Impulsive Waves: Model and Prototype Correlations

I M P U L S I V E

W A V E S :

P R O T O T Y P E

Department

A N D

C O R R E L A T I O N S

J. M . J O R D A A N ,

Pretoria,

M O D E L

of Water

JR.

1

Affairs

Republic of South

Africa

I. INTRODUCTION

261

II. THEORETICAL BACKGROUND

A. B. C. D. E. F.

263

Water Waves in General Nonimpulsive versus Impulsive Waves Impulsive Waves Properties of the Cauchy-Poisson Solution Solutions by Penney, and Kranzer and Keller Numerical and Graphic Methods of Analysis

III. PROPERTIES OF IMPULSIVE WAVE SYSTEMS

A. B. C. D. E.

Wave Traces and Profiles in Simulated Tests (NCEL) Bubble and Crater, Source Motions Finite Generating Depth and Shoaling Effects Range and Yield Effects Transformation and Interaction with Finite and Shoaling Depth

I V . EXPERIMENTAL DATA

A. B. C. D.

263 264 264 266 268 269 270

270 273 276 276 . 278 282

Scaling Relationships Full-Scale Test Results Simulation in Laboratory Analysis of Wave Propagation and Run-Up Data

282 287 290 293

V . DESIGN CHARTS AND PROCEDURES FOR PREDICTING IMPULSIVE WAVE PROPERTIES

298

V I . FUTURE INVESTIGATION POSSIBILITIES

300

MAJOR SYMBOLS

302

REFERENCES

303

I. Introduction Impulsive waves, also known as dispersive waves, are generated when energy is suddenly released at or near the water surface. The release of energy may be accompanied by a loss or gain in volume or a zero net gain. In either case the waves are initiated by the impulsive water surface displacement at the source of energy. After the source motion dies down the waves continue, Formerly with Harbor Division, U. S. Naval Civil Engineering Laboratory, Port Hueneme, California. 1

261

J. M. JORDAAN, JR.

idefinitely propagating the finite quantity of energy imparted by the sudden disturbance, radially outward into undisturbed water. In particular, the waves generated in deep water are in the form of a concentric series, displaying decreasing wavelength and period with reference to a stationary observer, but both properties are actually increasing with reference to the distance and time of travel. The wave crests and troughs are concentric circles in plan, and the wave heights decay inversely with distance. The entire wave system radiates outward in the form of a series of groups, bounded by amplitude envelopes which expand uniformly with time of travel. The waves accelerate through the envelope of the system, so that each ringshaped wave originates interior to the group and travels radially outward, through the group maxima and null points until it reaches the outside of the group to become a rapidly diminishing leading wave. The regions of maximum wave height and null points travel at constant rates away from the source. In water of finite depth, and especially where the depth reduces toward a shore line, the impulsive wave properties described above are modified to a state where wave period approaches constancy, and wavelength and celerity reach limiting values and subsequently reduce with shoaling depth. Under these conditions the wave amplitude gradually builds up again, and the leading waves tend to become significant and could even be vehicles of destructive energy. Only a small fraction of the original source energy is, however, reconverted into impulsive energy at the shoreline. Due to being expended over the perimeter of the water body surrounding the point of initial disturbance, as well as being dissipated over a length of time proportionate to the travel time of the wave system, the average energy level at the shoreline is considerably lower than at the source (Van Dorn, 1959, 1964). The principal difference between waves generated by sudden disturbances and those generated by a gradual process, is the presence in the first case of a well-ordered group of waves of ever-increasing, sometimes exceptionally long, wavelengths. The energy per wave per unit width of crest is comparatively high and this energy under suitable configuration of shore topography is again concentrated, to be released in a sudden process such as a localized impact or a run-up on the shore. An important result of the dispersive nature of impulsive waves is that, unlike some other waves, such as solitary waves or waves of constant period, these waves are not reversible in time. The partial recovery of instantaneously released energy, if spatially concentrated at an opposite pole of the globe or reflected to a focus, will be spread out in time and will not be instantaneous. In the following sections the significant properties and quantitative behavior of impulsive waves, such as generated by nuclear or high-energy chemical explosions, will be examined and compared to available theories and model experiments simulating high-energy disturbances (Kaplan, 1953; Van Dorn, 1961). 262

Impulsive

Waves: Model and Prototype

Correlations

II. Theoretical Background A.

WATER WAVES IN GENERAL

Water waves are gravity phenomena at the free surface governed by two equations. (1)

Continuity equation: u

a)

where u is the particle velocity, (j) is the velocity-potential function, rj is the vertical displacement of water surface from still water level, y is the water depth, x is the coordinate dimension in the direction of wave propagation, and t is time. (2) Dynamic equation or equation of motion: (2) where g is the acceleration due to gravity. Figure 1 shows the typical appearance of a general solution to the above equations; it is particularly noted that groups of waves bounded by an

9

-x

r

C g

(a) ~ \

SURFACE PROFILE

^ENVELOPE

S

W

L

I

(b) NODE^

•

SURFACE PROFILE

ENVELOPE-^

Xr

r\~T\^\

INITIAL ELEVATION

'jl \

ENVELOPE-v^'A ^"'T\

\ l TIME RECORD

INITIAL IMPULSE

FIG. 1.

Surface profiles and time records of water waves. 263

J. M. JORDAAN, JR.

envelope travel at group velocity c distinctively smaller than the phase velocity c. The excursion n of the water surface about the still water level (SWL), when plotted as a function of the coordinate x in the direction of propagation for a given time instant, is denoted the " surface profile," whereas when n, as observed at a fixed reference point, is plotted as a function of time, the curve is denoted as " t i m e r e c o r d " or wave recording. The measurement between successive crests is thus the wavelength, X, when portrayed as a surface profile, and the wave period, T, when shown as a time record. g

B. NONIMPULSIVE VERSUS IMPULSIVE WAVES

In dealing with waves generated by explosions, a behavior pattern as shown in Fig. 1(a), also known as an impulsive (or dispersive) wave system, is customarily found. This is in contradistinction to a simple harmonic system of waves, the monochromatic model solution of the above two equations, which is characterized by the invariant values of X, T, and c and the absence of grouping. The superposition of two or more simple harmonic solutions does give grouping and group velocities as seen in Fig. 1(b) for the simplest of such cases, the standing wave due to reflection of a simple harmonic wave system on itself, with the solution n = cos at cos kx

(3)

where a = gk and k = 2n/X, and characterized by the potential function 2

(f) = g(sin Gt\o)e

ky

cos kx

(4)

It is seen in Fig. 1(b) that the waves show grouping although each group contains half a wavelength and possesses apparently zero propagation velocity. The water surface motion at the source of an impulsive wave system, such as produced by an explosion, bears resemblance to the case shown in Fig. 1(b), whereas the motion far removed radially outward from the source is similar to that shown in Fig. 1(a). Wave systems characterized by single values of % are nonimpulsive, or nondispersive, as the energy is propagated at constant speed and uniform density, whereas systems having more than one value of T, or a spectrum, are impulsive, or dispersive, since the energy of each value of T, or element of the spectrum, travels with a different speed, i.e., group velocity, and the energy density reduces with time. C . IMPULSIVE WAVES

An impulsive energy release at the water surface, when concentrated at a point source, is characterized in the ensuing motion not by one value of wavelength or period, but by all values from 0 to infinity. A solution is 264

Impulsive

Waves: Model and Prototype

Correlations

obtained, for the one-dimensional propagation case, by integrating the solution for the case shown in Fig. 1(b) over all wavelengths from oo to 0, (i.e., wave numbers k = O-oo), = din Jo

( s i crt/cr) e n

cos kx dk

ky

(5)

which is the solution for impulsive waves generated by an initial unit displacement, or f J

/ ( « ) d* = 1

(6)

-oo

where / ( a ) is a unit impulse function. Substitution of Eq. (5) into Eqs. (1) and (2) yields - (3 • 5 ) " (gt /2x)

tl = (nxY {gt l2x l

1

2

2

+ (3 • 5 • 7 • 9 ) " (gt /2x) 1

3

2

5

- • • •} (7)

which is the solution (due, independently, to Cauchy, 1815, and Poisson, 1816) as presented by Lamb (1932). Any particular " p h a s e " of the surface disturbance, i.e., a maximum, minimum, or zero value of n, is associated with a definite value of gt /2x, and thus each crest or trough travels with constant acceleration in the x direction. Shown schematically in Fig. 1(c) and (d) are the time records for the ensuing motion at station x, and in Figs. 2(b) and (c), in dotted lines, the surface profiles at two instants of time t and t . In Fig. 2(a)-(c) are shown in addition to the Cauchy-Poisson solution [Eq. (7)], the practical time records and surface profiles obtained for the finite displacement, of halfwidth a, at the source. To return for the moment to the ideal case, an infinitesimal unit positive displacement (of length dimensions) given by Eq. (6), with wave record in Fig. 1(c), can be approximated by an alternative form to Eq. (7) as 2

x

2

2

rj - (gt l2xf 2

D

12

[cos (gt /4x)

(2n xY ll2

1

2

+ sin (gt /4x)] 2

(8)

where gt /4x is large. A corresponding but distinctive solution was obtained (Lamb, 1932) for the case of the wave system due to an initial unit positive impulse as 2

/. + 00

P(a) da = 1

(9)

J-oc

where P(a) is a unit impulse function, and due to downward pressure applied at the water surface as n -> (gt /2x) 2

x

1/2

[cos (gt /4x)

{tlx) (4n i px)x 2

x

2

where p is the water density. 265

- sin (gt /4x)] 2

(10)

J. M. JORDAAN, JR.

(a)

(b]

(c)

(d) SURFACE PROFILES AT t,

I

FIG. 2.

Practical time records and surface profiles for finite displacement.

Although the solution by Eq. (10) for the impulse case is not identical to that of Eq. (8) for the displacement case, the waves are comparable when viewed from a moving reference system; that is, an observer on a ship traveling at constant speed (dx/dt = — v) toward the source in the impulse case, Eq. (10), will experience a time record of wave motion in proportion to that which a stationary observer will experience in the displacement case, Eq. (8). >/i-*->/D-'/*-(2pr

1

(11)

Hence nJn = — (2pv)~ . Furthermore, the waves following a pressure impulse (no volume change) die away more rapidly with distance, than those following a displacement or change of volume. l

D

D.

PROPERTIES OF THE CAUCHY-POISSON SOLUTION

The analyses due to Cauchy and Poisson represent ideal cases of disturbances infinitesimally concentrated and having a positive sign: initial elevation above mean water surface, or pressure downward on water surface. For disturbances having negative sign, the records and profiles are merely inverted 266

Impulsive

Waves: Model and Prototype

Correlations

about the t and x axes, respectively. The real case of an explosion in a water body with a free surface presents a disturbance of finite width 2a at the source (whether it be a two-dimensional or a three-dimensional model), so that with respect to an observer at distance x, a/x is small but # 0. Due to the interference of waves created by elements of the disturbance, actually situated at a distance between x + a and x — a from the observer, the surface oscillations will not increase indefinitely with time according to Eqs. ( 8 ) and (10), but the sine and cosine terms cancel each other when gt /4(x

-a)-

2

t = [(2n/ga)(x

2

gt /4(x 2

+ a) = (In -

- a )(2n - 1 ) ] , 1/2

2

\)n

n = 1, 2, 3, • • •

(12)

since x > a,

t* n

[(2n/ga) (2n - 1 ) ]

1/2

x

(13)

which is the equation of the nih node in Fig. 2(a) traveling with constant speed. The Cauchy-Poisson solution versus the finite size solution as compared in Fig. 2(a) shows that the phenomenon of wave groups traveling outward with uniform speeds is entirely due to interference phenomena, and grouping would tend to disappear as a/x tends to vanish, e.g., for very large x. The initial oscillation at a point at great distance from the source would then also be indistinguishable from the undisturbed free surface, and a large number of waves will pass the observer before the node separating the first group from the next passes. The analogous profiles along the propagation axis of the waves (in a one-dimensional process) are shown schematically in Fig. 2(b) and (c), comparing the finite disturbance case with the pointsource solution of Cauchy and Poisson. The following properties can best be noted at this point. (1) The wave system stretches out in space as it travels due to the leading waves traveling faster and accelerating uniformly. (2) The wave system is bound into groups by a series of envelope loops; nodes between loops move outward at uniform b u t different speeds and hence each loop accumulates more wave crests and troughs with time as it stretches while traveling outward. (3) The group velocity of each loop is about one-half the phase velocity of the highest waves in the group, and as with time more and more waves populate each group, the highest wave in each group appears later in the series of waves of that group. (4) Eventually, at infinite distance of travel when a/x becomes zero, the waves form a single group, the leading portion of which tends to conform to the Cauchy-Poisson envelope. 267

J. M. JORDAAN, JR.

The cases dealt with above are for one-dimensional propagation (plane waves), but the properties are analogous for the more general two-dimensional propagation case of radial symmetry. In the two-dimensional case (Lamb, 1932, p. 429), plp = dct>ldt-gz

(14)

+ F(t)

wherep is the water pressure, z is the vertical coordinate, F(t) is an arbitrary function which may be merged in the value of dcfr/dt, and (j) is such that V cj) = 0. For the initial displacement case, the solution is 2

(15)

ri = cos(gt /2r)J (gt M 2

2

0

where / ( ) is a Bessel function and r is the coordinate in radial direction. Equation (15) may reduce to an alternative solution: 0

(16) Again, any particular trough or crest is associated with a value of gt /r and travels with uniform acceleration away from the source. The approximate solution is 2

cos

rj -> (gt l2 nr ) 2

5/2

3

(17)

For the case of the initial impulse, the solution is given by Cauchy and Poisson as n = (t/2pr) sin (gt /2r) J (gt /4r) 2

2

0

(18)

approximating to (18a)

E.

SOLUTIONS BY PENNEY, AND KRANZER AND KELLER

Where one desires to make predictions for the actual wave shape due to a specified distribution of energy at the source, a spatial integration of the Cauchy-Poisson solution for each element of the source distribution gives quantitative curves in the nature of Fig. 2, which are schematic. Solutions by Penney (1945) and Kranzer and Keller (1959) are available and are amenable to numerical computation. This will be further dealt with in the subsequent section. 268

Impulsive F.

Waves: Model and Prototype

Correlations

NUMERICAL AND GRAPHIC METHODS OF ANALYSIS

1. Numerical

Solutions

For the paraboloidal initial displacement, the solution by Kranzer and Keller (1959) is (19)

where t ^ r/(gy) and a = ky, £ = 2
1/2

[nP -i(y)-(2n+l)yP (y)] n

n

(20)

where s = h 4- r , y = h/s, P is the Legendre polynomial, g is gravity, and h is depth of explosion for optimum wave-making. 2

2. Graphic

2

2

Methods

If there are plotted on an x-t diagram the space-time paths or "characteristics" of crests and troughs, parabolas will be obtained due to constant gt j2x values associated with each " p h a s e " (crest, trough, or zero crossing) for Eq. (7) or (8); and straight lines of varying slopes for the space-time paths of nodes and group maxima for Eq. (13). When the waves have become sufficiently long to become affected by the finite depth y of the water (y > A), the properties of Airy waves are exhibited by the dispersive train of waves, which results in the following customary expressions for phase celerity and group celerity : 2

c = [(gX/2n) t a n h ( 2 7 r y / ; i ) ]

(21)

1/2

c = c(l + 2ky/s'mh 2ky)/2

(21a)

g

Both are functions of X(x, t) with c -> (gX/2ny , c - » c/2 for X < 2y, representing the deep-water case. In shallow water, X > 25 y, c -> (gy) , c -^ c for X > 25 y. The effect of gradual shoaling is to isolate the grouped, dispersive waves into a succession of nondispersive, shallow-water waves, each wave eventually becoming associated with a finite amount of energy when c -> c. This effect also distinguishes strongly the case of impulsive waves generated in shallow water from those generated in deep water. In shallow water where the wave group /2

g

1/2

g

g

269

J. M. JORDAAN, JR.

and phase velocities are limited, or c c = yjgy, the leading waves, traveling with the same speed as the group due to absence of dispersion of energy forward through the group, remain large as they travel outward. F o r the shallow-water (three-dimensional) case, the waves are nondispersive d n oc R~ , where R is the range or travel distance from surface zero. For the deep-water (three-dimensional) case, the waves are dispersive and rj oc R" , These proportionalities have been verified by prototype observations and experiments. Figure 2(d) illustrates the principal differences between these two cases. g

a n

1/2

max

1

max

III. Properties of Impulsive Wave Systems

A.

WAVE TRACES AND PROFILES IN SIMULATED TESTS ( N C E L )

Before discussing the properties of impulsive wave systems and their simulation in the laboratory, examples are shown next of experimentally obtained data. These serve to illustrate some of the features that will be dealt with in detail in the succeeding sections. The right-hand side of Fig. 3 shows wave records obtained at several locations along the center line of the Naval Civil Engineering Laboratory (NCEL) wave basin, due to the sudden forced immersion of a 14-ft-diam. paraboloidal plunger into still water. The top five traces are water surface oscillations; the sixth is a record of the horizontal motion of the water's edge at the shore. The plunger's own waterline moves forward horizontally as shown by the dotted trace at the top. The inset shows the configuration

DISTANCE FROM PLUNGER

IN FEET

TIME

AFTER

IMPULSE

I N SECONDS

\ _

5

/ \

To

Ts

WAVE RECORDS

INSET:

WAVE B A S I N

S?—

•64— 67— -70

FIG. 3. Results of wave shoaling and run-up produced by sudden drop of an impulsive wave-generator plunger. Inset shows location of full-round plunger as used in these early N C E L tests before modifications to eliminate reflections off back wall were made (Jordaan, 1964b). 270

Impulsive

Waves: Model and Prototype

Correlations

in plan and section of the plunger and wave basin. In later tests this configuration was changed to eliminate basin reflections of the wave systems. The left-hand portion of Fig. 3 shows the surface profile of the first wave of the wave systems, vertically exaggerated, which was obtained by graphic construction derived from the right-hand side. The inset in Fig. 4 shows the rearranged configuration of the wave basin and now semiparaboloidal plunger, situated close to the back wall of the basin and offset from the center line (CL), so as to permit wave-refraction studies in the larger right-hand compartment of the wave basin. The location of probe stations along the center line of the semiparaboloid is shown on the cross section at the left-hand side of the figure. The stations where profiles were computed are also shown. The main beach at a slope of 1 : 13.6 is flanked by the two spending beaches at slope 1 : 5. Wave absorbers, not shown, were aligned along these beaches. The central part of the figure shows the space-time paths of the advance of distinctive elements of the wave system which follows a forced withdrawal and immersion of the 14-ft-diam. plunger. The waterline recession and advance of the plunger/water surface interface is shown by a chain-dotted line. The details of breaking and run-up, where trough and crest characteristics lines meet, are shown for the first two crests. The profile of the leading waves in this case differs from that of Fig. 3, since the withdrawal followed by an immersion produces a depression wave, and hence a recession of the water's edge, initially. Figure 5 shows the terminal effects of the waves created as described for Fig. 4. The vertical travel of the plunger's lowest point is shown on the left, followed by a wave record at a 2-ft offshore location. Records of water surface elevation at the shoreline and at 2-ft, 4-ft, and 6-ft distances onshore are shown to the same time base. Superimposed on these is the trace of the water's edge motion. With reference to Fig. 4, the first two peaks above the zero line in the 2-ft offshore record are due to the first and third wave crests, the second wave crest having broken further than 2-ft offshore and producing only an insignificant effect. The first three excursions of the water's edge are thus the result of four or more waves and are not in one-to-one correspondence; so that a direct relationship between wave height and run-up does not exist. The chain-dotted trace below the wave records was obtained by attaching a sensitive force gage to a circular disk presented with its flat side toward the waves at the shoreline. It recorded the drag pressure history in the onshore motion of the water after the wave had deformed. The water particle velocity was calculated and is shown on a subsidiary scale. Comparing the drag pressure history with the wave record at the shoreline, it is seen that the peak of the drag pressure, and hence maximum particle velocity, occur at the instant when the water crosses the still-water shoreline. The drag pressure 271

USONfTId ONV NISVa 3AVM :±3SNI

ro

iCOMPUTED STATIONS'

o

—

o

Hf ' N v v c r a o r

CROSS SECTION OF WAVE BASIN RANGE,FT

PLUNGER

m

3903 S.U31VM •• r

60

I 'ON 1S3U0 -

PROBE STATIONS

I

IRUN-UP

BEACH

70 80 RANGE,FT

N0I1VIA30 1SUIJ -

ZLZ

QN3931

I *0N H9D0U1 -

Impulsive Waves: Model and Prototype

LEGEND

H O R I Z . TRACE WATER'S WAVE

EDGE

DRAG P R E S S U R E OF RUNUP

-I

i/> Z _;~

RECORDS

P L U N G E R STROKE

.VERT. TRAVEL IN FT

Correlations

TIME I N SECONDS FROM I M P U L S E

V

UJ >

O rl

i2_

. G A G E AT SHORELINE

FIG. 5. Near-shore wave records obtained at — 2 ft., shoreline, 4- 2, + 4 , and + 6 ft. (onshore) for the case of Fig. 4 (withdrawal and plunge): dotted line, water's edge; solid line, wave records; dashed line, plunger stroke; dot-dashed line, drag pressure of run-up. (Previously unpublished data, used by permission of U . S . Naval Civil Engineering Laboratory.)

reduces suddenly, develops a secondary peak and reduces to zero, which coincides with the point of maximum vertical elevation. Then, it becomes negative in the backwash and reaches a negative peak smaller than the preceding positive peak. It is seen that there does not exist a simple direct correlation between wave height alone at or near the shoreline and drag force, as the subsequent higher but shorter-period wave shown in Fig. 5 produces a lesser peak pressure, and smaller onshore run-up. B.

BUBBLE AND CRATER, SOURCE MOTIONS

An underwater explosion can originate a system of locally very high water waves. The distant effects, such as run-up on exposed shorelines will vary greatly, depending on the ocean topography traversed as well as on the magnitude, depth, and distance of the blast. FIG. 4 (opposite). Wave-propagation characteristics for the case producing maximum run-up onshore (withdrawal and plunge). (Previously unpublished data, used by permission of U.S. Naval Civil Engineering Laboratory.) 273

J. M. JORDAAN, JR.

Consider a chemical explosion at the optimum depth for making the largest possible waves at a desired location. Based on the work of Penney (1945), the detonation depth h for maximum wave-making efficiency should be equal t o Aj^Jl, Fig. 6(a), where A is t h e gas-bubble radius at its first maximum. Upon penetrating t h e water surface, t h e gas bubble momentarily transforms into a crater with an essentially parabolic section and with a waterline radius a, a n d effective mean radius a approximately equal t o A, Fig. 6(b).

I (d)

FIG. 6. Schematic illustration of wave generation by an underwater explosion showing formation of: (a) bubble and dome, (b) crater, (c)firstwave, and (d) second wave (Jordaan, 1965a).

This crater is the effective "initial displacement," giving rise to the impulsive wave train, Fig. 6(c) and (d). As time proceeds, these impulsively generated waves progress outward at an accelerating rate and the center becomes relatively quiet, new waves being formed continuously at the rear of the advancing wave group (Lamb, 1932). The period of waves of maximum height has the value T

= 2(2na/g)

il2

max

274

(22)

Impulsive Waves: Model and Prototype Correlations

275 (a)

(b)

(c)

FIG. 7. Theoretical water-wave propagation characteristics: (a) histories, (b) profiles, and (c) heights for an underwater explosion in deep water (Penney's theory) (Jordaan, 1965a).

J. M. JORDAAN, JR.

which is also equal to the pulsation period of the initial crater. The wavelength at maximum height has the value A

m a x

= 4a

(23)

The first wave's length equals 4R , the second wave's length equals (4/5)R , the third, (4/9)R , the «th, [4/(4n - 3)[R„, where R R , R , R denote the distance traveled from surface zero. Figure 7(a) shows the distance vs. time graph for successive waves emanating from an explosion near the water surface in deep water. Each successive crest accelerates uniformly outward and becomes the maximum of the wave train at successive uniform increments of distance. The phase velocity, group velocity, and distance-time relationship of the group maxima are dependent on the crater's effective radius a, whereas the individual phase accelerations are independent of a. Figure 7(b) shows the wave time-traces and profiles in space, following an explosion and Fig. 7(c) shows the contours of equal envelope height. The group maxima are inversely proportional to the travel distance. The wave heights at maximum are given by t

2

3

l9

2

3

n

(24)

H K(a) /R 2

MAX

C . FINITE GENERATING DEPTH AND SHOALING EFFECTS

The previous case presumed an effectively infinite water depth, i.e., at least greater than 2A. Since the wavelengths become longer continuously, the wavelength will eventually become greater than twice the depth and the bottom will limit the dispersion. If, subsequently, the sloping bed to the shoreline is encountered, the waves will be slowed down by shoaling and their properties will change as in the case of oscillatory waves. Figure 8 illustrates first the finite depth effect indicated by the short dashed limit, and then the shoaling effect. It is seen that the finite depth affects mostly the leading waves, whereas shoaling affects all the waves. Values for maximum wave properties under the limiting conditions are given. D . RANGE AND YIELD EFFECTS

Figure 9 shows the general wave-making effects at various ranges of various explosive yields, according to extrapolations based on the Penney theory. The main results are as follows. ( 1 ) The effective crater radius is proportional to the one-fourth root of charge yield B, a oc B

i/4

276

(25)

Impulsive Waves: Model and Prototype

Correlations

FIG. 8. Schematic illustration of water-wave propagation characteristics due to a shallow-depth underwater explosion: (a) section through water mass, (b) propagation characteristics (Jordaan, 1965a).

(2) At constant range R , wave height at maximum of envelope is proportional to square root of charge yield, H ^ K B

1

'

2

(26)

(3) F o r constant yield B, wave height at maximum is proportional to inverse of range, H ^ K R '

1

(27)

(4) The period of leading waves (whether sensibly high or not) increases as the square root of the range, T oc R T

277

1 / 2

(28)

J. M. JORDAAN, JR.

(5) The period of waves of maximum height is independent of range and is proportional to a , or 2 ? , 1 / 2

1/8

(29) period of first wave (sec) 100

4

1000

T

s

period

B

<->

YIELD ( T N T )

o

o

10

2T

of max height waves in group (sec)

T,

s

100

1000

R

2 mi 20 mi 200 mi

20,000 mi

RANGE

FIG. 9. Wave-generating capacity of underwater explosions in deep water at critical depth for bubble diameter (extrapolation by Penney's theory) (Jordaan, 1965a). E. TRANSFORMATION AND INTERACTION WITH FINITE AND SHOALING DEPTH

1. Attenuation

and

Amplification

Figure 10 indicates that simulated explosion waves, generated in the laboratory, follow closely the relationship of Eq. (27) up to the point of 278

Impulsive

Waves: Model and Prototype

Correlations

(c)

FIG. 10. Impulsively generated wave, amplitude vs. range, showing attenuation and amplification due to beach: (a) amplification of oscillatory waves by shoaling; (b) attenuation, followed by amplification, of impulsive waves; (c) section through basin, showing extent of beach. (Experimental, —• — . Theoretical: 7]ocb- y~ R ; rjaby~ \ , rjocR' .) Comparison between oscillatory (computed) vs. impulsive waves (experimental) generated by impact of paraboloid plunger in the NCEL Wave Basin (Jordaan, 1965a). l,2

llA

1

279

l,4F

1/2

112

J. M JORDAAN, JR

encounter with the toe of the beach slope, whereas uniform oscillatory waves would follow the horizontal dashed line. F r o m that point on, the impulsive waves deviate from Eq. (27), yet still decrease to a minimum some distance up the slope. Oscillatory waves would have increased according to Green's law (dashed curve), (30)

Hocb- y- * i,2

lf

where b is the horizontal spacing between rays or orthogonals. Experimentally obtained (chain-dotted curve, Fig. 10), in the shoaling of these impulsive waves until breaking, a modified theoretical relationship (31)

Hocb-^y-^R- '

1 2

is followed (dotted curve) (Jordaan, 1965a). 2. Comparison with Oscillatory

Waves

Figure 10(a) shows again the shoaling of oscillatory waves, increasing according to Green's law, Eq. (30) as compared to Fig. 10(b), showing the measurements of impulsive waves generated by sudden withdrawal of the paraboloidal plunger. F o r comparable wave heights and shapes just prior to reaching the shoreline, Fig. 10(c), it is seen that the history of the impulsive waves is entirely different from that of the oscillatory waves. The loci of the highest crest and lowest trough show minima and maxima over the sloping portion, while the oscillatory waves are monotonically increasing in amplitude. 3. Shoreline

Run-Up

Figure 11 shows the relationship for impulsive waves between relative run-up R U / i / and deep-water wave steepness H /L as measured at N C E L (Jordaan, 1965a). Kaplan's (1955) results for impulsive waves are shown for comparison. The run-up relationship obtained 0

0

RU/// =0.34(// /L )0

0

0

1 / 3

0

(32)

has the same exponent in the N C E L and Kaplan data for the beach slopes used, 1 : 1 5 and 1 : 30, respectively. The effect of change in beach slope on the run-up of impulsive waves is seen in the plot to involve a change principally in the proportionality constant. Finally, the run-up of tsunami-type explosive waves, i.e., nonimpulsive waves such as would be generated by very large underwater explosions where A > d/2, is illustrated in Fig. 12. 280

Waves: Model and Prototype

Correlations

relative

run-up,

He

Impulsive

,

wove

steepness (deep water)

FIG. 11. Relative run-up RU!H vs. wave steepness H /L (Jordaan, 1965a). Experimental curves: , Kaplan (1955), 1 : 30; Jordaan (1965b), 1 : 15; —, both (1 : 15, 1 : 30) slopes; Kaplan (1955), 1 : 60. 0

0

0

FIG. 12. Deep-water wave height H producing various values of shoreline (breaker) wave height H for very large underwater nuclear explosions (A > d/2) (Jordaan, 1965a). 0

B

281

J. M. JORDAAN, JR.

A first approximation is obtainable for the breaker height of waves originating as long waves, as follows: By Green's law, if H = H (Y /Y y^ (33) B

0

B

o

where H is the breaker height, H is the wave height in deep water, and F and Y are the corresponding local depths near shoreline and in midocean. If H = Y , then H = H$< Y*' (34) B

B

0

0

B

B

5

5

B

For example, for H = 1 ft in 14,500-ft depth, Green's law is satisfied up to H /H = 6 and hence H = Y = 6 ft. For a value H = 1 ft in 600-ft depth, on the other hand, the value H = Y = 3.5 ft is obtained. D a t a from Kaplan (1955) for H vs. H /L and from Jordaan (1965b) for R U vs. H , indicate that run-up of tsunami-type explosion-generated waves lies between two to three times the breaker height H . Hence from the data in Figs. 11 and 12 (taking the upper envelope rather than the line through the mean of the points), two meaningful estimates of the upper limit of run-up can be obtained: 0

B

0

B

B

0

B

B

B

Q

0

B

B

RU/i/ >0.5(i/ /L )O

o

1 / 3

O

(35)

and

RU/// >3(F /r )B

0

1 / 4

0

(36)

With H « Y and R U > 3 / / there results from Eq. (34) B

B

B

RU/7/ >3(F /// ) 0

0

1 / 5

0

(37)

In summary, the dominant wave properties are determined from the generating source magnitude and crater dimensions. The attenuation of impulsive waves and their subsequent amplification due to shoaling are determined from relationships given by Eqs. (27) and (31) for two-dimensional concentrically radiating waves, and by the conventional Green's law relationship for tsunami-type waves that already are shallow-water waves at their generating area. The maximum breaker height and maximum run-up on an ideal uniform slope is found for the latter case to be directly expressible in terms of the deep-water wave height, length, or generating depth, i.e., Eqs. (35)-(37). IV. Experimental Data A . SCALING RELATIONSHIPS

The scaling laws presented by Penney (1945) permit the use of experimental data to approximate the waves expected from a prototype explosion, assuming that the similitude can be extended to nuclear as well as chemical explosions. 282

Impulsive

Waves: Model and Prototype

Correlations

Part of the energy of an underwater chemical explosion is converted into the pulsating gas bubble, which in turn, upon reaching the surface, converts part of the energy into the generation of waves. The greatest waves from a given weight of explosive charge will occur when h, the detonation depth, is equal, roughly, to the maximum radius of the undeformed explosion bubble, A . This value of h (or ^ ) is found from Eq. (38), or Table I. It is estimated that about 40 % of the available energy E is used to displace the water around the explosion bubble during its expansion to the maximum size. The remainder is divided, about equally, between shock wave and thermal radiation, both irreversible processes with negligible wave-generating effects. If it is assumed that all of the work of displacement is converted into wave energy, then equating the work done to 0.4 E and setting A equal to h yields the following equation relating h to E: m a x

m!iX

$[4nh pg(h 3

+ Z)] = 0.4 E

(Maximum available wave energy)

(38)

in which Z is the barometric pressure in feet of water, and E is the available chemical energy. Knowledge of the relationship between charge weight and energy permits relating maximum bubble radius, A , to charge weight, as shown in Table I. mzx

TABLE I MAXIMUM RADIUS OF EXPLOSION BUBBLE,

A

mAX

VERSUS EXPLOSIVE CHARGE WEIGHT B

B

(TNT)

^max ( f t ) °

1.56 2.45 4.20 5.96 14.1 22.4 31.4 46.5 285

1 oz 4 oz 1 lb 4 1b 641b 300 lb 1000 1b 2 tons 2000 tons Extrapolation

13

20 200 2000 20000

kilotons kilotons kilotons kilotons

Penney (1945). "Jordaan (1964a).

a

283

537 960 1720 3060

J. M. JORDAAN, JR.

1. Scaling of Underwater

Explosions

Scaling the data from one event (experimental, observed, or theoretically computed) to another can be accomplished by scaling laws (Penney, 1945), which invlove only the charge ratio W and the critical explosion depth h. It is customary to express the charge ratio as n, with n = (W ) . In an air-burst case, the scaling ratio by similitude law is L = n. This is not the case, however, in an underwater burst, because whereas air pressure is not scaled, the hydrostatic pressure is. F r o m Eq. (38), assuming the energy ratio E equal to the charge ratio W , there results r

l/3

r

r

r

r

EJE,

[(h + Z)/(A, + Z)]

= « = (hjhtf 3

(39)

2

where the subscript 1 denotes the model and 2, the prototype. The energy ratio is a quantity greater than unity. F r o m Eq. (39) it is seen that for the underwater blast case, h /h or L can no longer be equal to n. The quantity h /h is the scale ratio specifically valid for underwater explosions and is denoted by m; h and h are taken as A and A from Table I. Equation (38) can be solved for given values of W and W ; hence, m and n can be determined. Similitude in an underwater explosion follows the scaling law: L = m (including wave height) and the time ratio T = m , in which m is the solution for h \h in Eq. (39). Thus, 2

2

1

r

l

2

t

2

x

2

x

T

1 / 2

r

2

x

rj(mr, m t) The celerity is proportional to m (a)

1 / 2

. Two simplified cases may be considered.

If a small-scale test in a laboratory is compared with a larger-scale test in nature, so that h <^Z<^ h , then, x

2

or

m = n \Zlh ) , 3/

1/4

1

(b)

(40)

= mn(r, t)

1/2

m = W

1 / 4 r

= constant

(41)

In the case of two small-scale laboratory experiments, h <^Z h <^ Z . Consequently, m « n x

and

2

2. Scaling of Above-Surface

(42)

Explosion

The distinction between air- and underwater-blast scalings is important. Similitude in the above-surface blast is in accordance with the following scaling law: L = n and T = « , which applies for all time dimensions t and space dimensions r, except the wave height which scales as rj = « , i.e., 1 / 2

r

T

1 / 2

r

rj(nr, n t) i/2

The celerity is proportional to

= n n(r, 1/2

n . 1/2

284

t)

(43)

Impulsive Waves: Model and Prototype 3. Application of Scaling

Correlations

Laws

The previous scaling relationships permit the application of experimental data to prototype situations. For underwater explosions in geometric similitude, it has been shown (Glasstone, 1964) that the linear scale ratio m between the prototype and a scale model is practically equal to W) , in which W is the ratio of the yields of the two cases. For geometric similitude, the depth of detonation, water depth, and horizontal dimensions must be so chosen as to be in the ratio W} as well. Hence, m « W} is not only a measure of the scale ratio of the geometry and of the ensuing wave system, but also of the geometry and properties of the explosion itself, e.g., the water depth, the maximum bubble diameter, and the explosion crater diameter momentarily formed after the bubble breaks the water surface are scaled in the ratio m. The wave histories of two geometrically similar underwater explosions of unequal yield will be to scale if compared at two homologous points, whose ranges from the surface location of the source are in the ratio m. The wave heights will (for two such homologous points) scale in the ratio m; the wave periods and arrival times will scale in the ratio m , as will the group velocities and phase celerities. The previous approximation, m « W] *, is valid only for comparing the explosions of a small model and a large prototype. The exact equation for a general case of two dissimilar underwater explosions is m = h /h in which h and h are found from Eq. (38) or Table I (h = A ), and will satisfy the relationship given in Eq. (39): lAr

r

IA

IAr

1 / 2

1

2

2

x

l

miix

(hjhtf

[{h + Z)l{h, + Z)] = W,

(44)

2

This equation is applicable only, as stated before, to the underwater explosion case, while in surface blasts and above-surface blasts, the scale ratio is equal to n = (W ) the scale ratio of the charge diameters. If the behavior at homologous points in two scaled nuclear-energy explosions, with respect to shock-wave phenomena (e.g., air-blast and underwater shock), is compared, the distances of the points from the source are found to be related by n as well. Hence, the ranges used for expressing the scaled effects of the 20-kiloton underwater nuclear explosion (Bikini-Baker) in Table II are related by «, because it is first necessary to investigate the effect of the air shock on a shore structure (Tudor, 1964). These ranges are proportional to n = W} , because for the case given, i.e., a shallow underwater nuclear explosion, the air-shock pressures are related by the same scaling as an airburst. The ranges given in Table III are calculated to be representative of a 2-psi overpressure in every case. According to Glasstone (1964), the wave height at range R,for a W-kiloton 1/3

r

9

lz

285

J. M. JORDAAN, JR. T A B L E II WAVE HEIGHTS FOR NUCLEAR EXPLOSION AT MID-DEPTH; FOR SHALLOW WATER VERSUS DEEP WATER

0

Shallow-water explosion Yield, W (kilotons)

Selected range, R (0.5 W miles)

Scale ratio of Mb model, m

b

113

Deep-water explosion

Max. water depth for shallowwater case r.(ft)

Wave height at given range R H (ft)

Min. depth for deepwater case Y* (ft)

Wave height at given range R H (ft)

85 180 319 569 1010

6.0 11.2 19.0 31.3 49.5

400 850 1500 2680 4770

18.0 33.5 49.0 71.5 99.0

d

s

c

0.50 1.35 2.92 6.30 13.60

1 20 200 2,000 20,000

62.5 133 235 420 740

d

d

Based on scaling of Bikini-Baker shot, 20 ktons at mid-depth in 180-ft water (Glasstone, 1964). For ranges other than that given in this column, height of waves will vary in inverse proportion for both H and H . For depths less than those given in this column, height of waves will be reduced proportionally. A constant depth Y or Y is assumed up to the range R for which the wave heights are given. Shoaling effects thereafter may increase the heights again. a

b

s

d

c

D

s

d

explosion at depth h, is equal to W times the wave height at range R for a l-kiloton explosion at depth h/lV *. The latter quantity is known as the scaled depth. This leads to the same result as the method given here: the wave height at range R,for a W-kiloton explosion at depth hjs equal to W times the wave height at R/W for a l-kiloton explosion at depth hj\V . 1/2

11

1/4

1/4r

4. Limiting

llAr

Depths

Glasstone (1964) refers to a " l o w e r " limiting depth defined by rf, < 85 W * lf

(45)

such that for smaller values of this ratio, the wave height is proportional to the depth. He refers to an " u p p e r " limiting depth by d >400 u

286

W

1/4

(46)

Impulsive

Waves: Model and Prototype

Correlations

such that for larger values of this ratio, the wave generation and propagation are not affected by the b o t t o m proximity. T h e term shallow-water explosion is used herein for cases where d/W > 85, and deep-water explosion for cases where d/W > 400. 1/4

1/4

B . FULL-SCALE TEST RESULTS

1. Underwater Nuclear

Explosions

Extrapolations of the data from an actual underwater nuclear explosion of the shallow-water type (Bikini-Baker) are given in Table II, based on data by Glasstone (1964). The expected wave heights at the given range R, for the given equivalent yields in kilotons, are compared for the shallow-water case and the deep-water case. At these ranges, the ratio of wave heights in the shallow-water and deep-water cases is from ^ to \ . The underwater explosion scale factor m between these prototype cases and the simulated 1-lb T N T model in the wave basin, is also given in Table II. Comparisons between theory and experiment are given in Table I I I , for wave height and wave period, for the simulated shallow-water type of underwater nuclear explosion. The theoretical results are compared to the wavebasin results at the N C E L (Jordaan, 1964a) for the simulated 1-lb charge (withdrawal of the 14-ft-diam. plunger from immersion of 2.5 ft). Also in Table I I I , the scaled-up values for 20 kilotons are compared with the Baker data. The theory gives wave heights nearly twice as large as those in the wavebasin experiments for the same depth of crater, but the experiments predict closely the reported wave heights (Glasstone, 1964) in Bikini-Baker, 24 ft and 19.3 ft, respectively. The agreement between experimental and theoretical wave periods is closer, although no prototype information on wave period is available for comparison. Table IV gives the predicted and measured properties of the dominant wave from a 20-kiloton shallow-water explosion. The table compares the N C E L prediction (Jordaan, 1966), scaled u p from model, m = 133, with the data obtained from Bikini-Baker. The agreement is good. In Fig. 13 waves generated in shallow water are compared to waves generated in deep water; both time histories are calculated from available theories. The effect of depth on the shape of the envelope is seen to be a shift of emphasis toward the earlier waves in the case of the shallow-water explosion. In a shallow-water explosion, the waves arrive considerably later than the waves generated in a deep-water explosion. Therefore, the waves in the shallow-water case are less dispersive than those in the deep-water case, and the effect of the bottom proximity is to conserve more energy in the leading part of the wave train. 287

T A B L E III WAVE PREDICTIONS FOR VARIOUS YIELDS OF SHALLOW-WATER

0

UNDERWATER NUCLEAR EXPLOSIONS

Height H (ft)

288

1 lb

4.07

expt. scaled up

KranzerKeller theory

expt. scaled up

Scale m

36

0.34

0.18

2.52

2.1

1

2.12

8.5

7.1

13.2

max

NCEL

NCEL

2 tons

46.5

412

3.85

2 kilotons

285

2520

23.6

12.6

21.0

17.6

74

20 kilotons

587

5000

44.5

24.0

29.0

24.2

133

1720

15200

76

51.6

43.2

420

2 megatons a

max

KranzerKeller theory

At range 8.8 A (ft)

Leading waves highest.

142

J. M. JORDAAN, JR.

Yield

Bubble A (ft)

Period (sec)

Waves: Model and Prototype

Correlations

1

•

Water Surface Displacement, »j(ft)

Impulsive

-0.2 I 0

•

•

•

•

5

•

•

•

1

•

10 Time, t (sec)

•

1

15

•

•

•

•

• 20

I

(b) FIG. 1 3 . Theoretical water-surface time records. Comparison of shallow-water and deep-water cases, (a) Shallow-water case (Kranzer-Keller theory for range 6 0 ft., effective crater radius 4 . 0 5 ft., depth 2 . 5 ft.), and (b) deep-water case (Penney theory for range 6 0 ft., effective crater radius 4 . 0 5 ft., infinite depth) (Jordaan, 1966, 1969).

2. Chemical Explosion—Field

Test

Figure 14 represents a comparison for the deep-water case between observed data and values predicted from models and theory. In Fig. 14(a), actual prototype waves generated by a 10,000-lb high energy (HE) underwater explosion (Van Dorn and Montgomery, 1963) are compared with waves predicted by scaling up model waves generated by a simulated blast. A 28-indiam. paraboloidal plunger was dropped 4 ft into the N C E L wave basin and it generated waves of 0.7-in. height and 1.0-sec period. The traces of the waves, here shown inverted, are considered according to Table I equivalent to those that would have been generated by a charge of about 1 oz of T N T exploding at mid-depth in the wave basin. The underwater-explosion scaling ratio, m = 55, was found from the ratio of the calculated crater diameters for prototype and model blasts. Although the prototype wave record is of limited precision due to a high background level, the two records are in fair agreement with respect to a maximum wave height, wave periods, and arrival times. The model record agrees more closely with the two theoretical predictions shown in Fig. 14(b). One is based on Penney's deep-water theory, and the other on the Kranzer-Keller theory where the actual water depth is accounted for. The theoretical wave periods and the maximum heights are somewhat larger than the experimental, but are generally in good agreement. Agreement between model and theory is higher with respect to the arrival times of the individual waves, the group maximum, and the nodes. 289

ro

o —

10

20

30 40 50 60 Time Since Disturbance, t (sec)

80

90

_

(a)

ro

HYDRA IIA reco nstructed envelope-

7

Penney theory

—

ro

Kranzer-Keller theory

//

\\

O

V

*p$f

/ga \

27T

\ Am

1 \

_

/'

V

? = 14 a

r " no

Water Surface Displacement, 77 ( f t )

Water Surface Displacement, 17 (ft)

J. M. JORDAAN, JR.

0

10

20

30 40 50 60 Time Since Disturbance, t (sec)

70

80

(b) FIG. 14. Deep-water case wave records: comparisons of measurements and theories. NCEL data for depth, d= 2.5 ft., paraboloidal plunger radius, a = 14 in. dropped 4 ft., crater radius approx. 2 ft. (Jordaan, 1966, 1969). (a) Comparison of field test (HYDRA IIA) and laboratory test (NCEL). Field test: 5-ton charge detonated at 15-ft. depth; 2d a 2A.

C . SIMULATION IN LABORATORY Figure 14(c) shows the case of a simulated deep-water explosion, the same case as was treated in connection with Fig. 14(a) and (b). T h e wave height at 42 ft from the source is a b o u t 0.8 in., and the period is 1 sec. T h e curvature 290

Correlations

Run-Up

1

Shoreline

^ ^ / \ / \ / \

010

too

/ \

/\_

ft"AZW^ 1

o 01

O

Range, R (ft)

Oro -g

Impulsive Waves: Model and Prototype

5

10

15

18

( c)

Range, R ( f t )

Shoreline

Time, t (sec) ( d)

FIG. 14.

(c) Wave histories,

(d) Propagation characteristics.

of the propagation characteristics in Fig. 14(d) indicates the initial dispersion effect (i.e., acceleration of individual waves), followed by a region of constant velocity where dispersion ceases, and finally, the retarding effect of the shoaling beach. The shorter waves at the rear of the train are more dispersive, and less retarded by the beach than the early waves. Next, in Fig. 15(b) the time records and wave-propagation characteristics are shown for the case of a theoretical shallow-water explosion. An initial crater is assumed to have the same dimensions as the 14 ft diam paraboloidal plunger when immersed to 2 ft. The results may be scaled up to prototype dimensions by the ratio a /a = m = 62.5 W and t /t = m . F r o m Table I, the model crater represents that due to a 1-lb yield at optimum depth in deep water. In Fig. 15(a), the experimental time records and wave-propagation 1 / 2

1/4

p

m

p

291

m

TABLE IV HEIGHT AND PERIOD OF DOMINANT WAVES DUE TO 20 kiloton UNDERWATER NUCLEAR EXPLOSION

Wave height (ft)

292 a b c d

Range (ft)

(D

2500

fl

Actual"

(1)"

(2)"

(3)

41

41

23

24

21

45

25

20

—

30

24

26

14

13

—

34

30

(2)

(3)

39

11

4800

16

8000

11

b

c

Glasstone (1964, p. 95). By Kranzer-Keller theory, N C E L R-330 (1959, pp. 58, 60). Scaled from N C E L test, 2.0 ft withdrawal of 5.7-ft radius paraboloid in 0.4 sec. Bikini-Baker, 20 kilotons at mid-depth in 180-ft water.

c

J. M. JORDAAN, JR.

Wave period (sec), predicted only

Predicted

Impulsive

Waves: Model and Prototype

Correlations

characteristics of a shallow-water explosion are shown for comparison with the theoretical results shown in Fig. 15(b). The experimental data were obtained by a sudden withdrawal of the 14-ft-diameter paraboloidal plunger in the wave basin from an initial immersion depth of 2.5 ft. The wave heights 76 ft from the source were about 0.2 ft before breaking; the maximum period was about 3 sec. The first wave remains the maximum throughout its history. From the shape and the slight divergence of the characteristic curves, it is evident that the waves are less dispersive in the early stages than for the deepwater case as shown in Fig. 14. The theoretically predicted wave-envelope for a crater 2.0 ft deep and 5.7 ft in radius agrees with the experimental records for this somewhat larger " c r a t e r " (2.5 ft deep and 6.4 ft in radius). The dominant wave heights agree reasonably well at each distance, with the experimental " c r a t e r " thus adjusted by a plunger immersion value 2 5 % deeper than the theoretical. Otherwise, the experimental waves are about 4 0 % smaller in height than predicted by the theory, and the wave periods are about 2 0 % shorter.

D . ANALYSIS OF WAVE PROPAGATION AND R U N - U P

DATA

Fihure 16, top, shows experimentally obtained wave records, for a sequence of tests, in which the magnitude of the disturbance (plunger stroke) was varied. The effect on the time record and on the run-up on the beach is clearly seen. In particular, for the larger wave systems, the second wave became steep to the breaking point, and, although higher, produced far less run-up than the first. In the lower portion of this figure it is shown how the initial mode of disturbance (up- or down-stroke) affects the sense of the leading wave-effect (depression or elevation). It also indicates that successive plunger motions tend to superimpose certain subcomponent phases of waves into very large single crests, producing a large run-up on the beach or a high breaking wave. A plunge is considered to simulate an above-surface or at-the-surface blast, and a withdrawal an underwater explosion. Plunge-and-withdrawal and withdrawal-and-plunge simulate those cases where high energy levels are combined with a minimum loss of volume such as very high altitude airbursts or deeper than critical depth underwater explosions respectively. It is seen that the initial sense of water motion at the source determines the leading wave type and greatly influences the terminal effects of breaking and run-up. Prediction of these effects depends upon the choice of the plungeroperation mode. N o data on waves and run-up of prototype large-yield explosions are available to verify these assumptions. Six underwater nuclear explosions are listed in the chronologic tabulation by Glasstone (1964). 293

Shoreline —.

/

82.3

Run-Up

76

_

vw

59

rr

Toe of beac R = 49 ft

J

• 49 o> c o

A

A

A

A

^

A

A

^/A vA • AA

K

A

~

A

vy\/\/v\ A

A

A +1-

35.5

-1-

\\ JA\ J AV A\ a j

21.5 14.5 Source motion ^ 0 Li \ \ 0

.

.

5

10 Time, t (sec)

15

Water Surface Displacement, TJ (in.)

vy v

69

20 * Point of breaking v

.....

— r

,

/

//

Si/-

Range, R (ft)


/ /

/ • /

•f/ t

" ?/ "//

^ */

/ /

7 /// 1

//

#

///<

////'

t *i

0

i

J

i

i

>'

/ ' /

/

/

/ / / s /

y////oy f'/'///// / / /

/ ,

*/*// /

/ / / / / / /

/<'/// ////////A

/ / / /

/////// '///////. //////X'Z: y/

/

/ / / / / / / / / ' / / / / / / 7 V 7

'

/ /// / / /

'/////

/ //////

/ /

/////

s's

v// / Xx , / /

////.

/

A/////

/////

w

. . . . 5

1.

10 Time, t ( sec)

1

1

1

1 15

1

1

1 20

( a )

FIG. 15. Theoretical and experimental wave motion for a shallow-water case, depth d= 2.5 A. (b): Kranzer-Keller theory for paraboloidal crater radius a = 5.7 ft., (a): NCEL experiments for paraboloidal plunger radius a = 6 . 4 ft. [(a) and (b): top, wave histories; bottom, propagation characteristics]. (Jordaan, 1 9 6 6 , 1969.)

Impulsive

Waves: Model and Prototype

Correlations

I•

Range, R (ft)

Water Surface Displacement, 7^ (in.)—

L

CI

Source motion

Range, R (ft)

20

( b) FIG. 15(b).

See facing page for legend. 295

J. M. JORDAAN, JR.

296 i|4d*Q UO;SJ«UJUJ|

i»6un\4 FIG. 16(a).

Effect of size of disturbance on wave form and run-up (Jordaan, 1966, 1969).

Time, t ( s e c )

af S h o r e l i n 0

/. - flo f»\

I J -3 (r-o2ft)

Effect of mode of disturbance on wave form and run-up (Jordaan, 1966,1969).

Correlations

J»6un|j

Run-Up Record

Waves: Model and Prototype

UOISJ»UIUI| FIG. 16(b).

ui d j Wave Record Near Shoreline (r = 7 7 f t )

(UI) U, '4U»UJ«3O| DSI Q EAOJJNC, J«*DM

d0

(tj) H* Q ... Wove Record at Toe of Slope (r = 5 0 f t )

Impulsive

(UT) U '*U»UJ»DO|DS!Q AOOJJN^ J»«D^

297 m Plunger Motion

J. M. JORDAAN, JR.

V, Design Charts and Procedures for Predicting Impulsive Wave Properties

l/3

Yield, W (kilotons) TNT

\

range for 2-psi air-blast overpressure (shallowwater burst): R = 0.5 W for other ranges, H oc R"

1

In Fig. 9 are shown the heights and periods in deep water, for underwater nuclear explosions of various yields and at various ranges, detonated at the critical depth of 1 bubble radius below the water surface. Figure 17(a), is a replotting of the information on expected wave heights presented by Glasstone (1964). In this figure, wave height is plotted as a function of nuclear yield and depth of water for the particular values of range given by the curve on the extreme left. This curve gives the range at which various magnitudes of a shallow-water underwater explosion will produce an air-blast overpressure of 2 psi. According to Tudor (1964), in a parallel study on impulsive wave forces, any coastal structure surviving this air blast will still be subjected to water waves, the maximum height of which is given

Depth of Water, d (ft) (a)

FIG. 17(a). Height, length, and period of dominant wave from an underwater explosion (Glasstone, 1964; Jordaan, 1966, 1969). See also alternative representation in (b). 298

Impulsive

Waves: Model and Prototype

Correlations

Fig. 17(b). The wave height at some other range can then be calculated from the relationship Hoc R~ . The effect on wave height of " d e e p w a t e r " versus "shallow w a t e r " at the explosion point is clearly contrasted. Figure 17(b) shows the period and wavelength of the dominant and maximum waves of the wave group, as a function of water depth at detonation. It is noted that the period of the dominant wave is constant, i.e., not a function of water depth, even though the position of the dominant wave in the group is different for the shallow-water case (d < 85 W *) and the deepwater case (d > 400 W ). (A = 10 d at the shallow-water limit, and X — 4.7 d at the deep-water limit.) The depth h of the detonation point below the surface has been assumed to be, in all cases, that which would result in the largest waves, i.e., the critical depth. F o r the shallow-water wave case, this is h = d/2 or mid-depth, and for the deep-water wave case, it is h = A, the radius of the explosion bubble at its maximum, obtainable from Eq. (1). Experimental and theoretical results (Figs. 6 and 14) show that the period of the dominant (i.e., highest) wave in the group is relatively insensitive to the influence of depth, whether it be the shallow-water wave or deep-water wave case. Moreover, it is insensitive to the range or distance of travel and l

l/

Equivalent Yield, W (kilotons) TNT

1/4

10

30

100'

300

1000

3000

Depth, d (ft) (b) FIG. 17(b).

See facing page for legend. 299

10,000

30,000

J. M. JORDAAN, JR.

remains a constant throughout the wave dispersion; it is associated, in turn, with the maximum heights of a continually moving progression of waves in the deep-water case, and with the first waves in the shallow-water case. This dominant period of the group was found by experiment (Figs. 14 and 15) to be solely a function of the dimensions of the area of impulsive loading, i.e., " c r a t e r " or " b u b b l e " diameter; which again is a function of the yield (Fig. 17). The shoreline effects, breaking-wave height, and run-up height, can be estimated with the aid of Figs. 12 and 11, respectively.

VI. Future Investigation Possibilities The waves emanating from a practically instantaneous release of energy at a point source have been dealt with in a fair amount of detail. The generation, dispersion, propagation, and termination of these wave systems have been quantitatively treated. On the other hand, the more gradual process of wave generation associated with tectonic changes in the earth's crust gives scope for further investigation. Where short-time releases of energy can be simulated by means of plungers in the laboratory, this could be extended to cases of pulsed energy releases, and gradual, time-based releases. Both these transfer mechanisms are far-removed from the continuous regular waves produced by simple-harmonic oscillators, in that the wave system is still a transient phenomenon that grows and decays. The case of a linear (two-dimensional) and a radially symmetrical source have been dealt with by the early investigators and conformed to by the laboratory and field tests herein described. The interesting combinations of two or more sources resulting in interference patterns and complex phase and polarity relationships have been demonstrated in ripple tank experiments to illustrate wave phenomena in general. Waves emanating from juxtaposed sources actuated in specified time and distance relationships to each other can be predicted to a certain degree as a superposable interference effect, and an exhaustive study of such patterns with relation to water waves if made would perhaps be rather academic. More important would be to determine the predictability of terminal effects: the maximum intensity of energy conversion of waves reaching the shore, their nonlinear superposition and reflection, and relationship to the habitable environment. The assessment of the damage potential of waves caused by explosions have only cursorily been attempted on a large scale. The extension to tsunamis invites further study, using simulators or controlled explosive energy releases in large test tanks. 300

Impulsive

Waves: Model and Prototype

Correlations

The time of propagation and expected time of arrival of waves are more closely predictable than the height and force of the waves, hence observations of water-wave effects at different terrestrial locations have been successfully employed to locate the source of disturbance. The damage potential of these waves, given a source energy and position, is (from wave-basin experiments) highly dependent on the rate of release of the energy. This aspect needs further study and documentation; treating the impulsively generated wave in effect as the upper limit of a family of tsunami-type wave systems in which the energy is applied at increasingly rapid rates. Finally, the concentration effect of the shoreline topography and slope of the foreshore, and the termination of the wave against some obstacle have been studied in the past for mostly three kinds of waves: solitary waves, oscillatory waves, and random waves. The impulsively generated wave of rising frequency has only recently become recognized as having a special behavior pattern at the shoreline, in that the extremely long wavelength leading waves may have energy levels per foot width of crest much in excess of the higher waves of short wavelength arriving later. This is the case particularly where the waves are generated in water shallow enough for the leading wave to remain of finite height. A series of experiments, where the effective radius of disturbance to generation depth is gradually increased and the proportioning of energy among the leading wave crests are quantitatively determined, would be very valuable in assessing the damage potential of waves generated by explosions and waves of seismic origin. The effect of very gradual foreshore slopes on steepening long-period waves has been studied, but might very well be extended to waves of gradually increasing frequency. The following further investigations are recommended:. ( 1 ) Use of simulators to study wave systems created by various rates of energy releases; (2) study of multiple, pulsed, and phased energy releases in wave tank simulators, and comparison of interference effects with predictions; (3) analysis of the distribution of energy in each concentric wave element in a propagating system for shallow-depth generation so as to assess the critical combinations producing high energy in the leading long-period waves; (4) the extension of study on more-predictable point-source-generated wave effects to transient wave effects created by line, areal as well as timebased energy releases, and application to seismic sea waves (tsunami); (5) investigation of the shoreline behaviour of impulsively generated waves to ascertain to what extent it differs from that of solitary, oscillatory, and random waves. 301

J. M. JORDAAN, JR.

Major Symbols Note: a

a

Other symbols are defined in the text; numerical dimensions are in the feetpound-second system, unless otherwise noted. Radius of crater formed by emerging underwater explosion bubble, measured at SWL, also half-width of disturbance Effective radius of crater formed by emerging underwater explosion bubble (a x A , a = a/\/2) Radius of explosion bubble (assumed spherical) when not deformed by bottom or water surface Width of orthogonal space, or distance between rays See W Celerity of wave element Depth of water at generation Energy available in an explosive charge (ft-lb) Function generating unit impulse Acceleration due to gravity Function, in general Depth of underwater detonation point Wave height (vertical distance from crest to trough) Bessel's function Wave number, lirlX Scaling length and wavelength Scale ratio or length ratio (m = L ) for underwater explosion only; m = h /hi max

A

b B c d E

/f(( ))

g 9 F h H

J k L m

l

2

n Cube root of charge ratio [n = (W ) ] P() Function generating unit impulse P Legendre polynomial r Radial coordinate dimension in direction of propagation R Range (horizontal distance from center of disturbance to arbitrary point) RU Symbol used for run-up above SWL t Time elapsed after event u Particle velocity T Time ratio: ratio of time interval in prototype to corresponding interval in model v Velocity of observer relative to coordinate system W Charge (equivalent yield of an explosive charge) in kilotons of TNT, also denoted by B (oz, lb, or tons) x Coordinate dimension in direction of propagation y Depth of water; second coordinate dimension Y Local depth z Third coordinate direction, special case for two dimensions in propagation theory Z Barometric pressure (34 ft of fresh water) 113

T

x

Greek Notation a A

Variable i n / ( ) or F ( ) Wavelength, also L Displacement of water surface from SWL (ft, in.)

Density of water, in slugs/ft (1.94 slugs/ft for fresh water) CT Wave frequency, ( g k ) ; or also ky T Wave period
3

3

112

Subscripts Property at point where wave breaks Critical Group of waves (e.g., c = group velocity) m Model n nth mode, or term P Prototype max Maximum B crit g

g

302

mm r

Minimum Ratio of some quantity in prototype to corresponding quantity in model:

1 Model 2 Prototype 0 Deep-water wave properties

Impulsive

Waves: Model and Prototype

Correlations

ACKNOWLEDGMENTS

The work herein reported was part of a program of research carried out at the U.S. Naval Civil Engineering Laboratory, Port Hueneme, California; and this research was supported by the Defense Atomic Support Agency of the U.S. Department of Defense. Permission by the U.S. Naval Civil Engineering Laboratory of the Naval Facilities Engineering Command of the U.S. Department of the Navy to publish original material is acknowledged. The contributions and support of members of the staff of NCEL with respect to the research on which this article is based is gratefully acknowledged. Permission was granted by the American Society of Civil Engineers to republish material appearing previously in ASCE publications, and is gratefully acknowledged. Figures 3, 6-17 accompanying this article were compiled from figures appearing in ASCE publications based on original figures in the NCEL reports. Figures 4 and 5 were compiled from the original, previously unpublished, NCEL data. Permission of the above two organizations to make use of original masters of these figures is gratefully acknowledged.

REFERENCES

GLASSTONE, S., ed. (1964). "The Effects of Nuclear Weapons/' revised ed. (prepared by the U.S. Department of Defense). U.S. At. Energy Comm., U.S. Govt. Printing Office, Washington, D.C. JORDAAN, J. M., JR. (1964a). "Run-up by Impulsively Generated Water Waves," Tech. Rept. R-330. U.S. Nav. Civil Eng. Lab., Port Hueneme, California. JORDAAN, J. M., JR. (1964b). Discussion on wave breakers on a beach and surges on a dry bed. / . Hydraul. Div., Proc. Amer. Soc. Civil Eng. 9 0 ( H Y 6 ) , 305-308. JORDAAN, J. M., JR. (1965a). Water waves generated by underwater explosions. Proc. Amer. Soc. Civil Eng. Spec. Conf. Coastal Eng., pp. 69-85. JORDAAN, J. M., JR. (1965b). "Feasibility of Modeling Run-up Effects of Dispersive Water Waves," Tech. Note N-691. U.S. Nav. Civil Eng. Lab., Port Hueneme, California. JORDAAN, J. M., JR. (1966). "Laboratory Simulation of Waves Generated by Underwater Nuclear Explosions," Tech. Rept. R-424. U.S. Nav. Civil Eng. Lab., Port Hueneme, California. JORDAAN, J. M., JR. (1969). Simulation of waves by an underwater explosion. / . Waterways Harbors Div., Proc. Amer. Soc. Civil Eng. 9 5 ( W W 3 ) , 355-377. KAPLAN, K . , (1953). "Pilot Study of Explosion-generated Waves," Rep., AFSWP-482. Beach Erosion Board, U.S. Army Corps of Engrs., Washington, D.C. KAPLAN, K . (1955). "Generalized Laboratory Study of Tsunami Run-up," Tech. Mem. No. 60. Beach Erosion Board, U.S. Army Corps of Engrs., Washington, D.C. KRANZER, H.C., and KELLER, J. B. (1959). Water waves produced by explosions. / . Appl. Phys. 3 0 (3), 398-407. LAMB, H. (1932). "Hydrodynamics," 6th ed. (1945), pp. 384-394 and 429-433. Dover, New York. PENNEY, W. G. (1945). Gravity waves produced by surface and underwater explosions. "The Gas Globe, II." U.S. Office of Naval Research (unnumbered bound report: " Underwater Explosion Research: A Compendium of British and American Reports," pp. 679-693, Washington, D . C , 1950). 303

J. M. JORDAAN, JR. TUDOR, W. J. (1964). "Uplift Pressures Under a Pier Deck from Water Waves," Tech. Note N-668. U.S. Nav. Civil Eng. Lab., Port Hueneme, California. VAN DORN, W. G. (1959). "Local Effects of Impulsively Generated Water Waves," Rep. No. II on Contract Nonr-233 (35), pp. 1-45. Scripps Institution of Oceanography, La Jolla, California. VAN DORN, W. G. (1961). Some characteristics of surface gravity waves in the sea produced by nuclear explosions. / . Geophys. Res. 6 6 (11), 3845-3862. VAN DORN, W. G. (1964). Explosion-generated waves in water of variable depth. / . Mar. Res. 22 (2), 123-141. VAN DORN, W. G., and MONTGOMERY, W. S. (1963). "Water Waves from 10,000-lb Highexplosive Charges," Final Rept., Operation HYDRA II-A, Rep. 63-20. Scripps Institution of Oceanography, La Jolla, California.

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