Compressibility effects in ship slamming

J.

'ARCHIEF

Compressibility Elfe

T. Francis

Introd u et ion

In most analytical treatments of tue ship slamming problem, the water is considered to be incompressible and nonviscous.

A potential problem is thus formulated, the potential being required to satisfy Laplace's Equation. as well as boundary conditions on the ship hull and the free surface. One result of such treatments is that if the tangent plane at the bottom of the ship is horizontal (as it is with most ships), the theory predicts infinitely high pressures at the instant of impact.

in order to correct this result it is necessary to drop the

assumption of incompressibility. In this process an already difficult problem would seem to become utterly intractable.

However, the time scale for compressibility effects is so grossly

different from the time scale for inertial and gravitational

effects that some useful results can be discovered.

Recently Professor R.. Timman suggested a manner of

idealizing the problem so that it reduces to a solved problem of supersonic aerodynamics [112). The solution of Timman's

idealization is described in great detail in G. N. Ward's mono-graph [2].

Essentially, Professor Timman's suggestion is this: If the problem is linearized in an appropriate way, and if the effect of the ship's hull on the water is replaced by a condition that

the fluid has a downward vertical velocity on a section of the free surface, then in two dimensions the problem can be made mathematically equivalent to the problem of steady supersonic flow over a lifting surface. The latter problem is the one found in [2], Chapter Six.

This paper starts with a mathematical statement of the

slamming problem. reduced immediately to two dimensions.

A linearization is carried out in terms of the parameter V01c,,, where V0 is the downward body speed at impact and C) is the

acoustic speed in the undisturbed water. The lowest order

approximatioll is then easily identified with the problem

con-sidered by Ward. The aerodynamic solution is adapted to the. present problem and is carried out explicitly as far as is

prac-ticable. Some interesting results appear:

The initial pressure at the center of the body is V0 c0, regardless of the detailed shape of the body. If the bottom is actually flat, the initial pressure has this value at all points of the hotoni, but this pressure drops very rapidly wien the relkf wave arrives from the edges of the bottom, the pressure curves passing through zero at time t = 2/c,, whete £ is the

half-width of the bottom. After this time the pressure possibly oscillates, with decaying amplitude, but in any case the prin-cipal compressibility effects are past at about this time.

This duration of compressibility effects is so short that one may expect the corresponding forces to act simply as impulses.

For the fiat-bottomed ship it proves to be possible to obtain this impulse, which is the multiple integral of the pressure:

S dt

.1 dx p(x, O, t),

Office of Naval Research, London Branch Office.

Numbers in brackets In the text refer to references Listed the end of the paper.

147

tab.

_{y. Scheepsbouwkunde}

Technische Hogeschooî

Detfi

cts in Ship Slamming

/

e4.ts_

r,,

/

j

/?j-

r

by an operational calculus method described in Section 8.11 of [2]. The result is .rV0 £2/2. It could be shown by dimen-sional analysis that the independence of the value of C)) iS

proper. Thus, although compressibility considerations are

paramount in the problem, the impulse applied to the impact.

ing bottom is independent of the characteristic parameter of

compressibility, c0.

The paper closes with a discussion of the validity of the

whole approach. and some possibilities for extending it to more

complicated situations are discussed. In particular elasticity

of the ship bottom can conceivably be accounted for, although simple analytical answers do not seem to be available.

Ogilvie')

Tite 1'1athetiintieal Problem

We assume that the water is compressible, but the flow is irrotational. The latter assumption allows us to write the velocity vector as the gradient of a scalar potential function,

V =

but p does not satisfy Laplace's Equation. The proper partial differential equation is obtained by substituting the equation

of state,

dp

= C-, (1)

into Bernoulli's Equation and Euler's Equations, and then substituting these into the equation of continuity. The equa-tion of state used here simply indicates that pressure, p, is a

function only of density, . c2 is an unknown function of den-sity (or of pressure). The procedure is carried out in [2],

where the result is given3) as Equation (1.4.6):

S 72 tPtPtt =

(7

+ 7

_{[(7 7) V] - g 7.}

3t

gis the acceleration of gravity.

On the body, the usual kinematic boundary condition holds:

=Vn,

(3)

an

--where n is a unit vector normal to the body surface, and V is the (vector) velocity of the body. It will be assumed that the first contact of the body with the water occurs at time t = O so that (3) has no meaning for t <O.

The free surface is taken initially as the x-axis, and for times t> O its shape is described by the equation:

yY(x,t) = O,

_x>(t),t>O.

(4)

Implied in this statement is the assumption that the body is symmetrical about the y-axis; £ (t) is the half-breadth of the body at the level at which the body surface intersects the free

surface. For t <O, Y (x, t) = O, for all x. See Figure I(a). The Last term of Equation (2) is not tnclitdc'd ¿n (2), because

the effects of gravity are there neglected from 11w start.

Sciuittstechnik Bd. 1O-1933HeIt53

On the free surface, we have the two usual conditions 131:

+ Y = 0;

(5)

gY +q + (Vp)2 = 0.

(6)

Equation (5) is the kinematic condition, Equation (6)

Ber-noulli's Equation.

Finally, there should be stated some conditions at infinity.

For the present we satisfy ourselves with the verbal statement

that there are no disturbances of the water other than those

due to the body impact.

We can have little hope of solving such a general problem;

so in the next section we carry out a linearization to give a tractable problem. However, one assumption will be made which is not part of the systematic linearization. This is the replacement of (3) by a simpler condition. We assume that the effect of the body on the water can be specified by the

following condition:

p =.-V(t),

Ix<L(t), t>0.

(7)

This is the most important of Timman's suggestions; if it is reasonable, the subsequent linearization is inevitable. Physi-cally it means that the water is required to have a specified velocity component on a section of the x-axis, rather than on the actual body, and this specified component is that which

is normal to the x-axis, not normal to the body. See Figure 1 (b).

-L

X

(o

-v

Figure 1 Geometry of the problem

(a) The physical problem (b) The mathematical idealization

The Linearization

Equation (7) is linear in q, but the differential equation and

the boundaryconditions are nonlinear, and the free surface is specified by an unknown function which also is nonlinear. To simplify this situation, we perform a conventional

linear-ization. Since compressibility effects may be expected to vanish quickly, we choose the units to ensure this result.

Let L, be a convenient standard half-width, V0 be the body

velocity at the moment of impact, and e0 the acoustic speed

¡n still water. We define new variables as follows:

x = 2 L .; y = 2 L,, Ti; e)

t = (2 ii c0) t

cl) q (x, y, t) = 2 £ V

tpt ('

t); e) Y (x, t) = 2 LYt (, t); :1) g = (V,,212 0) g*

Onlv a), b) and e) are completely obvious here, e) implies that

we are interested in events happening in times of the order

required by an acoustic wave to travel a distance 2 L,. d) ¡nl-plies that water velocities will be of the order of nIagnitude of V) and not, say, of c. The constant in f) is chosen chiefly to ensure that gravity does not affect compressibility effects,

which certainly is reasonable.

These new variables are substituted into (2), (5), and (6).

yielding:

=

co-V_O - V2

(7q)*)2 + ___7ç*. [(Vq)t 7) 7t*]

e,, T r

CO2

vu g*.7q*;

(8) (9) Yr* Vo co (p,t

+

(10) (p,* + () _* = O; c0 e0

g*y* +

o (Vq)*)20.

C,,

Equations (9) and (10) apply on i1Yt(, T)

0, for t>0.

The V-operator has the usual meaning, but in the (,

)-plane.

In a systematic linearization, one next proceeds to assume perturbation series expansions for the dependent variables. These expansions are substituted into Equations (8)-(l0), and

there results a sequence of problems for the successive approximations. However, in view of the drastic assumption

by which we replaced Equation (3) with Equation (7), it is questionable whether such a procedure is worth

writing out. We can at

this point simply neglect those

terms in the last three equations which contain the factor V,,/c,

and we set e2 = c,,. This gives the same first order approxi-mation and avoids the possible implication that the higher approximations are in any sense appropriate to the present

physical problem.

From Equation (8) we obtain for the linearized problem just the ordinary wave equation in two dimensions. In terms

of the original variables, it is:

y<O.

(11)

e02

Since initially the free surface is at rest, i. e., Yt (, 0) 0,

.we can integrate (9) after dropping small terms, obtaining for the first approximation

Y(x,t) = 0, x>L(t),t>-0.

(12)

Similary, from (10), since q

_{(. 1,0)}

0, we have

(p(X,0,t) = 0, IxI>-L(t),t>-0.

(13)

Thus the first order problem is reasonably simple: The

poten-tial is identically zero ori the undisturbed free surface, and

there is no free surface disturbance in this approximation.

The Corres pondin Aerodynamics Problem

We now wish to solve Equation (11), subject to the tions stated in (7) and (13). In addition we recall the

condi-tion that there are no disturbances besides those generated by

the impacting body. Tile sit ution is shown in iii ree dimi-n-riions in F'igiire 2.

Let us r'place the vilrial,it' t with ii flew viirii&bk z, i

t-

_________z,

Figure 2 Representation ¡n x-y-t space of the general impacting-body problem

where M is some constant greater than unity. Then the dif-ferential equation (il) becomes:

c'- + q - (W - 1) p = O,

which is just the linearized equation for steady high speed flow over an airfoil, the flow at infinity being in the positive z direction. M is a Mach number in the latter problem. The

boundary conditions, viz., (7) and (13), and the initial

condi-tions are easily seen to be appropriate to the airfoil problem too. Of course, the "airfoil' is a peculiar one, since the diord is infinite. 1f we take V (t) as a constant, then the "airfoil" is

flat, with constant angle of attack.

Since the general method of proceeding to solve this pro-blem is detailed in [21, there is no point in repeating it here. However, ¡f the bottom of the ship is flat and the sides verti-cal, simple analytical answers can be obtained. The results

are instructive and suggest some facts about the general

situation as well, and so this case will be solved in detail as

far as is practicable.

Solution for the Flat Bottomed Body

We now assume that £ (t) = L, a constant for t> O, and

that. V (t) = V. a constant. In general we might want to find the potential at an arbitrary point. (x1. Yi' t1), as shown4) in Figure 3. According to the theory of the hyperbolic equation,

(11), this point has associated with it a "domain of depen-dence", which is the conical region:

4) For easzer visualization we show the point (x1, y,, t,) such that y1>Q. This causes no difficulty, however, because the

problem formulated above has a symmetry about y O: Since

q o on part of the y O plane, r is odd with respect to y.

Solu-tion in the upper halfspace provides the soluSolu-tion in the lower half -space and vice versa.

Figure 3 Representation in x-y-t space of the problem of the impacting flat-bottomed body

ca t < co ti Vxx12 + (yy)

-Phenomena occuring at x Yt at time t depend only on events which occur at times and places corresponding to the interior of this cone. Since nothing at all happens in the fluid for times t <O, we can consider the domain of dependence as only that finite cone-shaped region shown iii Figure 3.

Ward [2[ shows that the value of the potential at (xj, yj, t) can be expressed as an integral of either cp or ç over the region

E shown in Figure .3. (E lies in the y = O plane and is bounded

by the line t = O and the hyperbola which resultis from the intersection of the plane with the cone.) Since ç. is known

over much of E, viz., over that part bounded by x = ± L,

we use Ward's formulation in terms of q:

ç> (x1 , y

, t) =

(14)

e0

1Í73\

dt dx

--Jr _{J J} 3y

)y=O Vc02(tti)xxi)2y12

As long as E includes only a region in which L <x < + L, this is the complete solution for the potential, since q:,.. is

known over this surface.

For points such that this is indeed the case, we can arrive at the value of tise potential in an easier way which is worth mentioning. If E is contained between x = ± f, disturbances at the ends of the, strip have not been felt at X>. y , t1. Then the local behavior must be the same as if thc whole x-axis were subjected to the same boundary condition. ç = V. In suds a case, there would be no variation of quantities with z, and the partial differential equation would be simply:

-

fu = O. The general solution of this equation is:

ç> (y, t) = q> (c, ty) + ç>.) (c0 t ± y)

But q must be identically zero for y <O and q:. identically zero for y> O, for otherwise effects would be observed at (x, y, t) arising from disturbances at later times. Consider

only y> O,

so that

q: = ç(c0ty). At y = + O,

ç =

ç>' (c0 t + O)

V, which is constant for all t>0. Thus

' (c0ty) = V for all

(y, t) being considered, and ç>

(c0 ty) V. Similarly, for y <O, q: = - (e t ± y) V.

Generally the region E extends beyond the strip L <x < + L. and the solution is more involved. However

the early stages of this solution are still fairly simple, and they will be carried nut next.

The information desired from the solution will all be

con-tained in q: (x1, + O. t1), for -L <x1< + L. t1 >0.

There-fore in Equation (14) we set y = + O, so thai:

(15)

o

1if\

dtdx

t _{J J} 3Y /J3=9 j/c02(t_t1)2_(x_x1)s

I

E is now the triangular area in y = O for which 0< c)t <

e,, t1 - xxj

Now perform a transformation of coordinate. to the

dlarac-teristic variables in the x-t-plane. (See [2[, Section 6.5.) Let

= c0 t1 - X1 = c0 t1 + x1

= ctx

Il' = c0t + x. Also, let

i

faq:.

N(,l1)=--

I - 23r \ _{3Y y=u} = ç>(xj, +0, t1). 149 Schiffstechnik Bd. iO-1963-Heft5

Then

=

Ç Ç N

'. i') d' di'

Ji V-'i-i'

_I

where is the region

I'i-Figure 4 shows ome of the lines const., const.,

on the xt plane. The particular lines = 0, O are those passing through the point x = t = O.

The potential is even in x, and so we concern ourselves with

finding it in the region O<x<E, t>O. We proceed step-by-step through the regions marked I, Ill, IV, V in Fig. 4. It will be necessary also to consider region II, although we are not

really interested in the solution there. To go beyond region V

introduces further difficulty, and a simple analytical result

does not seem possible, although procedures are available for obtaining numerical results.

Figure 4 Characteristics In the x-t piane

Figure 5 Domain of integration for (, ,) in Region I

Region I. The region of integration, , is shown in Fig. 5.

This case has already been discussed above, where it was shown that q) (Xt, + O. t1) = + Vc0 t1. The same result fol-lows from the integral formula, (15):

V (

d'

r

d1' I

-

I

-2t

J V-' J T'ii'

-t -'1 V

=----( + i) = Vc,t1.

2

We have used the fact that 9cp \

Y

)yO

FIgure 6 Domain of integration for (t. 'ì) in Region TI (16a)

Region II. Figure 6 shows the region . We note that:

a.) N (a', i') is identically zero for <- £

b.) N (', ') is unknown in Region II; c.)q) (,i) is identically zero ici

Re-gion II. We break the calculation of q) (' i) into two terms:

=

(17)

=

Ç

d'

N (a', ) di'

+

di't

= 0.

J V-' J

Vii'

2t J Vi-i' J

£ t'_04'

The quantity in brackets is identically zero for all _'. if, in

Il, we set:

1' +

V

2T21/n'_'__2f

Actually we do not use this expression for N (a', v'); all we

need is the knowledge that the sum of the integrals in brackets above can be set identically equal to zero in the region:

£>'>-e, i1>'+2L

Region IiI.

is shown in Fig. 7, where it is the union

of the two shaded areas. From the result in Region II, the

small shaded area in the corner contributes nothing, so that

the solution forq) is:

N(',i')

-1,i,'

_(17')

t

q,i)

-,1-21 + ç0t1 i Ve0t1 2 t '1 Ç dli'

'J V'n-ri'

i _{- -i}

c0t1 + 2x--2t

sin co ti

2v

+ - V -

x1) (e0 ti + Xl - L)

Figure 8 Domain of integration for (, ) In Region IV

Region IV. The problem here is similar to that in

Re-gion III. A result similar to that of ReRe-gion II may be proved for the left hand side of the figure, and then it is seen that the two small shaded areas in Fig. 8 contribute nothing to the integral over E. The solution is:

=

1+21 i 'i V _{J Ç}

d'

_di'

Ç

d'

Ç dii'

- 2t

_{I J v-'J}

Vti-ti'

J

i/

- + 2e t-2e

= -f- {

J/( ±xj)(c0tx1L)

(16e)

+ 2 j/(Ex1) (c0t1 + xjt)

1c0t1+2x1+2t

_{c,,ti+2xi-21)}

sin

l.

t

Figure 9 Domain of integration for (.t) in Region V

Region V. When we extend the arguments used in Re-gions III and 1V for eliminating parts of E from consideration,

we find in Region V (see Fig. 9) that we use the contribution

of E' twice. Thuk, we must subtract this contribution an extra lime, und we obtain:

(16b)

+ 2j/(x1)(c0tj ±x1L)

c0t1 + 2x1 + 2]}

___coti[sin1 c0t1 + 2x1-2 C

cot1 C0t1

We note that this is the same expression as that obtained

in Region IV.

Beyond Region V. The solution can in principle he con-tinued. However, the simple result from Section II, viz., that the bracketed part of (17) can be set equal to zero, no longer suffices to yield simple analytical formulas for The actual solution for N (a', i') given in (17') must be used

ex-plicitly. This procedure will not be carried out here.

Pressure on the Body

Bernoulli's Equation for unsteady flow of a compressible

fluid is:

p

+ gy1 + (7)2 + r

d

-J

In the linearized model, (p'), eau be replaced by its mean value, and the integral term becomes simply p/a. When the

whole equation is linearized, we are left with

P = - OWtt. (18)

Under the restrictions of the linearized theory, this holds

throughout the fluid, in general, and on the surfacesYi = ± 0,

in particular.

We have calculated p (x1, + 0, t1) for a range of t1, and so

we can now directly calculate the pressure in that range. Since

our interest is in the lower half-space, i. e. y <0, we again

call attention to the fact that ç (x1

, + 0. t1) = - (x1 -0, t1),

and then we use the previously obtained formulas to find

p(x1,-0,t1) for ß<x1<:

I _i

c0tj+2x1+2L

= QVC,, I--- sin1 L c,1t1

i

sin_i c0tj + 2x1 1 c0t1

All of the angles defined here are to be taken between t/2

and + t/2.

Figure 10 presents some numerical results for xj/ = 0, I. , 1. It is seen that p (x1, 0, t1) dianges sign as t1 passes

the value 2!/c0. This is seen to be generally true froni

for-mulae IV and V above.

I:

p (xj,-0,tj) =

(19a) III:

p(x1,-0,tj)

(19b) 1 -- 1

c0t1+2x-2

= Vc0

-- -

sin1 2 ;t Cuti

IV, V: p (x1, 0, t)

(19e) 151 Schiffstechnik Bd. iO-1963Heft5i

=

2e

2rt i J j/-'J Vii'

J j/-'J Viii'

V _{J Ç}

d'

Ç dii' Ç

d'

Ç dii' 21-1 -1' {

2 V1+ x1) (c0t1x1L)

(16b) co ti e, t1

9 .7 .6 .5 .4 s .1 o -.1 -:2

Figure 10 Pressure on the flat bottoni

All the curves shown have essentially the saine characteristic

shape. The pressure is constant, Vc0, until the pressure

re-lief wave arriyes from the nearer edge, at which time the slope of tite curve changes abruptly. There is a second abrupt change when the other edge makes itself felt. For x1 / £ = 0, the relief

waves from both edgés arrive at the same time, so that there

is only one slope discontinuity, occurring then t Lic0. The curves are all terminated at the times when the first

reflections of the relief waves arrive at the points in question. These times correspond to the lower boundary of Region V in Figure 4.

Physically it is apparent that the mathematical solution to the problem posed above must approach a steady state.

Mathematically in the solution for Region V above (see Fig. 9),

we see that the first integral is taken over a domain shaped like a parallelogram; neither the shape nor size of this

do-main nor the integrand changes if we increase the value of t1

Further changes in the value of q) come about only through the addition of integrals over domains farther removed from

(x1 , ti), and, because of the form of tile integrand, these have lesser effect on the value of the potential. q) approaches a

con-stant value, so that q approaches zero.

The Total Impulse

1f the flow at least in the neighborhood of the impacting bottom approaches a limiting condition quickly enough, the linearized theory can be used to find the total impulse which acts to decelerate the body5). It has been seen that the initial positive pressure period ends when t 22/c0. For a

slam-ining ship this would be a very short time indeed, and it may reasonably be hoped that there is not much net contribution

to the impulse after this time.

The Laplace Transform technique provides the means of performing this calculation. For any suitably behaved

func-tion, f (z, '.', t), we define the transform: 00

f

f(x,v;s) = s j' f(x,y,t)etdt

The inclusion of the extra factor, s, is arbitrary; we are here

following [2) (Chapter 8).

We begin by taking the transform of the differential equa-tion (Il), and of the boundary condiequa-tions, (7) and (13). It can

5) 0) course there wUt be a further vertical force when gravity free surface) effects become appreciable. They are ignored here.

be shown easily tile initial conditions are such that, after two

partial integrations,

= s4'.

Then, in place of (11), we have:

1'---

(x')

- tc J

(xx')2+y

C

-:

-F

J,dx'.

The second integral is a Cauchy principal value integral, as indicated by the i.'. This is an integral equation for 1(x). The general solution for the potential is obtained by solving this

equation and then inverting the transform . Fortunately, this is not necessary for our problem.

K (

.' £

I =

dt .1 p (x, 0, t) dx 0 -F L -. f

dx lins $ p (x, 0, t) et dt

-F

..o o

£ 00

00

Ç dx lins j' q)(x,O,t) etdt

-F s..+o o

-

Sdx(x,-0;0)

Thus, all we need to find is the transform itself, evaluated for s = 0.

In the integral equation, the first integral vanishes when s = 0, since K1(z) = (liz) + O (zlogz). All that remains of

the equation is:

s c

which has the solution

f(x) = VL2-x2.

We can immediately write down a solution whkh satisfies

(20) and (22), as well as the initial conditions,

¿(x,y;s) =

(23)

V(x - x')2 ± v ) dx',

where K1 is a modified Bessel function of the second kind. f (x) is an unknown function which must, be selected so that

(21) will be satisfied. To find the condition on f (x), it is neces-sary to differentiate (23) with respect to y and then let y

approach zero. The procedure is outlined in [2) and will not

be described here. We quote the result:

unI 4. =

V

y.-..')

Let us consider the formula for the impulse to the body:

+

-

_{(- )}

= 0.

(20)

The boundary conditions become:

= V, on y = O, x <L;

(21)

0, onyO, x>E.

(22)

x')2 1f(x') dx'

The transform of (Ç for s = O is (See Equation (23)):

e

(xyO)

yVx'

dx'.

t _J

(xx')2+y2

-C

We can now substitute this into the integral for I. The ex-pression is meaningless at this stage unless we keep y finite;

after carrying out the first integration we can let y-O. So

we write the expression for I and interchange the order of

integrations: ¿ É

r

r

yV2_x'2

1=

t -j

limidxt

, dx J

(xx)2+y2

-

-Ê e e

= -

_--

lii dx' y

= Vsgny

./ -C i

=

_{..?Y. hrn}

J

dx' [y

x'1

'

(tant

-C

ex'

- tant

y

$dx'

Vx'2

y2

--sgny.

2

Here, sgn y = + i if y> O and 1 if y <O. We are

inter-ested in the case that y O, so that

tQV2

2

This answer is quite reasonable in form. The lack of

de-pendence on c,, follows from two facts: The pressure is propor-tional to c5, hut the time delay before the pressure relief occurs is proportional to 1/c,. Similarly, the delay before pressure relief is proportional to L, and the pressure acts over

the span 2 L, so that the impulse depends on £2.

Finally, we note that if the initial pressure, Vc0, acted over the span 2£ for a period T and then were cut oli

sud-denly, the value of T would be re/4c0, which tends to confirm the previous assumption about the duration of compt-essibiliiy

effects.

Discussion of Valid ¡h-, Results, Extensions

The first and most important assumption lies in reducing the problem to two dimensions. One might he inclined to attempt to apply the results to three-dimensional bodies by

using a strip theory. But this is not likely to be generally suc-cessful. In order that a strip theory he valid all quantities must

to some degree depend mostly on the two space-coordinates used in the two-dimensional analysis. For the slamming pro-blem, this means that a large length of the ship must hit the

water practically instantaneously. This might happen

occasion-ally in an irregular sea, but not regularly.

However, the fact that it might happen occasionally is the key to the usefulness of the theory. It is reasonable to expect that the highest pressure loads will occur when a large part of a ship bottom does impact instantaneously, and in suth a case the two-dimensional analysis may be expected to give

dx

x')2 + y2

£ - x'

y

reasonably accurate predictions. In other words, the

calcula-Lion of pressure and total impulse by the two-dimensional pro-cedure will provide upper limits to these quantities.

In model tests in regular seas, where slamming may occur

more or less regularly, the theory presented here will probably

greatly overestimate impact pressure and thus impulse. Ex-perimental verification could he found in two possible ways:

( 1) from genuine two-dimensional tests, or (2) from

examina-tion of a great many records of bottom 'pressure on ships in irregular seas. In either case, high time resolution would he

needed to check the pressure history. From Figure 6 it is clear

that the interesting phenomena end after a time lapse of the order of magnitude of a few milliseconds at lull-scale or a

fraction of a millisecond at some reasonable model scale. Also, the pressure is very high, about 70 lbsuin.2 for an impact

velo-city of one ft./sec., or about 16 atmospheres for one meter!

sec. These facts suggest that verification will be quite difficult but not impossible.

After the reduction of the problem to one involving two

space dimensions, the most important assumption itivolves the replacement of the condition (3) by the simpler condition (7).

It does not appear possible to evaluate this simplification in any way except to compare final theoretical predictions with the results of experiments such as those described above. In other words. quantitative verification of the final theoretical

results would suggest good validity of the simplifications made

in formulating the mathematical problem. Otherwise, all that can be said is that the ship bottom should be quite fiat. This

will insure that it imparts to the surface water particles a

vertical velocity component which is approximately equal to

V.

Next we must consider the conditions required for validity of the general linearization. The most important check is to

compare the time scale of compressibility effects with the time ecales of other events associated with the general problem. The

ciiaracteristic time interval in the analysis of this paper is

2 E/e0 , whidi, as indicated above, amounts to a fraction of a millisecond on model scale and perhaps a few milliseconds

oil _{full scale ships. It seems reasonable to suppose that this}

time is too short for any gravitational effects to have any signi-ficance. The body itself will have entered the water a distance

of about 2LV(,/c0, which is very small compared with thc width 2 L for reasonable impact velocity, V,,, and the

de-crease in body velocity will probably also be extremely small, although this will depend on the mass of the impacting body. In the general problem depicted in Figure 2 it thus seems

rea-aonable to consider V(t) = V, a constant, and furthermore to

hold some confidence in-the validity of the linearizatioti approx.

i ma 'ion s.

There is another time scale in such problems whith might be more critical : the time scale associated with local vibrations of

the body. Certainly there will be no problem with vibrations

of the ship as a whole, because the corresponding frequencies will be so low that compressibility effects cannot possibly

mat-ter. But we must allow for the situation in whidi local

struc-tural vibration frequencies are high enough that there may be acoustic interactions with the water.

First, it should he mentioned that, for those modes of vibra-tion with low natural frequencies, tile total effect of the

com-pressibility stage can be summed up again iii ternis of

im-pulses applied to the body. However, these in general must be

modified to weight the pressure distribution according to the

usual methods of normal mode analysis.

If a natural structural frequency is high enough that water

compressibility must he considered, then the same arguments

used above for neglecting gravitational effects still hold. and

the formulation of the problem will be quite similar to the rigidbodv analysis presented here. The principal difference will be that the velocity function V (t) in Equation (7) will be

replaced by a function V (x, t) (in a two-dimensional analysis) which will he of the form f (x) g (t), f (x) describing the normal

mode shape and g (t) specifying the unknown time history of

the amplitude of this mode. Generally the integral in (15) will

have to be evaluated numerically. For each t = t,, the pres-sure p (x, t) will be found as a function of the unknown g (t) and the known f (x). Then the pressure function will be used to find g (t) from the ordinary differential equation des-cribing the time dependence of the normal coordinate.

This normal mode approach does not affect the solution

alread obtained for the rigid body, since the whole problem is linear in all the dependent variables. The solution to be

found for V (x, t) will represent a vibration superposed on the rigid body translation. Of course, sudi an analysis can be carried out for any number of modes independently.

There is one difficulty in principle with the above approach. Ordinary normal mode analyses break down if there is damp-ing present, and in this problem there certainly exists radiation

damping, even if structural losses can be overlooked. lt may be that this damping is small enough to introduce negligible

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coupling between modes. If not, however, the only remaining

approach seems to be to describe the structure by its basic partial differential equation and to solve this simultaneously

willi the hydrodynamics problem. Such an undertaking seems

rather formidable, but if the whole problem is to be solved

numerically step-by-step in time it may be feasible. This could be the subject of another paper.

Aeknowledgmen s

The author has a'ready mentioned the contribution of

Pofes-sor Timman to this work. In addition he acknowledges with thanks helpful comments from Professor G. N. Ward. Dr.

George Batchelor, and Dr. T. Brooke Benjamin.

(Received 15th May 1963)

References

R. Timman, Technological University, Deift,

Nether-lands, private communication.

G. N. Ward, "Linearized Theory of Steady High-Speed

Flow", Cambridge University Press, 1955.

J. V. Wehausen & E. V. Laitone, "Surface Waves", Handbuch der Physik, IX, Springer-Verlag, 1960.