Waves of long and short period

W A V E S O F L O N G A N D S H O R T P E R I O D

B Y I R . L . J . M O S T E R T M A N

PROFESSOR OF HYDRAULICS, TECHNOLOGICAL UNIVERSITY, DELFT

L O N G P E R I O D W A V E S

T h e movement o f t h e waves o f the water surface is a subject w h i c h has i n t r i g u e d b o t h physicists and engineers f r o m the beginning o f h y d r a u l i c studies. I t is reasonable and useful to deal w i t h long period waves and short period waves separately because o f t h e different fields o f application and d i f f e r e n t types o f mathematical treatment. M o t i o n i n l o n g period waves is essentially horizontal and accelerations i n the vertical direction are so slight that their eflfect can be neglected compared w i t h the eflfects o f t h e forces w o r k i n g i n the horizontal direction. The equations o f m o t i o n have therefore to be appUed i n only one direction, namely t h a t o f flow. T h e most i m p o r t a n t long-period waves occurring i n na.ture are the t i d a l waves. For ideahzed conditions they were intensively studied by the great hydrodynamists o f t h e last century. D u r i n g the design o f the m a j o r hydraulic undertaking i n the Netherlands i n the first h a l f of this century, the Zuiderzee works, i t became necessary to know more about the reduc-t i o n i n heighreduc-t o f reduc-t i d a l waves i n shallow wareduc-ters due reduc-to b o reduc-t reduc-t o m f r i c reduc-t i o n . T h e p a r t played by THIJSSE i n elaborating L O R E N T Z ' S theory o n the f r i c t i o n effect on t i d a l waves f o r practical use is described elsewhere i n this volume.

W i t h the increase i n scope and size of hydraulic constructions, long-period waves of a n o n - t i d a l character became o f increasing i m p o r t a n c e . T h e surge or translation wave, the flood wave on rivers, osciUations i n basins and the expansion wave, f o u n d i n cases where the compressibility o f the fluid has to be taken i n t o account, are essentially a l l long-period waves. T h e translation wave, a non-periodic phenomenon, was the first type o f

water wave subjected to mathematical treatment. I n 1781 J . L . L A G R A N G E published his w e l l - k n o w n f o r m u l a for the celerity of this wave c = Vgho. This expression was based on the assumption that the elevation o f the water surface due to this wave was small compared w i t h the original water depth {ho) (FIG. 1). Developments o f t h e theory by later physicists and engineers enlarged the insight into this phenomenon. Formulae were f o u n d describing wave celerity w i t h o u t the restriction f o r the height of

FIG. 1 Translation wave

the wave. The principle of superposition was proposed for use w i t h translation waves. W i t h the aid of this principle the interference o f several waves, the division o f the waves over branching canals, as w e l l as their total or p a r t i a l reflection at obstacles, could be studied. T h e v a l i d i t y of this principle for full-scale canals and rivers had not yet been proved, however, at the end o f t h e 1920's.

Also, the reduction i n height and the change i n shape o f translation waves caused by f r i c t i o n , problems similar to those studied b y L O R E N T Z f o r t i d a l waves, were not yet assessed by analytical study.

I n 1930 a number o f new navigation canals was under construction i n the Netherlands. Locks w i t h a relatively h i g h l i f t were b u i l t i n these canals. E m p t y i n g and f i l l i n g these locks had to take place i n as short a time as possible. By h y d r a u l i c model studies devices were developed that

insured r a p i d , and at the same time t r a n q u i l , f i l l i n g of lock chambers, so that the vessels were not Hable to damage d u r i n g this operation. T h e conditions i n the canals proved, however, to be l i m i t i n g f o r the speed o f locking. T h e e m p t y i n g and f i l l i n g generated translation waves i n the canal reaches near the locks. F i l l i n g the chambers caused negative transla-t i o n waves, e m p transla-t y i n g caused positransla-tive waves. Sudden changes i n watransla-ter velocity and i n water level were sometimes caused by these waves, partic-u l a r l y i f waves f r o m two sides of a canal reach were spartic-uperimposed. Vessels w i t h deep d r a f t ran aground, boatsmen lost control over their vessels and canal linings were influenced by the surging action.

Measurements were undertaken i n a few D u t c h canals by engineers o f the State Public Works Department. T h e m a i n reach of a large D u t c h navigation canal, the Twente kanaal, was near completion i n 1934. I t offered an o p p o r t u n i t y to take measurements on a scale that was w i t h o u t precedent. THIJSSE f o r m e d a measuring team that took simultaneous measurements on twenty different points along the canal d u r i n g a twelve hour period. Translation waves were generated by discharging inter-m i t t e n t l y up to 50 in3/sec through a lock w i t h 7.10 inter-m head difference. T h e variations i n water level i n the lock chamber were used f o r deter-m i n i n g w i t h high accuracy the quantities discharged. T h e water levels i n the gauging stations were read every 15 seconds.

T h e results ( L I T T . 1) conflrmed that the principle of superposition is v a l i d for the conditions encountered i n a canal of considerable l e n g t h w i t h several irregularities i n cross section and aUgnment. T h e f o l l o w i n g con-clusions were reached:

a. the celerity of a low translation wave can be described quite accurately by L A G R A N G E ' S f o r m u l a ;

c = Vgho

b. the relation between the q u a n t i t y of water displaced (Q) and the wave height can be described by the expression:

Q

c. for a wave of appreciable height, points on the wave f r o n t have celerities w h i c h vary as a f u n c t i o n of their elevation, so t h a t the wave f r o n t is deformed. T h e expression f o r the celerity at various elevations of the wave f r o n t :

I gho 1 + ' ^

2 Ao,

is not entirely i n accordance w i t h the observations at d i f f e r e n t levels but i t gives a good a p p r o x i m a t i o n ;

at changes i n cross section part of the wave continues w i t h the height: 2 ( é f ) i

Z2

Zl-( é c ) 2 + Zl-( k ) l

and another part reflects (FIG. 2) w i t h the height: ( i f ) i - ( é c ) 2

Z3

( k ) l + ( é c ) 2

Theoretical considerations guided THIJSSE to an expression f o r the damp-i n g o f a translatdamp-ion wave due to f r damp-i c t damp-i o n . He damp-i n t r o d u c e d as a parameter the length 7 over w h i c h a wave has to r u n before its height is halved. T h i s length can be f o u n d f r o m :

wC^hR y =

and the wave height after a distance x is:

y z =

zo—--x+y

where:

C = G H É Z Y ' S f r i c t i o n coefficient;

(p = d. cocfTicient v a r y i n g between 1 and 3/2;

Zo = the original wave height;

g = acceleration due to g r a v i t y ;

R = the hydraulic radius o f the cross-section.

^—

FIG. 2 Partial reflection of translation wave at canal transition

T h e data obtained f r o m the Twente canal measurements c o n f i r m e d the usefulness o f this f o r m u l a f o r practical purposes. This f o r m u l a f o r the reduction of the height of a translation wave still gives the best a p p r o x i m a -t i o n available.

I n 1940 one of THIJSSE'S collaborators, I r . D . M . D I E T Z , developed a

more accurate b u t also more intricate f o r m u l a for calculating the behavi o u r o f translatbehavion waves modbehavifbehavied by f r behavi c t behavi o n . A f t e r usbehaving de S A I N T -V E N A N T ' S assumption that the water velocity i n a cross section under the wave is a f u n c t i o n of the elevation of the water level only and t a k i n g into account the resistance influences, D I E T Z integrated the equations o f m o t i o n along the characteristics of the p a r t i a l differential equations. H e obtained an expression f r o m w h i c h both the celerity and the r e d u c t i o n i n height can be calculated ( L I T T . 2).

I n the design of the navigation locks b u i l t i n the Netherlands d u r i n g the last 25 years, simpler and more economical means of filling and e m p t y i n g have been used. Simple valves i n the gates are used instead o f special distributors o f water, systems o f aqueducts or dissipators of jet energy. T h e forces on the ship's hawsers are l i m i t e d by p r o g r a m m i n g the rate of opening o f the valves and by using favourable locations and d i m e n -sions o f the valve openings. T h e pertinent investigations are as a rule carried out i n a model. Study of the hydraulic phenomena o c c u r r i n g d u r i n g the filling o f the lock chamber has contributed considerably to the progress made. Essentially, the filling of the chamber does not take place by a continuous rise of the water level at a l l places simultaneously. Rather, filling takes place via translation waves travelling back and f o r t h i n the chamber that are d i m i n i s h i n g i n height and at the same t i m e increasing i n speed as the filling progresses. T h e static and d y n a m i c forces i n the vessels and the p u l l on the hawsers are determined by the surface slopes and water velocities caused by these waves. I r . J . B. SCHIJF, l o n g t e r m collaborator o f THIJSSE, worked out a method for the r a t i o n a l cal-culation of the forces on a vessel d u r i n g locking ( L I T T . 3). H i s analysis showed how the forces of most importance could be calculated. These were the f r i c t i o n exerted by the m o v i n g water on the ship, the force o n the ship due to the slope of the water level caused by the dissipation o f energy and the forces caused by the wave m o t i o n inside the lock. For a given filling curve o f the lock, the forces on the ship can be calculated.

This procedure shows once more the usefulness o f the theory o f long-waves for practical apphcations.

Floods on a river, caused by rain or melting snow, give rise to a specific type o f long waves that has been studied for a long time i n m a n y parts o f t h e w o r l d . Basically the celerity o f t h e flood wave (defined i n this case as the velocity of propagation of a section where a certain Q prevails) can be described b y :

= 1 ^

^ è dA

Here b is the total w i d t h o f fiow, i n c l u d i n g a f l of the bed area that takes part i n the storage o f the flood water. Most interesting measurements were taken i n the Netherlands after the coflapse o f the M ö h n e d a m i n the R u h r area, due to w a r action i n M a y , 1 9 4 3 . The wave that descended the R u h r after the d a m failed acquired the character o f a steep flood wave o n the Rhine River. The steepness o f this wave r a p i d l y diminished due to the considerable storage i n the river bed. T h e resultant reduction i n celerity caused a relatively slow progress of this wave. F l o a t i n g debris f r o m the disaster passed the lower reach o f the R h i n e i n the Netherlands long before a rise i n water level could be measured. T h e basic diflFerence between the propagation velocity of the flood wave and the velocity o f flow was clearly demonstrated by these observations.

M o d e r n technology demands an ever-increasing refinement of long-wave studies. T h e growing dimensions of works that infiuence the flow o f water i n rivers and canals for flood control, hydropower, navigation, drainage and i r r i g a t i o n purposes make an accurate study o f variations i n water level and velocity necessary. T i d a l calculations cannot always be l i m i t e d to the calculation o f t h e d a m p i n g o f a wave idealized by a sinusoid. O f t e n the d e f o r m a t i o n o f the t i d a l wave has to be studied. Non-periodic storm surges o f the k i n d that struck the Delta region o f the Netherlands so dramatically i n 1 9 5 3 , have to be studied i n detail where they enter the river mouths and meet protective structures.

A n analytical refinement of the calculations meets insurmountable ob-stacles. N o t w i t h s t a n d i n g the progress made by several o f t h e world's most famous scientists i n the last 1 5 0 years, LAGRANGE'S statement ( L I T T . 4 ) on long waves keeps m u c h o f its v a h d i t y :

„ P a r ces é q u a t i o n s , toute la m é c a n i q u e des fluides est reduite a u n

seul p o i n t d'analyse, et si les é q u a t i o n s é t a i e n t i n t é g r a b l e s , o n pour-rait d é t e r m i n e r c o m p l è t e m e n t les circonstances d u mouvement et de Taction d ' u n fluide m ü par des forces quelconques; malheureuse-ment elles sont si rebelles, q u ' o n n'a p u , jusqu'a p r é s e n t , en venir a bout que dans des cas t r è s - l i m i t é s . "

N u m e r i c a l and graphical methods w i l l have to be used. I n 1 9 1 6 , Prof. I r .

G . H . D E V R I E S B R O E K M A N , THIJSSE'S predecessor i n the chair at D e l f t ,

gave ( L I T T . 5 ) an outline o f a numerical method to be applied to storm surges o n the R o t t e r d a m W at er way. T h e magnitude o f the necessary computations prevented the practical application o f this method at that time, as electrical calculation aids were not yet available.

A f t e r 1 9 4 4 a graphical method came into use i n d i l f e r e n t countries simultaneously. This method was developed more than 5 0 years earlier by the engineer J . M A S S A U f r o m Gent w h o gave an excellent description o f the procedure w i t h various applications. His work was forgotten u n t i l its rediscovery b y the French professor A . G R A Y A i n 1 9 4 4 .

T h e method is based o n the f o l l o w i n g property o f p a r t i a l d i f f e r e n t i a l equations o f the hyperbolic type. G o n t i n u i t y and m o t i o n o f a long wave are described b y the equations:

dv dv dh S" ^ T

dv dh dh

h V V h — = 0 .

dx dx dt

T h e independent variables are time t and distance x. For the i n t e g r a l surface the f o l l o w i n g two equations o f these independent variables h o l d :

dv dv — dt -\ dx = dv dt dx dh ^ dh ^ — d,r H dt = dh dx dt

I n this set of four equations the unknowns are the derivatives:

dv dv dh dh

— , — , — and — .

dt' dx' dt dx

O

A n y o f these derivatives m a y have finite discontinuities that are prop-agated along two sets o f curves called characteristics. T h e directions of these characteristics are f o u n d f r o m the indeterminate f o r m of the above equations. C o n d i t i o n f o r indeterminacy is that the determinant o f the above equations is equal to zero:

1 V g 0

0 h V I dt dx 0 0

0 0 d,r dt

T h e two characteristics directions are f r o m this determinant: d,r

dt

O n the characteristics defined b y :

dx

— = y + V gh

dt ^

the v a r i a t i o n i n the expression v^lVgh is f o u n d to be equal t o :

v±Vgh

C^h dt.

and on the characteristic defined b y :

dx

the variation i n the expression:

y —2 V'^/z is e q u a l t o ^ / CVi / / B K • tl

ft

Principle of the characteristics method

Calculations are made using intervals o f finite length. Every p o i n t i n the

s-t diagram is determined along its two characteristic directions (FIG. 3).

As soon as he became aware of the possibilities offered by this method, THIJSSE started to apply i t to p r a c t i c a l engineering problems. H e suc-ceeded i n presenting this d i f h c u l t procedure i n a very lucid w a y i n his lectures. M a n y applications were studied by h i m and by members of his staff under his guidance. A m o n g the problems studied were the channel w i t h gradually v a r y i n g w i d t h and the channel w i t h depletion or a d d i t i o n of water over its entire length. Quite recently THIJSSE studied w i t h the aid o f this method the hydraulic phenomena that accompany the f o r m a -t i o n o f an ice deck on a river.

M . J . BOSSEN, who for many years was THIJSSE'S m a i n collaborator i n educational matters, applied the method of characteristics to the effects produced shortly after placing an object i n t o a river ( L I T T . 6 ) .

I f an obstacle such as a weir is placed i n a river, i t takes some time before the backwater upstream of the obstacle has developed head difference big enough to transport the original river discharge through the r e m a i n -ing open-ing. I m m e d i a t e l y after plac-ing the object, negative and positive translation waves are formed downstream and upstream respectively. T h e discharge is diminished locally for some time u n t i l permanent conditions have been established again (FIG. 4). T h e results of BOSSEN'S w o r k were not only applied to questions f r o m the practice o f hydraulic engineei4ng but also for c l a r i f y i n g a well-known event f r o m ancient history. T h e Garthagean general, H a n n i b a l , crossed one of the w i l d l y flowing alpine

FIG. 4 Sudden movement of a gate

rivers by p u t t i n g a row o f elephants across to f o r m a temporary dam. Ancient historians stated that the reduction of flow caused by this barrage lasted so long that the army could safely cross the river. By a p p l y i n g the characteristics i t is shown that this statement could be true.

By simultaneous application o f t h e characteristics rules and the c o n t i n u i t y and impulse-momentum equations, THIJSSE demonstrated how discon-tinuities i n the f l o w can be traced and passed. The recent i n t r o d u c t i o n of electronic computers for wave studies has extended the scope of long-wave calculations. Methods for advantageously using computers are studied at many places. T h e method of characteristics can be adapted to the possibilities and limitations of the computer. Finite difference approximations of the wave equations can lead to r a p i d and accurate c o m p u t i n g methods. I n wave studies for irregularly shaped n a t u r a l chan-nels the large quantities of data involved i n even a simple p r o b l e m make demands on the computer's memory that cannot always be met.

F R I C T I O N F O R M U L A E

T h e correct estimation of the influence o f f r i c t i o n a l resistance to flow was the m a i n d i f f i c u l t y that had to be surmounted by L O R E N T Z i n his studies of tides. I n most long wave problems f r i c t i o n is the m a i n l i m i t a -t i o n -to f u r -t h e r refinemen-t of -the calcula-tions. Because observa-tions show that the shape o f the velocity distribution is only changed very g r a d u a l l y by the progress of the wave, use is made of G H É Z Y ' S resistance f o r m u l a that belongs to stationary u n i f o r m flow.

L i t t l e was k n o w n h a l f a century ago about the physical laws governing t u r b u l e n t f r i c t i o n . T h e vast amount of measurements taken by engineers d u r i n g two centuries was summed up by a m u l t i t u d e o f e m p i r i c a l for-mulae. Arbitrariness and contradiction often obscured the insight i n t o the limits of applicability o f each of them. Go-ordinated observations and analyses by a number of scientists led to a coherent theory o f t u r b u l e n t f r i c t i o n i n the twenties and thirties of this century. Based u p o n these i n -vestigations W H I T E and GOLEBROOK proposed their formulae for resistance losses i n pipes and canals. These results d i d not have an immediate i n -fluence on civil-engineering practice, however. This could not o n l y be contributed to conservatism, b u t also to the simphcity o f some o f the older empirical formulae and to their accuracy w i t h i n the l i m i t e d d o m a i n

of their original data. None of them could be applied over a wide region of different dimensions, cross-sectional shapes and w a l l properties and they were lacking the inherent qualities o f v a h d i t y and beauty attached to physical laws. B o t h THIJSSE'S scientific idealism and practical sense were challenged by this situation. H e looked for a f o r m u l a w i t h a wide field of application, simple enough for practical use and adapted to the logarithmic velocity distribution that followed f r o m theory. I n 1949 he proposed that the I n t e r n a t i o n a l Association for HydrauHc Research

( L I T T . 7) adapt a simplified version o f W H I T E and COLEBROOK'S f o r m u l a

as the standard f r i c t i o n f o r m u l a for hydraulic engineers. A c c o r d i n g to this proposal the coefficient C f r o m G H É Z Y ' S f o r m u l a v = CVRI is deter-mined f r o m w a l l roughness and canal or pipe dimensions b y :

C = 1 8 1 o g i ^ ^ 4-5 i n w h i c h :

R = hydraulic radius o f cross section; d = thickness of laminar boundary layer; k = dimension o f w a l l roughness.

By THIJSSE'S teaching and influence this logarithmic f o r m u l a soon came i n t o general use i n the Netherlands. The experience i n this c o u n t r y has shown that i t gives sufflciently correct results for closed as w e l l as f o r open channels. F I G . 5 shows a diagram for finding GHÉZY-'S coeflicient for this logarithmic velocity distribution.

W I N D S L O P E

A subject on w h i c h THIJSSE d i d m u c h research was the b l o w i n g up o f the water surface by the w i n d , as a consequence o f w h i c h this surface gets a slope. T h i s phenomenon is not o f a wave nature, i t is so closely allied, however, to waves that i t deserves m e n t i o n here. F r o m the simple consid-eration o f e q u i l i b r i u m between the w i n d shear on the water surface:

üQaW^ (in w h i c h Qa is the density o f t h e air and w is the w i n d speed), and

the pressure gradient due to the slope of the water surface: gghi an expression f o r i can be f o u n d . T h e d i f f i c u l t y is i n the d e t e r m i n a t i o n o f the value of the f r i c t i o n coefficient a. Measurements on the Zuiderzee

and on lakes o f various dimensions i n f f o l l a n d and Friesland gave f o r a on the mean the value 3 X 10^3_ ^ h e so-called Zuiderzee f o r m u l a f o r the w i n d slope becomes then:

i = 0.4 X 1 0 - 8 — .

h

T h i s f o r m u l a is used f o r d e t e r m i n i n g the freeboard of dikes and dams and o f the banks of canals. I t has been apphed on a b i g v a r i e t y o f works all over the w o r l d . THIJSSE conducted several measurements i n nature and i n the w i n d tunnel to test its v a l i d i t y and to determine the coefheient.

A f t e r the enclosure of the Zuiderzee recording instruments were placed around the newly formed IJssel lake, where observations were easier be-cause the tides had disappeared. T h e coefficient a was f o u n d to have a smaller value f o r smaller w i n d velocities because o f t h e bigger smoothness o f t h e water surface. A w i n d that quickly increases towards a gale causes a higher set-up o f t h e water level t h a n the f o r m u l a indicates. I t was also shown that f o r smaller lakes and f o r canals the w i n d slope is considerably reduced because o f the sheltering effect of the banks.

S H O R T - P E R I O D W A V E S

W i n d action is responsible f o r the generation of v i r t u a l l y a l l short-period water waves that are of importance to the h y d r a u l i c engineer. V e r t i c a l accelerations have here the same order o f magnitude as the h o r i z o n t a l accelerations so that their influence cannot be neglected. I n 1 8 0 4 already F . J . V O N GERSTNER of Prague pubhshed a theory o f these waves. T h e wave shape and celerity as well as the m o t i o n o f the water could be described i n great detail. Later investigators, especially G . G . STOKES, removed inconsistencies f r o m GERSTNER'S theory and extended the theory. O n a few aspects most i m p o r t a n t to the hydrauhc engineer, namely the laws governing the waves' generadon, the deformation near structures and i n shoaling water and the forces by these waves on obstacles before

1 9 4 0 very little was k n o w n .

THIJSSE took already early an active interest i n wave studies. I n 1 9 3 5 a w i n d flume w i t h a length of 5 0 m (later lengthened to 6 0 m) was b u i l t at the D e l f t Hydraulics Laboratory. W i n d waves are f o r m e d by a strong current of air passing over the water. By measuring the velocity profiles i n the air, the w i n d slope of the water surface, the head loss i n the air c u r r e n t and the shape and the velocity o f propagadon o f the waves, the energy transfer f r o m air to water can be studied. By placing models o f hydraulic structures i n the flume the deformation o f the waves on or near these structures is measured and also the pressures by waves o n the structure. These laboratory facihties were unique i n the w o r l d for m a n y years.

D u r i n g the second w o r l d war the necessity to prepare large scale amphibic operations gave an impetus to wave studies unprecedented i n history. A group of scientists and engineers under the leadership o f t h e N o r w e g i a n

oceanographer H . U . SVERDRUP undertook wave studies, especially on the coast of California. A f t e r the war this work was continued and now the hydrauhc engineer has a considerable arsenal o f methods and data on short waves available. T h e deformation of waves i n shoahng waters, the breaking of waves near the shore, the refraction at a gradually sloping foreshore and the d i f f r a c t i o n of waves passing r o u n d an obstacle can be studied quantitatively nowadays ( L I T T . 10).

T h e mechanism of the energy transfer f r o m w i n d to waves was intensively studied b y THIJSSE. A f t e r the first disturbances o f t h e water surface occur, their size increases to ripples and then to waves by differences i n the air pressures above wave crests and troughs. I t was not quite clear whether the energy is transmitted to the water only by the f r i c t i o n between the two media or whether the pressure between w i n d and water is the agent of energy transfer. A paraffine model of a succession of wave shapes was made at D e l f t and a fluid was led to move over i t . T h e pressures along the wave profile and the velocity distribution i n the air over the wave were measured (FIG. 6 ) ( L I T T . 8 ) . I n the air leeward of the wave crest an eddy was observed. Due to this eddy the pressures at the leeward side of the crest were lower t h a n those at the w i n d w a r d side and consequently the energy transfer f r o m w i n d to water was bigger t h a n that f r o m water to w i n d . N o r m a l pressures are m a i n l y responsible f o r this energy transfer.

FIG. 6 Pressure and velocity distribution over a wave profile. Pressure in meters of water-column (with the pressure at the crest as datum)

T h e asymmetric shape of the wave is here a decisive factor. T h e under-pressure at the summit of the waves causes a gradual increase i n height followed b y instability and a collapse of these summits after w h i c h the wave takes again its original f o r m . Transfer o f t h e energy to the i n t e r i o r of the water causes an extension of the o r b i t a l m o t i o n i n the depth b y w h i c h the length of the waves increases. I t seems that an e q u i l i b r i u m condition is approached where a l l energy transferred f r o m the w i n d is dissipated inside the water. I t is not yet conclusively k n o w n whether there is a definite l i m i t to the g r o w t h i n height and length o f the waves. SVERDRUP published a graph showing the increase i n wave height [H) and wave length [X = 277:?) for deep water. T h e determining factors were beside the w i n d speed w, the time d u r i n g w h i c h the w i n d had b l o w n or the distance (fetch F) over w h i c h the w i n d had been acting o n the water. T h i s graph was made fit f o r use under various circumstances by t a k i n g as parameters the dimensionless quantities gFjw^, gHjw^ and grjw'^. T h e circumstances near the D u t c h coast made i t necessary to have data avail-able f o r the g r o w t h of waves i n water of small depth. I n shallower water dissipation o f energy is greater due to the f r i c t i o n along the b o t t o m . By the study o f observations i n various parts of the w o r l d combined w i t h measurements i n the w i n d flume THIJSSE could design a g r a p h f o r the g r o w t h of waves taking into account b o t h deep and shallow water c o n d i -tions. Also the water depth d was brought here i n the dimensionless shape

gdjiv^. T h i s graph has proved to be useful under a b i g variety of c i r c u m

-stances (FIG. 7 ) .

Special importance have i n the Netherlands the waves at the interface between two liquids of different density like salt and fresh water. T h e lighter water of less sahnity covers the deeper salty layers and at the inter-face short-period waves occur. Because of the small difference i n density between the two liquids these waves have a very small celerity and a considerable wave height. T h e wave crests become sooner instable t h a n surface waves and the resultant m i x i n g o f fresh and salt water brings the salinity f a r i n l a n d along the deep channels f o r navigation. T h e brackish water forms a danger f o r the h o r t i c u l t u r a l , i n d u s t r i a l and domestic use o f t h e water. Prof. D r . P. G R O E N o f t h e R o y a l Netherlands Meteorological Institute published a theory on this subject, that has been checked i n the D e l f t hydraulics laboratory ( L I T T . 1 1 ) .

O f special importance f o r the design of dikes or dams along seas and

V ?

FIG. 8 Wave run-up

reservoirs is the height of wave r u n u p . M a n y investigations i n the w i n d -f l u m e o -f THIJSSE'S laboratory were devoted to this subject. A wave passing the toe of the dike w i l l first lose p a r t o f its energy b y breaking. By the r e m a i n i n g kinetic energy a layer o f water w i l l be pushed u p w a r d along the dike's slope to such a height that the energy w i l l be dissipated or transferred i n t o potential energy (FIG. 8 ) . T h e water w i l l fiow d o w n then again along the dike's slope. The water rushing u p w a r d m a y meet o n its way the waters f r o m the preceding wave that flow d o w n again. This gives an i m p o r t a n t energy loss and reduces the run-up height. T h e en-countering o f successive waves on the slope is the determining factor i n r u n - u p . Both the time intervals between successive wave crests and the wave heights show i n nature a stochastic distribution. Therefore this p r o b l e m is essentially o f a statistical nature. Use of a w i n d flume where the waves generated have a d i s t r i b u t i o n of height and period similar to t h a t of sea waves is essential. F r o m the investigations at D e l f t the f o l l o w i n g f o r m u l a was f o u n d for the run-up height z that has a chance o f 2 % o f being exceeded:

z = m tg « (cos ,9 - B\l)

I n w h i c h :

z the 2 % run-up height;

I I wave height at the dike's toe;

a = angle of slope of the dike;

P = angle between wave crests and axis of d i k e ; B = w i d t h of a berm, i f any;

I = wave length.

T h i s f o r m u l a is v a l i d for a wave steepness H\X o f about 0.05. For other wave steepnesses the coefficient i n the formulae diflfers slightly f r o m the

value 8. Specially designed rough or permeable dike revetments give reductions i n the run-up height of between 10 a n d 3 0 % .

Reviewing the results of THIJSSE'S work on long and short period waves discussed here one encounters the features common to a l l his w o r k . Sharp observation o f nature, able experimentation. A scientific idealism that gives physical reasoning the p r i m a c y over empiricism. A n engineer's instinct that separates the non-relevant f r o m the useful. W i t h whatever amount o f pains a theory or method is once conceived, i t is m o d i f i e d or replaced when the advance of science and technology offers new insights or new possibilities.

T h e study of THIJSSE'S w o r k is not only a lesson i n hydraulics, i t is i n the first place an experience i n sound engineering.

L I T E R A T U R E

1. J . T H . THIJSSE: Onderzoek betreffende den invloed van golfbewegingen en langs-stroomingen in Jianaalpanden. 16de Internationaal Scheepvaartcongres, Brussel 1935.

2. D. N . DIETZ; A new method for calculating the conduct of translation waves in prismatic canals. Physica V I I I (1941) no. 2, p. 177-195.

3. J . B. SCHIJF : Calcul des forces agissant sur un bateau dans un sas d'écluse, pendant l'éclusage. Bulletin de l'Association Internationale Permanente des Congrès de Navigation 11 (1936) no. 22, p. 65-100.

4. J . L . LAGRANGE: Mécanique analytique. Paris, 1788.

5. G. H . DE VRIES BROEKMAN: Invloed van eb en vloed op benedenrivieren. De Ingenieur 30 (1916) no. 29.

6. M . J . BOSSEN: The effects produced during the first few hours after placing an object in a river. Proc. Kon. Ned. Ak. v. Wet. Series B, 56 (1953) no. 4, p. 392-402. 7. J . T H . THIJSSE: Formulae for the friction head loss along conduit walls under

turbulent flow. Report no. I I I - 4 , I.A.H.R., 3rd Meeting, Grenoble, 1949. 8. J . T H . THIJSSE: Dimensions of wind-generated waves. Report Gen. Assembly Int.

Assoc. Phys. Oceanography, Oslo, 1948.

9. P. GROEN : On the behaviour of gravity waves in a turbulent medium. Koninkl. Ned. Met. Inst, de Bilt. Med. en Verh. 8 (1954).

10. J . T H . THIJSSE and J . B. SCHIJF: Penetration of waves and swells into harbours; means of predicting them and limiting their action. Reference to model experi-ments. X V I I t h International Navigation Congress, Lisbon 1949.

11. P. GROEN: Contribution to the theory of internal waves. Koninkl. Ned. Met. Inst, de Bilt. Med. en Verh. 2 (1948) no. 125.