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Hydraulic study on Tsunami


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Coastal Engineeringin Japan, Vol. 6, 1963


Yoshiro Fukui* Makoto Nakamura** Hidehiko Shiraishi**

Yasuo Sasaki**


Up to present, the visitation of tsunami has inflicted great damage upon our àountry repeatedly. Especially, the fact that the Pacific Ocean coast suffered from the serious disaster due to the Chilean Earthquake tsunami. is fresh in our memory.

Tsunami is generated in the open sea as the long wave with very small

steepness and draws toward the shoreline. The coast where the great energy of

tsunami is transported without reduction has, generally speaking, fairly deep water

depth from offshore to the vicinity of the shore line. The Sanriku Coast,

north-eastern coast of Japan, that has been frequently attacked by tsunami has a typi-cal feature as mentioned above. Namely, the submarine ditch with 5,000-8,000

metres of water depth lies along the shore line and prevents the decrement of the energy of tsunami.

As far as the two-dimensional deformation of tsunami is concerned, the seiche phenomenon must be considered first.

In the case of a tsunami the period of

which is nearly equal to the specific period of the bay, the energy of tsunami is conserved in the form of seiche motion. Therefore it is accumulated in making the tsunami height inside the bay larger and larger. Secondly, the reflection and

contraction of thunami must be taken into consideration. Owing to these

pheno-mena mentioned above, the tsunami concentrates its energy in the most interior part of the bay. And the above phenomena become more conspicuous with the increase of the reflection of tsunami from the bay cliff.

On the other hand, inside the bay where the near shore water depth is shal-low, the tsunami is breaking and the reflected and concentrated energy is

dis-sipated there. Accordingly the tsunami induces a great force of destruction in

this zone.

Differing from the gravity wave, the breaker of tsunami is more akin to bore

because of ts very long wave length and large scale. Moreover, in the case

that the water depth becomes shallow abruptly, the above tendency is accelerated.

On the other hand, the tsunami that approaches to the entrance of bay in the shape of non-breaker transports its energy to the inside of the bay in the shape

of flow. In this paper, the tsunami is classified into two types; i.e. the one is

the "progressive breaker type" and the other is the "seiche type."

There are many difficulties in .establishing the planning of counter-tsunami measures; that is, how to estimate 1) the tsunami run-up height on shore-land and on dike, 2) the quantity of overflow discharge across a dike crown and 3) the

tsu-* Japan Engineering Consultants Co., Ltd.



nami pressure acting on a dike.

As to the progressive breaker type of tsunami that is considered to have the

strongest destructivepower against shore land, the authors have analysed its in. fluence upon dikes experimentally and theoretically. The authors obtained


conventional formulas nd the necessary data to design dikes against tsunami.


against tsunami is constructed on shore or in inshore

zone at very shallow water depth. Accordingly, the tsunami which strikes against those dikes is

one of the progressive breaker type and may be treated as a typical

bore. The tsunami of this

type has great destructive,power as Previously

men-tioned. The authors conducted the experimental

study on the tsunami behavior in the shape of bccre.

in this experiment two different water tanks, one is large size and the other small, were used in order to test the scale effect.

1. Laboratory, Apparatus for Small Scale Tests The

one side water tank as shown in Fig. 1 and Photo. 1was glassed. was made of steel and its The movable single slope

was set as a dike model and

three pressure gauges were attach.

ed on the slope. The bore genera Plane view D.M.EtCtnC oscillograph Strain meter 21.Qm Sido view (c)

(b) Bore generator (a)'

Wave essrire gauges

'Fig. 1 Experimental apparatus for a small

scale test.

the opposite side of water tank in order

after it is operied. After the water was

1, the flap gate'was opened in a mo-ment by thecounterweights as soon

as the wire fixing the gate was taken off.

2. Laboratory Apparatsg for Large Scale Tests

The water tank shown in ,Fig. 2

and Photos. 2, 3 and 4 was used for

large scale tests.

The system of

bore generator, was almost the same

as that for small scale tests and six pressure gauges were attached at

intervals of twenty centimeters along

tor of flap gate type was installed at the other end of the water tank. In order to obtain many kinds of the length and shape of bore, the quantity of water stored insi1e the

flap gate was varied by selecting

five positions of the flap gate. The flap gate was fixed vertically by a

,wire rope stretched obliquely to

,ward the storage tank,

and also

connected to two Counterweights

through two wiresstretched toward to keep the gate horizontally

in the air

stored as shown in the side-view of Fig.

Photo. 1 Experimental water tank for a small scale test.





(a). Bose genta1o. )b) Sde nev H Stran male,-)lar.eai _en-mailflg omellOgeap 3Q.er-I 30.5.' 25.5.-75.5', ( I. Oikm made) Stesin mate, Etectnc oscdtognma'

1Wave p,enswe gangeS

FIg. 2 Experimental apparatus for a large scale test.

Photo. 3 Bore generator for a large scale test.


Photo. 2 Experimental water tank for a large scale test.

Photo. 4 Large scale model of dike. the slope of dike model.

3. Measuring Apparatus

The water stage gauges were arranged as shown

in Figs. 1 and 2.' The

water stage .gauges installed in frontof the flap gate and the slope record the

change of bore shape and the velocity of bore travelling through these measuring points, and that installed inside the flap gate records the decrease of water depth in

Photo. 5 Recorder of wave gauges (large -scale test).




Photo. 6 Pressure gauges attached on



the storage tank. The length of bore

was determined from its two

suc-ceeding troughs. The principle of

the water stage gauges isas follows; the change of water level was

replac-ed with that of electric resistance of two parallel wires, and this system

was included in a bridge circuit. A

portion of recording system is shown in Photo. 5.. Strain gauges were

us-ed as pressure gauges, and Photo. 6 shows the reverse side of the slope attatched with pressuregauges. The recording system of the pressure

gauges for large scale tests is shown in 4. Experimental Results

A sample record of the electric oscillnr

is shown in Fig. 3. The curves of (a), (b) and



® I2O(g/cm)

-1 ,.ReIle, bore


Bore presowe SCa1>*. gauge


Pressure gauge Pressure gauge

Ware gauge 'j15

Fig. 3 Sample records by electric

graph (small scale test).


Photo. S Incident breaking bore in a small scale tank.

7 Part of pressure gauges on a

large scale.


aph in the case of small scale tests (c) were recorded by thewater stage

gauges. The curve (a) of the water stage gauge installed inside the flap

gate shows that the water depth

began to decrease just afterthe gate

opened and kept low water level

until the arrival of the reflexbore.

The curve'(b) of the water stage gauge installed in front of the gate shows the shape of the initial bore.

The curve (c) of the water stage

gauge installed, in front of the. slope shows first the shape of the incident


bore and next the

duplication of

the incident and reflex bore with

Photo. 9 Reflection of bore after

run-ning up the slope in a small sccale test tank.



sI ( Cl p


an ts bo in fir in cr w ar m Photo. Photo.


Photo. 10 Reflex bore.

the lapse of time. Photographies 8, 9 and 10 show the situation mentioned above. Figure 4 is a sample record of

water stage gauges by a pen-writing oscillograph in the case of large scale tests and has the same charac-- teristics as in the case of small scale tests. Photographies 11 shows the run-up phenomena of bore on the

slope in a large tank. The curves of

(d), (e) and (f) in Fig. 3 and the six curves in Fig. 5 were recorded by pressure gauges. It is considered

from these data that bore pressure operated impulsively on the slope at the moment when the bore struck the slope, and kept almost constant

value continuously. The height of bore run-up was measured with the

naked eye.


Photo. 11 Run-up on the slope in a large scale test tank.

C Sec - I Sec

Fig. 4 Sample records of wave gauges by pen-writing oscillograph (large

scale test).

- a

a a


0 1 2 3 4 5

Fig. 5 Sample records of wave pressure

by electric oscillograph (large

scale test).

In order to explain the hydraulic

and dynamic characteristics of the tsunami which Is classified

as the..

bore type, the velocity of bore and induced flow must b investigated

first of all. The symbols as shown. in Fig. 6 are adopted and the two cross-sections, II and between

which the head of bore is included, are determined. The following

for-mulas are introduced from the continuity condition of flow.





in which C and U are shown in the following formulas.




Uand ü in. the above formulas present the respective mean values of U and u. Euler's law of iiiomentum is adopted in regard to the portion of fluid between

the two cross-section, I-I and 11-11. The following formula taking in the direction of flow is led.


PSdV}_cc U2dS+ççS2pu2dS


PdScc rdl



where p is the density of water, r

is the frictional stress of bottom, V is the capacity and S is the area. Each term of Eq. (4) is simplified as follows:

,i555 pUdV=r1pUr1PCHU

where 1 crc


U pud V av2 at555v2 where where where and 555 51pU2dS=i1pU2Si=iiiPU2H


IU\2 dS 55 C 5


( ,c5)


pdS= _4w0h2

where wo is the unit weight of fluid.

Though the last term on the

right-hand side in Eq. (4) can not be discussed easily, the authors lead the following formula using the frictional stress.


(9) where r is the frictional stress of' bottom, U" is the friction velocity,

k is the

value showing the degree of roughness on bottom, z is the parameter determined by the scale of flow (the water depth etc.) and A is a coefficient of distribution

of flow velocity. The area of integral, l, must be determined in correlation to

h, H and the coefficient of the roughness of bottom, n. The following formula is led by using the mean value of r in the distance of 10.

55 tdl


(Ar+ - log


l0 /1 2.3 z\2

1)/ A+ - log



Therefore, it is considered that 7) ean be determined by using water depth, h; height of incident bore, C, and coefficient of roughness of bottom, n. This

correlation is shown by

These simplified expressions are put into Eq. (5). Therefore the equation of mo-mentu.m is


ripCCH_r2puchThPt12 + ij2pi2h =


(H2 h2)_0pH1J2


is defined, then

C(T1LIH_T2uh)(7U2H_112u2b2) (_)(H2_h2)

Equation (1) is put into the above equation, and hence the expression of U is

reduced as follows:



2(iHoC) -

I 2(jHrjC))


From Eq. (1) and (13), the bore velocity is obtained as

1 1 hH(T1+T2) ) - /( hH(1+2) 2_U + g2H(H±h)_2hH(T2h+V2C)ui2 (13)

U ±


2(71HijC) V 1 2(1Hi,'C)




74 (13), and (14). hü ±

/(__/\2 q2(H+h)_2h2

H -v H)



h hHü \ 2 gC2H(H+h)_2hH22


If ij is defined, Uand c can be calculated by using Eqs. (15) and (16), respectively. As for the signs, positive sign is used for the case that bore advances in the

direction of flow and negative sign is used, in the opposite direction. From Eqs. (7), (8), (12) and (13), j is defined as follows:

=*(*)8dH_F(n, -)


The first term of the right hand side is the value showing the degree of water disturbance occured by the advance of bore. Therefore, it may be named the

coefficient of disturvance. The second term .is a coefficient showing the resistance

of bottom. The former is nearly equal to zero in case when the water is disturb.

ed sufficiently. That is, if h/H-+ 0, then

1 (18)

The value of the latter' becomes larger in proportion to the scale of


i.e. as h/H becomes smaller, or it becomes smaller in proportion to the height of

bore. That is, if h/H-p 1, then

F(n,4H)-_0 (19)

Here, if the coefficient of disturbance is considered to be nearly equal

to 1 and

the resistance of bottom is neglected, is also nearly equal to 1. And C is given by

C=--{Cu± H2ü2+ gC2H(JI+h)

H2ü2 =u± J-J(H+h)


the approximate formula in case ofij 1. Hence, it has the following contradiction.

This formula is identical with the usual formula of bore velocity.

Equation (20) is

lim C -+

This means that Eq. (20) is not applied to all cases, and hence the limit of range of h/H whithin which Eq. (20) is applicablemust be given. Especially,

it is

un-reasonable to apply Eq. (20) to tsunamis, because the tsunamis have generally

small value of h/H. However, in case of h - 0, Eq. (16) is reduced to




That is, it has the finite value and is applicable to all cases. Moreover, ij is given experimentally. In this experiment, ü is neligible because of ü = 0. Therefore Eq. (15) and (16) may be reduced to the following forms

r r


I V 1:


(15) } (16)




/ g(H+h)


V 2H(Hrjc)

/ gH(H+h)



Equations (15), (16), (21) and (22) are the improved formu1s of borevelocity

revised by the authors and are general expressions including the usual formulas. The value of i obtained by the experiment is shown in Fig. 7. The plotted data are

obtained by the least square mean method. Figure 8 shows the comparison between

Fig. 7 Relation between resistance Co.

efficient Tj and h/H.

A (

-Fig. 8 Theoretical and experimental values of bore velocity.

the experimental data. and the curves calculated by Eq. (22).

The data in the

case of b=20 cm are plotted below the calculated curve. From this fact it may be

expected that the influence of the steel side-wall of water tank upon bore velocity becomes larger because of increase of water depth. This fact is proved by the

data of the large scale experiment in which the influence of side-wall is negligible. Hence, it can he said that thetheoretical formula agrees quite well with the

experi-mental data in general.


After striking the dike, the bore runs up on

its slope. Then the kinetic energy of bore changes itself into the potential energy. If the velocity of water

particle was observed microscopically, it may be recognized that the phenomena are very complicated. Especially the motion of .water particle at the moment when

bore strikes the slope is quite complicated. Though it is difficult to observe microscopically the energy of water particle, it may be considered that the

veloci-ty of water particle has a relation to the microscopic value of bore velociry, C.

Hence the following relation is assumed here:

vm=aC (23)

where a is a non-dimensional coefficient (a>1). If the flow is perfect fluid, the

horizontal kinetic energy of bore striking the slope changes itself into the poten-tial energy. The vertical height of run-up from still water level, R, is written by



0 013 roughness 8 I 6 5.,,


-Figure 10 shows the comparison of the experimental

and calculated value of R.

The relation between slope and R depends upon k of Eq. (25).

As k' is a factor

introduced owing to the effect of energy

dissipation, it is smaller than 1 and

de-creases in proportion to slope, a is larger

than 1 as it is shown by Eq. (23). V. CLAPOTIS HEIGHT OF

1.0 12 14


FIg. 9 Relation between k and slope Striking against the slope of dike,

of dike, bore begins to

run up the slope. In the

- wake of this, reflex bore

appears and the water depth in front of the slope in-creases in size due to the fact that the incident bore falls on the reflex bore.

This composite wave advances in the

reverse direction in the shape of bore.

Since the velocity of reflex bore depends upon the velocity of incident bore, clap-Otis

has a large wave height inan initial



period of the arrival of incident bore. Fig. 10 Relation between C and R. Accordingly, since the long period wave such as tsunami keeps a large clapotis height for a fairly long time, a large quantity of water overflows across dike crown the height of which is lower than the clapotis crest height.

Here, the


.. .1 .1, ' symi and the ing I Mor tion II is 3 2' : :


6 4



'.1 0.2 0.. I


Figure 9 shows theexperimental

the following formula is obtained;value of k. If Eq. (16) is introducedinto Eq. (26),

Whei R k f h

/(hHü \

C2H(H+h)_2hH1j2 head 2g

(Hri) U_VH_) +



(27) E If u=0, R_ k gH(H+h) kH(H+h) Zg

2(H) - 4(H)

(28) 76 R 2q (24)

As the energy is dissipated in reality, Eq. (24) is multipled by a non-dimensional

coefficient k. That is,

k' k'a2


2 2g In this equation,

k=k'a2 (25)

and k is determined as an experimental coefficient. Hence R is rewitten as



symbols as shown in Fig. 11 are adopted for use

and the condition of continuation of flow between the cross-sections I and Ills shown by the

follow-ing formulas;





= - U0 =


-Moreover, the equation of momentum for the por.

tion of fluid body between the cross-sections I and

II is as follows;

-j- pUodV- jV2 PUdV} pUo'dS


Where Vi is the volume from the head of bore to section I and V2, from the head of bore to II.

Each term is simplified as follows:




PU2H2Co pUo°dS =,iiPtj2Ho=i)jp.U2 ) JH0 where 1. f and

irr 'U'°

v= ()



As for the last term of the right hand side,

cc Ddl=sPHouo2=3Pçoo

J .100





-j--Therefore the equation of momentum is simplified as follows;



Moreover, the equation is rearranged as follows:


Fig. 11 Reflex bore.



pU2{2H2- 7)C0' 712CoH WQCO(H,H2)



_2H2 (._1)} =i(___,

tb-) -'}

Here, the ratio of (Ho/H)_m is defined and named "the ratio of duplication." The above equation is rewritten by using m as follows;

o(m_1)} =i(m_1)(m2_1)


-j-m(ml)(m2-1)+,j(m-1)+rj2m(m_1)__2m=0 (29)


2m(m.4)(m2_1)+m2_2m_1O (30)

Since Uand H are the known values, m can be obtained from Eq. (30) and Ho is given by H0 = mH. Figure 12 shows the relation between gH/2C2 and m. The the-oretical curve is calculated by

Eq. (30), hence the influence of bottom friction is neglect-ed. It may be proved easily

that each of Eqs. (29) and (30)

has only one real root under

the condition of m>1. VI. PRESSURE OF BORE


In this paper, the bore

II pressure is classified into the

the "impulsive pressure" and Fig. 12 Calculated and experimental values of tio the "continuous pressure."

of duplication. The former is the force which

operates impulsively on the slope of dike at the same time when bore strikes against the slope and the latter

is that which operates continuously and statically when the water depth in front

of dike rises up due to that the reflex bore fallson the incident bore.

1. ImpulsIve Pressure

It is considered that the factor which is closely related with impulsive pres-sure is the velocity of bore. Therefore, the relation between impulsive pressure Pk and C is plotted in a logarithmic section-paper as shown in Fig. 13. Though the data obtained by large scale tests are separated from that by small scale tests to some extent, the slopes of two straight lines are about 4 in common with each other. On the basis of this fact, it seems that the exponent of bore velocity which is related with impulsive pressure is about 4. After this is recognized as the experimental facts, the formula for P is led from dimensional analysis.. The


, 0 Sao 0 0 00

oSrnall scale test Large scale test





maximum impulsive pressure on the slope

of dike is represented by pa and the

fac-tors which are related to p,..,, are selected as follows: p=f(C4pgPCT) (31) Hence, Moreover, where ure." 3 .2 10' 8 .O a targe scat1

OQo a small scat



0 '8 100 80 60 40 12 .20 6



K0 is a non-dimensional coefficient named "the coefficient of wave

press-Figure 14 is obtained by rearranging the data of Fig. 13. In Fig. 14 the

difference between the data obtained by small

scale experiments and that by large scale ex-periments disappears and the theoretical line _____ is fit for the experimental data. The value

of Ka is obtained from Fig. 15.

The distribution of impulsive pressure is

shown in Fig. 16 and the maximum pressure is appeared nearly at still water level. The

formulas of the distribution of impulsive pres-sure are obtained as follows;

Pa R,

04 0.6080 20 40 6080100

1 A

(33) Porn

0. 02

C-Fig. 13 Relation between C and P.

Fig. 14 Theoretical and experi-



mental values of

impul-where Pa is the impulsive pressure at the

point of height R9 above still water level and A is a non-dimensional constant (Co1.4) determined experimentally. From Eqs. (26), (32) and (33), the following formula is led.

sive pressure.

02 04 06 0.8 10


-FIg. 15 Relation between ko and slope.



eon a smal scale 1ii



0 0.1 0.2 04 06 1.0 2. C' (rn/sect 2 1.4





As mentioned previously the water depth in front of dike increases consider-ably due to that the reflex bore falls on the incident bore. Here the duration of

the increased wave height is taken into consideration. It seems that the lapse of time, T, during which the height of reflex bore, Ca, decreases into the height of incident bore, C, is in proportion to the length of incident bore, 2, and in inverse.

proportion to the velocity of reflex bore, C0.

That is,


where K is a non-dimensional constant. Moreover the following formula is led by using the relation of Co= UHICO3 in which UH obtained from Eq. (1);

ih+CC CaC If ü=O, CC Co -CaC

Introducing the above result into Eq. (37), the following formula is obtained;

TcK2/( CaC)




where K is a Qn-4imensiQnal coefficient (0 2/3) determined experimentally. The




.-.- 4 04 06 Pk K0_woC4 A


= g2C kC2 (34) 2. Continuous Pressure

Since the pressure of this case is nearly equal to hydrostatic pres-sure of the clapotis height of reflex bore, the distribution of continuous

pressure is shown in Fig. 17 by using the clapotis height of reflex bore, Ca.

The formulation of Fig. 17 is



WoCa Ca

where p, is the continuous pressure at the point of height R9 above still water level and B is a

non-dimen-sional coefficient (=0.75) determined experimentally. Therefore, Ps=wo(Ca_fRs) (36) 0 0 o0 0 0 : 000 0


° ° g




0 '00 0 "0 00 . O C 0 0O 0* 0 0C)0/ .. , .0 0 :000 0

0oo:. P

Fig. 16 Vertical distribution of implusive


02 04 06 0.8 1.0


Fig. 17 Vertical distribution of continuous pressure.

0.0 1.0




experimental data and the corresponding values

cal-culated by using Eq. (38) are plotted in Fig. 18. Next, the quantity of overflow discharge across a dike crown is estimated under the condition of

C H<Ha<Ho as shown in Fig. 19. Under the other

condition such as H4>Ho>H or Ha.(H<Ho, it is not

necessary to be estimated due to the

following reasons. In the former case the bore runs up the dike only, hence the overflow discharge is negligib-ly small. While the latter condition is not accepted as the design criteria of dike.

If the initial overflow depth is represented by H0,

H0 = Ho. H4

The overflow depth after the lapse of t hours is given by

where p ( 1) is a coefficient depending upon the bore shape. The quantity of

overflow discharge for dt hours is given by dQ= ./EHw8/2dt

02 04 0.6 0.8 1.0

Fig. 18 Relation between Co/C and T/T.

Fig. 19 Overflow discharge of bore type tsunami across dike crown.



Therefore, the quantity of overflow discharge by a bore per unit length of dike, Q, is calculated by

ç wOHS/2dH

2 4r-ET H0h/2=2%'2Tc


pCo jo 5p



From the standpoint of preservation of shore structures, the authors classify

the tsunamis into two types as previously mentioned and have analysed theoretically and experimentally the behaviour of bore type tsunami. Especially, the scale effect

has been studied by using twc kinds of wave tank, one is small size and the

other large size. The hydraulic problems on tsunamis are under the rule of Froude's similarity law. The small scale model study has been carried out under the con-cept of Froude's similarity law, and the factors of viscosity and eddy viscosity,



which seem to be the important factors next to the gravity and inertia forces, are considered in the form of bottom friction. Although the necessary data can be obtained by the small scale experiment, the large scale .experiment has also been carried out in order to test the scale effect on the air mixing phenomenon at the head of bore and to increase the number of measuring point of bore pressure. Figures presented in this paper show the identity of results of both experiments. Accordingly, it seems that the small scale experiment is satisfactory to solve this

kind of hydraulic problems.

In the design of actual dikes, nothing can be obtained as the data of actual tsunamis except the traces left behind the tsunami attack. Therefore, it is very important to make a decision whether the traces show the height of incident tsu-nami, or the clapotis height of reflex tsutsu-nami, or the height of run-up on shore land, considering the features of fore-shore and hinterland. For instance, on the coast where there is a mountain close to the shore line, the traces show the cia-potis height, while on the coast with a gentle slope the traces show the run-up

height. In the case of tsunamis invading the inner part of hinterland, the traces near shore line may show the height of incident tsunami.


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