APPROXIMATE ESTIMATIONS OF CORRELATION COEFFICIENT BETWEEN WAVE HEIGHT AND PERIOD OF SHALLOW WATER WIND WAVES

From the fact that the marginal frequency distributions for wave height and period of complex sea waves both follow the Rayleigh type distribution and approximately exists a linear relationship between wave height and period, Bretschneider(l959) derived wave height and energy spectra of wave period, introducing the summation function of wave height. Then he estimated the correlation coefficient r between height and period of waves as a function of non-dimensional fetch f (= gF/U2). However, his estimation seems not theoretical but empirical, being derived mainly from qualitative considerations and observed data. In this paper, the author tries to derive theoretically the equation for r as a function of f, assuming the classical energy equation for significant wave is applicable to the individual wave in complex sea. Moreover, extending the same method, he intends to estimate the coefficient for shallow water waves as a function of f and non-dimensional depth d (=*gQ/U'z). As the results, coefficient r for deep water waves consists fairly well with that o£ Bretschneider and comparing with the author's observed data, the one for shallow water waves seems to be reasonable.


H/ 3 ~U0H, 7^« T/)-+a6or o)
After Bretschneider's fetch graph!. 1958) for deep water waves, significant wave height and period are approximately expressed by the following equations in terms of non-dimensional fetch f in the range of 1 <f < 2X10* .
where U is wind velocity, g the acceleration of gravity, F the fetch length.
Therefore, when the coefficient r is given by f, wave spectra for deep water waves are fully determined by (1) in terms of f through O) U) and ( 5).
As for r, Bretschneider assumed as follows!li) r= 0 for the upper limit of f , Ui) r = 1.0 for the lower limit of t, (iii) r decreases gradually from the lower limit to the upper limit of f, And moreover, from observed data of r for f ==10^ 10 3 , he empirically estimated r as a function of f.His estimations are, however, not yet ultimately determined but to be revised by the future accumulation of observed data.
The author tries to estimate the coefficient r using the energy equation and the relationships (4) and ( 5) in the region of 1 < f < lOf in which wave velocity is smaller than wind velocity.
InJihe case of significant wave, the transmitted wave energy P is equal to • §• C , where E is the total energy per wave length as given by jff J-H 2 and C is wave velocity.
In steady state, the space rate of change of P is equal to the supplied energy R-rplus &M from wind to waves, where Ry and H N are amount of energy supplied by tangential and normal stresses of wind, respectively.

RH-E^-J-^-^C' + f/ (8)
where K = ~fjH x , A = z/^!, (* = -y a , and f, fare densities of water *nd air, ^2 the friction coefficient of wind over sea surface, and s the sheltering coefficient after Jeffreys.overdrup and Munk suggested A == 6.5X10"^ and <^=2.5. (6) ( 7) and ( 8) are available to the significant wave and now we assume that they are also applicable to the individual wave.
Thus from (6), taking the averages for all the successive waves in any time interval, we obtain

(id
where the third term in the right hand side is neglisibly smaller than the second term in the region of f considered, and also 0.012r in the left hand side is small compared with 1. Hence, to the first order approximation, (l8') becomes l,o6 f' 6 ^ ojo3rn{\-o.337T)~^o 6H7'n^-0 -l 2 7T)f oM (19) from which r is obtained as followss

APPROXIMATE ESTIMATIONS OF CORRELATION COEFFICIENT BETWEEN WAVE HEIGHT AND PERIOD OF SHALLOW WATER WIND WAVES
Thus for m} 0, n)0 , r is a steadily decreasing function of f.
Therefore, m and n should be selected reasonably.
According to Bretschneider's estimation, r is nearly equal to 0.9 for f = 1.
From the figure, it is found that Bretschneider's estimation is.the best fit for n *= 0 .nftien n is zero, ra becomes 14..76 and A /; = 1A.76,oi.= 0, which means that supply of energy from wind to waves is mainly done 0y tangential stress and the friction coefficient )' i becomes twice the one given by Munk.
Such a result is somewhat different from actual phenomena, but its tendency is near to the facts that the sheltering coefficient may be much smaller than the one proposed by Jeffreys and also ^Z ma y become appreciably larger than 0.0026 in some cases.
Accordingly, taking m = 14..76 , n=0 , the first order approximation of r becomes r ^Z.t1~Z.6 ?f^* ( 22 ) And the second order approximation is obtained from (18') as follows: r=z ZJl-j^of ' -0.05^ j (?3 ) which consists fairly well with Bretschneider's estimation in the range of l<f <10 4 .Therefore, it might be not so unreasonable to assume that the energy equation for significant wave is applicable to the individual wave and in the range of f considered the energy supply by normal stress of wind may be possibly neglected for our estimation.

ESTIMATIONS OF CORRELATION COEFFICIENT FOR SHALLOW WATER WAVES
*s previously described in deep water, the correlation coefficient r gradually decreases from 1.0 to 0 with increasing f, that is, with the developement of waves.
While, in shallow water, Bretschneider suggested that by the effect of bottom friction the coefficient r decreases more remarkably and possibly tends to negative, that is, r varxates from + 1.0 to -1.0 with increasing f and decreasing d ( non-dimensional water depth as given by gD/U 3 , where D is water depth ).
From the assumption of Rayleigh distribution for wave height and period, however, r cannot tend to -1.0 but get to about -0.6*^-'-0.7 as its minimum limiting value.( Bretschneider did not show this limiting value.)Following to the above descriptions, we assume ior r in shallow water wind waves as follows: (i) r decreases with the developement of waves, ^.iij r becomes possibly negative but cannot get to smaller than -0.7.

DEVELOPEMENT OF SHALLOW WATER WIND WAVES
Now consider the case of constant water depth D.
The shallow water significant wave height Hs , period Ts , fetch length F, and water depth D are expressed non-dimensionally as follows: Bretschneider has shown the relations of he; , f and d from his observations and calculations of wave height change by bottom friction, of which the result is shown in Figure 150 on Page 28d of Technical Report No.4 entitled " Shore Protection Planning and Design ", issued from Beach Erosion Board.
Figure 2 in this paper is made from that Figure 15C, excepting the curves for tg v.s.f.It is seen from Figure 2 that for any fixed d, hs increases with f initially along the same curve as for deep water waves and from certain point of f ( say ft ) it begins to deviate and at the other point of f ( say fa ) attains a steady constant state.
The value of hgat point ft ( say |^t ) is expressed from the Figure 15 c of Bretschneider as follows: On the other hand, h t is the same for deep water waves at ft .Thus from (4.), ft is given as follows: ft = 2.020 at'' (26) Similarly the value of b$ at point f n ( say h«. ) is obtained from that Bretschneider's Figure.Hence we derive the relation of tgand f from the following considerations.
Similar to wave height b$, there must be ft , bellow which tsis the same as that of deep water waves, and fit, above which t$becomes steadily constant, and also ft and fu. are the same as those for wave height hg.That is, both wave height and period begin to deviate from the curves of deep water waves at point ft and get to constant at fu..The value of tgat point ft (say tt ) is obtained by substituting ( 5) with (26).
, 0,46* tt = 0.6/5 CL (29) Theoretically the effect of bottom friction begins to appear when wave length becomes twice the water depth.
The value of t$satisfying this condition vsay tt') is Hence, in the region of 0 < d <" 10, tt4s smaller than 11 and theoretically the effect of friction appears even when f <^ft.
After Bretschneider(I960), significant wave period Tg is not too critical and it is conveniently represented by wave height H s as follows: where Tg is in seconds, H s in meters.
Accordingly, it is easy for us to draw smooth curves of tsv.s.f on the fetch graph, which are tangent to those of deep water waves at ft> and asymptotically tend to the constant value at fu_.
Figure 2 is thus obtained.

ENERGY EQUATION FOR SHALLOW WATER WIND WAVES
For the steady state of shallow water wind wave3 being affected by bottom friction, the energy equation becomes as follows corresponding to (9).

_£|:« ]T T + ]T N -p f (35)
where Dfis the loss of energy by bottom friction, averaged for successive waves in any observed period.
The transmitted wave energy P is as.follows: p a fr , nT,u»p(it :yL )
'o HT and &N are energy supplied from wind by tangejntial and normal stresses, and similar, to the case of deep water waves, R^is considered to be neglisible to R-f.
As for R T in shallow water waves, Kishi(l955) proposed the next equation for significant wave.
where Ag corresponds to A for deep water waves, and its magnitude is considered to be in the order of (10 ^-20)X 10 -^ . Putting Fzis also a function of D/L c ( = 2tt)/g'f z -).

APPROXIMATE ESTIMATIONS OF CORRELATION COEFFICIENT BETWEEN WAVE HEIGHT AND PERIOD OF SHALLOW WATER WIND WAVES
where k is the coefficient of friction,.which is about 0.01 ~ 0.02 after Sretschneider.

(51)
where /o (52) Hence, substituting ( 35) with (40) (44) and t5l), putting BN = 0, and neglecting higher order terms than r 2 ", energy equation becomes where A, B and G are functions of r, t and d, and expressed as polynomials of r as follows: /»(4?' r )=Mw) + a <feW) r+fo (w) T ' where a, b, c, etc. have the following form, which are numerically integrable.
In practices, d varies from 0.02 to 1.0 and t from 0.1 to 0.8.
From numerical caleulations for d** 0.04, 0.06, 0.08, 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0, it is known that a OJ a,, a 2 and be, b|, ba are all the same order of magnitude, and at any given d, their variations for the change of t are small.c 0 , C|, c 2 and c 3 are all the same order of magnitude and for any given d , they are all increasing functions of t, and their rates of increase for the change of t are remarkably large.
Due to the above fact, higher order terms than r a can be neglected to the first approximation.
For convenience of calculations, a 0 , a ( , bo, b| are expanded in terms of t for each d.
As an example, for d= 0.06, t changes from about 0.15 to 0.25 and they are expressed in this region as follows: Replacing t by ts/Vl+"0.6r, and neglecting higher order terms than r* we obtain for d«= 0.06,

COASTAL ENGINEERING
waves.
Hence we take here A^ = 15X10 .As for d, 1 , it should be in order of 0.1 ^0.2 and be selected as reasonably as possible.
For the purpose of it, we tried rough calculations of r for each d, putting <jj*= 0.1, 0.15, 0.20, 0.25, and 0.30.Whenot = 0.1 and 0.15, r increases wit*! increase of f for d = 0.04 ~ 1.0, which is contradictory to the asummed properties of r.When 0.20, 0.25, and 0.30, r has the tendency of decrease with increasing f for all d, which becomes more and more remarkably for larger values of dJ and when.*= 0.30, r decreases beyond -0.7, the assumed lower limit of r.
In conclusions, the value of o< / suitable for the assumed properties of r should be in the range of 0.20 •~-0.25, which means from (50) that the coefficient of friction is to be about 0.015-^0.019.
(61) will be correct enough near r -0, but apart from it, it will become erroneous, and it is not always enough to use only (61) in order to obtain r.
Accordingly, we proceed as follows: (i) When f <(, f^, Dj becomes much smaller than k T and r should tend to that of deep water waves. From (44-) and ( 51),

FT '
A, B^rj Ut)-t b,(i)-T from which the value of f satisfying the condition D-f<C0.02RT ( say f 0 ) is calculated for each d, and at f a , r should be nearly equal to that of deep water waves.After calculations, f 0 is obtained approximately as a function of d as follows: which is shown in Figure 3. Compared with (26), f D is about a half of ft and still smaller than ft'(3l).
This may be because of the existence of longer individual wave than the significant wave.

(ii)
When f^) f } f t , taking A S = 15x 10"f <*'= 0.2 , the value of f where r becomes zero ( say f(r=») ) is obtained for each d from (56), of which the result is as follows: which is shown in Figure 3. Pig. 5. Relationship between correlation coefficient and non-dimensional fetch for shallow water waves.
As mentioned aboved, at f = f 0 , r begins to deviate from the value of r for deep water waves, and then gets to zero at f(Y=o>, and still continuously decreases down to the ultimate, constant value.
Figure 5 shows the curves of r thus obtained for each d.

SOME RESULTS OF OBSERVATIONS
The above-mentioned r for shallow water waves should be verified by a lot of observation data.
Wave observations are now carried on near the Port of IzumiOotsu on the east coast of Oosaka Bay by means of underwater-preesure type wave-meters, and correlation coefficients are calculated from records at the depthco'f-2.20meters below L.W.L., of whieh some results are plotted in Figure 5.
% to date, the amount of data is' not satisfactory but the above-estimated tendency of r is seen from the Figure in some degree.

CONCLUSIONS (I)
In the case of deep water waves, asumming that the energy supplied by normal stress from wind is neglisible to that by tangential stress and energy equation by significant wave is available to the individual wave, and taking A^ay^^isxio -6 , the correlation coefficient r for wave height and period is presented in terms of f in the region of 1 < f < 1C* approximately as follows: y t ss ZJ7-2-00 f ~ 0.053 j (II) In the case of shallow water waves, asumming similarly to the case of deep water waves and taking the coefficient of bottom friction as 0.015'^-' 0.019, the correlation coefficient r is given as a function of d and f, as shown in Figure 5. r decreases more rapidly than that of deep water waves and gets to a certain negative value for each d.
The above-mentioned estimation is, however, only derived by approximate'calculations with simple assumptions and so it should be necessarily verified by future investigations.
period, Bretschneider gave no curve.

(
iii) When f/" fu., dP/dF becomes zero and %-D.f..And then r should j^e constant.Such a constant value of r is obtained from (57), putting Df= iij , and is shown in Fiinire 4 as a function of d.