K) d = j 0 s d(f) df. (5) J e min

Transcription

1 CHAPTER 35 DIFFRACTION DIAGRAMS FOR DIRECTIONAL RANDOM WAVES Yshimi Gda, Tmtsuka Takayama, and Yasumasa Suzuki Marine Hydrdynamics Divisin, Prt and Harbur Research Institute Ministry f Transprt, Nagase, Yksuka, Japan ABSTRACT Cnventinal wave diffractin diagrams ften yield errneus estimatin f wave heights behind breakwaters in the sea, because they are prepared fr mnchrmatic waves while actual waves in the sea are randm with directinal spectral characteristics. A prpsal is made fr the standard frm f directinal wave spectrum n the basis f Mitsuyasu's frmula fr directinal spreading functin. A new set f diffractin diagrams have been cnstructed fr randm waves with the prpsed directinal spectrum. Prblems f multi-diffractin and multi-reflectin within a harbur can als be slved with serial applicatins f randm wave diffractin. INTRODUCTION Since the prf by Penny and Price [1] that the diffractin f water waves by breakwaters can be analyzed with Smmerfeld's slutin, wave heights behind breakwaters have been estimated with the aid f several diffractin diagrams [2^6]. The phenmenn f wave diffractin is a typical prblem fr which the slutin f velcity ptential can be applied with accuracy. Published as well as unpublished labratry investigatins have prvided the prf f the validity f wave diffractin thery. Disagreement between the thery and experiment if any is usually attributed t inaccuracy in labratry measurements. The nly exceptin is the appearance f secndary waves arund the tip f a breakwater wing t an excessive gradient f wave energy density there [7]. Such a success f thery, hwever, shuld be accepted with a cautin when the thery is applied fr sea waves characterized with irregularity. Mst f diffractin diagrams currently available are thse prepared fr mnchrmatic waves with a single perid frm a single directin. The irregularity f sea waves especially f directinal spreading prduces the pattern f wave diffractin quite different frm cnventinal diffractin diagrams. An experimental study by Mbarek and Wiegel [8] seems t be the first in demnstrating the applicatin f directinal wave spectrum t diffractin prblems, thugh they did nt present general diffractin diagrams fr engieers' usage. Being aware f these facts, Nagai [9,10] cnstructed diffractin diagrams fr sea waves in 1972, which have been utilized by harbur engineers in Japan. Figure 1 is ne f his diagrams, which shws the diffractin diagram fr sinusidal (mnchrmatic) waves in the left half and that fr spectral waves in the right half fr the pening width 628

2 DIAGRAMS FOR WAVES 629 f five times the wavelength; the difference between them is very clear. The directinal wave spectrum emplyed fr cmputatin was f SWOP type [11], which is primarily fr wind waves. In the present paper, recalculatin is made f diffractin diagrams f randm waves with a new prpsal f directinal wave spectrum, which is a mdificatin f the spectrum riginally frmulated by Mitsuyasu et al.[12]. Thugh these diagrams were previusly published in Japanese [13], slight crrectins have been fund necessary and they are duely crrected heren. The present paper als discusses the behaviur f waves reflected by breakwaters, which can be deduced frm Smmerfeld's slutin as prved by ne f the authrs [14]. With the abve knwledge, the prblem f multi-diffractin and multi-reflectin within a harbur can be slved numerically. SPECTRAL CALCULATION OF WAVE DIFFRACTION Randm waves in the sea are described with a directinal wave spectrum under the presumptin that randm wave prfiles are the result f linear superpsitin f infinite number f infinitesimal wavelets with varius frequencies and directins. Accrding t this presumptin, the spectrum f diffracted waves at a pint (x,y) is calculated as C e max S d (f[x,y) = S^f.e) K 2 d (f,e x,y) de, (1) J min where S-j(f,e) dentes the directinal spectrum f incident waves and Kd(fj6 x,y) is the diffractin cefficient at a pint (x,y) fr waves with the frequency f and the directin e. The spectrum f diffracted waves is given here in the frm f frequency spectrum nly, because the directinal spreading f diffracted waves is limited by the aperture f the breakwater gap lked frm the pint (x,y). The representative heights f incident and diffracted waves are derived frm the zerth mment f spectrum by the thery f Lnguet- Higgins [15]. Fr example, the significant heights are given by (H 1/3 ) i = 4.0/Ti^T, (2) (H 1/3 ) d = 4.0/[i^, (3) where, «Y e max nl Si(f,e) de df, (4) J e min K) d = j 0 s d(f) df. (5) Thugh the cnstant f 4.0 in Eqs. 2 and 3 is better replaced by that f 3.8 fr waves bserved in the sea n the average, it des nt affect the cefficient f diffractin fr randm waves, which is defined as (Kd)eff = C H l/3) d / < H l/3>i= / ("') d /(m) i (6) The representative perids f diffracted waves are nt necessarily the same with thse f incident waves. The change f wave perid by diffractin can be estimated by the thery f Rice [16] as

3 630 COASTAL ENGINEERING 1978 K d T " /(m 0 )./(m 2 ). ' < 7 > where, (m 2 ) d = f2 S d (f) df, (8) /» c e max (m 2 ), = f 2 S (f,e) de df. (9) J 0 J e min ' The effect f the wave spectrum upn diffractin cefficient is demnstrated in Fig. 2, which shws the diffracitn diagrams f a breakwater gap with the relative pening f B = 3L fr blique incident waves with the angle f apprach f 60. The diffractin cefficient is calculated with the apprximate methd by superpsitin f Smmerfeld's slutins fr tw semi-infinite breakwaters. The upper diagram is fr mnchrmatic waves and uni-directinal irregular waves (with frequency spectrum nly). The difference between them is small, thus indicating unimprtance f frequency-wise irregulariry. The lwer diagram is fr uni-frequency randm waves with directinal spreading and very randm waves with a directinal spectrum, which crrespnds t the case f Smax = 10 t be discussed in the next chapter. The difference between them is small, but the bth are quite different frm thse in the upper diagram. Thus, Fig. 2 indicates that the directinal spreading rather than the frequency-wise irregularity is imprtant in the diffractin f randm waves. PROPOSAL OF DIRECTIONAL WAVE SPECTRUM Functinal F-tm The directinal wave spectrum is generally expressed as the prduct f a frequency spectrum S(f) and a directinal spreading functin G(f,e), that is, S(f,e) = S(f) G(f,e). (10) The frequency spectrum S(f) is given the unit f m 2 -sec r its equivalent ne, while G(f,e) is nrmalized s as t yield the unit value withut a dimensin when integrated ver the full range f wave directin. The functinal frm f S(f) can be taken as Bretschneider's spectrum [17] mdified by Mitsuyasu [18] t satisfy the cnditin f Eq. 2. Thus, S(f) = H 1/3 2 T 1/3-4 f" 5 exp[-1.03 (T 1/3 f)- 1 *]. (11) This is a type f tw-parameter spectrum designated by an arbitrary cmbinatin f significant wave height and perid, H1/3 and T1/3. The mdal frequency r the frequency at spectral peak is set t satisfy the fllwing relatin:

4 DIAGRAMS FOR WAVES 631 This relatin was prpsed by Mitsuyasu [18] and has been cnfirmed t be representative f sea waves [19]. Spectral frms ther than Eq. 11 are als eligible as the standard spectrum, but the change f frequencywise spectral frm will affect little the diffractin f randm waves as suggested by Fig. 2. As t the directinal spreading functin, the frmula prpsed by Mitsuyasu et al. [12] n the basis f their detailed bservatins seems mst reliable at present. In a slightly mdified frm, it is written as where, G(f,e) = G c cs 2S (f) G = max nnn cs2s (f) J s max-( f / f p) ; Is (f/f r \ f i f P The term f G 0 is s intrduced t nrmalize G(f,e). The directinal cncentratin parameter S has the maximum value at f = fp and decreases at the bth sides f spectral peak. Stltatin {, S, max Figure 3 is a demnstratin f wave patterns, which shws the cnturs f surface elevatins abve the mean water level; the prtin f wave trughs is left as blank. This figure is a result f numerical simulatin by the principle f linear superpsitin with the spectrum f Eqs. 10 t 15. The maximum directinal cncentratin parameter Smax is subjectively chsen as 10 and 75, respectively. It will be seen that S max = 10 yields the wave pattern quite randm and smewhat resembling that f wind waves, while S = max 75 may crrespnds t the wave pattern f swell. The riginal prpsal f Mitsuyasu et al.[12] fr S max is t relate it with the nndimensinal frequency parameter as where U dentes th Equatin 16 is nt the design wave he t the wind speed methd suggests th assciated with th can be assumed t tin is supprted (13) (14) (15) S max =11.5 (2wf p U/g) -2.5 (16) e wind speed and g is the acceleratin f gravity, readily applicable fr engineering prblems because ight and perid are ften designated withut reference The knwledge f wave grwth depicted in the SMB at the increase f the parameter 2irfpU/g ( = U/Cn) is e decrease in the wave steepness H 0 /L 0. Thus, S max increase as the wave steepness decreases. The assumpby the example f Fig. 3 discussed in the abve. Frm the abve discussins, the authrs prpse the fllwing values f S max fr engineering applicatins: ->max MO : fr wind waves, \25 : fr swell with shrt t medium decay distance, [75 : fr swell with medium t lng decay distance. (17)

5 632 COASTAL ENGINEERING 1978 Thugh the abve prpsal is smewhat subjective, S max = 10 fr wind waves is nt withut grund because it yields the verall directinal distributin almst the same with the law f (2/ir)cs 2 e and the frmula f SWOP. Figure 4 shws the nndimensinal cumulative curves f wave energy calculated fr the directinal wave spectrum f Eqs. 10 t 15. The term f PE(S) is calculated by P E( 0 ) =^r f e f s < f > e ) df de O 8 ) m 0 J-TT/2 JO The diagram can be utilized t allcate the relative wave energy t several wave directins such as expressed in sixteen pints bearings. Calculatin f P r(9) als yields the apprximate relatin f I = 0.11 S max : I 1 2, (19) fr the type f G(f,e) f the fllwing: G(f,e) E G(e) = ^-i)!! cs 2 e, (20) where 2n!! = 2n-(2n-2) -4-2 and (2n-l)!! = (2n-l)-(2n-3) When applying the abve spectrum in shallw water, sme crrectin t S max is necessary because the phenmenn f wave refractin makes the directinal spreading t lessen. Calculatin f wave refractin in the water f parallel straight bathmetry has yielded the diagram fr the change f S max in shallw water as shwn in Fig. 5. The angle (ap) 0 dentes the incident wave angle t the bundary f deep t shallw waters. As the effect f (a p ) 0 is small, the diagram may be utilized fr waters f general bathmetry. RANDOM WAVE DIFFRACTION BY A SEMI-INFINITE BREAKWATER With the directinal wave spectrum specified in the abve, the cmputatin f randm wave diffractin is straightfrward s lng as the value f diffractin cefficient fr mnchrmatic waves crrespnding t spectral cmpnents are cmputable. The integrals in Eqs. 1, 4, and thers are t be evaluated in the frm f finite series. The number f frequency cmpnents des nt need t be great, but the number f directinal cmpnents shuld be carefully selected in cnsideratin f the trade-ff between the accuracy and cmputatin time. When the diffractin cefficient in the area far distant frm the breakwater is t be calculated, a large number f directinal cmpnents are required. Examples f the diffractin diagrams f semi-infinite breakwater are shwn in Fig. 6 fr the case f nrmal incidence fr waves with S max = 10 and 75. The diffractin cefficient f wave heights, r (Kd)? ff, is shwn with cnturs f slid lines, while the rati f wave perid is shwn with cnturs f dashed lines. A characteristic feature f Fig. 6 is that the diffractin cefficient takes the value f abut 0.7 alng the bundary f gemetric shadw. This value is abut 1.4 times the cefficient f mnchrmatic wave diffractin. In the sheltered area, the randm wave diffractin yields the cefficient far larger than that f mnchrmatic waves.

6 DIAGRAMS FOR WAVES 633 An verificatin f the superirity f randm diffractin analysis has been prvided by Irie [20] with the wave data at Akita Prt. Tw wave recrders f inverted ech sunder type were set utside and inside a lng breakwater as shwn in Fig. 7, and the simultaneus bservatin was carried ut in Nvember and December, The principal directin f waves incident t the breakwater were read frm the images f a radar with the wavelength f 8.6 mm, which is cmmnly emplyed in Japan fr detectin f wave directin since arund During the perid f bservatin, the principal wave directin varied in a narrw range f N85 W t N110 W and the directin f N106 w was emplyed in the calculatin f diffractin cefficient. The result f analysis is summarized in Fig. 8. The bserved data were classified by Irie int three categries f dispersive, median, and cncentrated wave patterns as judged n the radar image. In the cmputatin f randm wave diffractin, the maximum directinal cncentratin parameter f Eq. 17 was subjectively applied fr these wave categries with the crrectin f shallw water effect by Fig. 5. Thugh the scatter f data makes difficult the assessment f the accuracy f randm diffractin analysis, it yields quite reasnable estimates f wave heights behind the breakwater. The mnchrmatic wave anlysis, n the ther hand, yields the diffractin cefficient being ne half t ne quarter f the bserved value, thus revealing its inapplicability t the prblem f sea wave diffractin. RANDOM WAVE DIFFRACTION THROUGH A GAP OF BREAKWATERS Anther example f the effectiveness f randm diffractin analysis has been given by ne f the authrs [21]. Wave bservatin were carried ut at three statins in Nagya Prt frm 1967 t 1970, which is lcated at the recess f Ise Bay, Japan. As indicated n the inset f Fig. 9, the statins A and B were psitined utside and inside f a lng mle f caissn type. The diffractin cefficient fr mnchrmatic waves at the statin B is pltted in this figure fr predminant wave perid f T = 3 sec. The waves diffracted frm the east entrance penetrate t the statin nly when the incident wave directin is frm SSSW t SSE, and the waves frm the west entrance is appreciable fr the incident directin f NW t WNW nly. Thus the diffractin cefficient fr mnchrmatic waves is very sensitive t the incident wave directin. Observed wave recrds did nt exhibit such a directinality, as shwn in the example f Fig. 10, where the wave spectra at the statins A and B are cmpared. The wave directin is estimated as SW frm the wind recrd. As the diffractin cefficient fr mnchrmatic waves is abut 0.07, the spectral density f the statin B wuld have been abut 1/200 f the density f the statin A if the diffractin were t be calculated fr mnchrmatic waves. The bserved spectrum at B had the density f abut 1/10 t 1/20 f the spectrum at A, and it was nearly in agreement with the spectrum calculated as the randm wave diffractin phenmenn althugh the directinal spreading funcin f (2/w)cs 2 e was emplyed in the calculatin fr the sake f simplicity. The agreement f bserved and calculated spectra is an evidence f the necessity f intrducing randm wave analysis in diffractin prblems. Figures 11 t 14 are the result f the cmputatin f randm wave diffractin with the directinal spectrum described by Eqs. 10 t 15.

7 634 COASTAL ENGINEERING 1978 The left half f each diagram is fr the change f wave perid, while the right half is fr the wave height rati. The abscissa and rdinate are nrmalized with the pening width B instead f the wave length L. In applying these diagrams, apprpriate selectin f S max is t be made and interplatin f the diffractin cefficient frm the diagrams fr Smax = 10 anc ' 75 will be required. C0NBINATI0N OF WAVE REFLECTION AND DIFFRACTION In the analysis f wave tranquility in a harbur, wave reflectins frm quaywalls and ther slid structures ften becme the surce f trubles. Breakwaters with vertical faces may als cause additinal agitatin in the area utside the breakwaters. The height f waves reflected by a semi-infinite rigid breakwater can be estimated by means f the well knwn Smmerfeld's slutin. T illustrate the applicability f the slutin, it is rewritten in the fllwing frm: F d (r,a,f,e) = F id (r,a,f,e) + F rd (r,a,f,e) (21) where F(r,a,f,e) dentes the dimensinless cmplex wave amplitude at the pint P with the distance r and the angle a frm the tip f a semiinfinite breakwater fr the incident wave with the frequency f and the directin e (see Fig. 15). The cmplex amplitudes F-j d and F rd are expressed respectively as F id =^exp[i{krcs(a-6)+ }]x[{c( Yl )+l} - i{s( Yl )+ }] = exp[i{ kr cs(a-e)}] + 7=-exp[i{krcs(a-e)+ }]x[{c( Y i)- -} - i{s( Y i) - jjf}]. (22) F rd =^exp[i{krcs(a+e)+i}]x[{c( Y2 )+l} - i{s( T2 )+J-}] where, = exp[i{ kr cs(a+e)}] + ^exp[i{ krcs(a+e)+f}]x[{c(y 2 )- } - i{s( Y2 ) - J}], (23) Yl /i r cos<! i, 1 C(Y) = /"c0sfx 2 dx, (24) 0 ' f (25). Y2 = ; cs, j S(Y) = f Q sin^ In the derivatin f the secnd expressins frm the first nes f Eqs. 22 and 23, the fllwing equality is emplyed: {C + l} - i{s+l} = [{C-l} - i{s-l}] + [1- i] [{C-l} - i{s- } }] + /2 exp[-i ]. (26) As the distance r increases infinitely, the Fresnel integrals defined by Eq. 25 cnverge t the values listed in Table 1 depending n

8 DIAGRAMS FOR WAVES 635 the regins in questin. Referring t Table 1, it will be readily understd that F^ represents the sum f the incident waves and the assciated scattered waves, while F rc j represents the sum f the reflected waves and the assciated scattered waves. Mrever, the primary reflected waves exist nly in the regin I, whereas the regin III is primarily ccupied by scattered waves. Table 1. Behaviur f Fresnel Integrals at r =» Regin C(Yi) S(Tl) C(Y 2 ) S(Y 2 ) Primary Waves I II III 1/2 1/2-1/2 1/2 1/2-1/2 1/2-1/2-1/2 1/2-1/2-1/2 incident, reflected, and scattered waves incident and scattered waves scattered waves The abve decmpsitin f Smmerfeld's slutin leads t the calculatin f reflected waves by means f F rc j. If the reflective bundary is finite in its extensin, the amplitudes f reflected waves can be apprximately calculated by superimpsing the tw slutins f F rc ) fr the bth tips f the reflective bundary as in the technique f calculating wave diffractin thrugh a gap f tw semi-infinite breakwaters. Fr a partially reflective bundary, the cefficient f wave reflectin is intrduced t linearly reduce the amplitudes f reflected waves. Fr example, wave pattern arund a semi-infinite breakwater with partial wave reflectivity can be calculated by Fd(r,a,f,e) = F id (r,a,f,e) + K r F rd (r,a,f,e), (27) where K r dentes the reflectin cefficient. Equatin 27 remains as an apprximatin because K r usually des nt carry the infrmatin f phase relatin except fr the cases f K r = 1 and 0. An experimental verificatin f the abve analysis [14] has been dne fr the layut f mdel breakwaters shwn in Fig. 16. The breakwaters are made f vertical walls. Incident waves are diffracted by the right breakwater, but sme f them are reflected by the rear face f the left breakwater. Experiments were carried ut with uni-directinal irregular waves, which had the significant height and perid f H- /3 = 1.8 cm and T]/3 = 1.08 sec; their spectrum culd be apprximately expressed by Eq. 11. The result f measurements are shwn in Fig. 17 fr cmparisn with the theretical calculatin. Gd agreement between them is bserved except alng the line f x = 8 m. The difference is due t an assumptin emplyed in the calculatin that the surce area f wave reflectin can be specified by the principle f gemetric ptics in rder t simplify the prcedure f calculatin. The errr due t such simplificatin is expected t decrease when the directinal spreading characteristic f sea waves is intrduced. The analysis f the reflectin f diffracted waves can be prceeded fr much cmplicated harbur layut, even thugh the algrithm needs t be carefully established. Wave diffractin by verlapping breakwaters can als be slved with the knwledge f decmpsed Smmerfeld's slutin f Eqs. 21 t 23. An example f wave tranquility analysis fr a cmplicated harbur layut is shwn in Fig. 18, which represents

9 636 COASTAL ENGINEERING 1978 the Prt f Ykhama in a slightly simplified frm. Wind waves with the significant perid f T]/3 = 6.0 sec are cnsidered t cme frm the directin f SSE. The maximum directinal cncentratin parameter f Smax = ^ is emplyed in the cmputatin. The specificatin f wave reflectivity f the bundaries is made smewhat subjectively in rder t simplify the prcess f analysis f multi-reflectin and multidiffractin. Thugh the field data t verify the calculatin is nt available yet, Fig. 18 demnstrates the capacity f analyzing wave tranquility in a real harbur. It shuld be nted that the present analysis has n upper bund f applicatin with regards t the size f harbur relative t wavelengths because f the nature f the thery f wave diffractin. SUMMARY The present paper has discussed the diffractin f sea waves with directinal spectral characteristics. Majr cnclusins are as fllws: 1. The inapplicability f diffractin analysis by mnchrmatic wave apprach has been demnstrated by tw examples f field bservatin data, which at the same time have prved the effectiveness f randm diffractin analysis. 2. A standard frm f directinal wave spectrum is derived n the basis f the directinal spreading functin prpsed by Mitsuyasu et al. Thugh the selectin f directinal cncentratin parameter is left t smewhat subjective judgement f engineers, it can deal with varius stages f wind waves t swell. 3. Several diffractin diagrams are presented fr a semi-infinite breakwaters and a breakwater gap fr the case f nrmal incidence f directinal randm waves. The technique can be extended t the case f blique incidence as well. 4. Smmerfeld's slutin f diffracted wave amplitudes is decmpsed int the terms f incident, reflected, and scattered waves. The decmpsed slutins can be emplyed fr analyzing the behaviur f waves reflected by rigid breakwaters and ther reflective structures. 5. Wave tranquility in a harbur f large dimensin can be analyzed by serial calculatins f randm wave diffractin and reflectin. It is mentined here that the abve technique f randm wave diffractin analysis is daily utilized by harbur engineers in Japan with the aid f cmputer prgram peratable at the Cmputatin Center f the-prt and Harbur Research Institute, Ministry f Transprt. REFERENCES [1] Penny, W. G. and Price, A. T.: Diffractin f sea waves by breakwater, Artificial Harbur, Dire. Misc. Weapn Tech. His. N. 66, 1944.

10 DIAGRAMS FOR WAVES 637 [2] Blue, F. L. and Jhnsn, J. W.: Diffractin f water waves passing thrugh a breakwater gap, Trans. A.G.U., Vl. 30, N. 5, 1948, pp [3] Jhnsn, J. W.: Generalized wave diffractin diagrams, Prc. 2nd Cnf. Castal Engg., Hustn, [4] Wiegel, R. L.: Diffractin f waves by semi-infinite breakwater, Prc. ASCE, Vl. 88, N. HY1, 1962, pp [5] Mrihira, M. and Okuyama, I.: Cmputing methd f sea waves and diffractin diagrams, Tech. Nte f Prt and Harbur Res. Inst., N. 21, 1965, 60p. (in Japanese). [6] Takai, T.: The diffractin diagrams f sea waves by a breakwater gap, Tech. Nte f Prt and Harbur Res. Inst., N. 66, 1969, 42p. (in Japanese). [7] Biesel, F.: Radiating secnd-rder phenmena in gravity waves, Prc. IAHR 10th Cngress, Lndn, 1963, pp [8] Mbarek, I. E. and Wiegel, R. L.: Diffractin f wind generated water waves, Prc. 10th Cnf. Castal Engg., Tky, 1966, pp [9] Nagai, K.: Cmputatin f refractin and diffractin f irregular sea, Rept. Prt and Harbur Res. Inst., Vl. 11, N. 2, 1972, pp (in Japanese) [10] Nagai, K.: Diffractin f the irregular sea due t breakwaters, Castal Engineering in Japan, JSCE, Vl. 15, 1972, pp [11] Cte, L. J. et al.: The directinal spectrum f a wind generated sea as determined frm data btained by the Stere Wave Observatin Prject, Meterlgical Papers, Vl. 2, N. 6, New Yrk Univ., 1960, 88p. [12] Mitsuyasu, H. et al.: Observatin f the directinal spectrum f cean waves using a clverleaf buy, J. Gephys. Res., Vl. 5, N. 4, 1975, pp [13] Gda, Y. and Suzuki, Y.: Cmputatin f refractin and diffractin f sea waves with Mitsuyasu's directinal spectrum, Tech. Nte f Prt and Harbur Res. Inst., N. 230, 1975, 45p. (in Japanese). [14] Takayama, T. and Kamiyama, Y.: Diffractin f sea waves by rigid r cushin type breakwaters, Rept. Prt and Harbur Res. Inst., Vl. 16, N. 3, 1977, pp [15] Lnguet-Higgins, M. S.: On the statistical distributin f the heights f sea waves, J. Marine Res., Vl. XI, N. 3, 1952, pp

11 638 COASTAL ENGINEERING 1978 [16] Rice, S. 0.: Mathematical analysis f randm nise, 1944 and 1945, reprinted in Selected Papers n Nise and Stchastic Prcesses, Dver Pub., Inc., 1954, pp [17] Bretschneider, C. L.: Significant waves and wave spectrum, Ocean Industry, Feb. 1968, pp [18] Mitsuyasu, H.: On the grwth f the spectrum f wind-generated waves (I), Rept. Res. Inst. fr Applied Mech., Kyushu Univ., Vl. XVI, N. 55, 1968, pp [19] Gda, Y.: Statistical interpretatin f wave data, Draft Cntributin t the Cmmittee Rept. 1.1 t the 7th Int. Ship Structures Cngress t be held in Paris, [20] Irie, I.: Examinatin f wave defrmatin with field bservatin data, Castal Engineering in Japan, JSCE, Vl. 18, 1975^ pp [21] Gda, Y., Nagai, K., and It, M.: Wave bservatin at the Prt f Nagya (3rd Rept.), Tech. Nte f Prt and Harbur Res. Inst., N. 120, 1971, 24p. (in Japanese). [22] Bretschneider, C. L.: Wave variability and wave spectra fr windgenerated gravity waves, U.S. Army Crps f Engrs., Beach Ersin Bard, Tech. Mem., N. 113, 1959, 192p. spectral wave B/L=5.0 Fig. 1 Cmparisn f Mnchrmatic and Randm Diffractin Diagrams fr the Case f B/L=5.0 (after Nagai [9])

12 DIAGRAMS FOR WAVES 639 (a) mnchrmatic waves and uni-directinal irregular waves (b) uni-frequency directinal waves and directinal randm waves Fig. 2 Effect f Directinal Spreading Characteristic upn Wave Diffractin thrugh a Breakwater Gap

13 640 COASTAL ENGINEERING 1978 a) S m = 10 Wave Directin 0 L1/3 2ii. (b) Sma» = 75 «S3SD0<7/Hi/3<0.2 «^0.2<»?/HI/3<0.4 ««0A<n/Hui >A Wave Directin 1 I I 0 L1/3 2Li/3 Fig. 3 Surface Elevatin Cnturs f Randm Waves by Numerical Simulatin

14 DIAGRAMS FOR WAVES " -60" st*: 1 i ^ y ^ H V s. s 5 ^ m =5v<C f'w ^ ^ Fig. 4 Cumulative Curves f Relative Wave Energy with Respect t Azimuth frm the Principal Wave Directin c/? A \ ^ > V <v. _ 1 \ \ \ \ \ \ \ V N s\ > N <\, v \ S, \ \ N < :: -5i---i _' ^>m^k) = 75 ) ' u 4b 20 (a,x=0 ^ ^ ^ c h/l Fig. 5 Change f Maximum Directinal Cncentratin Parameter, S m= > Due t Wave Refractin in Shallw Water max

15 642 COASTAL ENGINEERING 1978 x. IS.-. i i Cd s- t d) 0) s- c CQ 0) -a Ol r- 7 i s. r- Q ^ N s. ^ --.f ^ s s ^> ""^^, 0' I " ' 0 r r' "^^ ^% V He. \ ^ c ^^.- *" i> 1 O' Vv i r= si k. 1 b «H 0 S q- a> > t t/i 3: e t E i- O>"0 IO c i- <0 Q ; C r O t r- c 4-> > O T- t 4-> s- u 14- Ol 4- s- X t Q Q «3 ^. Ol J / y 1' ic E!.7

16 DIAGRAMS FOR WAVES 643 AKITA PORT (1973) Wave Recrder (-20m) Wave Recrder (-8m) Fig. 7 Sketch f Akita Prt in Wave Pattern Dispersive Median Cncentrated Observat i n Calculatin (Randm Waves) (S mas = 15) (S m = 42) (S ax = 100) Observatin at AKITA PORT Nv.-Dec Fig. 8 Diffractin Cefficient Observed at Akita Prt in Cmparisn with Calculatin

17 644 COASTAL ENGINEERING 1978 Statin B 1.0 \ 200m 650 m 0.6 I : ^ I 3</\ ' == L \l000m WWW j L 2300m J f> 450m u i '^- Waves trm / West Entrance > -«-»-«-^-»-Tf"«"t-t-«-ll WNW w wsw sw ssw s Incident Wave Directin Fig. 9 Diffractin Cefficient by Mnchrmatic Wave Analysis fr the Statin B in Nagya Prt _ i MM 1 i I in r A Wave 18:00 / i Directin: SW : / \ : V / li V - /' a [ h ~ A Diffracted Spectrum V Vl ] b - _ Waves \ _ at St. B I I Hill I 'III; / (Hz) Fig. 10 Wave Spectra Observed at the Statins A and B in Nagya Prt (Wave Directin f SW)

18 DIAGRAMS FOR WAVES 645 H K) "S " ^^Nv O ii a > r O i: ::! 1 *«">r ^> A*.TT ^ 1 :,^^ 'c ' in t ] a / ' rj... ^/ LO 3 x r II r X r s- Q) (/) -l-> r --* 5 CM -^ - r ai s- CQ r Og O F <1) r () i- c n> (11 r - u c e > r F r i- s- 4- ^ 4- i 4- Q O

19 646 COASTAL ENGINEERING 1978 > t E c u,<=l M- +-> 3 CL r S- O ni!- ca r lf tn f- Cl) m u!- e en <u tc T3 10 s- s- Q O CM T

20 DIAGRAMS FOR WAVES 647

21 648 COASTAL ENGINEERING > O t- n X re s- ca a F OJ m O s- C Cl) <0 T3 rn O c c 1 1 1!-> r O F (O i- i- M- ii.j_ M- O O

22 DIAGRAMS FOR WAVES 649 REGION I REGION III Fig. 15 Definitin Sketch Dig. 16 Experimental Setup fr Measurements f Wave Diffractin and Reflectin

23 1 650 COASTAL ENGINEERING ' Kr ' _\ 3^Dj=5&- V j x - 2m NJ ^~"" c - 4 a a [ _ 3 (1.. Cl. Fr Bretschneider's Spec. Cl. Fr Measured Spec. Experiments 1.0 x a 4 m a O ; n n x 8rr 1 ~- _7 """" 4 5 y/l Fig. 17 Measured and Calculated Cefficients f Diffractin by Mdel Breakwaters Water Depth : 10 m Waves <- T 1/3 = 6.0 sec '20 l 1/3 Fig. 18 Estimated Equi-Cnturs f Wave Height Rati in Ykhama Prt fr Wind Waves with T1/3 = 6.0 sec frm SSE