CHARACTERISTICS OF WIND-INDUCED WAVES IN A SHALLOW WATER ZONE

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1 Acta Geodyn. Geomater., Vo. 3, No. 4 (44), 4-49, 006 CHARACTERISTICS OF WIND-INDUCED WAVES IN A SHALLOW WATER ZONE Tamara SPANILÁ* and Kare JAHODA + Intitute of Rock Structure and Mechanic AS of the Czech Repuic,V Hoešovičkách 4, CZ-809 Prague 8, Czech Repuic, Te.: , Fax.: *Correponding author e-mai: tamarapania@eznam.cz (Received March 006, accepted Novener 006) ABSTRACT The parameter determine wave energy in haow water zone that pronounce the crucia infuence on araion of oth natura and artificiay paved ank. The effort to re-deveopment of the reation wa found a aoutey neceary for wave energy cacuation. Sutantia enefit of the work i found not ony in enaing the ue of computer whie avoiding timeconuming and difficut appication of diagram, ut namey in recent recognition that the cacuation reut howed a rik of underetimate the rea impact of wind-induced wave. In ome cae, the cacuation repecting the aove tandard produce ower vaue of wave height and time-period and thu ao ower vaue of wave energy. KEYWORDS: height, ength and time-period of the wave, energy of wave, haow water zone. INTRODUCTION There i a Czech tandard ČSN vaid for cacuation of wind-induced water wave, however, in time when the tandard wa introduced; modern computer had not een widey avaiae for project deigner. Thi i why auxiiary diagram had een ued for cacuation needed for oving of rather difficut cacuation reevant to the proem. The aove tandard had introduced diagram ued for cacuation of wave time period and height originating from former Soviet Union a een on Fig. 3 and 4. Jut very recenty it wa recognized that, mot proay, oth diagram were overtaken from iterature ource (Jahoda, 997) without referring to reevant formua, giving reut in rea coordinate for wind veocitie 0 and 30 m. -. Ao, it ha not een poie to find what procedure wa originay ued for paing from the diagram to the dimenione coordinate of wave height, time-period and ength. Actuay, unknown reation on whom the aove-mentioned diagram were deveoped hamper the ue of computer. Thi concern namey of wave parameter in haow water. The parameter determine wave energy in haow water zone that pronounce the crucia infuence on araion (Lukáč and Aaffy, 980; Kratochvi, 969) of oth natura and artificiay paved ank. Whatever compicated and/or time-conuming, the effort to re-deveopment of the reation wa found a aoutey neceary for wave energy cacuation. Sutantia enefit of the work i found not ony in enaing the ue of computer whie avoiding timeconuming and difficut appication of diagram, ut namey in recent recognition that the cacuation reut howed a rik of underetimate the rea impact of wind-induced wave. In ome cae, the cacuation repecting the aove tandard produce ower vaue of wave height and time-period and thu ao ower vaue of wave energy. The preented tudy i therefore focued on cacuation of main wave parameter in haow water zone.. WAVE PARAMETERS IN DIMENSIONLESS NUMBERS Height, ength and time-period of the wave occurring on water urface i dependent on reervoir ize, wind veocity, water depth and reervoir ottom hape (Spaniá et a., 999). In order to avoid the infuence of ottom hape that i impoie to decrie y numer, a meaurement had een performed on deep water. According to the aove tandard, thi concern any depth igger than one-haf of the wave ength. Due to genera expenivene of the meaurement, method were earched for, how to generaize and ue the meaurement reut not ony oca ut ao foreign author. Thu, for exampe, infuence of the reervoir ize and wind veocity i decried y a dimenione numer and the wave height y a numer ν. The foowing reation define thee dimenione characteritic:

2 4 T. Spaniá and K. Jahoda gh gt gh gd v=, t=, =, = πw w w w () The meaurement of wave ength i not performed a far a it wa hown that the meaurement itef i not ony extremey difficut ut ao much e reiae than that of the wave timeperiod and height. The determination of the waveength mean vaue aed on cacuation i more reiae. According to the aove Czech tandard, the cacuation procedure i a foow: For given water depth H, the ength of wind route D and it veocity w, characteritic numer and are cacuated for which reevant dimenione characteritic ν = gh/w and t = gt/πw are found on diagram in Fig. 3 and 4, eading further on to cacuation of wave height h and time-period T. The waveength cacuation uing progreive-approach method i decried in the tandard. At the initia phae - earching for reation, according to whom curve were contructed in the mentioned diagram, the ony requirement wa conidered, e.g. technicay ufficient fitting of deveoped formua with curve in Fig. 3 and 4. Finay, the foowing reation were found: t t () t v v For the wave height δ v, δ 0. = 0, 3 (3) Symo ν i ued for the dimenione height, whie ymo t for dimenione wave time-period in deep water, theoreticay for H. Reevant curve in diagram in Fig. 3 and 4 can with ufficient accuracy e decried a the foowing function of parameter : og = (og ) (og ) (og ) (4) og t= (og ) (og ) (og ) Logarithm ued in the aove equation are decade one.) Fitting of the reation () and (3) v. curve in Fig. 3 and 4 i reativey good, however, the accordance ha not een reached in the whoe extent of the curve. Thi concern namey partia curve hape for = gh / w 0. 04, that i for 000 unikey fat. 3. WAVE PARAMETERS CALCULATIONS IN SHALLOW ZONE The foowing genera reation (ČSN, 985) i vaid for the wave motion on water urface of the fina depth H: gt π H λ = (5) π λ Theoreticay, the former reation i vaid for reervoir with a fat ottom, without an exceive error it can e appied for mid oped ank, araion of whoe i moty anayed. For wind-induced wave on deep water, theoreticay for H, practicay for depth interva of aout 7 to 0 m, when the ottom hape infuence i diappearing, the foowing reation i vaid etween time-period T and waveength λ : gt λ = π (6) The meaurement reut are expreed a dimenione numer (Spaniá et.a., 999). Whie ti miing in reation (), for the dimenione characteritic of the wave ength the foowing reation wa derived: = gλ w Now, genera equation (5) can e tranformed into dimenione form, e.g. π = π t (7) The equation i decried in Fig. for deep water with curve, then for ony a itte maer depth with curve -, and for haow water with curve = A procedure i indicated in upper right corner of the Fig., how to hift from the depth to that of - which i expained in the next indent: Reation (7) i to e modified a foow: t = π (8) π The foowing operation are focued on finding what change undergoe the time-period t and ength when a ma depth change from to - occur. A far a the formua can e oved with numerica method ony, the differentia are to e dt t ( t t) d ( ) d ( ) (9)

3 CHARACTERISTICS OF WIND-INDUCED WAVES IN A SHALLOW WATER ZONE 43 The indice (one) eong to the depth =-. The procedure then continue o that out of known parameter t and for deep water () parameter t and are tep-y-tep cacuated for depth ony a itte igger ( ) - i.e. each depth change poe a negative ign. In our cae the cacuation are to e terminated when depth = 0.0. When the depth i diminihed y, the period i changed y t and the ength y. Interreation etween the change i otained when differencing of equation (8), thu t = F + K (0) where t t F = ; K = () The new parameter t, and have to ie oth on ine (0), and on ine - =, the further decriing reation etween the new parameter - ee point X on the right upper corner of Fig.. At the ame time, eide equation (0) ao the foowing equation ha to e atified t = π () π Referring to reation (9), finay it i derived that (3( for aready known depth the earched parameter t, are otained y oving of the foowing equation F = F π π π π K (3) The oution primariy ead to ength, whie the time-period t i determined y the expreion found under the radica on the eft ide of the equation - ee ao reation (). The expreion F and K are ighty more compicated - they are partia derivation of a compoed function. Their initia reation i to e written a foow: t = h( u, v ) (4) where u h = ; u = ; v = ( z) ; v π z π = The foowing reation are vaid for expreion F and K: F = + (5) where = = z ( z) = π. z ( z) ( z) ; ; z = π coh z K = + (6) Whie / = 0 and h / i aready known from the precedent equation, the foowing needed to e determined in addition: z = coh ( z) It remain now to determine, what vaue might e conidered a deep water, and how ig iteration tep i to e choen in order to reach the mot poie accuracy of the cacuation. According to the mentioned tandard, deep water incude each depth greater than one haf of the waveength. More tringent criterion wa choen in our tudy - any depth that i igger than twice waveength characterize deep water (Spaniá et a., 999) Seection of uitae iteration tep wa determined empiricay - ee Fig.. The arget difference in the time-period t cacuation can e found in the cae when = 00, even in rather different iteration. In the firt cae for / = 0.05, in the econd one for / = 0.5. Detaied anayi howed that optima eection ie etween 0 and 0 percent - here it i conidered 5 percent eading to / = 0.5. Cacuation of wave height i aed on the aumption that enderne ratio of wave expreed a λ/ h remain practicay unchanged with change of water depth. Thi i ao vaid for ratio of reevant dimenione characteritic and v, thu v π t = (7) v Dimenione time-period t and height v eong to vaue meaured on reervoir deep water characterized y parameter. Ratio (7) might e derived from reation () and (5) to (7). In introduction it wa tated that in ome cae maer wave height and time-period are otained according to the tandard, however, moty it i true ony for one of the two parameter, i.e. either height, or time-period (Kranožon and Sidorova, 96). Whatever, thi or that, the rik of wrong determination of wave energy i in neither cae diminihed. It i neceary to decide how to modify or amend the tandard according to the actua tate of computer technoogy. Beide utituting of diagram in

4 44 T. Spaniá and K. Jahoda Fig. 3 and 4 for diagram in Fig. 5 and 6, there are generay two poiiitie. The firt i to find for t and v reation imiar to thoe given ad () and (3) expreing with ufficient accuracy reation given in Fig. 5 and 6. The econd i to preent required reation in the tandard o that any deigner i ae to prepare reevant computer program. The author of thi paper conider the firt poiiity a more uitae one. Determination of equation (3) root i decried in Fig. 7. The root houd e earched for within an interva e then / Thi expain that ony a very approximate oution coud e found in the time when the tandard wa introduced. 4. CONCLUSIONS Cacuation of ank pavement - eventuay of their araion when not paved - are aed on wave energy in ittora area, i.e. moty in haow water. Cacuation performed according to the tandard (ČSN 75055, 985) need extremey time-conuming earch in uppementa diagram. A numer of aternative cacuation have to e performed for dimenioning of ank pavement, o for reevant cacuation uitae computer program houd e deveoped. Thi further require knowing reation on which drawing of uppementa diagram preented in Fig. 3 and 4 wa aed. The tandard wa initiay prepared on data from former Soviet Union, and thu, effort (Kranožon and Sidorova, 96) to acquire the origina data faied. Nothing ee remained than to try to deveop the miing reation once more. After evera faiure there came a ucce that i hown in thi tudy. Sutantia enefit of the work i not ony in enaing computer cacuation ut ao in reut demontrating poie rik of underetimating of wind wave impact. In ome cae, the cacuation repecting the tandard produce ower vaue of wave height and/or wave time-period and thu, ao, ower vaue of wave energy. LIST OF SYMBOLS D - wave coure ength [m] H - water depth to ottom [m] w - wind veocity [m - ] h - wave height [m] λ - wave ength [m] T - wave time-period [] - dimenione depth [-] v - dimenione wave height [-] t - dimenione wave time-period [-] g - gravity gradient [m - ] REFERENCES ČSN : 985, Cacuation of wave impact on water reervoir and ackwater tructure (in Czech). Jahoda, K.: 997, New computing method of wind wave parameter, Acta Montana, Praha, Serie A, No (04), Kranožon, G.F. and Sidorova, A.G.: 96, Wave tranformation in haow water. AS of Ruia, (in Ruian). Kratochvi, S.: 969, Ociation progreive wave in water reervoir. Journa of Hydroogy and hydromechanic, No 4, Bratiava, -4. Lukáč, M. and Aaffy, D.: 980, Wave motion in reervoir, it impact, and anti-araive - meaure, Bratiava. Spaniá, T. et a.: 999, Technique of data recording and data proceing of wind wave ampitude, Journa of hydroogy and hydromechanic, No 3/999, Bratiava, ACKNOWLEDGEMENTS The author acknowedge financia upport for thi reearch provided y Grant Agency of Academy of Science, Suppementary Agreement No IAA Ing. Kare Jahoda paed away uddeny in June 003. The author acknowedge upport for thi reearch and preparing for thi paper.

5 CHARACTERISTICS OF WIND-INDUCED WAVES IN A SHALLOW WATER ZONE 45 Fig. Cacuation of change in time-period t and ength v. ma change of depth from to = -. Fig. Accuracy of cacuation v. iteration tep equa to. A) / = 0,05 t time-period B) / = 0,5 depth

6 46 T. Spaniá and K. Jahoda Fig. 3 Graph for determination of characteritic wave height h in deep and haow zone a dependent on parameter = gh / w and = gh / w (according to tandard ČSN ). Fig. 4 Graph for determination of characteritic wave time-period t in deep and haow zone a dependent on parameter = gh / w and = gh / w (according to tandard ČSN ).

7 CHARACTERISTICS OF WIND-INDUCED WAVES IN A SHALLOW WATER ZONE 47 Fig. 5 Graph for determination of characteritic wave height h in deep and haow zone a dependent on parameter = gh / w and = gh / w (Accurate cacuation). Fig. 6 Graph for determination of characteritic wave time-period t in deep and haow zone a dependent on parameter = gh / w and = gh / w (Accurate cacuation).

8 48 T. Spaniá and K. Jahoda Fig. 7 Scheme to approximation of time-period dependence t = gt/(πw) and height v = gh/w on parameter = gh / w for water depth = gh / w, when x = og, y = og t. Fig. 8 Approximation of dependence when cacuating wave height v = gh/w, a hown exacty on Fig. 5 =gh / w, = gh / w.

9 CHARACTERISTICS OF WIND-INDUCED WAVES IN A SHALLOW WATER ZONE 49 Fig. 9 Approximation of dependence when cacuating wave time-period t = gt/(πw), a hown exacty in Fig. 6 =gh / w, = gh / w.