On the interaction between waves and currents

Ocean Engng, Vol. 17, No 1/2, pp. 1-21, 1990. 0029-8018/90 $3.00 + .00

Printed in Great Bn'tain. Pergamon Press pic

ON THE INTERACTION BETWEEN WAVES AND

CURRENTS

R. E. B A D D O U R and S. S O N G

Faculty of. Engineering and Applied Science, Memorial University of Newfoundland, St. John's, Newfoundland, A l B 3X5, Canada

Abstract—The interaction of a current-free plane surface wave train of fixed frequency and a uniform wave-free current nomial to the wave crests in the same or opposite direction to wave propagation is described. Neglecting'dissipation, the combined field after the encounter lis assumed stable, uniform, steady and irrotational. The parameters describing the wave-current field generated by the interaction, namely the wave height, wave length, stream current and water depth are calculated numerically by solving a set of nonlinear equations. These equations are obtained on satisfying, to second order, conservation of mean mass, momentum and energy flux, as well as a dispersion relation on the free surface of the after-encounter field. Possible solutions are identified a posteriori. Contrary to other approaches the additional contribution to the change in the mean value of the current and mean water depth is not neglected in this work. Numerical results are presented which illustrate the changes in the wave length, wave height, current speed and water depth. A comparisoa with the numerical and experimental results reported by G . P. Thomas [/. Fluid Mech. 110, 457-474 (1981)] is also provided.

1. I N T R O D U C T I O N

T H E E F F E C T S of a following or opposing uniform current on the propagation of surface gravity waves were first discussed by Unna (1942). O n neglecting the interaction between the waves and the current, the changes i n wave length and wave height were determined by the following two equations: C = Cr± Uo and E ^ Q o = E{Cgr ± Uo); where C, Cg, E, and U are respectively the wave celerity, group velocity, wave energy density on still water, and mean current speed; with the subscript o indicating the parameter before the wave train and current meet, and subscript r indicating the relative value o f the parameter. Longuet-Higgins and Stewart (1961) showed that the above equation of energy transfer assumed that no coupling between the wave train and current took place.

Since 1960, the theoretical and experimental aspects o f the interaction between gravity waves and a current motion have received increasing attention,, covering a wide spectrum of problems ranging from studies on the combined wave-current field to changes in wave amplitudes etc. The works by Longuet-Higgins and Stewart (1960, 1961), Whitham (1962), Peregrine (1976) and Jonsson (1978), to name but a f e w , are already classics. The mechanism of the interaction is connected with the so-called radiation stress (Longuet-Higgins and Stewart, 1960, 1964), wave action (Bretherton and Garrett, 1968) as well as mean energy level (Jonsson, 1978 and Jonsson et al., 1978).

The problem of waves propagating through a known slowly varying, depth-independent, horizontal current is reviewed by Craik (1985), I n their report, Peregrine and Jonsson (1983) present an overview of wave-current interaction, and include a comprehensive review of references until December 1981.

1 TECHNISCHE UNIVERSITEIT Laboratorium vc»r Scheepshydromechanlca Archief Mekelweg 2, 2628 CD DeFft Te*,; 015 - 788873 - F t a 01S • 781836

2 R . E . B A D D O U R and S . SONG

When waves moving through still water encounter a current at an angle with the wave direction, the waves are refracted, undergoing a change in length, steepness and direction. In this paper the two-dimensional version of this situation in water of finite depth on a constant horizontal bottom is considered, i.e. periodic waves on still water propagating into a wave-free, irrotational, colHnear current. Neglecting dissipation, the process of the interaction between waves and a current in the same or opposite direction of wave propagation is assumed to form a stable, uniform, irrotational wave-current field. The physical situation considered here corresponds to the laboratory experiment performed and reported by Thomas (1981).

In contrast to the approach of Thomas (1981), the wave-free current mean speed and mean water level in the present analysis are allowed to change due to the encounter and the interaction of the wave and the current. I n this paper, a new numerical attempt is made to calculate the changes in wave height and wave length by considering the changes, due to the interaction, in the mean current speed and the mean water depth.. The objective is to investigate the relationship between the current-free wave length Lo, wave height Ho, wave-free current speed f/^ and mean water depth before the interaction and their counterparts, denoted without subscript of the wave-current field. The transfer of momentum and energy from the waves to the current is accounted f o r , and a new method for calculating the resulting wave-current field parameters, namely L , H, U and d, is presented. This method is based on solving, simultaneously, the fundamental conservation relations f o r the mean rate of mass, momentum and energy transfer f o r the considered fluid flows, with no compensation by a known vertical upwelling f r o m below or known horizontal inflow f r o m the sides. The conservation o f wave crests and the coupling of the wave and current in the wave-current field are taken into account. Numerical results are presented for the variation in L / L „ , HIHo,

VIUo and did^, respectively the wave length, height, current and depth ratios f o r

different wave and current combinations. A comparison between the present approach, and the numerical and experimental results reported by Thomas (1981), is included. The present method is not a reproduction of other investigations and could be easily extended to higher order i.e. steeper waves. However the analysis presented herein serves as a prelude, to a study of the three-dimensional problem of a wave train encountering a current at an angle (Johnson, 1947). Computations based on a theory f o r arbitrary angle of encounter will be reported in the f u t u r e .

2. F O R M U L A T I O N

We consider a current-free wave train o f wavelength L ^ , height Ho =^ IÜO and period

T propagating on the surface o f still water of depth do and a horizontal u n i f o r m

wave-free current Uo of depth do. When the wave train meets with the current, interaction, neglecting dissipation, is assumed to take place forming an irrotational, u n i f o r m wave-current field defined by the parameters / / , L , U and d which are unknown a

priori.

The conservation of wave crests is assumed. This assumption implies that the wave period T does not change after the interaction. The schematic configuration of the system is shown i n Fig. 1, in which T denotes the control volume o f fluid bounded by the control surface 5, the free surface and the horizontal b o t t o m . The surface 5 is assumed to be made up of the vertical surfaces 5^^, S^o and 5 w , extending f r o m the

Interaction between waves and currents 'S

bottom to the free surface such that S = S^o + •^co + S^.^. Through S^.^, S^o and 5»,^ fiow the current-free wave, wave-free current and the combined wave-current fields, respectively. The angle 7 in Fig. 1 denotes the relative direction between the current-free wave and the wave-current-free current. The limiting cases 7 = 0° or 7 = 180° correspond to the coUinear wave-current interactions which are considered in this paper. We pose the following two-dimensional problem i.e. the wave train propagates in the same on opposite direction of the current. Given the current-free wave parameters Ho, Lo, do and the wave-free current parameters Uo, do, it is required to find the combined wave-current field parameters H, L, d and U. A solution is presented in Section 5. The physical situation corresponds to the laboratory experiment of Thomas (1981). I t is also easy to imagine in an oceanic context a current-free theoretical wave catching up a wave-free current i n the same or opposite direction to generate a combined wave-current field.

3. PROPERTIES OF T H E W A V E - C U R R E N T F I E L D

The characteristics of the wave-current field are reviewed f o r completeness and f o r reference in the subsequent sections. The wave-current field with parameters L,H, U and d is assumed to be irrotational and u n i f o r m .

Adopting a system of coordinates, with 2 vertically upwards, x horizontally in the direction of the wave propagation and the origin on the undisturbed free surface, with the horizontal bottom at z = -d, the velocity potential $ ( x , z , 0 of the irrotational wave-current field, to first order in wave elevation a, is (cf. Peregrine, 1976):

^{x,z,t) = Ux + <C-U)'^^^^^^\n{Kx-^t) + O(aa^) (1)

where C is the absolute surface wave celerity, K = l-ulL is the wave number and

(J = KC denotes the wave angular frequency such that the Doppler relation

o^KU+a, (2)

is satisfied, with U the mean stream-like velocity, and a, the relative angular frequency given by

(Tr = {Kgt?inhKdY\ (3)

Equation (2) becomes

^ ^ t Z - f f f t a n h / C r f r (4) The periodic free surface elevation, to first order in a, is expressed as

r\ = a cos {Kx-aS) ^ Oid^y. (5)

The X and z components of velocity V = V $ i n the combined field are to 0{Ka} terms given by

4 R . E . B A D D O U R and S . SONG and w -gaK smh K{z+d) ün{Kx-(jt) + 0 ( f l 2 ) . (Jr cosh Kd

The X and z components of particle acceleration are given by:

(7)

(8)

(93 The pressure distribution as needed to second order of approximation. I t is written i n the forme gKa' + pga 2 slnhlKd cosh K{z+d) cosh Kd ^ p ^ ^ ' ^ i h ^ h W [cosh2K(z+d)- 1] cos(Ajc-CTr) cosh 2K(z+d) sinh^ Kd cos 2(Kx-cTt}. (10)

The corresponding mean value of P with respect to time, to second order, is given i n Longuet-Higgins and Stewart (1960). The particle trajectory is given as:

where _ a s i n h A : ( f - f ^ ) sinh Kd fl cosh K(z+d) ^ ~ ' sinhATrf (11) (12) (13) and i t is assumed that the particle initially at {x,z) moves to (x+X, z+Z) i n time t. The mean mass, momentum and energy flux densities of the wave-current field could be expressed to the second order in the wave amplitude a as follows. The subscript wc is used to denote a wave-current field quantity. Using Q^^, M^^ and E^^ to denote respectively the mean rate of mass, momentum and energy transfer across a vertical surface fixed i n space in the wave-current field, we can write f o r a two-dimensional flow (Whitham, 1962): ^ " ' ^ 2 l T 1 {^^ f-^ p dz

de

Interaction between waves and currents

2 ^ 1

dz de (16)

where Q = Kx - at and the free surface is z = -r^(r). The mean value with respect to time is taken after performing the integration in the z direction..

N o w , to order a^, Equations (14), (15) and (16) become: J _ 2 l T J _ 2-rr ^,dz + -n{x,t)^,{x,o,t)

de

+ -n(x,/) lp(x,o,t)

$ . ( x , o , o )

de. (19)

The t e r m of order Ka^ in P lin Equation (10) namely: P gKa^

2 sinh 2Kd [cosh2K{z+d)-l]

has a contribution of order in the values of M^c and E^c- On substituting Equations (1), (4) and (10) into the above equations, the fluxes of the first order wave-current field (1) can be obtained to order a'^ in the f o r m :

{2„,c =^a^K{C-U)cothKd+pdU D fl 2Kd \ o Is = ^ga' ~+

.77,^^

E^^c as expressed by Equation (22) is easily recognized as in Equation (35) of Longuet-Higgins and Stewart (1960). The four terms in Equation (22) above are identically equal to their R^, R2 and R^ respectively. Moreover, E^.^ could easily be written as the following sum:

(23) where:

E^, = E^Cgr + {E^+S:,) U + EyUd + e^U + EuU

^ R . E . BADDOUR. and S . SONG

denotes the mean wave energy density per unit horizontal area i n the absence of stream;

(25) denotes the group velocity (cf. Lamb, 1932) relative to still water;

4Kd

(26) C - d c x . _ l / 2Kd \

^ ' = 4 P ^ ^ M ^ ^ i n h 2 / ^ ^

denotes the radiation stresses defined by Longuet-Higgins and Stewart (1960);

Eu =

\pdLr-denotes the mean current energy density in the absence of the wave;

2

C4

(27)

(28)

(29) denotes the relative wave drift velocity; and

1 ga^

2 a,

with p g aV{2(j,) representing the so-called wave action.

4. C O N S E R V A T I O N E Q U A T I O N S A N D M E T H O D OF S O L U T I O N

From a general point of view in a non-viscous fluid, the mass, momentum and energy conservation relations are respectively given by (cf. Lamb, 1932):

p ( V - n ) d 5 + f , p d T = 0 '5' 01 p F dr - P n d5 - p(V • n) V d5 - - p V dr = 0 is is dt P + p - ^ - + p ^ z ) ( V - n ) d5 = 0 (30) (31) (32) where n denotes the unit outside normal to the fixed control surface S in the fluid domain enclosing a volume T o f fluid, w i t h the z-axis vertically upwards,

V, F and P denote, respectively, the velocity vector, the fluid body force per unit volume and the pressure. For an incompressible, inviscid fluid under gravity and f o r the system considered i n Section 2, the above equations take the f o r m :

p (V • H ) d5 = 0

p F d T - PndS- p ( V - n ) V d S = 0

• JS Js

Interaction between waves and currents 7

(35)

where T and S are as shown in Fig. 1, and F = -g the acceleration due to gravity. The integrals over the control surface S could be split into the sum of integrals over the component surfaces S^^r S^o and S^.^ making up 5 , implying that

+

+

For the Hmit'situation of two-dimensional wave and current encounter the angle 7 tends to zero and equations (33), (34) and (35) give per unh width and in the x direction

Uo dz + Uodz- udz = 0 }-d -d^ [ f , „ + p [ ^ J d 2 --d„ [ P ^ , + pu^-]dz = 0 (36) (37) do -Pco+l^o + PgZ U„dz -d L P^, + ^u- + pgz udz = 0

where TI,, is the current-free wave surface elevation, d^, denotes the water depth of the free and wave-free fields, and and u are the particle velocities in current-free wave, wave-current-free current and combined wave-current fields respectively. Taking the time average these expressions become simply

(39) (40) (41) Q^o + Qco = öwc

where Q^^, M„o and are the mean rate of transfer of mass, momentum and energy across a vertical pl-ane i n the pure wave field in the absence of current before interaction, and Qco, M,o and E^o are the corresponding mean values f o r the wave-free current field. They are given to order al, by:

Q.o=\pal {gKoCOthKodoY'^ M 1 2^0 I. , 2 K o d o sinh 2 Kodo (42) (43) (44)

R . E . B A D D O U R and S . SONG C o . l - o r d o , H o Known p a r a m e t e r s O U T W a v e - c u r r e n t field U , L , H , d Unknown p a r a m e t e r s Curreni field

Control surface S : vertïcol cylinder t>ounded by free s u r f a c e ft tiorijontal solid bottom

Uo.do

KnDv^n par£imet,ers

FIG. L System configuration defininj controli volume T and control surface S,

Qco = P doUo

E,o = ^pdoUl

(45) (46) (47) where the subscript o is used to denote the value of a parameter before the

interaction-The quantities M^.^ and E^^ were presented in Section 3 above and are expressed

by Equations (20), (21) and (22). The quantities in Equations (39), (40) and (41) use the horizontal bed as their datum.

The left-hand sides of Equations (39), (40) and (41) include the sum of the fluxes o f the current-free wave and wave-free current which are assumed to flow through S and respectively. The flux into the control volume is hence considered as the sum of current-free wave flux and wave-free current flux. However, the right-hand side o f the above mentioned equations deal with the fluxes of the combined wave-current field leaving the control surface. The fluxes of momentum and energy in such a combined field should not and are not calculated by summing the fluxes induced by the current-like term component and wave-current-like component making up the combined field flowing

through S^,c, ö^vc, MH-C and E^.^ of the wave-current field are calculated f r o m expressions

(17), (18) and (19) leading to (20), (21) and (22) respectively. The parameters H, L,

d and U are released and unknown, a priori.

Now, the following non-dimensional variables are introduced:

A = alldè; B = UJQ; D = LJdo; W^did-(48) (49) (50) (51)

Interaction between waves and currents and Y^iLILoY'^; Z = a^ldl ( 5 2 ) ( 5 3 ) ( 5 4 )

I n terms of the above variables, the dispersion relation ( 4 ) and the conservation relations (39), ( 4 0 ) and ( 4 1 ) become:

T ~ = X + Y [coth {l-nlD) tanh {ITXWIDY^-)]

BD + TTA coth ( 2 7 T / D ) = DWX + ^ [coth (2TT/Z)) coth {2TrW/DY^)Y'^-,

l + A 1 2Tr 2 ^ D 1

\

V

= - WXHanh^ + 3Zr~^ tanh ^ coth ^

2^1 ^2TTW'V^2 2TTiy : y coth — t a n h - ^ 1 Ï + . , 2Triy , 2TTty s m h c o s h + ZX\ 3 + 4TrW DY^ . . 2TTW . 2TTW sinh -^TTTT cosh DY^ DY^i ( 5 8 )

The set of Equations ( 5 5 ) , ( 5 6 ) , ( 5 7 ) and ( 5 8 ) establishes the relationship between the current-free wave train and wave-free current parameters before the interaction and the wave-current field parameters. Given the current-free wave height Ho, length Lo, wave-free current speed Uo and mean water depth do, the variables W, X, Y and

10 R . E . B A D D O U R and S . SONG

Z are obtained by solving numerically the above algebraic equations. The wave height, length, current speed and mean water depth of the combined field are then obtained as: H = 2DoZy^ (59) l' = LoY^ (60) U=CoX (61) and d^doW. (62)

In Equation (4), or in its non-dimensional equivalent Equation (55), the positive root is adopted. However, the value of t/„ in Equation (49) is taken as positive when considering the interaction o f a wave with a following current, and negative when considering an opposing current. I n this fashion, all possible distinct solutions as described, f o r example, in Peregrine (1976) could be taken into account. Once the values of the wave-current field parameters (59)-(62) are obtained, one of the f o u r solutions could then be identified a posteriori,, according to the following classification:

(i) t / > 0 (63) (ii) -Cg,<U<0 (64J (iii) - C < f / < - C , . (65)

(iv) U<-C, (66) where Cg, and C are given by Equations (25) and (3), respectively.

Cases (i) and (ii) are of engineering interest. Cases ( i i ) , (iiii) or (iv) are possible solutions when waves interact with an opposing current. Discussion of their physical interpretation is given in Longuet-Higgins and Stewart (1960) or Sarpkaya and Isaacson (1981). I t must be noted, however, that i n an opposing current the assumption of small amphtude Hnear wave theory implied in Equation (1) and in the present model could fail for relatively small negative ratios of UJC^, due to the increase in this case of the predicted height H o f the surface disturbance of the wave-current field.

5. C O M P U T A T I O N A L C O N S I D E R A T I O N S A N D N U M E R I C A L RESULTS The set of nonlinear Equations (55)-(58) is solved numerically f o r W, X, Y and Z , using a Newton technique. I t is worth noting the convergence and stability behavior o f the iteration. For given wave data a,, L „ , 4 and current velocity the solution is found f o r any set of arbitrary initial guess values of a, L, d and U. The wave before interaction should only satisfy the small amplitude wave theory criterion. I t seems that general sufficient conditions f o r the uniqueness of the solution of the nonlinear system of Equations (55)-(58), f o r a given ratio UJC^, are not available. However, the range of values chosen f o r the parameters in this study produced the only solutions presented herein. I n the present solution, a negative value f o r represents a current flowing i n the opposite direction to the wave propagation. The case when both the wave speed and the wave energy are swept upstream, but at a slower rate than i n the absence o f

Interaction between waves and currents 11 1.20 h 1.00 o.soh oteoh a4ti!. _{- 0 . 2 0 -OJ15 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0} J L

F I G . 3. Wave lieiglit ratio HIH„ vs current speed ratio UJC,, for plane wave parameters a„ = 0J5 m ,

L„ = 100 m, d„ = 25 m.

the current, fs of engineering importance and refers to case ( i i ) above. Case ( i i ) or (iii) or (iv) is identified a posteriori, once the post-interaction wave-current parameters are calculated.

A s an example, the plane current-free wave defined by the parameters = 0.75 m and Lo = 100 m , on still water of depth i i ^ = 25.0 m , is considered to interact with u n i f o r m streams of current ratios UoICo varying over the range of values between - 0 . 2 0 and +0.20. The wave length, wave height, stream current and water depth after the

12 R . E . B A D D O U R and S . S O N G X 10" 1.00 < Ü - 1 5 0 - 2 . 0 0 1 1 1 1 1 1 - 0 . 1 5 -0.10 - 0 . 0 5 0 0 0 0 . 0 5 0.10 0.15 0 . 2 0 U o / C o

F I G . 4. Current change ratio {U- U„)/C„ vs current speed ratio UJC.„ for plane wave parameters a = 0 75

L,. = m m, d,, = 25 m.

-0.15 -0.10 - 0 . 0 5 O O O 0 0 5 0.10 0,15 0 , 2 0 U o / C o

F I G . 5. Depth change ratio {d-d„)ld„ vs current speed ratio UJC„ for plane wave parameters a

L„ = 100 mi, d„ = 25 m. = 0.75 m„

interaction are found on solving Equations (55)-(58). The wave length ratio L / L ^ , wave height ratio ƒ / / / / „ , current change ratio ^UICo and depth change ratio ^dldo, where

MJ = U-Uo denotes the variation in U and Ad = d-d„ denotes the variation in depth,

are easily evaluated. These quantities are plotted respectively, in Figs 2-5 against UJ

Interaction between waves and currents 13 As expected, the length of the disturbance on the surface of the water decreases when the wave train encounters an opposing current, while increasing when the current

Uo is in the same direction (Fig. 2). Also, not surprisingly, a reverse behavior is noticed

in the height of the wave (Fig. 3 ) , increasing in magnitude with an opposing current and decreasing when meeting with a following current. Qualitatively, these results compare favorably with a number of other researchers' findings, f o r example Longuet-Higgins and Stewart (1960) and Jonsson, Skougaard and Wang (1970). Quantitatively, it would be difficult to make any comparison with other results, since the present approach is a departure f r o m other investigations in that it does not include a k n o w n ,

a priori, upwelling or horizontal inflow f r o m the sides, nor does it assume a given

change in the bottom configuration, however small. Moreover, in the above-mentioned references the subscript o is used to denote the value o f a parameter f o r the infinite depth case. However, a comparison with the results of Thomas (1981) is presented in Section 6 below.

The change in the mean value of the current is presented in Fig. 4 as (U-Uo)/Co. I t shows an increase in the magnitude of the average stream velocity U over Uo when the wave and current propagate in the same direction, and a decrease otherwise. The relative change i n the mean water depth due to the wave-current interaction is given in Fig. 5 as {d-do)ldo. This figure shows a decrease in the mean value of the wave-current field depth d relative to its wave-free or current-free value do, when the wave and current propagate in the same direction. A n increase is noted otherwise. Both changes are relatively small quantities and depend on the velocity ratio f/o/C^.

It is noted that the solution set of the nonhnear system developed here is found to be independent of the initial guess used i n conjunction with the Newton iteration technique and the considered range of values of Uo/Co- However, there is no point in graphing the relations beyond the value of Uo/Co = - 0 . 1 6 since the small amplitude assumption will fail then, due to the increase in wave height and decrease in wave length. The method is easily extended to deal with steeper waves. Accordingly, higher order terms in the corresponding expansions will have to be kept. A negative value given to the ratio Uo/Co represents a current in the direction opposhe to that of wave propagation. For the considered range of values -0A6<Uo/Co< 0 it is easy to find that the corresponding range of U is such that -Cgr<U<0, with Cgr given by Equation (25).

, Numerical values f o r the ratios L/Lo, H/Ho, AUICo, and Ad/do are given for a range of values of the ratio of the wave-free current and the current-free wave celerity, Uo/

C^. These are presented in the Appendix. The different values of the ratio of the

current-free wave length to water depth Lo/do and the current-free wave steepness Ho/

Lo, in Tables A 1 - A 4 , ensure that the theory of linear waves over finite depths is valid

in this case ( L e M é h a u t é , 1976).

Tables A l and A 2 show respectively the wave height and wave length ratios. Table A3 shows the current variation ratio (U-Uo)/Co, while Table A 4 displays the depth variation ratio {d-do)/do. I t is found that the changes in the mean depth and the mean current are less pronounced in comparison to the changes i n wave height and wave length. Table A 3 shows that the mean value of the current term in the combined field increases in a following wave and decreases when flowing in an opposite direction to the wave propagation. Table A 4 , on the other hand, shows that the depth of the wave-current field also changes slightly f r o m its wave-free or current-free value do.

14 R . E . B A D D O U R and S . SONG 2 . 2 0 1.80 i.-io 1.00 o Thomas (19811 experiment Present Itieory J 1 I L - 0 . 1 2 - 0 . 1 0 - 0 . 0 8 - 0 . 0 6 - 0 . 0 4 - 0 . 0 2 U o / C o Wave Height c h a n g e s - 0 . 0 0

F I G . 6V The predicted and experimentally obtained values of the non-dimensional wave amplitude a/a,, plotted against the current ratio U,JC„ where C„ = 1.8014 m/sec.

1.00 0 . 8 0 0 . 6 0 OAQ o — O T h o m a s ( 1 9 8 1 ) experiment — P r e s e n t theory 1 1 1 1 1 - 0 . 1 2 - 0 . 1 0 - 0 . 0 8 - 0 . 0 6 - 0 . 0 4 - 0 . 0 2 - 0 . 0 0 U Q / C O Wove Lengilh c h a n g e s

F I G . 7. T h e predicted and experimentally obtained values of the non-dimensional! wavelength L/L„ plotted against the current ratio UJC,„ where C„ = 1.8014 m/sec.

T A B L E 1. T H E MEASURED A N D PREDICTED V A L U E S OF T H E NON-DIMENSIONAL WAVE AMPLITUDE a/a

Predicted Measured Predicted

U„ (mm/sec) \(Jjc„\

Thomas (1981) Thomas (1981) present method

U„ (mm/sec) \(Jjc„\ a/a,, ala„ a/a„ Difference (%)

59.7 0.0332 1.067 1.085 1.068 1.57

116.2 0.0645 1.156 1.156 1.147 0.78

i59.8 0.0888 1.242 1.267 1.222 3.55

203.0 0.1127 1.315 1.309 1.315 0.46

The cxperimenlal and predicted data of the third and fourth column arc taken from Thomas (I98I). The predicted values in the fifth column are derived using the method developed in Section 3 of the present paper. Percentage difference between measured and predicted values arc also given. C „ = 1.8104 m/sec.

The values in the four tables are within the pre-assigned convergence criterion chosen in conjunction with the numerical iteration used, and within the precision kept f o r a l l variables. The accuracy of the results is checked in an ad hoc fashion, by running the computer program several times using different precisions in the computer calculations. Double-double precision, i.e. typically 3 3 decimal digits, is the degree of precision used for all variables i n the calculations

Interaction between waves and currents 15 (a) 0 . 0 0 (c) •o.zo\ - 0 . 4 0 - 0 . 6 0 - 0 . 8 0 -1.00 — P r e s e n t Sheory o G.P. T h o m o s e x p e r i -- m e n t a l results ( i 9 8 H . (b) 0 . 0 0 - 0 . 2 0 - 0 . 4 0 - 0 . 6 0 - 0 . 8 0 10.00 3 0 . 0 0 5 0 . 0 0 7 0 . 0 0 9 0 . 0 0 u>„(r) Ua = - 5 9 . 7 mm/s -too — P r e s e n t theory 0 G.P. T h o m a s e x p e r J -• - m e n t a l results (1981). I lO.OO 3 0 . 0 0 5 0 . 0 0 7 0 . 0 0 9 0 . 0 0 ~ „ ( z ) Uo ^ - 1 1 6 . 2 m m / s 0 . 0 0 - 0 . 2 0 h - 0 . 4 0 - 0 . 6 0 - 0 . 8 0 - 1 . 0 0 — P r e s e n t theory o G.P. T h o m a s e x p e r i --mentol results (1981). J I I 0 . 0 0 - 0 . 2 0 - 0 . 4 0 - 0 . 6 0 - 0 . 8 0 - 1 . 0 0 ' — P r e s e n t theory o G.P. T h o m a s e x p e r i -- m e n t a l results (J98I). 10.00 3 0 . 0 0 5 0 . 0 0 7 0 . 0 0 9 0 . 0 0 7„(2) Uo = - 1 5 9 . 8 mm/s 10.00 3 0 . 0 0 5 0 . 0 0 7 0 . 0 0 9 0 . 0 0 Uo = —203.0 m m / s

F I G . 8. Comparison between present theory and experiment of Thomas (1981) for the amplitude « ( 2 ) of the wave-like horizontal particle velocity under a regular wave train interacting with a constant <:urrent V„. All

velocities are in mm/sec.

6, C O M P A R I S O N W I T H T H O M A S ' (1981) RESULTS

The interaction between a regular wave train and an adverse current in two dimensions is studied experimentally and numerically by Thomas (1981)- The constant current provided by Thomas has a value equal to the depth-averaged current profile. The experimental data are taken f r o m Table 1 in Thomas (1981). The measured values,, and those predicted by Thomas, of the non-dimensional wave amplitude and wavelength are given in Table 2 of Thomas (1981)^

I n the £ r e s e n t notation Uo, and Lo correspond to Thomas' values of the free-wave currents U, the current-free wave amplitude Oo and wavelength X,,, respectively. N o w ,

R . E . B A D D O U R and S . SONG

T A B L E 2. T H E MEASURED A N D PREDICTED VALUES OF T H E NON-DIMENSIONAL WAVE LENGTH LIL.,

U„ (mm/sec) \UJC,\

Predicted Tliomas (1981) LIL„ Measured Thomas (1981) LIL,. Predicted present method UL,. Difference (%) 59.7 116.2 159.8 203.0 0.0332 0.0645 0.0888 0.1127 0.946 0.893 0.849 0.804 0.954 0.894 0.844 0.810 0.940 0.882 0.835 0.786 1.47 1.34 1.07 2.96

The predicted and experimental results of columns 3 and 4, respectively, are ilakcn from Thomas (1981). The predicted values m column 5 are derived using the method developed in Section 3 of the present paper. Percentage difference between measured and predicted values are also gfven. C„ = 1.8014 m/sec.

T A B L E 3. P R E D I C T E D V A L U E S OF Ad/d,. A N D MJ/C,„ USING T H E PRESENT M E T H O D , WHERE = d-d A N D AU U - U„

U„ (mm/sec) _{Adld., A C / / C , ,}

59.7 116.2 159.8 203.0 0.0332 0.0645 0.0888 0.1127 0.040 X 10-* 0.111 X 10-* 0.203 X 10-" 0.342 X I O - * -0.386 X 10-* -0.877 X I O - " -1.395 X I O - " -2.095 X 10-"

for a wave of amplitude - HJl = 0.00918 m , wave length L„ = 2.261 m and water depth (/^_= 0.57 m , the changes ala^ and LIL^ are predicted f o r the different values o f

Vo, (or V in Thomas' notation), using the method described in Section 4 above. The

comparison of the results with the experimental data o f Thomas is given i n Tables 1 and 2, and i n Figs 6 and 7.

Tables 1 and 2 show that the maximum difference between measured and predicted values is of the order o f 3.55% f o r the wave amphtude and 2.96% f o r the wave length. It must be noted that this difference is not uniform over the values tested. These results could be considered surprisingly accurate.

The present method also predicts the changes in the current V and the water depth d after the interaction between current and waves. These are presented in non-dimensional f o r m i n Table 3, f o r completeness. These changes in the current V and the water depth d are neglected by Thomas (1981) and other authors.

Table 3 shows that the method presented in this paper predicts an increase in the water depth when waves interact with an adverse current over a horizontal bed. The maximum increase Ac/ is estimated as 0.0019 cm f o r a water depth of 57 cm, and a ratio

UoICo = -0.1127. I t also predicts a decrease in the mean current velocity. The

maximum decrease A t / is given i n Table 3 as being equal to 0.38 mm/sec, f o r a velocity ratio VJCo = - 0 . 1 1 2 7 .

The measured amplitudes o f the wave-hke horizontal particle velocity (Thomas, 1981) in a wave-current field and the corresponding theoretical predictions given by ü{z), the amplitude o f the periodic part of u{z) i n Equation (6), are presented i n Figs 8a-d, f o r a sequence of increasing current U^. I n each of the figures the experimental results o f Thomas are shown by a circle, and a predicted profile using the present approach by a solid line. The predicted profile of Thomas (1981) is not included. Good agreement

Interaction between waves and currents 17

could be seen between the measured and predicted velocities. I t must be noted that the method presented in Section 4 could be easily extended to include a distribution of vorticity or higher order waves if necessary, and will be presented elsewhere.

7. C O N C L U S I O N

The interaction of a following or opposing uniform current and surface gravity waves on water of finite depth on a horizontal bed are discussed. The resulting u n i f o r m , stable wave-current field is described by four parameters H, L, d and V. The values of these parameters are computed on solving a set of f o u r nonlinear equations obtained on satisfying the dispersion relation on the free surface and the conservation of mean mass, momentum and energy transfer to second order, before and after interaction. Wave-current interaction terms are taken into account. A p a r t f r o m the usual expected changes in the wave height and wave length, i t is f o u n d that the mean value of the current and mean water depth also undergo changes which are computed in some numerical examples. Qualitatively, the changes in wave height and wave length compare positively with previous findings which neglect the changes i n the mean current and mean depth due to the interaction. The method is easily extended to higher order.

Acknowledgement—This research was supported by the Natural Sciences and Engineering Research Council

of Canada.

REFERENCES

B R E T H E R O N , F . P . and G A R R E T T , C . J . R . 1968. Wave trains in homogeneous moving media. Proc. R. Soc.

A 3 0 2 , 5 2 9 - 5 5 4 .

C R A I K , A . D . D . 1 9 8 5 . Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge. JoH.NsoN, J . W . 1947. The refraction of surface waves by currents. Trans Am. geophys. Un. 2 8 , 8 6 7 - 8 7 4 . JONSSON, I . G . 1 9 7 8 . Energy fiux and wave action in gravity waves propagation on a current. J. Hydraul.

Res. 1 6 , 2 2 3 - 2 3 4 .

JONSSON, I . G . , B R I N K - K J A E R , O . and THOMAS, G . P . 1978. Wave action and set down for waves on a shear current. J. Fluid Mech. 8 7 , 4 0 1 - 4 1 6 .

JONSSON, I . G . , SKOUGAARD, C . and W A N G , J . 1970. Interaction between waves and currents. Proceedings of

the 12th Conference on Coastal Engineering, Washington, D.C-, pp. 4 8 9 - 5 0 8 . American Society of Civil

Engineers, New York.

L A M B , H . 1 9 3 2 . Hydrodynamics. Cambridge University Press, Cambridge.

L E M É H A U T É , B . 1976. An Introduction to Hydrodynamics and Water Waves: Springer, Berliirr.

L O N G U E T - H I G G I N S , M . S . and STEWART, R . W . 1960. Changes in the form of short gravity waves on long waves and tidal currents. / . Fluid Mech. 8 , 5 6 5 - 5 8 3 .

L O N G U E T - H I G G I N S , M . S . and STEWART, R . W . 1961. The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 5 2 9 - 5 4 9 .

L O N G U E T - H I G G I N S , M . S . and STEWART, R . W . 1964. Radiation stress in water waves: a physical discussion,. with applications. Deep Sea Res. 1 1 , 5 2 9 - 5 4 9 .

P E R E G R I N E , D . H . 1976. Interaction of water waves and currents. Adv. appl. Mech. 1 6 , 9 - 1 1 7 .

P E R E G R I N E , D . H . and JONSSON, I . G . 1 9 8 3 . Interaction of waves and currents. U . S . Army Corps of Engineers Coastal Engineering Research Center, Report No. 8 3 - 6 .

SARPKAYA, T . and ISAACSON, M , 1 9 8 1 . Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, New York.

THOMAS, G . P . 1981. Wave-current interactions: an experimental! and numerical study. Part 1. Linear waves. ƒ. Fluid Mech. 1 1 0 , 4 5 7 - 4 7 4 .

U N N A , P . J . 1 9 4 2 . Waves and tidal streams. Nature, Lond. 1 4 9 , 2 1 9 - 2 2 0 .

WHITHAM^ G . B . 1 9 6 2 . Mass, momentum and energy flux in water waves. J. Fluid Mech. 1 2 , 1 3 5 - 1 4 7 .

A P P E N D I X

Numerical values for the ratios LIL^, HIHg, LUICo and Adldg are given for a range of values öf the ratio UJCg. The results are presented in Tables A 1 - A 4 respectively (see discussion in Section 5 above).

T A B L E A l . V A L U E S O F WAVE HEIGHT RATIO HIH„ UJC,. -0-15 "0.10 - 0.05 0.00 0.05 O.IO 0.15 24 0.00053 0.00091 0.00129 0.00149 1.28395 1.26565 1.26112 1.26003 1.15954 1.15464 1.15346 1.15318 1.07029 1.06973 1.06960 1.06958 1.00000 1.00000 1.00000 1.00000 0.93862 0.93911 0.93923 0.93925 0.87914 0.88293 0.88386 0.88409 0.81604 0.82847 0.83154 0.83229 0.74424 0,7.7335 0,78044 0.78220 16 0.00079 0.00137 0.00194 0.00224 1.29890 1.28038 1.27581 1.27472 1.16692 1.16199 1.16080 1.16052 1.07319 1.07263 1.07251 1.07248 1.00(K)0 1.00000 1.00000 1.00000 0.93661 0.93710 0,93721 0.93724 0.87568 0.87946 0.88039 0.88061 0.81152 0.82389 0.82694 0.82770 0,73893 0.76789 0.77495 0 77669 10 0.00127 0.00219 0.00310 0.00358 1.34050 1.32144 1.31678 1,31568 1.18680 1.18182 1.18062 1.18035 1.08086 1.08030 1.08018 1.08016 1.00000 1.00000 1.00000 l.flOOOO 0.93143 0.93191 0.93202 0,93204 0.86683 0.87056 0.87148 0.87169 0,8(H)02 0.81222 0.81522 0.81596 0J2550 0.75403 0.76097 0.76269 4 0.00317 0.00548 0,00775 0,00895 1.58871 1.56829 1.56423 1.56376 1.29008 1.28521 1.28426 K28415 1.11808 1.11758 1.11753 1.11755 1.0(X)00 1.00000 1.00000 1.00000 0.90775 0.90817 0.90825 0.90825 0.82710 0.83038 0.83116 0.83133 0.74926 0.75992 0.76252 0.76314 0.66740 0.69226 0.69829 0.69976

T A B L E A 2 . V A L U E S OF WAVE LENGTH RATIO LIL„ 24 16 10 4 H J L „ - 0 . 1 5 - 0 . 1 0 - 0 , 0 5 0 . 0 0 0 5 3 0 . 8 4 5 6 1 0 . 8 9 7 3 3 0 . 9 4 8 7 7 0 . 0 0 0 9 1 0 . 8 4 5 5 7 0 . 8 9 7 3 1 0 . 9 4 8 7 6 0 . 0 0 1 2 9 0 , 8 4 5 5 1 0 . 8 9 7 2 8 0 . 9 4 8 7 5 0 . 0 0 1 4 9 0 . 8 4 5 4 7 0 . 8 9 7 2 6 0 . 9 4 8 7 4 0 . 0 0 0 7 9 0 . 8 4 0 1 8 0 . 8 9 4 0 4 0 . 9 4 7 2 7 0 . 0 0 1 3 7 0 . 8 4 0 1 4 0 . 8 9 4 0 2 0 , 9 4 7 2 6 0 . 0 0 1 9 4 0 . 8 4 0 0 8 0 . 8 9 3 9 9 0 . 9 4 7 2 5 0 . 0 0 2 2 4 0 . 8 4 0 0 4 0 . 8 9 3 9 6 0 . 9 4 7 2 4 0 . 0 0 1 2 7 0 . 8 2 5 2 7 0 . 8 8 5 1 1 0 . 9 4 3 2 1 0 . 0 0 2 1 9 0 . 8 2 5 2 2 0 . 8 8 5 0 9 0 . 9 4 3 2 0 0 . 0 0 3 1 0 0 . 8 2 5 1 4 0 . 8 8 5 0 5 0 . 9 4 3 1 9 0 . 0 0 3 5 8 0 . 8 2 5 0 8 0 . 8 8 5 0 2 0 . 9 4 3 1 8 0 . 0 0 3 1 7 0 . 7 3 0 2 3 0 . 8 2 9 8 4 0 . 9 1 8 5 3 0 . 0 0 5 4 8 0 . 7 2 9 9 7 0 . 8 2 9 7 4 0 . 9 1 8 5 0 0 . 0 0 7 7 5 0 . 7 2 9 5 7 0 . 8 2 9 5 9 0 . 9 1 8 4 5 0 . 0 0 8 9 5 0 . 7 2 9 3 0 0 . 8 2 9 4 9 0 . 9 1 8 4 1 UJC,, 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 I.OOOOO 1 . 0 5 1 0 5 1, 10197 1 . 1 5 2 7 6 1 . 2 0 3 4 6 1 . 0 0 0 0 0 1 . 0 5 1 0 6 1, 1 0 1 9 8 1.15277 1.20348 1.00000 1.05107 1 .10199 1 . 1 5 2 8 0 1 . 2 0 3 5 0 1,00000 1 , 0 5 1 0 8 1, 10200 1.15281 1 . 2 0 3 5 2 1 . 0 0 0 0 0 1 . 0 5 2 3 4 1 . 1 0 4 3 6 1.15611 1 . 2 0 7 6 6 1 , 0 0 0 0 0 1 . 0 5 2 3 5 1, .1 0 4 3 7 1 . 1 5 6 1 3 1.20767 1 . 0 0 0 0 0 1 . 0 5 2 3 5 1 , 1 0 4 3 9 1.15616 1.20770 1 . 0 0 0 0 0 1 . 0 5 2 3 6 1 ,10440 1 . 1 5 6 1 7 1 . 2 0 7 7 2 1 . 0 0 0 0 0 1.05577 1 .11072 1 . 1 6 5 0 2 1 . 2 1 8 7 9 1 . 0 0 0 0 0 1 . 0 5 5 7 7 1 . 1 1 0 7 4 1 . 1 6 5 0 4 1.21881 1 . 0 0 0 0 0 1 . 0 5 5 7 9 1 . 1 1 0 7 6 1.16507 1 . 2 1 8 8 4 1 . 0 0 0 0 0 1 . 0 5 5 7 9 1 . 1 1 0 7 7 L 1 6 5 0 9 1.21886 1 . 0 0 0 0 0 1.07631 1 .14874 1 . 2 1 8 1 5 1.28514 1,00000 1.07633 1 . 1 4 8 7 7 1 . 2 1 8 1 9 1 . 2 8 5 1 9 1.00000 1 . 0 7 6 3 6 1 . 1 4 8 8 2 1.21825 1 . 2 8 5 2 6 1 . 0 0 0 0 0 1 . 0 7 6 3 8 1 . 1 4 8 8 5 1 . 2 1 8 2 9 1.28531

T A B L E A 3 . V A L U E S O F C U R R E N T crtANCE RATIO {U-U,)IC„ X lO" UJC,, H„IL,f -0.15 =0.10 -0.05 0.00 0.05 0.10 0.15 0.20 24 0.00053 0.00091 0,00129 0,00149 -0.15298 -0.42401 -0.83093 -1.10246 -0.07698 -0.22300 -0.44215 -0.588235 -0.03110 -0.09253 -0.18471 -0.24619 0,00000 0,00000 0.00000 0,00000 0,02382 0.07093 0.14162 0.18876 0.04422 0.12917 0.25662 0,34161 0.06318 0.17979 0.35474 0.47139 0.08152 0.22567 0.44192 0.58610 16 0.00079 0.00137 0.00194 0.00224 -0.16637 -0.46271 -0.90765 -1.20457 "0.08330 -0.24169 -0.47942 -0.63800 -0.03355 0.09986 -0.19936 -0.26573 0.00000 0.00000 0.00000 0.00000 0.02557 0.07617 0.15209 0.20272 0.04736 0.13850 0.27523 0.36641 0.06753 0.19256 0.38015 0.50523 0,08702 0.24156 0.47340 0.62799 10 0.00127 0.00219 0.00310 0,00358 -0.20756 - 0 5 8 1 9 2 -1,14412 -1.51928 -0,10203 -0,29711 -0,58995 -0.78533 -0.04064 -0.12105 -0.24173 -0.32222 0.00000 0.00000 0.00000 0.00000 0.03047 0.09083 0.18140 0.24180 0.05604 0.16430 0.32673 0.43504 0.07942 0.22756 0.44980 0,59799 0.10189 0.28470 0.55895 0.74182 4 0,00317 0.00548 0.00775 000895 -0,65398 '1,88354 -3.73721 -4.97927 -0.27335 -0.80579 -1.60619 -2.14097 -0.09986 -0.29840 -0.59657 -0,79559 0.00000 0.00000 0.00000 0.00000 0.06749 0.20173 0.40320 0.53759 0.11932 0.35291 0.70346 0.93725 0.16332 0.47582 0,94473 1.2.5744 0.20351 0,58241 1.15092 1.53003

T A B L E A 4 . V A L U E S OF DEPTH CHANGE RATIO {d-d,)ld„ x 10' UJC. LJd.. 2 4 1 6 1 0 HJU, - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 0 0 0 5 3 0 . 1 2 6 0 2 0 . 0 5 8 1 1 0 . 0 2 1 3 3 0 . 0 0 0 0 0 - 0 . 0 1 3 9 6 - 0 . 0 2 5 1 9 - 0 . 0 3 6 6 5 - 0 . 0 5 0 3 7 0 . 0 0 0 9 1 0 . 3 3 6 8 1 0 . 1 6 4 3 7 0 . 0 6 2 9 4 0 . 0 0 0 0 0 - 0 . 0 4 1 0 6 - 0 . 0 6 9 6 4 - 0 . 0 9 1 5 3 - 0 . 1 1 0 5 5 0 . 0 0 1 2 9 0 . 6 5 3 3 0 0 . 3 2 3 8 7 0 . 1 2 5 3 9 0 . 0 0 0 0 0 - 0 . 0 8 1 7 3 - 0 . 1 3 6 3 3 - 0 . 1 7 3 8 7 - 0 . 2 0 0 8 4 0 . 0 0 1 4 9 0 . 8 6 4 4 9 0 . 4 3 0 2 8 0 . 1 6 7 0 5 0 . 0 0 0 0 0 - 0 . 1 0 8 8 4 - 0 . 1 8 0 8 0 = 0 . 2 2 8 7 8 - 0 . 2 6 1 0 4 0 . 0 0 0 7 9 0 . 1 2 3 3 9 0 . 0 5 6 1 0 0 . 0 2 0 3 2 0 . 0 0 0 0 0 ^ 0 . 0 1 2 9 8 - 0 . 0 2 3 2 1 - 0 . 0 3 3 5 9 - 0 . 0 4 6 1 1 0 . 0 0 1 3 7 0 . 3 3 0 0 7 0 . 1 5 8 6 7 0 . 0 5 9 9 6 0 . 0 0 0 0 0 - 0 . 0 3 8 1 4 - 0 . 0 6 3 9 0 - 0 . 0 8 3 0 4 - 0 . 0 9 9 3 1 0 . 0 0 1 9 4 0 . 6 4 0 3 9 0 . 3 1 2 6 4 0 . 1 1 9 4 4 0 . 0 0 0 0 0 - 0 . 0 7 5 8 9 - 0 . 1 2 4 9 6 - 0 . 1 5 7 2 3 - 0 . 1 7 9 1 2 0 . 0 0 2 2 4 0 . 8 4 7 4 8 0 . 4 1 5 3 5 0 . 1 5 9 1 1 0 . 0 0 0 0 0 - 0 . 1 0 1 0 7 - 0 . 1 6 5 6 7 - 0 . 2 0 6 7 0 - 0 . 2 3 2 3 3 0 . 0 0 1 2 7 0 . 1 1 7 0 1 0 . 0 5 1 0 3 0 . 0 1 7 7 7 0 . 0 0 0 0 0 - 0 . 0 1 0 5 1 - 0 . 0 1 8 2 6 - 0 . 0 2 6 0 0 - 0 . 0 3 5 6 0 0 . 0 0 2 1 9 0 . 3 1 3 7 4 0 . 1 4 4 2 6 0 . 0 5 2 3 9 0 . 0 0 0 0 0 - 0 . 0 3 0 8 1 - 0 . 0 4 9 5 9 - 0 . 0 6 1 9 7 - 0 . 0 7 1 5 2 0 . 0 0 3 1 0 0 . 6 0 9 2 0 0 . 2 8 4 2 1 0 . 1 0 4 3 5 0 . 0 0 0 0 0 - 0 . 0 6 1 2 7 - 0 . 0 9 6 6 1 - 0 . 1 1 5 9 3 - 0 . 1 2 5 4 1 0 . 0 0 3 5 8 0 . 8 0 6 4 2 0 . 3 7 7 5 9 0 . 1 3 9 0 1 0 . 0 0 0 0 0 - 0 . 0 8 1 5 9 - 0 . 1 2 7 9 7 - 0 . 1 5 1 9 2 - 0 . 1 6 1 3 4 0 . 0 0 3 1 7 0 . 0 9 8 0 5 0 . 0 2 9 4 0 0 . 0 0 6 6 0 0.0(X)00 - 0 . 0 0 0 0 9 - 0 . 0 0 2 8 0 - 0 . 0 0 6 0 0 - 0 . 0 0 8 4 2 0 . 0 0 5 4 8 0 , 2 6 9 5 2 0 . 0 8 2 9 7 0 . 0 1 9 2 8 0 . 0 0 0 0 0 - 0 . 0 0 0 6 8 - 0 . 0 1 1 2 5 - 0 . 0 2 6 7 7 - 0 . 0 4 4 5 5 0 . 0 0 7 7 5 0 . 5 2 8 0 7 0 . 1 6 3 5 1 0 . 0 3 8 3 3 0 . 0 0 0 0 0 - 0 . 0 0 1 5 6 - 0 . 0 2 3 9 3 - 0 . 0 5 7 9 5 - 0 . 0 9 8 7 7 0 , 0 0 8 9 5 0 . 7 0 1 3 4 0 , 2 1 7 3 3 0 . 0 5 1 0 5 0 . 0 0 0 0 0 - 0 . 0 0 2 1 5 - 0 . 0 3 2 3 8 - 0 . 0 7 8 7 5 « 0 , 1 3 4 9 3