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人q

uo110gl∂uuDリコDID爪01」6u王ID＝⊃SO

人1lD⊃拉」；）∧D L川八S∂人P〟し」∂lP八lO uO！lD」∂u∂9叫⊥

d ■ON l召Odヨ召

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」○

1召Od］召

THE GENERATION OF WATER WAVES WITⅡA VERTICAIJIJY OSCIIJIJATING FIJOW AT

A CHANNEIJ：BOTTOM：

By YoshimiGODA半and Tooru KIKUYA珊

CONTl∃NTS

SYNOPSIS

Ⅰ．INTRODUCTION

よⅠ．THEORY OF WAVE GENERATION WITH A VERTICALLY OSCILLATING

FLOW

ⅠⅠⅠ．SOME RESULTS AND DISCUSSIONS OF EXPERIMENTAL STUDY…H …15

ⅠⅤ．CONCLUSIONS

ACKNOWLEDGEMENT

REFERENCES

APPENDIX：Water Surface Profi1esin a Multi・Partial－Renection System．．．，……‥23

LIST OF TABLES AND FIGURES

Tablel．Magnitude ofI）isturbancein Terms of Wave Amplitudefor hh＝hb＝1‥”12

19

6

9

13

.16

Table2．Experirr）entaland TheoreticalDriving Force for Piston

Fjg．1．I）e負nition Sketch of Coordinate System

Fig．2．Integration Contours

Fig．3． Variation of H／2e versus h／L for b／h＝0．5，1，and2

Fig．4． Sketch of Test Channel

Fig．5． Comparison of ExperimentalValues of印2ewith Theoretica10nes．． ”．17

Fig．6． Variation of Driving Force for Piston versus Wave Period…‖ ．．．18

Fig．7． Definition Sl（etCh of Multi－Partial－Reflection System

Fig．8． Variation of St且nding Wave Height versus Relative Depth‖‖． ．”．21

儲Chief of ModelTest Section．Hydraulics Division，Sc．M． 琳Breakwater Laboratory，Hydraulics Division．

－ 1－l

ーー Z －

∴・ －い∴… ∴J・：∴∴、√．・：・∴－：ご：ニー、、－－＝∴小・、、・：∴・．、l

箪笥屡‘引立静謝賂若 く＝川宰款相場捗珂瑠剖悶ぽα門型許関目甲拍感r

馴塞尋萌超丁“一∠ミ牢1／｛ィ′≠拝謝軒αモ‘首そ｝＼へGつ4糞貸さ苛｝しアブ樫庄αコ

∴∴ト、∴：ゴ∴ －・・十．：－∴∴ ∴∴

軍票f‘坤暮雲融マ％9‖駅路拍機種知恕賦酌折行雄稲塚‘・1￥壁軍持∈9社寺市￥卑肇1

し㌣部捌墾炬覿鋸＝パ用托H嘩佗噸悌‘封む苛宜丁6祖ユ鼻嚇車 0管管ミ軍等′￥管壌要そ

・こ－、・さ：ニー・；丁、：－ ざ∵∴－・∴こ－∴・：・・∴、∴∴ニ∴－、－：■∵－・・十∴：て‾．、・∴

＼√ 二、∴・ ン、；－、・い

“ユて1し切望庄軍票9了剥製曜＃誕尊、、

●ApnlSユapUn S∂Jnlコnユ1SI∂pOu工

pt陀∂lppl≡d a＾1〃h atIIUa∂叫aq UO！7つ叩∂・I・∂A一三Jh－ⅢnuIJO Stu∂tqOld∂qltIl！瓜p∂lqnOll

u∂aq S叫qつⅢ血 判UIZl∂A一三瓜 te叩∂叩J∂dx3 tle Olp印93q OlauO8uts叩OJd Aユ∂A

甘S！JO叩J∂U∂33Ae凡 才O ∂dAIS！ql‘叩墾つ切∂0つ UO！lつ∂冨∂J瓜OISⅢ1JO∂Snl三つ∂宜

●l申！∂q∂Al〃止JO SuJ∂1U！叩aつユ3d9t∂q OlpUnOJOStl≧Se瓜JO叩ユ3U∂方∂A一三爪∂ql

JOiU∂！つ闇孤つ UO！P叩∂ユ∂qエ ●s∂UO t一三つ！豆1官ulト靴甲ユ媚加tlU∂つJ∂d o乙∂ⅦOS凱甲q

uoIS！dJ3叩Jhlapun ut？3A！1POlpal！nb∂1つつJOIaqlpUIZ S∂UO tl三つ！1J（tt”t2JOIU∂つ・Iad

O8凱甲q S川和叫3A官瓜田叩ユ∂U∂冒3ql：p∂dot∂AapAJO∂ql叩‡t11！瓜iu∂u∂∂J加ユ叩J

瓜OtISユ0叩ユ∂u∂ヨ3A官八手0∂dAIS叩りOJIS∂I∂t12つSlteuS甘JOS叩S∂Ⅱ ●p∂叩∂S∂Jd∂ユ官

∂JnSS訓戒1ⅢOl叩q pt陀S∂柑0ユd∂A℃瓜JO手SUO！SS∂ユdxa pu官‘p∋叫官1qO u∂∂qS官q S∂A∇瓜

p∂叩ユ∂u∂万一瓜叩ⅤつnSユOJl叩u∂10dA里つOl∂A手OtlOT叩lOS3qエ ■uO抑〕ql∂Ut陀qつで1召瓜叩

3叩甘tt！〇SO」（tt叩1月Aセql叩甲1以∂U∂ヨ∂q U官つS∂Ae瓜ユ∂叩凡才OtI！叫ユ℃ln8∂JV

SISdONAS

LIST O下1SY－MBOLS蓑

AD ＝amplitude of orlglnalprogressive wave

b ＝half－Width of wave generating slot

B ＝Width of channel

e ＝amplitude of piston movement

f（x，Z）＝function for velocity potential（see Eq．4）

F（u，・2＝）＝Fourier transformation of f（x，E）（See Eq．7）

F ＝force required todrive the piston

g ＝aCCeleration of gravlty

g（w）＝integralld function for G（E，Z）（see Eq．19）

G（E，E）＝nOminalfunction for f（x，．2・）（See Eqs．17and18）

h ＝Water depth

H ＝generated wave height

Hs ＝Standing wavelleight

Bも ＝height oforlglnalprogressive wave

ヱ ＝imaginarynumber（＝Jニロ

L ＝WaVe number of27T／L（see Eq．24）

ん ＝imaginary wave number（see Eq．25）

K，t ＝reflection coe流cientin terms of wavelleight

l ＝distance between a verticalwalland the source of partialreflection

L ＝WaVelellgth

フ之 ＝i11teger（＝1，2，3，・．．．）

♪ ＝bottom pressure

P ＝tOtalpressure over the wave generatlng area

R（α）＝reSidue of g（てU）at pole of w〒α

∫ ＝time

T ＝WaVe period

u ＝integration variable for f（x，ヱ）

U ＝maXimum ve10City of vertically oscillatingflow

W ＝COmplex val．iable fol－theintegration o土g（w）

W ＝Weight of water mass movingln the same phase with the piston

Ⅵ′ ＝aVerage rate Of work done by oscillatingflow

E ＝horizolltalcoord■inate measured from the center of wave generation

Z ＝Verticalcoordinate measured upward from the channelbottom

＝Subscript refel’rlng tO theinside of wave generating area

＝Subscript referI・1ng tO the outside of wave generating area

｛＞Only the most commonly used notations are defined her？・Notations not generally

used throughout the study are de丘ned onlyin their place of usage．

－ 3 一

・‾ ト’‾

■S∂AIZ凡p∂叩・l∂ua男心叩JOJl叩ualOd AIPOt∂A‡O uO了叩lOS＝

tlZ‡叩aIOd直甲Ol3A＝

ヱ／粟ZJOノ（つu∂nb∂ユJ・柁In凱柁＝

ユ∂l甘瓜‡O AllStT∂p＝

（…■t■691テ1■E）1t柁1SuOO＝

（gl■b王∂3S）？－エ■JO？＋ごJOトガロ叩U印SJ∂】∂ⅦeJ官d＝

3190Jd∂At2瓜aA†SS∂J眉oJd〒

∂柑OJd∂∧官凡3人！SS∂J窟0ユl∂J＝

（侃）βJO∂10d＝

∂Ae此方u叩u官1S OI凱【lユJ∂J∂Jld！JOSqnS＝

¢・や 〇一d 〃一号了比 弓↓ p 倉

Ⅰ．INTRODtTCTION

In the planning alld desigllOf harbol・StruCtureS alldcoast protection works．

modelexperimelltS areilldispensable tooIs fol・engllleerS．The stability of a

proposed breakwater against示aveactions，thecrowllheight of a sea wallto

PreVent the overtopplllgOf waves，and the mostefEective arrangement of brealく－

WaterS tO Shelter a harbor from waves，are a few exalTlples of the problems

wilich can be solved successfullythroughsome scaletests．Thesemodeltestsalld

Other fundamentalexperimentsal‘e Carried out in a test貝ume or tanlくWhichis

PrOVidedwith some wave mal｛er．

One of the problemsillherent to modeltestsis thesimulatiollOf the open sea

Where alloutgolng WaVeS die awayinits vast area．The modeltest has to be

Carried outilltheboullded area of a wave tank．Waves propagating towarda

Verticalwallof the wave tank are renected at the walland tend to disturb the

Orlglnalwave system，ullless some devices of wave ellergy dissipation are pro・

Vided．For this purpoSe wave absol’bers areinstalled at the end of a wave tank

and，in some cases，along the side walls too．Butin the caseOf the stability

test of a vertical－Wallbreakwater，incident waves may be re8ected total1y by、

the modeltreakwater．If the wave makeris made of a vel－ticalplatelTlOVlng

t）aCk and forth，Or up alld dowll．the wavesl・e鎖ected by the modelbreakwater

Willbe re負ected again by the wave makel‘and disturb theincident waves to the

t）reakwater；hence the test can not be continued anyll10re．

The current technlque tO aVOid this multi－re8ection of test wavesis eithel．

to complete the test befol・ethe alTivalof re日ectedwaves or toinstallwave丘Iters

illfront of the wave maker．Butlleither methodis satisfactory；the fromel‘for

its short time available and thelatter士or stiu remaining re且ected waves．

A promislng method to soIve this multi－re8ection problemisto generate the

WaVeS With averticallyoscillatillg負owat thebottomof awavechanllel．Thiscan

t）edonewithanullderwaterpistonwhichmovesup and downin averticalcylinder

connected to the challnelbottom．Since this type of wave maker does not have

a verticalplate which causesthe wave re負ectioll．itis considered that therewi11

belittle re8ection from the wave maker．Actually a similar typeof wavemaker

has been constructed atKyotoUniversitylll1960and has been used for thestudy

of wave overtoppings for sea wa】1s（Iwagakiet．al．1962），Butit seems to the

authors that tllere has beellnO allalysis of the wave genel・ation mechanism and

littleinformationis available for the efBciency of wave generatiollalld the power

required forit．

－ 5 －

‾ 9 ‾

一（6S■d）096Tユ∂司01SJ鞘官∞耳＝∬叩S∂Ae瓜∂A！SS∂J岩OJdJO t∬uOJ

叩1∂AIZq tl甘qSl叩u∂10d AIPOt∂A∂ql‘s！1甘qlfs∂uO∂AOqe凱IlOluO印puO〇3JOtU

∂UO ppt20Slt2 ＾t2uJa瓜 一atr［！1S！プ pttt2‘LL凱甲q pO！Jad aAt2此∂tIlt印JVLL／yzJO

A⊃u∂nb∂JJJeTnjヲu官印1S！β‘此0甘凱Ⅰ！1甲l！つSOJO∂p叫！u冨甘u叩1S∂10u∂pβqつ叩瓜叫

1′＝Z］岬0＝β＋

0＝Z叩ヴ＜叫JOJ

。＝叩酬J。‥βS。。月ご‡＝還…

u∂1S」（S∂叩Il！pJOO〔＝O11〇1呵S uO！l叩リaq ■t■方！止

†♪POつ∩ 瀬畑川i

q一

7■∑

l

㌔

：（T’B！且aaS）st2P∂SS∂Jdx∂∂Jt2SuO叩puOつ人相punOq aql‘？＝r Olq：＝r uIO．1壬

uollOql∂uu叫つ∂qlJO叩tS叩1叩 プβSOつ月JO瓜0甘l甘つ！リ∂A凱Il叩11！つSO AltuJOJ！un

JO pue uO！10uIapnⅢdt∬t2 ttt2uISJO SUOT］dtunsst2aqlⅦ！此 ■0＝2：‘tuollOq T∂utt12tP

∂ql飢10lt2ptte Ly＝2：’aっeJ・InS・I∂叩此 aql叩 uaATB 3，It2 SuO印puO？J（Jt2punOq ∂tIエ

0＝▲＋

：uO耶nb∂S。∂つ甘td甘1

∂ⅥS叩S！叩S tIつ川瓜‘¢‘l官印柑叩d」叫つ01∂A3qlJO∂コtt∂1S！Ⅹ∂∂ql∂unS∂ユd snl∂1

冒Ⅶ0！I！pⅦ0〇Aユ叩ⅥnO圧pⅥ℡Ⅵ0耶叫柏t甘叩au叩ⅥnO止

血01dll虻ⅠエアllI〇SO

AllV〇ⅠエⅦ宜A V耳エⅠ〟L NOIエⅤⅥ闇人指に）ヨAV瓜且O AⅦ0Ⅱ耳エ ●ⅠⅠ

’℃叩p t甘叩∂uリ∂dx∂∂tuOS tIl！瓜Au訓つ皆∂飢Ⅰ甲ユ∂u∂3∂A吼肌l叩1∂JO叫1JO UOSト柑dしuOコ

∂qlS∂SSnコS叩pt柑 AJO∂qlaⅥJOlu∂udolaA∂p叩1Ⅵ叫Sl甘∂plJO由」叩∂S∂Jd∂qエ

●飢IIJ∂∂uI凱柑l甘1S甘0コ‡Opla甘叩1u！S∂叩叫St∂pOtUJP甲u＝）∂∂つOJdpu甘∂つU∂pりt10つauOS

Ⅵ！血J∂彗官u∂AIZ瓜∂t11u男！S∂p uロコSJ∂∂u侶u∂1甘qlOSJ∂司甘u aA甘凡才O∂d」く1S川＝0壬

uo耶J∂tl∂3∂A℡八っqlJO」（JO乱n e dolaA∂pOl∂p甘u u∂∂q S叩1dtu叩甘耶∂つu∂H

Since we are seeking thesolution for the velocitypotentialof oscillating for血

in two dimensionaldomain，the velocity potentialcan be writtenin the following

form：

¢（ヱ，ヱ；f）＝一′（∬，ヱ）cosJf

Then the function f（x，3Z）must satisfy the Laplace’s equation and the boundary

conditions of

for ■∬l≦∂

（富）ど＝。＝イ㌃ 可＞み

′恒）＝0 （雷）乙＝乃－÷

Fonrier Transformatiom of f（∬，ヱ）

The solution for f（x，ヱ）can be obtained by means of theFourier transforma－

tion．Let F（u，ヱ）be the Fouriel．tranSform off（X，3Z）such that，

勒ヱ）＝／00β－よ・こゾ（∫，ヱ沖

′（ェ㌧Z）＝去／÷岬（“，ヱ）dz‘ The Laplace’s equatiolland boundary conditions for f（x，ヱ）are then rewritten

forダ（〟，ヱ）as

一岬〟，g）＋＝0

∂ダ 2こJsin〟ゐ

∂z′‘；0 〃

（富）－＝九＝手鞠九）

Since the generalsolution of Eq．9is glVen by

ダ（〟，g）＝Cl（〟）β】ほ＋G（2‘）g‾〟ど

thesimultaneousequationsfor Cl（u）and Cら（u）are obtained fromEqs．10andll

aS：

廻 cl（〟）一C2（〟）＝ こ

JJ

（〟一子）♂油Cl（〟）－（〟＋号）β一叫G恒）＝0

By determining the parametric constalltS Cl（1L）and C盲（u）from Eqs．13and14，

the function F（u，R；）is obtained as：

Usill〟ゐ［（〟訂＋J2）β〟（‘‾ル）＋（昭一J2）β‾“（‘‾九）］ ダ（〟，ヱ）＝ ‥ （15）

u2（0－2coshuh－ugSinhuh）

－ 7 －

‾‾ さ’‾

一uo！1〇∂S S叩1nOtl茅noJqlpa！lddl≡S！SttB！S∂tqnOp凱IllO‡UO！1

－℃lOd s叩！0ヲヨユりu宕！SJ∂爪0†叩pu甘0＜ヨユOJSTOZ・bヨ：才0叩TSpu叫叩印∂ql

u！StIBてS alqnOp∂ql‡O tTBts・1addrL aqL －S3tOdJOJ∂qtunu叫1S！ut pt陀‘～Ep＝侃1t2

∂lOd∂qlJO∂np！S∂・T∂qlSて（彗P）ど‘・∠JO Sn叩甘ユ官ql！血管pt陀0‘グー＝m∴岬SJnO13p

∂qlJOト等 pul≡ Og‘守一∫‘y一岬no∬耶ト17 ■uOユJ∂一つコロー叩∂Sl≡ S∂10tI∂p ygⅥ〇川瓜叫

l＝－7J

（■1P）首エア〟Z亭＝ 一代

∩紳）βyJ＋姉巧＋頑呵β叫J＋侃梱β♪ ．1＋守⊥ l・・一 Vs

∩紳）β■‘コ 叫（叫β／申紳）β叶ノ伊納βソ＝の7 打‾寺

：Sl∋

u∂1叩瓜∂ユS！∠t－bヨ：JO UO耶ユ冨alu！∂ql‘u∂ユ0叫1∂np†詣ユS。AtI〇n℡〇∂qlヨuTA【ddv

‘S！Ⅹt！－n∂qluO SalOd atllp！0へt：01paP！AOld os

∂ユ甘SユnOl∂p叩エ ■Z●ヨてd u！p∂q〇1∂司S S甘‘0ヲヨJOJJl電qユ∂瓜Ol3ql叩pu℡0＜ヨユOJ

au官Id 〟一花JO 甘肌【ぷddn∂qlu！γ p【柑‘0‘す一＝皿叩SユnOl∂p ql！瓜al〇Jロー叩お℃

S甘口∂1℃1SてuO！叩」冨∂1叫∂ⅥJOJnO叩0つ∂qエ ■皿‘∂lq叩℃A X∂tdtHOつ∂qlAq p∂つ℡ld∂J

凱甲q‘乃‘．∂lq町潤A uO！1電ユ冨∂1叩∂qltⅢ瓜∠t‘b3JO pulヲユ三bltT！∂qlS！（侃）β甲四帆甲

∂

皿押）β／＝舟7

：∂u甘td

〟一プチ∂ql【Ⅰ！‘nl‘∂l甲！ユ℡A Xaldtuoつ凱Ilql！瓜uO！叩ユ云blu！凱Il爪0【lO巨∂t11ユ∂p！SuO〇Sn

l∂1 ■uO！叩ユ冨∂叩！Ⅹ∂1dⅦ0つ‡O pOtll∂u凱IIAq p∂叩nl℃A∂aq甘酢）UO耶ユ苫∂叩！叫エ

や‘ご）、／才0ⅦJOJIPTldx∂

ulZ u！電1qO OISIZ OS エt ●bヨ：叫 UO耶ユヨ∂叩∂ql∂叩nltヲA∂ 01S！u∂tqOユd∂qlSn¶エ

［（巧－ご）〇－（再＋如］＝（芸卯

：S甘（Z‘ヨ）〇1弟l爪l声SS∂ユdx∂S！（g‘ご）′uopつunJ∂qlu∂ql

（γ児qu！Sβ乃一曾7ZqSO〇乙β）乙乃 ∞‾ ′ ′＿．‘‘。、〈

J＝（ズ‘∋）〇 （∠T）・‥ ・フェア ［（γ一ご）プ7qu！Sz♪十ぐリーg）乃qSO〇餌お”｝∂

（g‘∋）〇uOrPunJ凱Ⅰ！爪OllO巨∂ql∂つnpOユ1叫∂凡打

（9t）…

フェア【（？－〇）れ亨∂一（叶ご）71I∂］

［γ乃qtt！S伽－γ乃q∽つ石♪］z乃

（1／一方）乃qu！Sこ』十（γ一三）乃qSO〇β？Z

［‡／JZqu！Sβフ才一γ7ZqSO〇石♪］z乃

’JJ’

［（ツー可7Zq里Szβ十（【／一之）柁qSOつβ呵マフZu！S葺ケー芦∂

：S電9’bヨ：‡Ot10！1el∂ユ∂qlql叫p∂泊AO〇∂一tS！（g‘ご）′Ⅶ叩つunJ

∂ql‘sT－b王U！p叩uユ叩p凱甲q（ニ‘乃）guo！1つunJuO耶Uユ0声u叫凱11昭！瓜

Ⅶ0耶ユ眉a叩ⅠⅩaIduIOこ〉℃岬no叩l（Z‘∬）ノJOlエ0耶nt℡A：彊

l⊥

、（。

号＜○

Fig．2．ImtegratioれC（Int（Iur5

Asit can be seen readily，the part of the complexintegrationalong the

uTaXisfrom pA to Ais the samewiththeintegratiollinEq．17，prOVided the

radius A being extended toin丘nity and thel‘adius r being reduced to zero．

Therefore G（E，R：）canbe obtained from the following equation：

A→∞ Gほ，g）＝悪（±2痩馳卜左9（ひ）dひ－／9（てU〉dひ むた 一存（ひ）dひ一左伊（w）ゐ‡

The terrnsin the righthand sideof theaboveequation are evaluated onebyone

as shownin the followlng．

First，the singularpoints of g（w）are obtained as the zeropolntS Of thedeno－

minator，i．e．，

ひ＝0

J2＝9ひtanhひん

Equation23has thesolution o土w＝rk and尾，andてU＝－ikn andikn such that

J2＝9尾tanh長鳥

㌔＝一夕見ねtaIlゑ％ゐ

These equations determinethelocationsof the poles of g（叫）．The condition of

【 9 「

】 Ol‾

：S甘u訪日リ瓜∂ユ甘Sl叩S乱Ia甘エ ■9Z■bコ叩∂SOtp Sマメセ瓜∂u甘S8Ⅵ

叫p∂叩nl払∂∂J官比ぎターpu甘℡′ぎブ＝nユ叩S叩OdJモーn3u－S叩S∂np鱒ユ∂ql‘Ⅶ皿0且

て／智q耶1ぎ （雪ぎZ＋γぎZqu！S）￥ 二・、∴・ （6Z）・t・ ‥［（ ズーV）＋ T 如SO〇ズぎqSO〇予

▼く、ロヾ †ハ紳）βJ＋榊）り＋彗卯－）巧 澤t

‘∂〇u∂H ●（侃）β‡0ユ0叩叫山OU∂p叫1

JO叫叩Aてユ∂p叩S！（血）ノロpu℃（ハエ）β享OJOleユ∂unu∂qト10‡SpU叫S■（侃）〃甲‡q仏u‡ 0＝恥

禦㍗拍一方）＝ ［

（雛柁＋γ冒Zqu‡S）ぎ q‾＝1托

一＝ ［焉箸］＝（グー）冴 方ぎqSO〇i耶－∂Z

町卑m［1宗］＝（ぎ）冴

（写ぎZ＋ヤぎZqu！S）ぎ

gぎqSO〇∋那∂Z

：Sl…p∂叩tn〇Tl≡〇∂J官S∋np！S∂J∂S；〉qエ

‾ ［（軌（0）冴・（㌢瀞±＝†n子紳）り中州β0ソ中州gソ渾 ，＼ヾ

：S官（す）首pu官‘（0）首‘（ぎー）冴S∂np野ユaql甘け肌

凱IOl官SuO耶ユ鮎叩！∂∂∫甲‘pユ叩ユ p∂1でnt甘A∂ ∂ユ℃．ぎ pt陀‘0‘ぎ一＝仇 叩 SユnOl∂p

瓜ク（侃）り声 」l－ 0＝ヲ 羊苛盲ヲ ト∴▼ミ、

〔〃－l・’

‘∂〇u∂H 一石叩n p∂∂〇Ⅹ∂10u S∂Op

pul≡J鮎叩‡∂AOq甘；呵1JOJO叩ユ∂Ⅶnu 叫l‘0ヲヨJOJ瓜‡O t堅uOpJ∂瓜Ol∂ql叩pue

O＜ヨユOJ′つユJO t準uOpユ∂ddn∂ql叩u∂司官1S！uO耶ユヨ∂1u！叩l‡0ユnOluO〇∂ql∂〇u！S

J 00←y Ⅶlt≡；

1／乃ユqu！S岩仇

（γ一方）∩ユqSO〇抽l∂

侃ク（呵β’シ匝 ［T一（γ侃quqβ鶴）／石β］γ侃Ⅵu！S石nl

［β侃／（ヤーズ）侃qul≡1石β＋l］（甘一昔）nユqSO〇沖∂

：S官t泊SS∂ユdx∂S！

刀←y t印凡UO叩頭叩卜叩＝01叩！t叩l‘s∂S甘∂ユ〇里ハエSl≡叫un tI〇官OJd加てヤー方）侃

q耶1p肌＝押＝叩勘‘翫0！1〇UnJ∂ql∂〇u！S 一句！u甘U！01p∂pu∂lX∂S‡y S℃OJaZ OlO方

01u爪OqS S！Ⅴ一OlyⅦ0ユトドg叩ユn∂方ユ叫叫l凱IOtl：uO耶ユ方∂1里aql‘pⅥ0〇∂S

－SuO！1〇∂S・柑叩t u！u；〉∂S SIZ－1；）qlHnu；仏で瓜叩1；）q Ol

Ⅵ瓜OtIS S！すA耶叫〇Ⅴ －7JO tIl凱旧t∂AIZ爪atn ql！瓜S∂A℃瓜ユ∂叩瓜∂pn里1dlu官tl官ⅦS

aqlユOJ7／影Z‡OJ∂qunu∂A官瓜aql岩u叩∬S∂p 叩ql二S官 ∂ueS aqllSnFS！予Z・bコ

e封■れ〔cos鳥乃ヱ 属（±f鳥兜）＝2

足先（Sin2ゑ侃／と＋2点乃ん）

Substitutingtheresultsofthecalculations，Eqs・26，29，and30，into Eq・21，

we obtair）the士unction G（E，3＝）in the following form：

β耳七兜∈cosんIZ COShk£COSkE

Cほ，g）＝如才iチ，邑 一4 尾（Sinh2銑＋2紬） 尾m（Sin2ん訪＋2み誼）

＋＋（エー ん）

Asstated before，theuppel・SlgnS Of thedouble signsin therlght handsideof

the aboveequation arefor E＞O and thelowersigns fol‾E≦0，

Thefunctionf（X，ヱ）isnowcalculatedfrom Eq・1Bwith G（E，R＝）obtainedin

Eq．31．Thecalculation must bedone separately for three rallgeS Of x：i・eリ

エ＞あ，み＞エ＞－あ，and－あ＞エ．

For thefirst rallge Of二ご＞b，

e－L’，Lllsinl1knbcosknh 一 仙ヱ）＝4坪票謡㌶こ；誌㌢意

Forあ＞エ＞－∂，

… （32）

足先（Sin2んめ＋2尾，iん）

胸ヱ）＝4叫亘e芸憲…£㌶謡ヱー4C豊1認諾㌍

1 1 ＋（ヱ‾ん）‾十亮tふ云ii盲元r

For一あ＞ち

ふ・ヱ）＝－4Ui豊…浩子軍 讐 el：71缶sinhknbcosk創文 ト・・ （34） ＋∑

左±1‰（Sin2ん諭＋2点，▲ん）

1tis shown that the above solutions off（x，R：）satisfy theIノaplace’s equation and

the boundary conditions of Eqs．5．and6．

With the function f（x，R；）evaluatedin theabove，the velocity potentialis

describedillEq．4．

The S（〉lution of Generated Wave Profi1e

Since the Laplace’s equation and the boundary collditions arelinear with

respect to¢，Wemay Superimposea velocitypotentialof the followingform，4）s，

inorder to obtaillthe velocity potentialfor progressive waves：

小慮＝ム（ヱ，エ）sinJ∠

where二

sinkbcoskxcoshkヱ ム（ェ，ヱ）＝4Lr

点（Sinh2抽＋2抽）

ThelleW Velocitypotentialthus formedis written fol■therange oflェl＞b as：

－11－

■ －Zt‾

：St∋paSSつユd）：∂U3qIS！

陀相計叩即卸媚す昭uOユ‡A℃爪官3190ユd3人℃瓜3qエ ●』∂ql！仙p∂01…t由JS＝つ尤甲Ol∂A

叫l‘』‡O AつU訓1由J‡J官In節柁∂Ⅵpt陀∂さ0∂p口和d一江e u官ql！凡J∂Uu官Ⅶl官p！OSnU！S叫

A叩2〇！叩A 岩utAOtU∂1呵d 匹叩OZuOq官 t押坤p∂〇npO．Td s！瓜叩如P官IT！〇SO別口JI

ⅦnS ■柑PU！召Ⅶ3J S＝て‘ l Z＝“

Ⅶユaエ1Sユ1d／uJaエPuOつ∂S

ーr＝ヴす＝リグJOJ

apnⅢdttIV∋入電瓜；O StUJ3エU！aつUl≡qJnlS！uJO∂Pnl！U万甘N ●T‡n柑Ⅴエ

▲∬JO∂S協ユ〇U！∂qlt鞘瓜ul・t∂1ptIO〇∂S aqlJO∂S閣ユつ∋p p吋セユOqIS∂叩ユ1Snll！

坤℃∂l〇 ∂ldtuでⅩ∂ 鞘エ‘T＝鞘＝γ曾‡O t10叩pt10つ ∂qlユ0‡u山勘 pUOO∂S pU甘1S巧

a111JO tIO甘ent12A∂le〇！ユ〇untIe享01lnS∂ユ叩1S爪OtlS T∂tql三エ ●uJ∂11S・Ig∂qlJOleql

q］てJh UOSut2duro〇uち p∂1〇∂lB∂U ∂qll！瓜 ∂〇ut2qln呵p S仰JO apnl！tT812ur aqlS∂St2〇

1SO山uI’甘∂J甘加一昨J∂U∂冒乱IluOJJJ官JS∂0ヨ1！Sl≡ムーp亨d甘ユ∂p叫甲帽切∬Sl！S凱阿叩叫p

∂コU甘qユn呵p訳Ⅰユ ■甘∂ユ℡ 3u耶・l∂U∂3叫1pUnOJe p∂J∂1tI∂〇∂〇u甘qnJIS叩甘 SltI∋S∂Jd∂J

S；〉！ユ∂S朝里増田！Ulヲtl…叫〇℡Uf S亨q〇Ⅲ瓜uJ∂1ptTO〇∂S；〉qエ ■∞－＝＝ニr pUl…∞＝ニニr p北肌01

甘∂・Tl∋ 凱Ⅰ卿・1∂u∂冨 ∂qluDJJAE血管 a叩ヨedoJd pulヲ p∂叩ユ∂u∂冨 ∂J甘 ∂P叫！ldtu甘 aM電S

乱11t11叫S∂Al…爪‡O St準ユIO」叫 叩qISつ叩つ叩U！ムーユ甘∂lつ gC■b五‡0 Ⅲユ∂llS叫叫エ

†叩S亘立≡葦 十（押」半可射SOコ

1／す買＋γすZtIU！S

碑U！Sγすq叩S

†‘＝才

［芸］て＝血

：St＝l12亨叩∂10d j叫⊃Ol∂A S！tIluIOIJP∂U叩qO u∂qIS！a【gOld∂⊃t2JJnSl∂叩血∂qエ

！・卜・．こ・∴・ごい・．・‥トトこ‥：－・；il、ト・・・こ、・ご・・・・、丁・川∴・‥、・・：・・い吊い・：．．∵‥∵．● ‥・∴・．・、卜・∴い：・・ミ√

JO Salり0ユd∂qI∂S11甘〇3q‘uo耶！ユ電人声皿SS3ユd∂叩

S！9＞＼ご＼‡03如官ユ∂里ユ0‡l町叩胡Od 郎〇Ol∂A

ユ0‡uO！paSlX∂u叫IUT p∂1U∂9Jd

爪凱【凱ロ コO‡UOTSSヨユdx∂∂qユ

r＝祝

■ こて＋

（て／鮎ぞ乙＋γ！‘すZU！S）！‘す （め………………‥†プ♪ヲ0⊃

が隼SO〇？’‘ぞqU‡S酔‘；卜∂よ

（叫ご伽s・（等認諾欝） 月ア蠣＝

叩U！S（Z‘∬）ゾ＋7・pSOつ（g‘ご）て「＝や＋け¢＝P¢

sinh尾ん Sinkbcos（klxl で＝4g

Sinh2紬十2紬

Thewave generatinge伍ciency，Orthe ratioofthegenerated waveheight to the

Stl．Oke of the unJdel●Water piston，is then obtained from Eq．31as：

ガ 4sinh尾ゐ sin尾∂

2β Sinh2兢十2肋

It willbe readily seenin Eq．40that this type of wave generatoris very e伍cient

illthe shallow water butlosesits e伍cieny as the watel・becomes deep．This

Characteristicis the opposite to the ordillary Vertical－piston－type Or flap・type

generator which has a highe伍ciencyillthe deep water．Another characteristic

isa peculiar nature tllat thewave height becomes nullwhenever theslot width2b

L2

t．1

l．0

」⊥0・9

2e 0．8

0．7

0．6

0．5

0．4

0．3

0．2

0．1

0

0．O1 0．02 0．05 0．1 0．2 05 1

名′L

Fig．3．Variation of月ソ2e TerSllS町エf小一わ／鳥＝0ふ1，and2

－13－

ーl・t －

：Aq

U∂A！岩s！‘∂JO事∂J叫1‘∂つJOJ凱1！Aup∂qエ ■仲－bヨ：JO∂JnSS∂ユdl甘10l∂qlOIUO甘押p甘u！

SS甘uユ∂叩瓜∂里享0∂つJOJ【叩ユ叫∂qllSu！嘘e u∂Aリp∂qlSnt山ⅥOIS！d叫エ ●uoIS！d

e‡0叩∂u∂AOul∂qlql！瓜∂S12qd∂u甘S乱Pl里∂叩Il！つSO Olp∂つJOJS！司ulヲ1∂p！S∂qlU！

pu12・laPu！TAつItZ叩J∂A atllu！・l∂叩JhJO SSt2u叫1‘sastz3∂S∂qluI●・l∂pu！一人〇∂qlOl

p∂13∂uuOつ】叩佗1∂p！S一三u！・101∂p叫IJ（つl叫U∂A t2ulpaつtZld3qlSnul瓜0甘飢Ⅰ耶tt！つSO

叩1∂叩J∂u∂君 OluOl墾d で‘J∂A∂瓜OtI‘sJOl電ユ∂u∂君 ∂A℃瓜JO Su冨！S∂pl電叫つ電 uI

■uOllOql∂uul∃qつ∂q101JIZ∂u

ム【∂A p3つモーd uo呵d官JOユOl山0叫Oq t∂UⅥ叩つ凱IfAOtu℃JO∂つJOト凱一丁A！Jpunl叫Ⅹ電Ⅶ

3ql∂A！BII叫 ∂・lnSSaJd tunt叫Ⅹt2tLttzIPnS’9P Pt陀1師‘Sb丑 tIlfJhぎJputZ7dPJO

uo耶nl電人∂∂qlJ∂耶Pケ■b互1∬OJ‡p∂1ndtuoつ太一！S甘∂S！∂JnSS∂JdllZ叩1Ⅶntu！ⅩeⅦ∂qエ

（γすZ＋望柁月u！S）γす解 朋凱γ鴫＝gJ

9￥岩tT！S

（弼‥・・・‥∫旦竺竺里＋卜 （膵叶紺叩叩）昭甲石－ t l 々す乙叫S

（9郡曾Z＋9雅ぎZu！S）γび粥7‘す乍碧 ）堅飢γ吸＝甘 ：・

（。－′甘岩＿∂一l）

qlp！瓜t∂uu12qつ＝g ニ∂J凱I爪

q▲－

㌻PSOつ甘＋プ叩S甘＋γ9がdz＝∬〆。Jg＝J ：S12u；朋り瓜

S‡uO耶J8∂1uf∂ql享0れnS∂J∂qエ ●9ご∬019－＝∬uOユJCP －b王手O uO耶ユ苫31u！∂里

J（qpau！t21qOu∂qlSf‘J‘瓜OtIB叩tZlt！3SO叫l‡O12a・ltZ aqll∂AO a・一nSS∂1dlqOl∂qL

プ♪SOつ（0‘∬）少d＋7♪ufS（0‘∬）少d＋写財＝♂

：StZ31nSS∂JdlnO粥Oq叫1Spl∂！J（lP●b3rOluト抑●bヨ：‡O uO！1n叩SqnS∂qエ

プ♪叫Sゾ＋7♪SO叫㌧＝坤＋ゆ＝7の

■■3●！：gE－b互Aq u叫岩8¢JO

pt陀∈C■bヨ：叫 u∂A！冨ゾql！瓜坤才O t山nS叫1S！∂S甘つS叩ユOJl叩u∂10d叫つOl∂A∂qエ 0三’

［芸仲一呵d＝♂

：S電uO耶nb∂S‘引l叩Otu咽p∂ZfleJ∂u∂ヨ∂qluOJJp∂叩1n〇teつ∂q u甘つ（0＝g

pt柑マラl一可）電a・陀凱1耶ユ∂u∂君∂A電瓜叫1JOt∬0粥Oq∂ql叩uO！1叩電A∂JnSS∂Jd∂tTエ

℃∂ユⅤ別Ⅰ！叩ユ∂Ⅶ∂〇∂⊥lV瓜叩1叩∂ユmSS∂ユdlⅥ01lO往

●γ0●石pし陀‘γ0－l‘7／g■0＝ヴ才O S∂SⅨ）別口JOJ

AつU∂！つg∂飢Ⅰ耶ユau∂3∂小官瓜∂qlJOし【0！呵ユ℡A a里S瓜OtlS g∂Jn君‡且 ■uO耶ユ∂u∂君∂A電瓜

∂A叩君抑ql‡爪uO耶J∂u∂8 ∂A鼠仏3A印SOd‡O uO！1Ⅵt∂〇u12つ っqlO13爪p S12p∂u！召tdx∂

∂q血u s叩∴p∂1アユ∂u∂眉叫 01qlヨu∂l∂A電瓜∂qlJO∂ldⅢnut 電 ql！瓜 田pfつu！0つ

F＝Pscosqt・（如誓w）sinqt

where＝ W＝Weightofwatermassmovlnginthesamephasewithpiston・ Someexamplesofnumericalcalculationsby Eq・47aregiveninTable2in the next chapter．

Allother interesting problem is the average rate of work done by the oscillating丑owtothewaves．Theaveragerateofwork，町iscalculatedas：

2む 面＝去C√♪ト貰］ヱ＝。ゐdf ＝去蔦

2

＝与J： 2

SillCetheilltegrationsof coso－tand sin20－tOVer OneWaVe Period become zero， the aboveintegrationis rewritten as：

面＝封ユ肋OS2♂fdf＝4Jぴ2β函諾告両＝‥ ・・・ （48）

UsingtheresultofEq・39andtherelationof U＝eO－，theaverage rate of work Of the oscillating貝owisfurtherrewrittenas：

βf才2エ 2烏ゐ Ⅳ＝2×p9

T

Thisisexactlytheaveragerateofwaveenergy propagationintotwodirections・

ⅠⅠⅠ．SOME RESULTS AND DISCUSSIONS OF EXPERIMENTAL

STUDY

ExperimentalApparatus

Inorder toinvestigatethe validity of the analysis developed，a Smallscale

test has been carried out．Asmallwooden channel，3．50mlong and O．20m wide，

was constructed with a slot of 20cm square atits center as shownin Fig．4．

The slot was connected toa side tank of20cm square at a depth of36cm below

thechannelbottom througha duct of5cm highand20cm wide．The side tank

WaS prOvided with a piston of20cm sqlユare Whichwas forced to oscillate sinusoi－

dallyin the verticaldircction with a crank wheeland connecting rod mechanism．

The oscillation of the piston thus produced a sinusoidally pulsating丑ow at the

channelbottom．At the ends of the channel，WaVe absorbers of crushed stones

Withslopes oflto4were also provided．

The pl．0fi1es of generated waves were recorded on an electro－magnetic oscillo，

－15－

‾9l‾

poo方u‡StA〇u∂‡〇慰∂叩1JO uO押町柑A飢lエ 11ヨu！pユ0〇〇官乙師■00190乙■OIHOユJp叩官A

‘∂Z uOIS！d∂qlJO ∂斗0月S ∂Ⅶ 01川和叩 ∂AIZ几∂qlJO O！叩ユ ∂qlユ0‘A〇u∂！〇W∂凱It

一叩ユ∂u∂方∂A甘瓜叩エ Ⅶつ96■ZOl予Z■ltuOユJp叩℃A川和叫∂A吼肌p∂叩J∂u∂3∂qエ

ー・、・、、、∴：：・・∴、‥∴川十：卜l・ごLl・

JOしadA叩d呵d」（ユ℡urPユO t陀Aq p∂叩ユ∂u∂岩∂SOtIISIZ∂Ⅶ甘S∂qllSnr；〉ユ∂瓜S∂加ガ∂S∂ql

ql！瓜p∂pユ0〇∂JS∂AIZ瓜 p∂叩J∂u∂ヨJO．S∂旧0ユd ■J∂1u∂〇 ∂qIuユOJAe瓜官 Ⅶ〇09 q〇セ∂

やuul≡qつ 叫1叫1∂S；〉ユ∂瓜S∂ガマ居 ∂A甘瓜0瓜エ ■SpuO〇∂S Z19－0 01ZO‘Z uOJJp準用A

SIZ瓜 uO叩tl！つ印∂qlJO pO！ユ∂d訓串 p耶Ⅶ〇C＝∂ 叩1∂S Sで瓜1u；〉Ⅶ∂AOⅥユuOl墾d訓弟

JO∂pnlHdtn電∂qエ ■u〇91JO叩由pユ∂叩瓜∋ql叩p∂1〇npuO〇S一三瓜1S∂11Sユg∂qエ

S∂A官瓜p叫Ⅵユ∂Ⅵ∂D‡O SIq眉！∂H pl川S∂柑0ユd

■Sl！3！pJnOJOldn

l∂∂q瓜さ叩でユ〇∂qlJO uO耶10ユ∂uOユOJp叩nb∂ユ∂叫1∂qlp∂1unOつqつ円瓜J∂1unO〇－∂Ⅶ！1

〇！uOll〇∂t∂tTtZⅥ！Jhp∂1nSIZ∂uI∂1aJhSpO！・I∂daAIZJh∂tIエ ■pOl凱1！1〇乱IuO〇aqlJO∂tpp！uI

∂ql叩p∂1ユ∂St工！S甘瓜q〇門瓜∂dAl∂居甘万一叫甘ユ1SJOlt∂〇p一三OtlZ V甘仙1準pユ0〇∂ユOSl一三∂ユ∂瓜

Ⅵ0呵d∂qI∂A叩01p；〉J叩b∂JS∂〇ユOJ∂甘エ ーS∂方官ガ∂A甘〟L∂dAl∂〇ulZ呵S∂JtIl！瓜qd甘ユヨ

l∂uu叫DIS∂ユJOll叫鋸帽 ■予t8！且

］

0．02 QO5 0．t 0．2 0，5 l

罰′L

Fig．5．Comparison of ExperilllelltalValues ofBT2e▼I▼itllT】1COrCticalOlleS

agreement with the analytical 011e Of Eq．40 as showllin Fig，5，Whel－e the

efBciellCy月ノ2eis plotted agai11St the relative depth h／L The wave generating

e餓ciellCy，both experimentalalld analyticalones，increases as thel－elative depth

increases up to h／L＝0．2，and then shows a decrease with theincrease of h／L（Or

with theshortellingo士wave period）．Theexperimentalwavegelleratinge侃ciency

is about80per celltOf the allalyticalonein theaverage．The smallness of the

experimentale伍cieny may be explained as due to the ellergy dissipation caused

by eddies around the edges of the slot alld cllallnelconlleCtion．The丑uctuation

oftheexperimelltale8iciencyis considel・ed as the result of partialre丑ectit）nS

from the wave absorbers which were not e伍ciellt enOughbecause of srnallspace

available；iftheslopeofcrushed stonecouldhave beenlnade asgentle aslto

lO，there would havel〕eenlittle伽ctuation of the experimentale侃ciency．

Driving・Force for the Pistoll

ThelneaSuremellt Of driving forces for thepistonwasmadefor theamplitude

Of3cm for the pistonlnOVemellt at thewatel■depthof15clll，The measurement

WaS also madeforthe caseofan emptytank．The drivingforce for tlle piston

【 17 「

‾ゞl’‾、

101官ユ∂u∂B aAl≡瓜tZ‡O aSOdlnd atIllOd ●SpO！ladl10tIS 叩 β也1nD‡O a〇10Jll…即au！

aqlOlp乱柑duIO〇ttt：uIS Al∂A∂uIO〇aq Gc［pul≡7c［‡O S∂1nSS∂1d 〇！ul官uAp atIltP！tI瓜

u！Z∂lq一三エu！p∂叩ユ1SnH！OSll≡S！1〇腎‡S門エ ●1u腎u！uO甲Jd凱甲q SSl…uユ∂1冒瓜叩1・‡O

aつユO才一叩ユ∂叫∂Ⅵ S∂1官〇！pu！S門エ ●po！ユ∂d aA一三瓜‡0 ∂ユ腎nbs ∂Sユ∂Au！∂qlOIA耶t10‡1

－ユOdoユdlSOtul官S∂S官∂ユ〇u！uO呵d叩1ユ0‡∂〇JO‡3叫Aドp∂Ⅵ‘9‘男！且u！u∂∂SSV

■SSl≡uJ∂叩瓜‡0叩3！∂瓜aA！p∂＃∂∂甲8uts朗J〇u！SntIl‘p∂叩Il！〇SO

u∋∂q ∂A印卜血Ⅷ10lS ∂qlJ関ul∂utI叩〇 ∂Ⅵu！J∂叩瓜‡O ttO！1JOd ∂uOS ごユ官∂t〇

10u S！S∂つlo‡t腎叩∂叩ユ∂dx∂pt陀t腎叩∂JO∂qlu∂∂叫∂q∂つu∂ユa誹p叩＝0∂Sn腎〇叫エ

●uOP且柑p！SuO〇01u！u∂耳管13u甲qlつnp∂qll甘叫つOt∂人手O∂St∋∂Jつu！aqlql！瓜∂S℃つS門l

ユ0‡叫S一肝Sl≡瓜SSl引エ【J3l甘瓜∂qlJOlq8甲瓜訊Ⅰエ ■9句且put∋Z∂lq一三エu！Ⅵ謁S Sl≡

1u∂つ1∂d os olOTJOlunOuJIZ atIIAq s∂つJO才一l三つ！1∂JO∂t［1∂ttlut2qll∂BJl≡T∂1∂瓜S∂つ10‡

苫u！Aリpl∂u∂S叩エ ●S∂つJO‡1∂tl叩1pt∂！A olユ∂叩爪ql！瓜 p∂l旧 刊u官1叩1u‡∂SOt11

uIOJ‡PalつtulqnS∂1∂瓜耳ut：1AlduJ∂atllu！Ⅵ01S！d aqlJOJS∂つ10JBu！＾！lp atIl‘1∂叩瓜

叩OtIl！瓜pu一三ql！爪S∂Sl≡つ0叫JOJ∂p官u∂J∂瓜SIS∂1∂ql∂つu！S ■∠ウ‘b互Aq tto！叩tnつI一三〇

atIl‡O SltrLSal凱IIS一三TT∂爪St：9－B！d u！u＾OtIS pt陀 Zalq官エu＝つalS！t∂1t：（Sa〇JOT

uo！SS∂1duIOつ pu一三 uO！Sual‡O S∂nll払 耶∂d‡O SaBtua＾1…）sa〇10‡ヨu！＾！1p tunuJ！Ⅹt：uI

atTl‡01≡叩p TtZlu∂uJ！1∂dx∂ atIエ ’∂つu官Jt：∂ddl≡1uauOduJO〇 〇！uOtlut2q puOつ∂S ∂甲

JO tTO！叩つ！pu！叩1！t S一三瓜 al∂tIエ’aul！1011つ∂dsaJql！A ATTq）！OSnu！SISOuJTl三pa！1t2＾

po！エad a▲℃瓜SnSユa▲Ⅷ01昏！dJOJa）ユ0慮加！A！ユqJOⅧ0！叩！ユ℡A ●9●眉！丘

（〇8S）⊥ POり∂d■∂∧D州

0’E O’Z 引 0■l∠－0・⊆■0

ロ「ぞぎg rOrCe fOr Pis一〇ヲ「へkg）

Z

Z

∈

⊆

∠ 0

0

design，thedynamic pressures of Pt and Ps may be neglected for the estimation

ofmaxilnumdrivingforce，PrOVidedtheeffectiveweightofthewater massbein壷’

estimated reasonably．

TAB工．E2．ExperimentalarLd TheoreticalDriving Force for Piston．

Note：♯凡is the driving forcein a tank with water

紬j㌔is the driving forc巳in an empty tank

Ref呈ection from Wave Generating Are

Thelast of the tests was concernedwith the wave reflect：ion from the wave

ge11eratlng area；unless alow re允ection had not been proved，the present wave

generator couldnot besaid to be a practicalone．Since a direct measuremellt Of

wave re且ection from the gellerating area was di餌cult，allindirect method

utilizing the multi－reAectionsbetween thewave generating areaandaverticalwa11

WaS employed to estimate the magnitude of the wave reflection．

When a progressive waveisintroducedin a region which has the complete

WaVe reaeCtion at one endand a partialwave re且ection at the other endina

distance oflasshowninFig．7，the standingwave height at a verticalwall

（x＝l），Hs，is exI）reSSed as（see the Appendix for detail）：

Fig．7．Definition Sketch of加ulti・Partial－Renection System

一・19－

‾OZ一

触l叫でu∂J乱Ilp甘巳1郎uで王丸1耶ユ∂u∂3aA℃爪乱Il叩 p∂l〇∂甘∂ユ Sで瓜A3ユ∂u∂∂A℡瓜∂ql

JOIU∂〇ユ∂d9■Z叩叩S∂叩〇！pU！lU甲田∂0〇uOIP粥匹Ⅰ訊ロJOlunOtH℃S！qエ t∂ヨ℃J∂AIZ

叩Iu！lu∂〇ユ∂d9T S℡∈S ■bヨ：ⅦOJJp∂叩1n〇ド〇 Sl…瓜℡∂北 訊叩℃J∂u∂富 ∂AIZ瓜∂qlJO

IU墾〇里∂0〇uO！I〇∂甘∂J叩l‘甲d叩∂A！叩t∂J∂Ⅶ官S乱n」OJAつtl叩切∂ヨu！咋J∂u∂富∂A℃仏

訳nJO S∂nl官A tⅡntⅡ！叫Ⅶ ptI官 Ⅶntu！Ⅹ官Ⅶ∂qI3utユnS閤ⅦAq pu℃1引帽pllZIU∂叫J∂dx∂

孔【IJOl三叫u！u pUマ マⅦ！Ⅹ官u 叩I凱Ⅰ！I〇∂UttO〇 S∂UH q叩OtnS O瓜l飢Il瓜℃Jp A江

’セ∂ユ官

凱Ⅰ耶ユ∂u∂3 ∂A官爪印l叩 uO！l〇叩∂Jt叩J℡d叩I∂Sn官〇 Olp∂ユ∂p！SUO⊃S！叩IS叩IJO

pu∂J∂lnO訓n叩Il即A訓n‘知OJ∂ユ孔【エ ●9＝αq方noエ甲乙＝〃JOJ乙／7Z／＝7JO SltmOd

司t！∂dl12〇！1Att！ut！∂qlql！JVL tt∂JVLJtl∂A 柑 t？叩p tt！叩∂叩1∂dx∂ atJIJO S叩詔d u∂＾∂S

叩I‘7uO！方∂JuO押印∂⊥t叩ユでd一口lntⅡ∂甲‡0甲訊Ⅰ∂t叩1S官U∂司官1S！∂S℃つIu∂S∂Jd凱11

ユ0‡tu〇S9T S！tP！q瓜10tS 孔n‡O ptI∂ユ∂1nO∂ⅥIpu℡ t；〉uu叫〇 ∂ⅥlJOll吼帆ll：つ！串∂A

孔【Iu∂∂叫aq∂〇ut？IS！P atTIJI●∂1nBg atT‡u！u∂∂S ATlt？叩S！S】lt！∂d u∂＾∂Sl川瓜t！叩p

lで叩∂叩J∂dx∂叩l‡OuO耶nl〇叩〇！pO叩d℡‘A〇u叩里∂t叩l∂ユ0叩1叩1W爪叩Ⅷ緋加

pooガマS瓜OtIS官叩pl官1u∂叩J∂血∂叩IJO∂dol∂Au∂J∂ddn叩1q3not川Ⅴ’ll吼肌l官つ即さA

孔【11t？1tIB！atT∂At！瓜凱IIJO凱Ⅰ！TqnOp孔nJO∂Sn官〇∂q uaSOtPOS St！瓜∂E：JO pt！alSut

A〇u∂！〇里∂ 加耶ユ∂u∂方aA官爪印1ユOJ∂予JOユ0叩u叩OtI∂p 叩エ ●7／7／ql由p∂人世l∂J

叩11Su！官如p訂HOld s！∂予／首A〇u墾〇里∂加耶ユ∂u∂富∂A官仏訳【l∂ユ叩瓜9■万！且u！u仙）qS

∂ユ官Iu∂u∂加S瞑Ⅷ【別口 ヨO S叩S∂ユ叫エ ■∂．TOJ叫S官u〇C S蝕Iu∂u∂AOtu uOIS鱒

aql‡0∂pnl！tduIt？qtIIput！u〇ZT叩1∂S St？JVL tIldap・l∂叩瓜∂qエ’1aqlOSqt？∂At？JVLt！

叩OtIl！JVL tt？3！IlaAl由3T S吼帆pu∂1aqlO aqlごl∂uut？tP孔【1‡O pu∂∂uO叩Atuo pa〇t？Td

・l∂qlOSq12 ∂At？瓜凱Ilql！JVL p∂1〇npuO〇SE〟LIS∂1atTl‘sTSA【t！ut：∂＾Oqt？atTluO p∂St？江

■pO！ユ∂d aA℡瓜叩1011〇∂ds∂ユql！瓜1q3！孔【aAで爪飢叩u官1S aql‡OuO耶卜柁A孔【13uI

－1nS甘訓∬【＾qlua！〇里aO〇uO！13a甘∂1atIIJO∂pn叩加uI∂昭alt2tu！1S∂01Satqt？uaSS叩aql

－ユ∂A∂ul！‘Ⅹ叩u∂ddv∂qlu！u∂∂S S官凱tO∂叩叩Ⅹ0ユd血u電甲StSAl官u℡ S叩 q3not川Ⅴ

叩（∫」首）＋El‖エ（∫月■）

、・1J u叩（g」首）－Ⅹ℃lU（ざ月■）

：S℃∂Alヲ八一eu一万tユ0明り01岬！∂qlu官1Sし10つaq13utt∬nSSl…Z与pt柑丁与●Sbヨ：uOJ‡

p∂U叩qO u叩IS！uOPつ叩∂ユーl叩ユ官dJO ∂つJnOS 訓帯1Ⅲ0ユJIU∂！〇里∂0〇 uO！1〇∂甘∂ユ孔【エ

りー＋T ＝叩（ミ首）

8打Z

コl．y一丁 ＝ユ∵CてU（ミ首）

○打Z

：S官u叩！ユ瓜∂Jでで叫町叩pm＝犯叩柑u3S∂ql叩Sl申！孔【∂Al…爪

印エ ■ユ∂あ叩u官S了－‘甲叩瓜u！‘予／7（1＋〟Z）＝7u∂q爪S∂nt電AtunutⅡ叩p甘e岩／7α＝7

u叩瓜S∂nl℡Al∬nl叫Ⅹ℃tu S∂耶Igガ1岬！叩∂Aで瓜冨u！pu叫S∂ql‘∂ユ0押ユ叫エ・1U墾〇里∂0〇

uO！1謁甘∂J叫IS！打身pu℃∂A官爪∂A！SS∂ユ飢）ユdl官Ul訊ユ0凱Il‡01岬！∂qaqlS！ヌガq〇門瓜u！

；／T［7すZSO〇ⅤガZ一言打方＋T］ （0与） ＝∫〃

？首Z

0．3

Q2

0．1

0 0．O1 0．02 QO5 0．1 a2 0．5 l．0

≠／L Fig．S．VariえtioT10fStandi11gW8Ve HeiglltVerSuS Relative Deptll

97．4pel・Cent Of thewave energy traveled acrossit toward the other side of

the challllel，although some wave energy might have been dissipated by the

turbulence aroundit．The tranSmission coe伍cient of thewave generating areais

estimated as 98．7per centif the amount of energy dissipationis assumed to be

negligible．

The highvalue of the tranSmission coe缶cient of the wave generating area

WaS COll丘rmedin the observation of the wave profi1es at the other side of the

Challllelwhich had the wave absorber．The wave heights at this pal－t Of the

Channelbecamelessthano．5cmat thewaveperiodsof O．63，0．82，andl．34seconds．

Thesedecreascs of the wave heights are considered as the result of the superposi

tion of two waves with their phase angles being different by180degrees．The

COnditio11for the opposite phase angles for the twowaves，One generated and the

Othel‾ mOVlng backward after the re且ection at the verticalwall，is glVellby

l＝（2n＋1）Lβin whichlis the distallCe Of the verticalwa11and thelocation of

SuPerPOSition．Iflis taken as185cm，this condition glVeS the wave periods of

l．300．79，and O，62 seconds forlZ＝1throughn＝3，Showing a fairly goodagree－

ment with the observed periods．

－21一

‾ ZZ 【

一t？Tt！つSO p∂つnpuIa＾t？血”：Z961：－・tf．H●Ⅶ‘ut2A！llnS put2‘●d‘u∂ttP！t？苫‘●J．－Ⅴ‘u∂ddI

●√・‘0押ノ叩p了∫プチ1／‘か〃年0一軍巧打‘√劫78αγク軋乙

β ∂プ昭子7∫エ‘J∫ププ∂∫乃Yク汐∫∫Z甥甘‘－J∂♂∑甘 P∂押7？プZ♂町】“‘叩駅唱p3SOt〇℃U！ユ列1‡d官

‡O UO！SS叩SU℃ユエ‡01U∂！つ閑∂0〇叩1∂JnS甘∂川01pOtIl∂m V”：Z96T叩門SOA‘℃pO9

SⅡ〇N彊ⅦⅡdⅦⅡ

■1！01SuO！1S∂富男ns∂t甲nl甘A

pul≡1d！ユつSnt柁u∂ql‡0瓜∂！A∂Jl官つ印ユつS叩JOJ‘山0112JO叩TJ∂1甘血判甘∂咽∂Ⅵ1‡OJ∂叩〇

‘01Ⅰ叩nA叩SOA■刀川 Oisuo！1里つ∂Jd血ユ！∂qlSS∂ユdx∂ 01t阿此OSll≡ SユOqln甘∂qエ

‘！ul≡叩1！）Ipul≡∇司℃Ulヨユ℃H’S・ISS∂m‡O

S3つUl≡1S！SSIZ叩1S∂呵つ∂ユd伽ムー∂J∂つ叫S」叫∂ユ叫SユOtI叩甘∂qエ ●叩℃叩1！H O12耳甘ユ’ユ川Aq

pal〇nllSuOつSl≡瓜Puu叫つ1Salaq‡pul≡‘閏t：ul≡Jt？H Ol∂司nS ◆・Im＾q p∂1t：（bld aJ∂几

p∂AotduJ∂SUO！叩1U∂Ⅶnユ1S叫 ∂qエ’∂叩叩SUIqつユ℃∂S∂Ⅶll≡つ叩qつ∂ユニmOqユ℃H tIUl≡1ユOd

∂qlJO uO！S！A！q Sつ！tn12エP」（H叫1叩甲l〇npuOつ詑凡t甲・l∂q p∂1u∂開ユdノこpnlS叩エ

エN富江耳9q宜7血ONX〇Ⅴ

’ApnlSユ∂pun

S∂1nlつruISl∂pOuIpt陀1aさTl≡tn aAl∋瓜℃ tt∂a爪1aq tTOflつ叩∂1－！1lntnJO tH∂tqOJd

叫1∂AlOS Ol∂UO飢ItS叩0ユd」（・t∂Al≡∂q OISしu∂∂S SntIlntO1lOqlauUl≡qつ℃ 叩

瓜0甘凱Ⅰ甲Il！つSO」（耶つりユ∂Aセql叫UO耶ユ∂u∂冨s∂Al≡凡・t∂叩爪‡O pOtIl∂Ⅶ叩エ ■与

‘10lSl12つ！1ユaA叩1JO

pu∂JalnO訓串：叩Itl≡瓜叫1Aq甲Sll甘つaq OISⅦ∂∂S t10！1〇3甘∂ユ∂qlさ1u∂つユ∂d9T

lnO甲∂q OIS甘p∂叩t叫1S∂S12瓜甘∂J∇如！叩ユ∂U∂冨∂A甘爪印IuOユ‡uOtP叩∂J∂qエ ■予

■S∂uOll≡つ甲JO叩l机11ul叩り成加11U∂つ－t∂d oz：3uOS∂ユ∂瓜

S∂つユ0‡凱t！A！ユp一世叩∂叩J∂dx∂∂qエ ●pO！ユ∂d∂A甘瓜∂ql‡0∂・柑nbs∂SJ∂Au！叫101

l召UO甲Odoユd」（l∂叩Ⅶ！Ⅹ0ユdd甘凱甲q‘∂つJO‡l叩ユ∂uり0郎叩ⅦS！瓜叩飢Ⅰ！1閏l‡つSO

∂qlS∂つnpOJdtIつ門瓜 ttO呵d ユ∂叩瓜ユ叩un t柑 ∂AリpOlp叩nb∂ユ ∂つJO‡∂qエ ーC

■∠Z■O tI方no叫1

90■O uOユ‡ql由p∂A！1叩J∂ql‡0∂飢Ⅰ官J3qlJO‡S∂nll≡Al叫1∂ユ0叩1∂ql‡Olu∂つ

J∂d o9：押0甲 ∂J∂瓜SltI訊叫 aA甘凡p∂叩ユ∂u∂3‡O S∂nl甘Al℃1u∂Ⅶ！J∂dx∂∂qエ ●Z

■112卿∂IOdノこ］！つOlaA∂qlⅦOJJp∂A！ユap

am：t泊∬巳 二計Ⅰ！叩ユ∂u∂方∂A甘瓜∂t111甘∂ユnSS∂ユd uollOq pu℃ S∂l！才0ユd ∂A且舶L」OJ

SUO！SS∂ユdx∂pUt2‘p∂u閂qO U∂∂q S叫Ⅶ0110ql∂utI叫つ℃ 叩肌0甘訊叩明1fつSO」（l

－l甘つりユ∂Al≡ql！瓜p∂叩ユ∂u∂方s∂Al≡瓜ユ∂叩瓜JO‡l叩U∂10d舟坪甲AJOu叩nlOS∂qエ ●T

：よpnlS叩∂S∂ユd∂甲‡O SuOてSnlつuOつユ0匝u

∂ql∂J甘S訊I一肌01lO‡叫1‘suo！1つ∂S 凱りp∂つ∂ユd 叩1‡O S叩1S副：訓れ・訊ITZ！ユ官uunS

SNOISn7〇NO〔）●AI

tionsin Harbors：Effect of Ellergy Dissipatorsin Coupled Basin System，”

t∴．．・．∴、さ・∴、∴∴・・ ∴ ‥‥、、ご●・∴● ∴．‥∴ ∴∴ ∴・・・・．∴・・．・

入b．52，49pp．

Iwagaki，Y．，Tsuchiya，Y．，alldInoue M．1962：‖Study on the Wave Over－

toppings for Sea Wallsin a Wind－Wave Channel（1），”Proc．9th Co，Lf．

（おβ∫ねg∬〃g．わzJ‘ゆαクZ，ノざCE，pp．153－158．

Stoker，J．J．1957：“Water Waves，”h7te）‘SCLe］ZCe Publi5hers，New Yorlく，567pp．

APPENDIX

Water Surface Pro丘1esin a鳳Iulti－Partial－Reflection Systcm

Allallalysis of the water surface pronlesin alnulti－partial－re負ection system

has been developed by the author（1962）and preselltedin the report byIppen，

Raichlen，alld Su11ivall（］962）for the ptlrpOSe tO eStimate the transmission

COe伍cient of EIWaVe Blterin a closed basin．The allalysIS Can eaSilybe modiBed

to thesystelTIShownillFig．7so as to describethevariation of the waveheights

in the system，aS Showllin the followlng．

In tlle analysIS the fol10Wlng aSSumptiollS areimposed：

i）Smallamplitude wave theory applies．

ii）Waves are reflected completelyatユ、＝land re8ected partially at x＝O with

the reflection coe伍ciellt Of Kll．

iii）The n－thl－eflected wave callbe supel‘pOSed ollthe originalprogl■eSSivc

WaVe．

Tl－e Orlglnalprogressive waveilltrOducedillthe systemis expressed asニ

り1＝Aocos（ケ才一点エ）

This orlglnalprogressivewaveis reaected completely at the verticalwallof E＝l，

followlllg the丘rst retrogressive wave：

ど1＝Aocos（Jf＋点工一2以）

Whellthis retrogressivewave retul▲nS tOthe origin，a partOf thewaveis re8ected

and the followlng SeCOnd progressive waveis formed：

で2＝‰Aocos（ケ才一尾エー2尾g）

As before thisprogressive waveisl・eflected at x＝l．The resultant second retro－

gressive waveis described by：

ど2＝定点Aocos（Jf＋尾エー4烏g）

Illa SilTlilar manller tWO families of progressive waves（77，L）and retrogl・CSSive

WaVeS（E，＆）may bede丘ned，their amplitudes decreasing pl▲OgreSSivelywithincreas－

1ng71due to the multi－partial－1‘e8ectiollS at the orlglll．

Thewave system which ultimately exists between the source of partialre貝ec－

－23

－∇Z －

ー1Ⅹ∂1∂qlu！0与－bコS！q〇！q瓜

岩／直す乙SOO打方Z－

8打Z

岩打方＋t］ （OltV） ＝‘〃

：S∂Ⅶ0つ∂qlq冨！∂q∂A甘血

u

O

‡01q∂！∂q∂qlS！0vz＝0打qつⅢ孔明

凱巧pu甘1S凱P ヨO tuJO手l甘叩∂qエ ●suO耶IndTu甘

aAOql三∂qIuO・巧 p3A！ユ；〉p u∋tIIS！lauul三qつ∂qlJ

訊Ⅰ叩耶1S∂qエ ー∂A巳〟L3A！∬∂J方0ユdl甘utj∃Tユ0∂ql

粥昭SOつ打方Z一岩打曳＋l ⊥ （6’Ⅴ）・・‥ … （（押＋プ♪）so〇打方ー（押－7♪）soつ） ＝才＝ロ」血

0打

：－∂’！‘9－v tb耳1里j＝ご乱丁問at Aq Aldt叫S p∂u！召1qO

S！耶瓜Ⅰ閻！1ぷ人3ql叩粥0ユd3A甘瓜∂qエ・Ⅰ∂hu℃qつ∂qlJO耶瓜l叩1ユ∂A訓串ptl甘

uoIlつ∂甘∂Jl叩ユ甘dJO∂つユnOSaqlua∂叫3q uO耶〇0IAueユ0事∂Ⅶ！1‡OuOロ〇un「巳S12l∂A∂t

ユ∂1甘爪u甘∂ⅦtuOユJ∂〇甘JJnSJ∂叩爪∂qlJO叩∂Ⅶ∂〇etds叩∂qlS∂SS∂ユdxa uo！叩nb∂STqエ

（g‘Ⅴ）・‥1（訂ぎ＋プ♪）soつチ才一（プすZ＋∬γ－7♪）犯〇平方－

（～すZ一∬汁珂犯叶（∬ぎー呵S。。）裡竺塑里ゴ旦！ユ＝友 O y

：引∋u柑1SAs uo！pa甘3⊥t叩ユ甘d－！1lntu∂qlu！

∂柑0ユd∂A電爪3qlユOJuO！SS∂ユdx∂；呵11耳即qO∂瓜‘∠●Ⅴ■b耳JOlユでdl甘∂一丁叫18u！軍eエ

（ム●Ⅴ）

［プす7Z－］dx3打方ーt ユ ［肋ぎ乙一軍＋呵がⅩ叶陶－プ項］dx∂）

］ t甘3Ⅶ去ん

Oy

：01S；I8ユ∂AUOつ

S∂！J∂S SⅢl‘∂uO tI叫1SSatlu；〉！〇甘∂0つ tTO！1つ叩∂JaqlJO；〉ntl≡A；〉lnlOSqe ulZ ユ0且

（9●Ⅴ）……………… ［ト＋［ブ獅一］dx∂岩惣＋廊Z一］dx∂惣＋て）×

胸Z一軍＋勒］d苧叶［（方省一勅］dx柵］l電∂Ⅶ＝友

：S電p∂∬∂Jdx∂∂q血tu s●Ⅴ‥bⅡJO∂A12瓜1uりInS∂J∂ql

宜押（l一〝）プZ－】dx∂×［（訂す一匹）7］dx∂Oy，＿㍑打方）l電∂Ⅶ＝

［押（l－7J）z一∬γ－一押］soつOyト，捏打方＝咄

：’3●T

‘sl∂qulnu X∂1dulOつJO SIJtZdl官∂J∂tIIStZp3SSaldxa∂q OSl∇J（tztu sa＾tZ瓜∂Saql∂つuTS

‥・＋†んj＋咄＋…＋乙j＋砲＋tj＋叱＝友 （与’Ⅴ）

●S；〉AIZ凡p；〉lつa甘∂ユptI甘

∂A†SS∂ユ80ユd aqlllE；O tHnS∂qlS甘p∂SS∂ユdx∂3q u∂qlJセul11℃爪l12つ印3A pt陀 uOり