow, time t, and density

p, viscosity ~, and specific weight y of the water. (The possibilities of viscous and gravitational effects cannot be excluded a priori.)

The nonuniformity of the flow is not considered, which is justified as long as both k/L and s/L are small; L denotes wave length. Thus,

Dp

=

f(k, s, T, Urn' t, p, ~, y)

Grouping the variables into dimensionless parameters yields:

U T pU k f(~, m i., m

s 1 ( ' T -~-' PUyT m) (20)

433

Measurements of Keulegan and Carpenter (1958) showed that the drag coefficient CD of flat plates in oscillatory flow was essentially constant throughout the cycle except for a narrmv time interval centered around

the moment of reversal of flow direction, when CD showed a sudden increase. This deviation from constancy will be neglected here, which is justified by the fact that the contribution of the drag in that interval to the

total time-integrated drag of one cycle is negligible (because in that interval U(t)/Umax is very small and passing through zero). For the

present problem the time-integrated drag is what counts (see equation 16), rather than instantaneous values of the drag. It follows that the time t need not be retained as a parameter, and equation (20) may now be written:

where use has been made of equation (11).

PUm)

yT (20a)

Expression (20a) can be modified slightly by considering that the quantity UmT is proportional to the amplitude of the horizontal orbital

motion, a'~. Assuming that the gravitational effects, if any, depend

'~Keulegan and Carpenter (1958) call UmT/k the "period parameter." It would seem to be more appropriate to call it the "amplitude parameter."

on the motion near the surface*"', we will take the surface value of Urn in the parameter pUm/yT:

where H is the ~vave height 2h.

Expression (20a) can no~v be re~vritten:

a k Re l!)

f(k"';'

' L

where Re stands for the Reynolds number pUmk/\l.

(20b)

The importance of the various independent parameters and the expected functional trend of CD as the dependent variable will be discussed next. Effect of a/k

The trend of CD as function of a/k as well as a physical explanation for this trend is given by McNmvn (1957), Keulegan and Carpenter (1958) and McNmvn and Keulegan (1959). Briefly, the reasoning of these authors relevant to the present problem is as follows: The wake formation in unsteady flow mayor may not be complete. This depends on the period of the oscillation (T) in relation to the time required for a wake to grow to its full extent (proportional to k/U), or, in other words, on the ratio a/k. For very large a/k, the conditions are quasi-steady and the wake development is complete, i. e. the ~vake will grmv until it is 1 ike tha t

for steady flmv. For small a/k, however, a wake is only partially formed and destroyed. The size of the wake affects the drag. The larger the ~'lake becomes, the higher the pressure within the wake and the smaller

the coefficient of drag. For very small a/k (very small wake) the pressure in the wake is very lmv, resulting in a high value of CD. A theoretical limit of CD is infinity. This unrealistic value is a consequence of the assumption that the pressure over the entire down -stream face is the same as the pressure at the edge, no matter how small the vortex. Actually, the low pressure extends over only a short dis tance initially (McNmvn, 1957). Nevertheless, CD should be large for

**The meaning of this statement can perhaps be clarified by referring to the case of drag in oscillatory flow in absence of a free surface, as would occur, e.g. if the present study were to be carried out in a pulsating water tunnel (Lundgren and Sorensen, 1958), and also, essentially, if the roughness strips in the present study would have been placed on the bottom. The different behavior of CD in presence and absence of a free surface, other things being equal, is due to gravitational effects.

w

AVE ATTENUATION

small a/k and decrease Hith increasing a/k until it approaches the

steady-state value for corresponding Reynolds number. The measurements of Keulegan and Carpenter (1958) fully confirm this. Their minimum a/k

is 0.54 Hith a corresponding Cn

=

11.55, considerably in excess of the

steady-state Cn value of 2.

Effect of k/s

435

The dependency of Cn on k/s, the roughness height over spacing ratio,

can be estimated in a rather elementary Hay: for very large k/s, the

roughness elements are sheltered by each other with, in the limit, full protection and no drag at all (remembering that only the broad side of

the plates \~aS considered), \~hereas for very small k/s the shelter effect disappears so that then Cn is independent of k/s; in that case, Cf is proportional to k/s, see equation (13). In other words, Cf tends to zero for small kls and for large k/s. For an optimum value of k/s, Cf

\~ill attain a maximum value. Johnson (1944) gives some experimental curves shm'ling this effect. HOHever, those results are not directly applicable to the present conditions because of a great difference in

geometry. His roughness elements are either square or flat Hith the

broad side against the wall while those employed in this study are flat

with the broad side perpendicular to the wall. Moreover, his data were

for steady conditions. The effect of k/s on Cf will undoubtedly be different in the unsteady state, because of incomplete \~ake development for small a/k. Thus the shelter effect will be less pronounced in the unsteady state than in the steady state, so that the k/s value correspond-ing to maximum Cf is greater in the unsteady state than in the steady state; the difference increases as alk decreases.

Effect of Re

The influence of the Reynolds number PUmk/~ on Cn may be estimated

from the Cn - Re relationship for flat plates in steady floH, where Cn is constant for Re

>

103 • The range of Re in the present experiments

is from 2.10 3 to 1.105 • It may be surmised, therefore, that for these

conditions the Reynolds number is of no importance. This was confirmed

by the measurements.

Effect of H/L

The importance of gravitational effects is less clear. At least ~~o distinct processes seem to be of some influence.

1. Surface wave resistance;

2. Transversal momentum and energy transfer due to transversal

surface slopes.

1. The existence of surface \~ave resistance was concluded from the

fact that energy is scattered by the roughness elements in the form of

cross waves, as will be explained in Section IX. Failure to take this

It was decided to evaluate the error thus made experimentally. It was reasoned that the process involved was restricted to the free surface. That portion of the roughness boards which extended above the troughs of the highest "laves to be used ,~ere, therefore, in effect taken a,~ay.

This was achieved by covering these portions with a metal strip and by closing the resulting chambers bet,~een two successive boards, the side ,~all and the metal strip, with wood blocks. With this arrangement no cross ,~aves ,~ere visible in the channel.

The ,~ave attenuation under these conditions was compared to the

attenuation when the roughness boards were exposed through the full depth. Allowance was made for the difference in depth of the exposed roughness by using the proper limits of integration in expression (16). The results shovled that the contribution of the cross waves to the total energy dissipation was negligible.

2. The effect of gravity on transport of energy and momentum perpendicular to the side ,~alls can be explained as follows.

A velocity defect near the side walls would immediately result in a corresponding wave height defect because of the kinematical relation-ships bet,~een U, H, and T. Transversal slopes of the free surface would develop and these would be counteracted by gravity. As a result,

the velocities near the roughness elements tend to be higher than in absence of this gravitational effect, and the drag coefficient likewise (because U in equation (11) refers to the average in the cross section). Thus the effect of gravity in this case is similar to the effect of turbulence on steady pipe and open channel flow: a more uniform distri-bution in the main profile, higher gradients near the wall and a higher rate of energy dissipation by wall friction than for laminar flow. It would have been possible to evaluate this effect experimentally by attaching the roughness to the bottom of the channel instead of to

the side ,~alls. Such experiments were not performed, however, because the s tudy ,~as aimed specifically at the effects of roughened side ,~alls

on wa ter ,~aves.

It should be emphasized that the above reasoning is qualitative only, both as to the absolute magnitude of the effect and as to its magnitude relative to the momentum and energy transfer by the trans-versal flm~ as described in Section VIII. The latter is apparently predominant because measurements failed to indicate a consistent relationship betl'leen CD and H/L.

The results of this section may be summarized as follows:

(a) CD f(a/k, k/s) (21)

is a statement valid for the test condition of this study. (b) as a corollary of (a): as far as its effect on CD is

concerned, the oscillatory flm'l is sufficiently decribed by the amplitude "a," thus by UmT; neither Urn nor T as such are important.

WAVE ATTENUATION

437

(c) CD decreases with increasing a/k; for large a/k it approaches

its steady-state value.

(d) CD goes to zero for very large k/s (considering the normal pressures on the broad sides of the strips only), ~~hile it becomes independent of k/s for very small k/s.

(e) Cf goes to zero for both very small and very large values of k/s. It attains a maximum for some intermediate value of k/s.

A consequence of the fact that CD = f(a) for certain k and s [thus,

CD

=

f(h)] is that the amplitude attenuation equation (8) is valid only on intervals of x where CD is essentially constant. On such intervals, the reciprocal of the amplitude should increase linearly with x. The plot of l/h vs x will be curved whenever i3 varies with h. In that case, the coefficient i3 may be determined as a continuous function of h from the gradient of l/h with respect to x, as follows inunediately from

equa tion (7): AIM 1 dh h 2 dx d(ft) dx VI. EXPERIMENTS (22)

The aim of the experiments was to investigate the processes by which

energy was transferred to the side ~~alls, and to measure the rate of this

transfer. These processes of transfer were to be linked to transversal currents and to the transversal ~'lave height distribution, which was ex-pected to show a decrease from the center line to the side ~~alls. The rate of transfer ~~as to be determined indirectly by measuring the

attenuation of the ~~aves with distance of travel. The underlying assump-tion was that only the energy dissipation at the rough side walls Has significant. The validity of this assumption was checked by measuring

the ~~ave attenua tion in the smooth channel, i. e. the reinforced concrete

channel without roughness elements. The measurements in the smooth

channel therefore ought to be looked upon as a calibration of the channel.

EXPERIMENTAL SETUP AND PROCEDURES

The measurements were carried out in a 10 ft. wide, 125 ft. long and 21. ft. deep rectangular wave channel with a wave generator consisting

2

of a 7.5 hp elec tric motor driving a bulkhead which can move either as a piston or as a flap hinged at the bottom. A variable speed drive allows a continuous range of wave periods from 0.7 sec. to 3.0 sec. The stroke of the bulkhead can be adjus ted so as to give the required ~'Tave height. After generation the waves pass through a l6-layer filter of wire mesh which partially eliminated higher harmonics. Subsequently, the Haves pass through the actual test section ~~hich is about 90 ft. long. The remainder of the wave energy is absorbed almost completely by a 1: 10 beach of rubble and ~~ire mesh.

The roughness elements consisted of wood strips or boards attached to the vertical side walls with the broad side perpendicular to the direction of wave propagation, thus perpendicular to the horizontal component of the orbital velocities (see Fig. 1). The boards extended through the full depth of the l~ave channel.

The fol101ving combinations of width (k) and spacing (s) were employed:

k(cm) s(cm) 5 10 5 20 10 40 20 40 20 80

Different values of k and s with the same k/s ratio l'lere used in order to extend the range of measurements of alk and to check on scale effects.

Measurements of Amplitude Attenuation. The wave amplitudes were measured by means of parallel wire resistance type gauges. The output signal was fed into a "Van Reysen" 24-channel amplifier and recorded on a dual channel Brush recorder. Static calibrations were performed before and after each run. The period of the waves was determined by means of a stop watch.

For most of the measurements the channel was too short to allow a determination of the shape of the attenuation curve. It lvas, therefore, decided to extend the decay distance by the following artifice.

For certain water depth, channel roughness and wave period, the

l~ave generator was adjusted so as to give the largest wave height to be

used. Then the attenuation was measured and H(x) was plotted vs x. (For H(x) the average H in the cross section was taken --- see Section

IX). For the subsequent run the wave generator \Vas adjusted so as to give a wave height slightly exceeding the wave height measured at the end of the channel in the previous run. The attenuation was measured, and so on. The individual plots of H vs x for each run were then con-nected in such a way that a smooth, continuous attenuation curve was obtained without discontinuities in either H or dH. By this means,

dx

the length of travel of the l'laVe has been extended to a considerable number of actual channel lengths. An example of a plot of H vs x obtained in this way is given in Fig. 2. A disadvantage of this method is the regeneration of higher harmonics with the start of each successive run.

SYMBOL RUN

•

64 14

.

x 65 k = 5 cm 0 66 s = 20cm 12

..

67 T = 1.98 sec.

9

68 10 69 70 E B (,) J: 6 4 2 0 0 25

50

75 100 125 150 175 200 x (m)

Measurements of Transversal Currents. Transversal currents, due to the wave motion in the roughened channel, were measured using a de,:,ice consisting of two neutral density beads, about

1

in. in diameter, Ln a

2

"tube" of eight stainless steel piano wires. A frame consisting of three tubes and scales was installed in the channel perpendicular to the ~hannel axis. Thus only the horizontal component of the velocities perpend~cular to the direction of wave propagation was indicated by the beads. PLctures of the moving beads '''ere taken a t an interval of 5 sec. or 10 sec. and the readings were obtained by projecting the negative on a screen. The.actual velocities will be higher than those thus measured because of frictLon between bead and wires.

VII. RESULTS OF MEASUREMENTS IN SMOOTH CHANNEL

The attenuation measurements in the smooth channel did not give conclusive results as far as

a

is concerned. The linear theory predicts a maximum attenuation of only 3 per cent for a wave of T = 0.80 sec. traveling the full length of the channel 'vith d = O. 6Om. For the longer ,,,aves, the attenuation is even less. The measurements do show a. decrease of amplitude with distance but not a monotonic decrease. This i s

probably due to parasite waves of t'·10 types: higher harmonics a.nd .

transversal s tanding waves (refer to Sec tion IX). Both may give rl.~e to a limited apparent increase of amplitude of the main wave wi t:h dLs-tance of travel.

The usefulness of the tests in the smooth channel was twofold: 1. It was confirmed that the damping in the smooth channel was very small. Such knolv1edge is of importance in the next phase 0 f the study where the energy is partly dissipated by turbulent frictiOn along the side walls and partly by friction along the bottom, the int:e rnal friction being negligible.

2. It appeared that certain parasite waves were an inhere o t feature of the wave channel. Thus, in the tests with the roughe ned side walls, these waves could not be ascribed to the roughness eleme o t:s.

VIII. RESULTS OF TRANSVERSAL CURRENT MEASUREMENTS IN ROUGH CHANNEL

The transversal currents shOl"ed a definite pattern. Near t he surface the current was directed tOl"ards the channel center, wi.. t h velocities gradually reducing from a maximum close to the wall -co zero close to the center of the channel. Near the bottom, the direc -cion of flolV lVas toward the sides of the channel. It started slowly ne.;3~ the center of the channel and gradually sped up. In the

1

width re g LOn

4

close to the side, the current reached its maximum. It reduced sharply close to the side wall. Fig. 3 gives a schematic picture of th.42' trans-versal currents.

WAVE ATTENUATION

Figure 3. Schematic Transversal

Current Picture

DISCUSSION

The observed transversal flow pattern '-las the result of the deflection of the flow tm-Iards the channel center caused by the wood strips at the side walls. As is well known from elementary fluid mechanics, the flmv pattern around such strips is not symmetrical with respect to the central axes of the strips because of flo,v separation at the edges. On the upstream side of the strips, the flmv is deflected imvard rather sharply but down -stream it does not follow the same path towards the wall. Thus, the flow past the strips maintained an inward direction and generated a vortex at the downstream face of the strips near the edge. After reversal of the flow direction, the same inward deflection of the flm-I occurred. The entire vortex generated in the previous half cycle was carried alVay from the wall by this deflected flmv, while at the same time a new vortex was generated at the other face of the wood strip. Fig. 4 gives a clear picture of this phenomenon. This photograph was taken just after the flmv had reversed direction and started moving from right to left. The "old" vortex is expanding and is moving to the upper left in the photo-graph. Once outside the region of the wood strips the original vortex would very soon lose its identity.

The cause of the transversal circulation pattern is the combination of the tendency of flmv to be directed away from the wall as just described, and the fac t that the velocity amplitudes -diminish from the surface dmm-,yards. The rate of flolV tmvards the channel center is, therefore, highest near the surface and lowest near the bottom. The superposition of the return flow (which is necessary for reasons of continuity) on this pattern gives a flmv pattern as shmm in Fig. 3.

The importance of the observed currents with respect to the trans-versal flux of wave energy lies in the fact that they continuously

transport water tm-Iard the center ,yith very little momentum in the direction of the undisturbed orbital motion, thereby retarding this motion. In other words, there is a strong momentum exchange beDveen the wall regions and the central region, similar to the turbulent momentum

exchange. A difference is that here the velocity fluctuations are not random, but periodic with period.!. T.

2

IX. RESULTS OF AMPLITUDE ATTENUATION MEASUREMENTS IN ROUGH CHANNEL

Before proceeding to a presentation of the 'vave attenuation data proper, the effect of various disturbances of the primary wave on the amplitude measurements IVill be evaluated.

SECONDARY HAVES

The primary wave is defined as the ,vave which is genera ted by the wave generator and which is traveling away from the generator without change of form except insofar as it decreases in amplitude due to friction, and which has a period equal to the period of the wave generator.

WAVE ATTENUATION

443

Secondary ~vaves are those Haves disturbing the primary ~vave: a. Free-traveling higher harmonics emitted by the Have generator; b. Haves reflected from the end of the channel;

c. Transversal standing Haves, and

d. Haves scattered by the roughness elements, to be called "cross Haves" hereafter.

Haves of type (c) and (d) are parasites inasmuch as they have to HithdraH pOHer from the primary wave in order to maintain themselves.

The secondary waves (a) and (b) may be assumed to be laterally uniform to the same degree as the primary Have. In other words, the lateral wave height distribution is not altered by their presence. However, because Haves (a) travel in the same direction with a different velocity, compared to the primary wave, ~vhile waves (b) travel in the opposite direction Hith the same velocity, a combination of beat and partial clapotis will occur with a resulting longitudinal variation of the wave height. A local apparent increase of wave height in the direc-tion of travel of the primary Have is even possible. The variation due to reflected waves does not, on the average, alter the rate of attenua-tion of the primary wave. The same is true for free higher harmonics. But both tend to introduce inaccuracies. In order to minimize errors arising herefrom, a l6-layer filter Has installed between the wave generator and the test section of the channel. Reflection from the end of the channel was minimized by installation of a 1:10 slope of rubble covered with several layers of wire mesh. The influence of reflected Haves ~vas imperceptible ~vithin the accuracy of the measurements. This

fact 'vas established by recording the surface time history in a few points after starting the Haves from rest. Points ~vere marked on the record Hhich indicated the computed arrival time of reflected waves, if any. In no case was a trace of change of amplitude noticeable. These measurements were repeated for waves of varying period and steep-ness in points at various distances from the beach.

The transversal standing Haves (type c) Here a source of disturbances of the transversal wave height distribution. The magnitude of these disturbances depends on the height of the standing Have as well as on the phase difference between standing wave and primary ~vave at each point. The phase of the primary wave depends upon the dis tance from the v.<lve generator but not upon distance from the side walls. Hmvever, for the transversal standing wave the phase does depend on distance from the side Hall; it differs by 180 degrees on different sides of a node. For a binodal transversal standing wave for instance, the measured wave height at a certain distance from the generator may be larger near

the Halls than in the center, the difference being t~vice the height of the s tanding wave if the t~vo waves are in phase near the Halls and 180 degrees out of phase in the center. At another distance from the genera-tor, the situation may be exactly reversed. It is obvious that ~vaves

WAVE ATTENUATION

of type (c) are very undesirable because the transversal distribution of

the ,~ave height is of eminent importance for this study. Attempts to

eliminate the transversal standing waves failed. Guiding vanes parallel

to the direction of propagation of the main ,~ave improved the situation

but at the same time prohibited flux of energy perpendicular to this

direction. Efforts to obtain a detailed picture of the transversal

pri-mary wave height distribution had to be given up finally. Instead, the

,~ave height was measured in eight equidis tant points in each cross

sec-tion and the average of those heights was taken to represent H(x). Throughout the remainder of this report, H denotes the average wave

height in the cross section.

Without such averaging, the transversal standing waves also disturb

the longitudinal wave height distribution, to the effect that often the decrease of wave height in the direction of propagation, measured in points at a constant distance from the side walls, is not monotonic. By taking the average H in each cross section, the effect of the standing waves on the longitudinal wave height distribution is canceled.

The secondary waves of type (d), the so-called cross ,~aves, were the result of scatter of ,~ave energy from the roughness elements. The

front of a wave scattered from one roughness board would be joined by similar fronts emitted by neighboring boards, as it traveled a,~ay

from the ,.,all, to form a straight wave front propagating obliquely

across the channel. The two systems of cross ,.,aves emanating from the two side walls constituted a diamond pattern superimposed on the pri-mary wave. The resulting peaks and troughs moved parallel to the side

walls with the velocity of propagation of the primary ,.,ave. Evidently, the transversal primary wave height distribution is disturbed by the

presence of cross ,~aves. But the longitudinal distribution is not,

provided the average of wave heights in each cross section is taken in

order to cancel the effect of cross waves.

The height of the cross ,~aves appeared to decrease at a higher rate than the primary ,.,ave. :For lm.,er steepness of the primary wave no

cross waves could be observed.

ATTENUATION DATA AND DRAG COEFFICIENTS

Table I gives a summary of the test data. The largest and smallest wave heights of each series of runs are designated by H(O) and

H(i)

respec tively, where

f

stands for the total virtual length of travel of

the wave. The number of runs used for eacq series (see Section VI) is indicated in the last column. The length

i

depends, of course, on

this number and on the overlap of individual plots necessary to obtain a smooth curve.

Table 1. Summary of Test Data

Roughness Wave Distance Number of _runs

k(cm) s(cm) T(sec) H(O)(cm) H(!)(cm) l(m) 5 10 0.80 5.5 0.7 38 2 5 10 1.00 10.5 1.3 70 4 5 10 1.43 13.9 3.3 124 6 5 20 0.80 10.6 1.3 38 2 5 20 1.00 9.8 2.2 54 3 5 20 1.43 14.6 2.8 152 8 5 20 1.98 11.0 4.0 177 7 10 40 0.80 10.6 1.5 27 1 10 40 1.00 9.0 1.0 54 2 10 40 1.12 12.0 1.7 77 3 10 40 1.43 16.6 2.2 128 5 10 40 1.98 13.5 4.0 108 4 20 40 0.80 attenuation insignificant 1 20 40 1.00 9.2 2.6 27 1 20 40 1.20 12.0 1.2 27 1 20 40 1.43 14.8 2.8 27 1 20 40 1.98 15.2 2.6 74 3 20 80 1.43 13.2 2.0 49 2 20 80 1.98 15.4 3.8 54 3

The data ~~ere plotted in t~~o ways: .!. vs x on linear scale, and H vs x

H

on semilogarithmic scale. A linear H vs x plot would give no direct

clue as the whether equation (8) fits the experimental data or not.

The.!. vs x plot is more useful in this respect because it should give

H

a straight line ~'lhen f3 is independent of H. This ~~as indeed the case

in some runs. Fig. 5 gives an example. In most runs, however, the

.!. vs x plot would be straight for smaller values of x (high values of

H

H) and gradually curve up~"ards ~~i th increasing x (decreasing H). This

indicates an increase of Cf with decreasing H, which is in accordance

1-1ith the results obtained by McNown (1957) and Keulegan and Carpenter

H(em) 16 10 8 6 4 2 k=20 em. s" SOem. T= 1.9Ssee. I

...

----

.

.

. ----.?

.

.

-

..

----~-

...

-~(m-I) 30 20 10 I I I

1

0

o

10 20 30 40 50 60 - x(m)

"Figure 5

,

Exanrple of

H.yperbolic Attenuation

~

>

-<

_trJ

>

_>-,3 >-,3 trJ Z

:;

>-,3

(3

Z

>I>-~

All

1

vs x plots were reduced according to equation (22): H

(22)

From ~ the values of Cf and Co ~yere obtained using equations (19) and (13) respectively. The results are given in Figs. 6 and 7 ~Yhere CD is plotted vs a/k; s/k

=

4 and 2, respectively; a is the average of surface and bottom value of "a". With respect to Figs. 6 and 7, it may be remarked:

1. that the CD values obtained are very high compared to the

steady-state value of a single flat plate of infinite length,

which is about

2;

2. that CD increases strongly for small values of a/k;

3. that the scatter in the experimental points is considerable; 4. that the effect of k/s cannot be established with any accuracy

because of the large scatter in the points and because only

two different.values of k/s were used. In Fig. 7 (k/s = 0.50)

the CD appears to be smaller than in Fig. 6 (k/s

=

0.25), as

it should because of the shelter effect, but no quantitative conclusion can be drmvn.

Part of the scatter is undoubtedly due to the fact that in the integration of

1. U

with respect to y, CD has been taken as independent

of y. It nOly appears that the variation of CD ~Yith a/k can be con

-siderable, particularly for the 10lver values of this parameter. The

waves employed in this study were not long waves and the horizontal

orbit axis is therefore decreasing downward from the surface. The actual CD then will increase dowmvard. Using only one value of CD for each cross section amounts to averaging actual CO(y) values, ~yeighted

with U(y)3, thus with cosh3m(d+y) compare equation (16)

o

_

~-d

CO(y) cosh3m(d+y) dy

CD = (23)

(O

cosh3m(d+y) dy

J-d

The error resulting herefrom depends on the variation of CD from

surface to bottom. The error will, therefore, increase with increasing

md and also with decreasing a/k inasmuch as the variation of CD as

function of a/k is much larger for small a/k than it is for large values of a/k. Most of the runs were made at the same depth (2 ft), so that indirectly the CD values do depend on T. This of course is not contrary to conclusion (b) of Section V, ~Yhich refers to the local resistance coefficient.

70 60 50 4 CD

130

20 10

\

.'\.

"-

"

-

"

'¢"

'.

.

i('4 --.~--.-- --.~--.-- --.~--.-- --.~--.-- --.~--.-- T-o.eo .• c . - - - T = LOOsec. _ . _ . _ . - T-1.12 sec. _ .. _ .. _ .. - T = 1.43 sec. _ .. ·_ ···-·T·1.gesec.

0. • DIRECT DRAG MEASUREMENTS

01 I I I I I I I

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 l20

- - . [

k

Figure 6. Experimental Results: CD vs. a/k with s/k 4

~ ~

<:

tIj ~ ....:.l ....:.l tIj

Z

§2

....:.l

(3

Z

~ ~ ~

50

40

20

10

\

\

\;

\

\

\

\

\

\

,,

\

'.

_"-

\ _. ..

"

_{... ~} / RESONANCE

~

~

.

"

,

'

.

...

'::::,:.

'

... -::::--..

_

..

-

~

~...:::::::::

.

:--- "'- -

-

...

-

.

-

..

~---

..

-

..

_

.

-- -- -- --T = O.80sec. - - - -T= LOOsac. - '- -'- -'- T= 1.I2aec. _ .. - -.. - .. -T = L43sec. -"'- "'- '" T = L98sec.

-

..

-

.. -- .~~

0

~----(

0~10

;---

aa.2WO

~---o

Oj,

.

30

~---C

0~4~0

~--

~0

.

;50

~==~~

0

~:6~O

~=-~

'

~

'

~=~~';-;-=-~

₀

_.

₇₀

"

~=-=-~

₀

"

_.

₈₀

~=-=-~

'

~~=-='

₀

_.

~

₉₀

-=-==~

'

~-=-=-~

"

I.

--

0

-~

k

w

AVE ATTENUATION

In a study aimed at determining CD(a/k) more accurately, the

rough-ness elements should be placed at the bottom in order to avoid such

inaccuracies.

Another possible source of error is the use of the linear I~ave

theory for the computations of horizontal orbital velocities. Some of

the I'Taves employed had a considerable initial steepness (see Table I)

and a certain error will be involved in approximating the actual velocity

fields by the linear theory. This error is augmented by raising the

horizontal velocities to the third pOl'Ter. On the other hand,

instan-taneous values of the rate of work done on the strips are of no

impor-tance, but, rather, their time integral, and the relative error (due

to nonlinearities of the orbital velocities) in the time integral of

U3 Ivill be only a fraction of the maximum relative error of its

instantaneous values, because such errors are partially canceled in the

process of integrating with time.

It was decided to evaluate the maximum resulting error

quantita-tively; for this purpose a I,lave I"as chosen from the experiments (Table I)

I~hich would show the larges t deviation from linear theory.

The depth in all runs was the same, while five different periods

were used. The shorter I.aves had a greater maximum (= initial) steepness

than the longer waves and it was therefore not immediately clear for

which combination the nonlinear effects would be maximum. The choice

was based on the relative magnitude of higher order orbital velocity

terms which were determined according to Stokes's finite amplitude

theory (Skjelbreia, 1959). The results are indicated in Table II:

Table II.

Maximum Relative Higher Order Terms

[F2

+

I

F31 ,~ T(sec) Fl max 0.80 2.3 x 10- 5 1.00 LOx 10- 3 1.12 3.7 x 10- 3 1.43 2.6 x 10-2 1.98 8.5 x 10- 2

*Fl and F2 are all.ays positive, but F3 is not.

I.here Fn is the nth order dimensionless orbital velocity coefficient.

It appears from Table II that nonlinear effects are strongest in the

longest wave (T

=

1.98 sec).

The integral

(TO dt T](_d(t)

} ) IU (y, t)13 dy,

,.,here 1'] is the height of the free surface above M. W. L., was evaluated numerically for the 1.98 sec. wave at its initial steepness, using the third order approximation, and compared to the corresponding value based on the linear theory. Results are presented in Fig. 8.

The error in CD due to using the linear theory instead of the third order theory (,.,hich represented the actual wave motion very well compare theoretical and measured surface profile) appears to be

(..2d

-1)

10.9 -13.8%

which is not negligible. But i t should be remembered that this is a maximum possible value. Not only is the error far less for the shorter waves employed (see Table II), it also diminishes rapidly for all the waves with distance of travel as the wave amplitude attenuates.

x.

DIRECT MEASUREMENTS OF DRAG COEFFICIENTS

In the preceding section the drag coefficient of the roughness strips ,.,as determined from amplitude attenuation measurements. A check on the values thus obtained can be made directly by measuring the force on the strips as a function of the relative velocity of undisturbed flo,., and strip. A truly direct force measurement by means of strain gauges was outside the scope of this study, hmvever. A simpler method was

therefore devised which required no installation or use of equipment not already in use for the study.

Consider the expression

(24)

which at first glance may look rather trivial but ,.,hich has significance for our purpose.

The left-hand member is the integrated differential pressure on the board due to drag and inertia forces. Under a crest or trough the inertia term vanishes and then

25 "'0 Q) U> '<t ... E u

"'

0

-...

-5 -0 0-

t:.

7](t) I(t) =

J

I

u( y.dldy -d ~ 3

:::::::::-

-4 5 51 ,-order theory d 3-order theory measured 6 d T 3 order:

J

-

5 4 2

Idt=10.9XIO em /see

f

-

--o

T ~ I order

:

J

_{I dt = 9.4}

_x

₁₀5 _{em 4/see}2

o

= -

-8 9 surface profile H = 14.50 em T = 1.98 sec d = 60 em fO II 12

..

30 I T - - - - . . ; : : -d

Figure 8. Stokes's 3 Order Theory VS. Linear Theory

----

-

---

_~

>

<

tz::l 13 14 15

_>

>-3 >-3 tz::l Z

c:::

>

_>-3 ... 0 Z

--

-

-

-Using the linear wave theory, ~.,e find for A, the total drag per unit width: A or A 2 2 nCDPw h 2 m tgh md

where h is the average wave amplitude in the cross section.

(25)

The right hand member of (24) represents the net force per unit width on the board due to differences in phase and/or amplitude of the ~vater level fl uc tua tions on the upwave and do~m\Vave sides. This force, which must equal the drag and inertia force on the board inasmuch as the pressure on the free surface is constant, can be evaluated from the difference in surface elevation by using the knOlm vertical pressure

distribution in a progressive or standing small amplitude wave:

p _pgy

+

pgh cosh m(d+y)

cosh md sin wt

It fo1101-IS that the total force per unit width on the strip is

A =

JO

-d pg6h cosh m(d+y) dy cosh md pgAh tgh md m (26)

~.,here 6h is the (fluctuating) difference in surface elevation on upstream and dOlms tream s ide of the strip.

Elimination of A ben-Ieen (25) and (26) gives

2 tgh md

n

6h _{cr, tr}

mh2 (27)

where 6h_{cr tr is the difference in} elevation on the two sides of the

strip at the time that a crest or a trough of the undisturbed wave passes the strip; it was measured by recording the surface fluctuations in the proper points simultaneously. Results of five such measurements are

plotted in Fig. 6. Hithin the scatter of the data, there is agreement

with the results from the amplitude attenuation measurements, in partic-ular as far as the sharp rise of the CD curve for low a/k is concerned. This increases the confidence that may be had in these measurements.

1.0 0.8 0.2

o

WAVE ATTENUATION Figure 9. Resonance at s

=

~

L 10 15

_ A

L

20

(

5

,1

0

)

etc

.

=

(

k,s

) i

n cm

.

T=0

.

80sec

.

Figure 10. Influence of Resonance on Damping

455

XI. COMPARISON OF ROUGHNESS STRIPS TO RESONATORS

AS WAVE DAMPING DEVICES

Several types of wave damping devices have been designed which employ

the principle of resonance (Valembois, 1953; Valembois and Birard, 1955;

Bruun, 1956; Lates, 1963). An inherent disadvantage of such damping

de-vices is the rather narrow frequency range of high effectiveness; i t is

only for frequencies very close to the resonance frequency that these

resonators perform satisfactorily.

The roughness elements employed in this study functioned properly

for the complete wave spec trum with two exceptions: ,dth s =

1

Land 2

k =

i

L, resonance was observed to occur with a standing wave 'vithin

the compartments. IVhen this occurred, the efficiency of the roughness

elements dropped sharply. (These cases evidently do not conform to the

assumption made in Section V, that both k/L and s/L would be sufficiently

small to have no effect on CD.) In the first case (s

= 1

L) a uninodal

2

oscillation came into existence with its nodal line perpendicular to the

wall. Fig. 9 shows an example. Such an oscillation consumes only a

minor part of the wave power. The wave motion is very nearly in phase

and of equal amplitude on the two sides of the roughness boards with

the result that very little energy is exerted on the board. This appears also in Fig. 7 where the CD value computed for T

=

1.00 sec. and Snet

=

0.78m. are seen to be decidedly lower than the others.

In another case where k

= 1

L, an

4

nodal line parallel to the wall at the

oscillation is possible when s is less

oscillation occurred with its edge of the strips. Such an

than

1

L (Valembois, 1954).

2

For larger values of s/L, the standing wave is not guided sufficiently

and the main wave actually propagates parallel to the wall in the space

between two strips. In the present study, the standing wave did not

occur for s/L

=

1, but it appeared when s/L was changed to

1.

As in 2

the oscillations mentioned above, a comparatively small power is needed

maintain these oscillations. There is very little interchange of 'vater

between neighboring compartments and vortices, if any, are very small.

In Fig. 10, the H vs x plots are given for four tests with T

=

0.80

sec. The results of three of them are about equal but the fourth, with

k

=

0.2Om and s

=

0.4Om, shows a strikingly small attenuation due to

the oscillations just described even though there was no perfect

resonance with k = 0.20m and T = 0.80 sec. (L = l.oOm), as was also

evidenced by the existence of beat.

It thus appears that the roughness elements work well as wave

damping devices except when there is resonance, while resonators work

efficiently only ,.,hen resonance occurs. It would seem, therefore,

that roughness strips may well constitute a more adequate wave damper

than resonators, depending on the width of the spectrum of the ,.,aves

to

WAVE ATTENUATION

XII. CONCLUSIONS

1. The flux of energy perpendicular to the direc tion of I~ave prop-agation takes place mainly by a large-scale momentum exchange caused by

transversal currents which, in turn, are caused by friction elements, e.g. strips at the side \Valls which deflect the flow from a direction

parallel to the wall to a direction nearly perpendicular to it.

2. Vertical strips at the side Halls as used in this study proved

to be very effective as Have damping devices.

3. The drag coefficient of the strips was found to be a function of amplitude and roughness spacing, both in relation to the roughness

height, or CD

=

f(a/k, s/k).

4. The unsteady-state drag coefficient may be considerably in

excess of the steady-state value for the same Reynolds number, the ratio depending upon a/k.

5. The effec tiveness of s trips as I~ave dampers decreases sharply Hhen resonance occurs in the space between the strips.

6. Strips may provide a good alternative for resonators as Have

dampers inasmuch as they are effective over the whole width of the

spectrum except near resonance, while just the reverse is the case for

resonators.

7. To establish the relationship Cf

=

f(a/k, s/k) more fully and accurately than was possible in the scope of this study, further

investigations are desirable.

ACKNOlIT.EDGMENTS

The \Vork described herein \Vas sponsored by the National Science

Foundation under Grant Number G 19939. The I~riter expresses his thanks to Dr. Per M. Bruun, who initiated this study and I.ho gave continuously of his advice \Vhile it was being carried out. Thanks are also due to J. v. d. Kreeke for his valuable criticisms of the manuscript.

REFERENCES

Bagnold, R. A. (1963). Mechanics of marine sedimentation: The Sea,

edited by M. N. Hill, Interscience Publishers, pp. 507-528.

Biesel, F. (1949). Calcul de l'amortissement d'une houle dans un liquide

visqueux de profondeur finie: La Houille Blanche, pp. 630-634. Biesel, F. and Carry, C. (1956). A propos de l'amortissement des houles

dans Ie domain de l'eau peu profonde: Commentaires et Discussions, La Houille Blanche, No.6, pp. 843-853.

457

Bruun, P. (1956). Destruction of wave energy by vertical Halls: Journal

Eagleson, P. (1962). Laminar damping of oscillatory waves: Journal of the Hydraulics Division, Proceedings A.S.C.E., Vol. 88, No. Hy3. Grosch, C. E. (1962). Laminar boundary layer under a wave: The Physics

of Fluids, Vol. 5, No. 10, pp. 1163-1167.

Grosch, C. E. and Lukasik, S. J. (1961). Attenuation of shallow water

,~aves: Bulletin Am. Phys. Soc., Series II, Vol. 6, pp. 210-211. Grosch, C. E., \~ard, L. IV. and Lukasik, S. J. (1960). Viscous dissipation

of shallow water waves: The Physics of Fluids, Vol. 3, pp. 477-479. Hunt, J. N. (1952). Viscous damping of waves over an inclined bed in a

channel of finite width: La Houille Blanche, pp. 836-842.

Inman, D. L. and Bowen, A. J. (1963). Flume experiments on sand transport by "laves and currents: Proceedings of Eighth Conference on Coas tal Engineering, pp. 137-150.

Johnson, J. IV. (1944). Rectangular artificial roughness in open channels: Transactions A.G.U., pp. 906-914.

Keulegan, G. H. and Carpenter, L. H. (1958). Forces on cylinders and plates in an oscillating fluid: Journal of Research, National Bureau of Standards, Vol. 60, No.5, pp. 423-440.

Lamb, H. (1932). Hydrodynamics: Sixth Edition, Cambridge University Press.

,

Lates, H. (1963). Recherches hydrauliques de laboratoire sur 1 'efficacite

de quelques types d'ouvrages de protection des petits ports maritimes contre la penetration des vagues et des alluvions charrites: I.A.H.R. Congress, London, Paper 1-4.

Lundgren, H. and Sorensen, T. (1958). A pulsating water tunnel: Proceed-ings of Sixth Conference on Coastal Engineering, pp. 356-358.

McNmm, J. S. (1957). Drag in unsteady f1m~: IXth International Congress on Applied Mechanics, Vol. III, Brussels, pp. 124-134.

McNo,m, J. S. and Keulegan, G. H. (1959). Vortex formation and resistance in periodic motion: Journal of the Engineering Mechanics Division, Proceedings A.S.C.E., Vol. 85, No. EMl, pp. 1-6.

Skj elbreia, L. (1959). Gravi ty ,~aves, Stokes' third order appr oxima tion, tables of functions: Council on IVave Research.

Valembois, J. (1953). Etude de l'action d'ouvrages resonants sur la propagation de la houle: Proceedings Minnesota International Hydraulics Convention.

Valembois, J. and Birard, C. (1955). Les ouvrages resonants e t leur applica tion a la protec tion des ports: Proceedings of Fi f th Con-ference on Coastal Engineering, Berkeley, pp. 637-641.

WAVE ATTENUATION 459

LIST OF SYMBOLS

A Net horizontal force per unit width on roughness boards parallel to sidewall.

a Horizontal semiaxis of particle orbit.

if Average of "a" at bottom and at surface. CD Drag coefficient of roughness boards. CD Weighted average (through the depth) of CD.

Cf Coefficient of boundary resistance.

Cg Group velocity.

d Mean ,~ater depth.

E

Ivave energy per unit area. e Base of natural logarithms.

F Energy loss per unit time per unit length of channel.

Fl, F2, F3 1st, 2nd, and 3rd order orbital velocity coefficients.

g Acceleration of gravity.

H Ivave height (averaged through the channel width). h Ivave amplitude =

t

H.

k lVidth of roughness boards.

L Wave length.

2

Effective total travel length of wave for attenuation measurements.

m Wave number

=

21l/L.

n Ratio of group velocity to phase velocity.

P Ivave power.

~ Differential pressure on roughness boards due to drag and inertial forces. Re Reynolds number related to roughness height.

s Spacing of roughness boards.

T Wave period.

t Time.

U Horizontal component of orbital velocity (averaged through the channel width).

Urn Maximum value of U.

W Channel width.

x Horizontal distance, positive in the direction of travel of the primary ,qave.

y Vertical distance above mean water level.

a

Coefficient of laminar damping.

~ Coefficient of turbulent damping.

y Specific weight of ,qa ter.

1] Height of free surface above mean ,qa ter level. ~ Dynamic viscosity of water.

v Kinematic viscosity of water. p Density of water.

T Apparent shear stress on side walls. w Angular frequency of wave

=

21f./T.