Original surface disturbances from wave records

Original surface disturbances from wave records TARANG K H A N G A O N K A R AND BERNARD LE M E H A U T E

Division of Applied Marhw Physics, Rosenstiel School of Marhw and Atmospheric Science, Universi O, of Florida, 4600 Rickenbacker Causeway, Miami, Florida 33149, USA The general formulation of water waves generated by an instantaneous axisymmetric free surface disturbance on an initially quiescent body ofwater is recalled. The initial conditions characterizing the localized original disturbances are defined by the sum of a static departure of the free surface from the still water level and a free surface velocity distribution. A general solution is presented in the form of Fourier integrals which are solved by the fast Fourier transform in preference to the more limited method based on the stationary phase approximation. Based on this theoretical approach, the inverse problem is defined, allowing the determination of the initial disturbances from the wave records taken at a distance from the origin. The method is applied to explosion generated water wave records in shallow water. Analytical linear disturbances are defined as an equivalent mode of generation to the complex nonlinear dissipative physical process resulting from underwater explosion. A predictive mathematical model is presented in a parametrized form, which could be related to the explosion yield, depth of burst and water depth.

INTRODUCTION The problems associated with impulsive generated water waves generally consist of determining the time history of the free surface elevation at a distance from the origin from the knowledge of the nature of the original disturbances. In other words, the effects - the water waves, are predicted from the causes - the free surface disturbances. However, in many cases, the original disturbances need to be determined from the effects such as given by the time history of the generated waves at a fixed distance from the origin. This would be the case, for example, of a tsunamigenic wave record. Since the free surface elevation at the origin is nearly equal to the vertical sea-floor displacement in the frequency range of interest, one can conceive that a realtime analysis of the wave record could eventually allow the determination of the permanent vertical earthquake displacement. It is also the case of water waves generated by underwater explosions. However, in the latter case, the initial conditions have to be defined not only by localized departures of the free surface elevation, but also by an initial upward velocity of the free surface. Then, a predictive mathematical model of explosion generated water waves can be developed from numerous experimental results in which the variable parameters are the explosion yield, the depth of burst and the water depth. From a theoretical viewpoint, the first problem, the determination of the wave pattern as a function of the original disturbances has been abundantly investigated in the past 1'4. The general solution is given in the form of a double integral. The first integral is a Hankel Transform of the original conditions. The second integral represents the summation of the wave number components. It is resolved using Kelvin's method of stationary phase. The solutions are only valid at a distance from the origin and they are not applicable to Paper accepted February 1990. Discussion closes December 1991. O 1991 Computational Mechanics Publications

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Applied Ocean Research, 1991, VoL 13, No. 3

the leading waves 5. The leading wave solution is obtained using a mildly dispersive relationship. The solution is then given by an Airy function which is not valid for the trailing wave 6. Numerical integrations are also difficult due to the oscillatory nature a n d generally slow convergence of the integrand as the wave number increases. The solution is presented here in the form of Fourier transforms'* which is easily solved by fast Fourier o'ansform ( F F T - 1). This method is not subjected to the constraints imposed by the previous approaches. Furthermore, it leads to the formulation of the inverse problem, i.e., the initial conditions can be determined by Fourier analysis of the time history of the free surface elevation at a distance from the origin. It is the purpose of this paper to briefly recall the general form of solution, resulting from the two previously mentioned types of disturbances, develop the methodology of the inverse problem and apply it to experimental wave records from underwater explosions in shallow water. A predictive mathematical model for explosion generated water waves is then proposed. GENERAL FORMULATION The water motion is defined with respect to time t* and space (r*, z*, 0) in a cylindrical system of coordinates centred at the middle of the original disturbance, Or ground zero (GZ). r* is the radial distance; z* the vertical distance from the still water level and the motion is axisymmetric with respect to the 0z* axis (0/00 = 0). The water depth D* is constant, the free surface elevation is q*(r*, t*). The motion is nondimensionalized with respect to the water depth D*. The following dimensionless quantities are defined

r, z, tl=(r* , z*, q*)/D*

(I)

t = t*(g/D*) 1/2

(2)

Oriohml surface disturbances from wave records: T. Khangaonkar and B. Le Mehaute y is the gravity acceleration. The solution satisfies continuity and linearized free surface boundary conditions. The motion is assumed to be irrotational allowing the definition of a potential function

49= c~*/(9D'3) '/2

The above integral can be rewritten making use of Fourier integral transforms. For a particular r, consider the Fourier transform pair defined by

q(t) =

(3)

The general solution is given by a continuous linear superposition of harmonic components ~

x [A(k) sin at + B(k) cos ot]k dk

(4)

~---

k tanh k

(5)

where A(a) denotes the complex Fourier coefficients defined by

oO

which is

q(t) =

(8)

w~(ro, 0)=

+-[Fz(a)cos (at)-Fx(a) sin(at)] da (19) 2~ _cO

(9)

Jo(kro)a2B(k)k dk

(10)

Jo(kro)qo(ro)ro dro=H,(k)

Since q(t) is real, the Fourier coefficients follow the symmetry defined by A ( - a ) = A * ( a ) where A* is the complex conjugate of A. This implies that

Fl(a) is even;

or F l ( - a ) = F l ( a )

and

fo o do(kro)aA(k)k dk

Inverting equations (9) and (10) by virtue of the Fourier-Bessel theorem and introducing the notation H(k), yields -A(k)a=Jo

[F 1(a) cos(at) + F2(a) sin(at)] da --cO

(7)

Differentiating equation (4) with respect to time, with z = 0 , and inserting equations (7) and (8) at time t = 0 , then one finds qo(r o, 0)= --

[Fl(a) e -~' da+iF2(a) e -i*t da] (18)

q(t)=

and an initial free surface velocity

w,(ro, 0)= -G(ro, 0)

(1 1)

(17)

Fl(a) is the real part and F2(a) is the imaginary part of A(a). Putting equation (17) in (15),

(6)

The coefficients A(k) and B(k) are defined from the initial free surface condition at time t = 0 . Initial free surface elevation,

qo(ro, O)= --~bt(r o, O)

(16)

A(a) = F 1(a) + iF2(a)

is related to the dimensionless wave number, k=k*D* by the dispersion relationship a2

q(t) e~' dt --cO

where the dimensionless frequency a=a*l(oID*) ~12

(15)

and A(a)=

Jo(kr) cash k(l +z) cash k

~b(r, z, t)=

A(a) e -~' da --cO

(20)

F2(ff ) is odd;

F 2 ( - - a ) = --F2(a)

Therefore, the second part of equation (19), [ f 2 ( a ) cos(at)-Fl(a ) sin(at)] d o = 0

(21)

--CO

and [Fl(a) cos(at)+F2(a) sin(at)] da (22)

q(t)=g-s --O0

B(k)a 2 =

Jo(kro)w~(ro)ro dro = Hw(k)

(12)

R is the radius of the initial disturbance. Note the H(k) are the Hankel transforms of the initial conditions. Introducing these into equation (4) and since '1= --4,

q(t) = 2 x - 2~z

[F1 (o') cos(at) + F2(a ) sin(at)] do(23)

This being true for a time series recorded at any r,

(13)

q(r, t)=

we obtain, I1(r, t) =

which can be written using equation (20) (see Ref. 3),

Jo(kr)[H~(k) cos at + Hw(k)a- I sin at] k dk

(14)

(F l(a) cos at + F~(a) sin at) da

(24)

Inserting

da = dk. V(k)

(25)

Applied Ocean Research, 1991, Vol. 13, No. 3

111

Orighml surface disturbances fi'om wave records: T. Khangaonkar and B. Le Mehaute ,l(r, t)=-_

EFt(a) cos(at)+F2(a) sin(at)] V(k).dk (26)

numerical reasons. Finally,

Ilo(ro) =

- ElF 1(a)] g(k)Jo(kro) dk

(35)

w,(ro) =

- aE[F2(a)] V(k)Jo(kro) dk

(36)

Comparing equation (26) and equation (14),

F t (a) = nkJ~ H~(k) V(k)

(27)

and

APPLICATIONS

rckJo(kr) Hw(k) F2(a) = - -

V(k)

a

(28)

Thus if the initial Hankel transforms Hw(k) and H,(k) are known. The solution to forward problem is simply equation (15) oO

q(r, t) = ~--~nf -oo A(a) e -i`' da where the real and imaginary parts of A(a) are given by equations (27) and (28).

The hwerse problem The inverse problem consists of determining the original condition qo(ro) and w~(ro) from a wave record ll(r, t). Referring to equation (16) A(a) =

I

oO

--

II(r, t) e ~' dt

(29)

r

Then F t (a) = Re A (a)

The previous theory has been applied to water wave records obtained from underwater explosions. The experiments took place in a shallow water pond at the Coastal Engineering Research Center, Waterways Experiments Station, Vicksburg, MS, USA. The basin is rectangular (100'x 140') and surrounded by levee with 3/2 slope. A series of tests were performed varying the yield, the depth of burst and the water depth. The water depth is adjusted by a pump. The explosions took place on a 30'x 30' thick concrete slab so as to prevent the projection of mud which would add significant noise to the wave record. The rest of the bottom of the basin was a mixture of silt and fine sand. Thewaves were recorded along the radius at various distances from ground zero by water shock proof pressure sensors. Figure 1 is an example of a series of wave records obtained at various distances from a 10 lb TNT explosion at mid depth in 80 cm of water. The nonlinearity of the waves appears evident particularly for the wave records close to origin or GZ ( G Z = g r o u n d zero). Therefore, the theory is applied by making use of the wave records at the larger distances. Initially, it is verified that the solution is unique, i.e., the initial conditions are found to be identical for different wave records from the same explosion. This is achieved by comparing the envelopes of the F(a) curves from

(30) g

F2(a ) = Im A(a)

(31)

-I -~J -,

which are obtained from the forward Fourier transform of the wave record ll(t ). Given Ft(a) and F2(a ), the Hankel transforms, H~(k) and Hw(k) can be obtained from equations (27)and (28). Inverting equations (11) and (12) and inserting expressions for the H(k), gives:

'e _eg

II 8 g

;/o(ro) =

Ws(ro) =

I ~~Ft(a)V(k) Jo(kro) dk

3o 7rJo(kr) I ~ aF2(a)V(k) Jo(kro) dk 3 0 rCJo(kr)

~

r = ~0.'2-

, A.H

, 8.$;I

, IZ.00

, |6.gll

, Zll.g0

, 24.H

28.ee

,

9 ell

8 gll

t Z 011

16 00

Z | 00

24.1HI

2~.e~l

(32) r = 32.7

.~

(33)

Theoretically, the values of k for which the denominator Jo(kr) is nil corresponds to the values of a for which the F(a) are nil. However since the integrations are done numerically and F~ and F2 are calculated from the data, this coincidence is fortuituous. The problem is resolved in noting that Jo(kr) is the modulating function of F(a) SO that the envelope E can be defined as

"e.eg

4.00

8.ell

|Z.|l

16.90

.

I Z*,. 811

0 ZS.eP

4.Bg

8.egl

IZ.ell

16.115

211. e l

24.e8

zs.e;I

eg

e $0

IZ ee

16 I 0

21~ ee

2~.eo

zs.es

=

e O~

t

ElF(a)]= J~ [Jo(kr)] 2 + e

(34)

where e is a very small parameter introduced for

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Applied Ocean Research, 1991, Vol. 13, No. 3

Fig. 1. Typical wave records 9enerated by underwater explosion at various distances from GZ (orighO. Note the nonlinear feature of the free surface profiles as the distance from GZ decreases

Original surface disturbances from wave records: T. Khangaonkar and B. Le Mehaute various records obtained from the same experiments (equations (30) and (31)). Figures 2 and 3 present the functions Ft(a) and F,(a) for the three last gauges. Note that the two last gauges give practically the same functions F(a), whereas more differences do exist with the closest gauge as a result of nonlinearity. Now using the envelopes E[F(a)], qo(ro) and w~(ro) are determined (Fig. 4). The initial velocity is upwards in the form of a dome, which can physically be related to the initial underwater bubble expansion. The initial free surface elevation is in the form of a crater with lip which nearly satisfy continuity. It could physically be related to the bubble reaching the free surface with a small phase difference with the initial velocity9 Even though ~lo(ro) and w~(ro) can be related to the physical phenomena, they are the only linear equivalent of complex nonlinear convective and dissipative processes. In particular, since energy dissipation is neglected, the physical crater is much larger for it actually reaches the bottom - while its mathematical equivalent does not. Similarly, the free surface velocity is smaller than the initial velocity of the plume generated by the explosion.

%'= r

=

,50.2

gI

81.8~

'e

l e( . e o

Cr

r = 45.2

0"

:~

r = 32.7

PREDICTIVE M A T H E M A T I C A L M O D E L First of all, it is realized that neither qo(ro) nor w~(ro) alone are able to simulate the original wave record (Figs 5 and 6). It is seen that the ~%(to) is mostly responsible for the leading wave whereas qo(ro) contribute more to the trailing waves. The theoretical model is obtained by fitting idealized

~ I 'e.ee

z.eo

I

810.

6.co

~.ee

! 18.~0

O'

Fi9. 3. Fourier transform (imaginary part F2(o')) of three wave records of the same experiment as presentedin Fig. 2 .2"

^a e

r = 50.2

,illl

8 . e8

I Z.OO

# ~.e8

618e

8lee

I |.II

,t,,

,t,,

,t,,

II.eO '

I Z' e e

I

I Ig.eg

|2.1De

1' 4 . | |

'o

i xe.e8

~t

4 84

m A r = -t,5.2

!=

'e.o8

2[80

. meo

6180

8~

~

~, 'e.go

r

.l Z e@

, ,.oe

1. 6 eo

, 8.00

I

r 10.80

=

Z.II

32.7

, io.8o

(7

Fig. 2. Fourier transform (real part Fl(a)) of three wave records of the same experhnent. Note the similari O, of the two curves at the larger distances

I

4.eO

I

6.|1

%

8 gO

t

I

14.$0

Fig. 4. Distribution of the initial free surface elevation Ilo(ro) and initial free surface velocity w~(ro) obtained from the analysis of wave records mathematical functions which can be Hankel transformed analytically, with the numerical curves obtained for both qo(ro) and w~(ro). Then it is found that a quartic such as (Fig. 7)

=A r4-(r~ 2 l(ro~ 4 11 'l~176

"L3 \ R . J - 3 \ R - ~ / -- _J

Applied Ocean Research, 1991, Vol. 13, No. 3

(37)

113

Oriinal surface disturbances from wave records: T. Khangaonkar and B. Le Mehaute r

=

ft,, D

,50.2

=

4.0 ft.

where the two parameters A..=0.803 and Rw=0.8. In the latter case, the Hankel transform is kR w 2

A,.R,,.2

(40)

O

~1 II. Oil

t 10.00

I 20.00

I 30.1}g

i 40. ~;I

i $;I. 0;I

I 60.0;;

t

Fi. 5. Time histoo' of the fi'ee surface elevation at a distance r=50.2 due to qo(ro) alone and comparison with the wave record =

,G/

=

,50.2

ft.,

D =

4.0 ft.

Other mathematical functions have also been tried successfully9 Then it is seen that the four parameters A,, R,, Aw, R~ can be parametrized as a function of explosion yield, depth of burst and water depth from the large number of experimental values. Inserting these into equations (38) to (40) and the corresponding Hankel values into equation (14), a predictive mathematical model is obtained. The best results are obtained by combining the two effects with a small phase difference9 Figure 9 presents an example of comparison of the results of this mathematical model with the experimental records. The discrepancies which remain can be explained by a number of facts. First of all, the nonlinearity of the problem is not accounted for by the theory. Second, a part of the bottom of the pond made of silt and mud was not quite horizontal. Thus, the wave motion, being affected by refraction, did not exhibit perfect cylindrical symmetry, as the theoretical JO(kr) would imply9 Finally, the leading waves were partly reflected by the levee and interferred with the direct trailing waves.

m m

'o:oo

I Ill.gill

I 2(11.1111

I 30.00

I 40.08

I 50.E6

t 60.09

A.

t

=

0.803,/'~

=

0.8

Fig. 6. Time history of the free surface elevation at a distance r=50.2 due to ws(ro) alone and comparison with the wave record A,~ = 0.403,

~

]/n = 1.25

Ii

v

6,gO

,o

'0~. O0

i Z9

| ~9

i 6.6~

i 89

i 10.00

i lZ.OO

Fi9. 89 Comparison of the mtmerically obtained initial disturbances w~(ro) with analytical fimction in the form of a Gaussian distribution

~0

A,~ = 0.403,

Comparison of the mtmerically obtained initial disturbances qo(ro) and with attalytical fimction hi the form of a quadric

F/~.

R n = 1.25

A . - 0.803,//~ = 0.S

7.

Z

which satisfies continuity, where the two parameters A,=0.402 and R~= 1.25, fit the numerical results best. The corresponding Hankel transform is H,(k) = - - ~

J.;(kR,)

(38)

Similarly, w~(ro) can be fitted by a Gaussian distribution, such as, for example (Fig. 8)

d[

'g 9

i 19.00

I 2g.0|

t 30.90

t 40.00

t 50.06

t 66 9

t

r

2

L \R.]j 114 Applied Ocean Research, 1991, Vol. 13, No. 3

(39)

Fi9. 99 Comparison of the predictive mathematical model with experflnental wave records

Oroinal surface disturbances from wave records: T. Khangaonkar and B. Le Mehaute CONCLUSIONS A methodology has been developed which allows the determination of initial condition of axisymmetric impulsive generated water waves from the Fourier analysis of wave records. The method is applied to explosion generated water waves. It is found that the initial conditions are given by the sum of free surface velocity distribution in the form of a dome, and a free surface elevation in the form of crater with lip. A parametrized solution is presented which can be related to explosion characteristics and water depth.

ACKNOWLEDGEMENTS This investigation has been sponsored by the Coastal Engineering Research (CERC), US Army Corps of Engineers, under subcontract with the Defense Nuclear

Agency. The experimental work was done at CERC under the direction of Doug Outlaw. Lt. C. Carlin of DNA is acknowledged for her support, encouragement and guidance. Drs Shen Wang, Frederick Tappert and Michael G. Brown participated in many discussions in the course of this investigation and significantly contributed to its solutions. REFERENCES 1 Lamb, H. Hydrodynamics, Cambridge University Press, 1932 2 Le Mehaute, B., Wang, S. and Lu, C. C. Spikes, domes and cavities, J. lnst. Assoc. Hydraulic Research, 1987, 5, 583-602 3 Papoulis, A. The Fourier bltegral and its Applications, McGraw-Hill Book Comp. Inc., New York, 1962 4 Stoker, J. J. Water Wares, Wiley Interscienee, New York, 1965 5 Wang, S., Le Mehaute, B. and Lu, C. C. Effect of dispersion on impulsive waves, J. ofMarflw Geophys. Res., 1987, 9, 95-111 6 Whitham, G. B. Lhwar and Nonlinear Wares, Wiley lnterscience, 1974

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