ReportoftheNatiom1ResearchCenterfor

DisasterPrevention，No，5，March1971

551，466：551，465：551，556．8（265．5）O▽ershooti皿ginWina・G㎝emtea

Waves

By

YukioFujinawa∬〃α壬∫〃肋B7α〃c乃，Nα〃o㎜1R2∫θακ乃Cθ〃〃∫oグDゐα8加7Pプω2〃〃o〃Abstract

When waves are generated under the iniuence of wind，non1inear e任ects
are1ikely to become signi丘cant． Recent1y Bamettθ左α1．asserted that wave
components overshoot and undershoot，as said in terms of eIectronics，
after the phase of exponential growth．The present paper con丘rms this state−ment from a number of observations．

1．

Introduction

Since Je伍reys theory of sheltering coe冊cient，the generation of wind waves
has been one of the most attractive prob1ems of oceanography，but after morethan forty years since him we have no satisfactory theory．Wind waves are
generated and deve1oped under the iniuence of wind b1owing over the ocean
surface，and strict1y speaking，we cou1d not solve the prob1em without a perfectknow1edge of the wind structure over the ocean surface．But the wave−causing
wind is in a1most a11of the cases turbulent，which is neither homogeneous nor
isotropic，but a turbu1ence on the Hexib1e boundary．
In the ie1d of theoretical study of turbu1ence
since Prandt1 s mixing theory，but most of them are

there are many investigations
concerned with more speci−
ied turbulence such as homogeneous or isotropic one，and the turbu1ence near a
wa11is hard1y said to be attacked．So it is not strange that we have practica11yno investigations of wind structure over a Hexib1e boundary，the most general
type of turbu1ence．But the wind waves are generated in fact and cause a great

iniuence on the human1ife，and by some suitab1e assumption or from the ex−perimenta1resu1ts we can proper1y describe the wind fie1d over the sea and canresearch the mechanism of air−sea interaction，and inverse1y we cou1d infer thewind structure from the wave motion．There are many investigations of thewind−wave deve1opment both in theory and observation or experiment．But usu−
a11yinmany

factors

oceanography

which

therearein刊uence

nodefinite

eachcase

resu1tsofofobservation．

observation，and

these

There

extra

aresoterms

cannot be contro11ed by our force．In the laboratory we can not construct theexperiments with an equa1simi1arity and can on1y infer qua1itative1y，incapab1eof quantitative comparison with the theory．These circumstances may be a maincause which hinders the smooth deve1opment of oceanographyl
O．M．Phi11ips（1957）proposed the resonance mechanism of water waves under
the inf1uence of the turbu1ent且uctuation of the air且ow．This theory predicts
the1inear growth of wave component in resonance in time at the initia1stage ofwave

generation．

morede丘nite

There

are

severa1observations

supportingobservation

andconsideration

be

一97一

shou1dthistheory，butmade．Mi1es（1957）calcu一Reportof

theNationa1ResearchCenterfor

DisasterPrevention，No．

5，March1971

1ated

the e伍ect of air iuctuation induced by the waves to the amp1iication of
the waves．He assumed that this disturbance is in a neutra1state，and used thestress formula after C．C．Lin（1955）．But this assumption does not give serious

modiication

andon1ychanges

thenumerica1constant（0．M．Phi11ips，1966）．

Besides，there have been many observations of the amplification factor of windwaves．Some of them agree with Mi1es prediction，and some give several times
larger amp1iication rates．It is necessary to a㏄umu1ate more experimenta1andobservationa1results in regard to the wave deve1opment and to the innuence of
air iow on wind waves，before we can conc1ude whether or not his mechanism
iS

COrreCt．

Now，Mi1esmechanism

predicts

thatwave

components

grow

exponentia11y．

←CAPACITANCEN

丁警Y0

M−M
HIRATSUKA

IZU

PLANE

OBSERVATIONBOSOSENSORROOMPEN．
∈PEN、

CAPAC1TANCESENSOR

■寸一M
MEAN

W灯ER

LEVEL

；≡OSHIMA
140＾

＼220。
210。 、。ぴ／。8150

160．

O。

190。

1ηゴ

σ

PACIFIC

OCEAN

oFig．1．

Map

aroundtheObservation

Tower．

12

Fig．2．

SketchofMarineObservationTower．

10；5
ゴ02｝一5；
一10

3

Water

Fig．3．

Ca1ibration

curveof

4

5

Leveし（m）

capacitance
一98一一

wave

measuringsystem．

0vershooti㎎

ψO．

inWind−Generated

25March1969竈。：μ20。

lm后〕

ワ：Wind＼。＼」。ノ；

5240
210

0180

1400

1500

in time，but they can not growindeinite1y and are1imited by thestabi1ity

Oirection

1510

Fujinawa

Waves−Y．

1600コST

criterion．The

breaking

arenum−is

ca11edrange，and

thethepower

spectra1densityψ（〃）is：
ψ（〃）：βσ2〃■5，of20mfromthemeanwater1evel．

Marchequal，

equi1ibrium

Fig．4．1 Wind speed and direction at the height

25

wave

ber range，where the energy input
from wind and the output by thewhere〃isthegravitationa1

acce1eration

frequency，gand

the

βa

constant（0．M．Phil1ips，1958）．But

1969

recent1y，Mitsuyasu（1969）asserts
thatβis not a constant but depends
on the fetch or duration． Re▽ea1−

l06

ing
、oτ＝3．3s㌧卜8105＝■』』τ三2．5softhisfact

wi11suggest

mechanismofequi1ibrium

which

wasusual1yfrom

dimensional

treatedanalysistherange
on1y

exceptLonguet−Higgins（1969）．

τ＝5．OsThus，aswaves

grow，non1inear

effect wi11emerge and a new fact
wi11occur．Recent1y，Barnettθ左αZ．

婁どl04

1400

1600JST

1500

Fig．4．2 Wave growth curve． Parameterτ
shows the wave period．

1061420JST
25March（1968）reported that waves do notreach the equi1ibrium va1ue grad−ua11y，butandunder・irmed
paper

this resu1t． In the next
we wi11try to exp1ain thisfact，

takingproperties

1969

overshootshoot ，as said in terms of elec−
tronics． Our observation has con−2．

ofaccountwaveofnonlinearie1d．

Observation

薫105

A11theseries

have

Observation

observations
been

done

Towerofatof

thistheMarine

Nationa1Re−

0search Center for Disaster Preven−tion．The tower is situated at a1−

ξ104丹most the center of the coast ofSagami Bay of the Paciic Ocean（Fig．1）．Mean

waterdepth

atthesiteofthetowerisabout20mand
l03l0tower

from

the1

Period（s〕

measuring

spectrum．

wave

Fig．4．3Wave

0．2

the2

一99一isabout1km

distantcoast（Fig．2）．The
detector

measuring

is

awave

capacitancesystem，whosesen一

ReportoftheNational

ResearchCenterfor

DisasterPrevention，No． 5，March1971

sor

12Ju1y1969

has

position∈i）
2rO昌
■
■／80ち
0gO ・一 L0∈

ざWi．dOr・ectio・

喝15
冴10／糾
μ20。

…5；

タ1500

1600

1『O01800

1900

2000JST

Fig． 5．1Wind speed and direction at the heightaof

directionmeasuredareiswind

shown

speedbyinandtheaero−vane shown also in Fig．2．Ana1og
output vo1tage has been digitizedand transmitted to the laboratory
throughcableandpaper

received

puter

asCalibration

of

ing

systemshown

there1ation

vo1tage

by

com−

tape．

is

and

the

wave

in

betweenmeasur−Fig．3，and

thethe1ength

output

of

the

probe

water

is

fairlyObservationa1resu1ts

areshown

Fig．4

there

immersed
12Ju1ジ969

of5mm．Its

insta1lation

Fig．2．Means

of20mfromthemeanwater1evel．

＿105

diameterofinthe1inear．

う一／へくへ…

．吾

竃104
＝ll05

3．

I己esu1ts

from

Fig．4．1to

are

shown

γ，3．5sτ＝4．3sIn

theFig．7．3．

igurestheofobservationa1resu1ts

of25May1969．Fig．4．1shows；￡7■・6s
l04

1600

1『O0

1800

Fig． 5．2Wave

1900

growth2000JST

curve．

the

wind

speed

at

the20m

from

the

mean

water

σ20エ。andits

direction，when

began

blow

re1ative1y

and

to

b1ew

constant1y

concerned．

from
106

Fig．2andFig．4．2is

1800JSTFetch
theheight

of

leve1
wind

sudden1yduringthecaninferredbewindgrowthtime

direction．
curve

ofthreewave components whose wave peri・ods τare respective1y2．5，3．3and
5．0sec． The wave component of

12Juい969

τ＝3．3sec does not increase in theear1y stage when wind speed is not

亀105

strong，andas

it

exponentia11y

theincreases

windapPearance

attains

toof

thenear1y身strongandtakes

overshootandmaximum

severa1times．On

烹
4
上10

bIows

and
other

hand，

componentτ＝2．5sec

remains

constant

1ibriumtheminimum

and

may

state．And

in

equi−

the1onger

be

com−

ponent τ＝5．0sec does not deve1op
so fast，and seems to increase mod・

l03

6040

12Per…od

Fig．5．3Wave

64

erate1y．Power「．2（s）

spectrum．
＿100一

spectra1density

wave

ie1d，a1mostatthetimethecomponent

T：3．3secof
when

over一

Overshooting

inWind−Generated

迎Eξ

21Ap・π1969

ざ∈

σ20m

Fujinawa

shoots，isshown

shows

that

ponenthasthan

←15

奄10身5

Waves＿Y．

2『Ol。

in Fig．4．3．This
overshooting com−theaperiod

a1itt1eanoptimum

wave．

The

iguresofshorter

Fig．5show

theδWindresu1ts of data of 20 Ju1y 1969．
180で90妻 Wind direction is near1y constant，
but fetch is rather sma1l．Fig．5．2
1600こ！ST
represents the time change of fourD1rection

妻1000

120Q1400

Fig． 6．1Wind speed and direction at the heightof20m from the mean water level．

wave

components

Twol06

ごノ㌧τ、。．。、

〆・㌧・・

分；

ε10 4wa▽ecomponentsofτ＝4．6

andτ＝4．3sec show1arge overshootand the component of T＝4．3sec
has a signiicant undershoot．Other

21Ap・iし1969

喜105

ofτ＝6，τ＝4．6，

τ＝4．3andτ＝3．5sec，respective1y．

1000

1200

Fig．6．2Wave

1600JST

1400

growthtwo

components

remainratherstantbecauseastateof

lowenergyriumorwind

field．Fig．5．3shows

wave

very

spectrumovershootisattheevident；

con−equi1ib−

input

from

the

wind

time

wheninthiscase

two o▽ershooting components have
1onger periods than the optimum
component has．
The igures of Fig．6show theresu1ts obtained on 2Apri11969．
The

curve．

ofwind

blew

in

a

very

constant

direction，but wind speed tended
to be stronger（Fig．6．1）．GrowthlO筍

curvesoftwoshowninFig．6．2．The

wave

components

arecomponentof T＝4．3sec deve1ops in a 1inear
manner in time at the ear1y stage，

1240JST
21Ap・111969

and then grows exponentia11y andovershoots． On the other hand，

ご

shortercomponents

fasterreach

theモ105equi1ibrium range and take theirmaximum1arger than the va1ue atothetimestage．

wave？104⁝⁝andovershooting

spectrum
canspectrum￡

ofFig．6．3
bethe1ate

thewindat12h40m

JST，

totypical

said

withinshowsabeasma11swe11com−

ponent．

The五gures of Fig．7show theresults obtained on1l May1969．

l03
60

40

126

Per1od（s）

On that day the wind began to b1ow
sudden1y as is seen from Fig．7．1，

spectrum．

andFig．6．3Wave

41．2

一10！一thefetch

was1ongest

intheReport｛oftheNationa1ResearchCenterforllMaジ969∈

2r03

E90三・η；、；：㌧＆1O5；

1000

1200

1400J

ST

from

the

mean

water

level．

1lMay1969

つl06

101
へ；「二5．5s〆

戸豆；

εl04

1。・

。／
！／
1000

Fig．7．2Wave

1200

growthdinatearehasacomponent

rather

and

1400J

ST

curve．Right−hand

shows

the

component

10・or−ofτ：30s．

wind

as

Such

aonce

inofτ＝5．5sec

time

of

over−

later

a1arger

is

seen

from

shou1dsituation

theFig．7．1．

beoccurredseriesanddata

a㏄umu1ated，if

wewishthisisdecideanatura1phenomenon．
theornotFig．7．3

spectra1density

timeofevident

4．

Conc111sion

attheovershoot．

severalobservationsde−scribed above，we might say thatovershooting and undershooting inthewavedevelopment

rea11yexist．

Observations show that the wave
component，which is thought to be
a

state

of

remainequi1ibrium，does

notconstant

iuctuate

but

time

under

the

iniuence

Usua11ythisHuctuation

of

is

a1−

wind．

treatedas a noise，but if we take accountof the overshooting and undershoot−圭105

ing，this

f1uctuation

wi11become

physicalIy

signiicantsomeandgestion wi11be obtained as
equi1ibrium constantβ．
＼㌔
捌04、1）Phillips，O．M．（1957）：Oneration￡of

1．FZ〃づ∂ハwaves

byationof∫．FZ〃ゴ〃l03

1264

1．23）

（s）

Fig．7．3Wave

thethegen−turbulentwind．

θo乃。，2，417＿445．

2）Miles，J・W・（1957）：On

Periodsug−

toRefe爬mes……

6040

on1ymore

whethertoin
l1March1969

wave

Fig．

The very long wave component of
τ＝30sec shows an extraordinarily
large growth rate that can notbe exp1ained by Mi1es mechanism．

ways
1230JST

twoinovershoot can be seen．However，
this may be due to the weakening

in

1Ocof

sma11amount

someFrom！

800

represented

shows／＼み・・

さ101

curvescomponents

ofFig．7．1 ，Vind speed and direction at the heightof20m

series．Growthshoot；
800

Prevention，No． 5，March1971

180評 7．2．TheW1nd口irection

呵15i

Disastersurfacewaves

ハ4；θo力．，3，Lin，C．C。（1955）：bytheg㎝er−shear

f1ows．

185−204．

τ肋θτ灼θoび助〃oφ伽〃o∫〃舳妙． Cambridge
Univ．Press，London，155p．

spectrum．
一102一ヴOvershooti㎎inWind−Generated

Waves−Y．Fujinawa

Phi11ips，O．M．（1966）：τ肋Dツ伽〃c∫oグ伽吻ク〃0閉〃．CambridgeUniv．Press，London，261p．

Phillips，O．M．（1958）：Theノ

equi1ibrium

rangeinthespectrumofwind

generatedwaves．

．1η〃｛∂ハ4セo乃．，4，426−434．

Mitsuyasu，H．（1969）：Onthe（；α尾冶α｛K9冶δsん勿．

∫ゐ勿冶ゴ

τα4尾α61（δθ〃development

Longuet−Higgins，M．S．（1969）：Ongeneratedwaves．

waveofwindwave

spectrum（2）．州伽o〃Kα妙δ

breakingandtheequi1ibrium

spectrumofwind1〕700．Roツ．∫06．λ，310，151＿159．

Bamett，T．P．andSutherland，A．J．（1968）：Anoteonanovershooteffectinwind−generatedwaves．∫．Gθo助ツ∫．肋∫。，73．6877＿6885．
（Manuscriptreceived1O

December1969）

風浪における行過ぎ量
藤縄幸雄

国立防災科学技術セソター平塚支所

波浪の発達において初期段階でフィリプス（O．M．Phillips）の，また主発達段階ではマイルズ（J．W．

Mi1es）の機構が，量的にはともかく，作用しているらしいといわれている．主発達段階から平衡領域に近づくにしたがって波浪場の非線形性が顕著になると、思、われるが，最近，電子工学の用語でいう行過ぎの現象があるのではないかといわれはじめた．ここでは観測からその存在を確かめた．

一103一