A ak ENTE
HYDROMECHANICS
THE EXCITING FORCES ON FIXED BODIES IN WAVES
Oo by AERODYNAMICS J.N. Newman @) STRUCTURAL MECHANICS @) HY DROMECHANICS LABORATORY RESEARCH AND DEVELOPMENT REPORT ne LIED MATICS
May 1963 Report 1717
THE EXCITING FORCES ON FIXED BODIES IN WAVES
by
J.N: Newman
Reprint of a paper published in the Journal of Ship Research, Society of Naval Architects and Marine Engineers, Vol. 6, No. 3, December 1962.
May 1963 Report 1717
i
yy ui
The Exciting Forces on Fixed Bodies in Waves By J. N. Newman’
General expressions, originally given by Haskind, are derived for the exciting forces on an arbitrary fixed body in waves. These give the exciting forces and moments in terms of the far-field Velocity potentials for forced oscillations in calm water and do not de- pend on the diffraction potential, or the disturbance of the incident wave by the body. These expressions are then used to compute the exciting forces on a submerged ellipsoid, and on floating two-dimensional ellipses. For the ellipsoid, the problem is solved using the far-field potentials, and detailed resuits and calculations are given for the roll moment. The other forces agree, for the special case of a spheroid, with earlier results obtained by Havelock. In the case of two-dimensional motion the exciting forces are related to the wave amplitude ratio A for forced oscillations in calm water, and this relation is used to
compute the heave exciting force for several elliptic cylinders.
Expressions are also given
relating the damping coefficients and the exciting forces.
Nomenclature
A = wave amplitude A = wave-height ratio for forced oscillations (ai, @2, a3) = pemiene of ellipsoid ; = damping coefficients 4 = nondimensional roll exciting-force coefficient D; = virtual-mass coefficients, defined by equations (18) and (19) g = gravitational acceleration hk = depth of submergence t= —1 j = index referring to direction of force or motion 1 x(z) = spherical Bessel function, jx(z) = (5) ie Jn 1/2(2Z) K = wave number, K = w?/, P; = functions defined following equation (17) R = polar coordinate v; = velocity components (z, y, 2) = Cartesian coordinates a; = Green’s integrals, defined by equation (20) 8 = angle of incidence of wave system @ = polar coordinate p = fluid density ¢; = velocity potentials w = circular frequency of encounter Introduction
In order to determine the exciting forces on a ship in waves, it is necessary to know not only the hydrody- namic pressure in the incident wave system, but also the effects on this pressure field due to the presence of the ship. In the linearized theory the undisturbed pressure of the incident-wave system is well known for a given plane progressive wave system, but. the diffraction or dis- turbance of this incident system due to the ship is gen- erally very difficult to evaluate, and in fact it is neglected in the so-called ‘‘Froude-Krylov” hypothesis.
Recently Haskind [1]? has derived expressions for the
' David Taylor Model Basin, Washington, D. C 2? Numbers in brackets designate References at the end of the paper.
exciting forces and moments on a fixed body, which do not require a knowledge of the diffraction effects men- tioned in the foregoing, but depend instead on the velocity potential for forced oscillations of the body in calm water. Moreover, it is easily shown that the asymptotic characteristics of this velocity potential for large distance from the body is sufficient to determine the exciting forces for a given incident-wave system. For many problems this asymptotic potential is rela- tively easy to obtain, compared to either the near field forced-oscillation potential or the diffraction potential, and thus Haskind’s relations are extremely vdluable. For example, it is known that for a submerged body, such as an ellipsoid, the potential in the far field can be obtained, to first order of approximation, in terms of the singularity distribution for the same body in an infinite fluid, but the near-field potential requires the ‘‘image”’ of the free surface inside the body. With this in mind, asymptotic far-field potentials were recently used [2] to study the damping of an oscillating submerged ellipsoid; to study the near field potential for the same problem would probably require expansion in Lamé functions, and would certainly be extremely difficult. Thus it is apparent that Haskind’s relations permit the determina- tion of exciting forces for bodies which would otherwise be highly untractable.
Since Haskind’s relations are not well known, we shall first present an outline of their derivation. As one il- lustration of their use, we shall utilize the far-field ve- locity potential of a submerged oscillating ellipsoid, as derived in [2], to obtain the six forces and moments acting on a fixed submerged ellipsoid in oblique regular waves. For all but the roll moment, these results reduce to expressions obtained by Havelock [8] for the special case of a spheroid, or an ellipsoid of revolution. The roll moment, which of course cannot be obtained for the
spheroid, due to axisymmetry, is then studied in detail, for the general case of an ellipsoid. As a second illustra- tion we present the exciting-force amplitudes of various two-dimensional floating elliptic cylinders which are easily obtained from the corresponding damping charac- teristics. In the special case of a circular cylinder, the results obtained check with direct calculations made by Dean and Ursell [4], and for more general bodies, the method is consistent with the extensive calculations pre- sented by Grim [5].
Haskind’s Relations for the Exciting Forces
We consider two independent problems involving a floating or submerged rigid body; 7.e., the diffraction problem of regular incident waves moving past the fixed body, and the radzation problem of forced sinusoidal oscillations of the body in otherwise calm water. In both cases, assuming small disturbances of an ideal fluid, there exists a velocity potential ®(z, y, z, t) satisfying Laplace’s equation and the free-surface condition
Ob
a + 9S? =0 on z=0 (1)
Here (z, y, z) is a Cartesian-coordinate system, with z = 0 the plane of the undisturbed free surface and the z-axis positive upwards. For incident waves of fre- quency w or forced oscillations with the same frequency, we can write
B(x, y, 2, t) = v(x, y, z) (2)
where the real part is to be taken in complex quantities involving e’. From (1), the potential ¢ satisfies the condition
Ke =0 on z=0 (3)
where K = w?/g. For the diffraction problem, with the incident wave system given by a known potential, vo, the total potential
e = got ¢7
must satisfy the boundary condition of zero normal velocity on the body, or
2 («, + 61) =0 onS (4) where n is the unit normal vector into the fluid and S de- notes the submerged surface of the body. For the radia- tion problem, there are six degrees of freedom, including surge, heave, sway, roll, pitch, and yaw, the velocities of which we denote by
ve" G ad i 2, 3, 4, 5, 6) respectively. For oscillations in the jth mode we can write ge = PH; and in general the velocity potential will be of the form
6 g= x vy9;(z, Y, 2) (5)
where, due to the presence of the free surface, the po- tentials y, will be complex. On the body, the potential ¢; Must have the same normal velocity as the corre- sponding mode of the body, or
OP) 1G, y, 2) on S (j = 1, 2,... 6) (6)
on where fi = cos(n, z) fe = cos (n, y) fs = cos (n, z) fs = y cos (n, z) — z cos (n, y) fs = zcos (n, x) — x cos (n, 2) fe = xz cos (n, y) — y cos (n, 2)
Finally, the radiation potentials 9;(j = 1, 2, ...6) and the diffraction potential ¢; must each sat*sfy the radia- tion condition of outgoing waves at infinity, and must vanish at infinitely large depth in the fluid. In view of these conditions and the free-surface condition (3), which must be satisfied by each potential independently, it follows from Green’s theorem that
1, 2, 16) ce)
Now we consider the six exciting forces and moments, which we denote by X,, following the same designation of index as for the velocities. Thus, X; is the surge force, X, the sway force, X, the roll moment, and soon. Then
x,=— | {pas (8) Ss
where the hydrodynamic pressure p is given by the linearized Bernoulli equation
o® 9 i cob Deli Se Oe (9) For the exciting forces on the fixed body in waves, e=ntenr and thus X,=
1wpe* ff (go + oa) f,aS Ss
: D inne f{ (o + e) or dS ‘Ss
However from (7) and the boundary condition (4) for g7 on S,
(10)
O¢o
(egies |e c= = Illor
Substituting in (10), it follows that
ed O¢; fo) X,; = twpe™ iG = = 65 5) dS (12) S
Thus we have found the exciting forces in a form de- pending only on the incident-wave potential go and the radiated-wave potentials ¢;, and which is independent of the diffraction potential y;. Finally, we note from Green’s theorem that if S, is any control surface in the fluid outside S, then since ¢ and ¢, both satisfy the free- surface condition,
JJ (moe -
S+So
dS (11)
Ovo
(13)
and therefore, X; may be evaluated from a surface in- tegral at infinity :
) X; = —iwpe™ fy («52 ej =
Thus X, depends only on the asymptotic behavior of 9, at large distances from the body. (Haskind reached this conclusion by the introduction of Kochin functions.) We note that in (14) the direction of the normal n is inward, or from the control surface into the fluid.
If we take as the control surface S, a vertical circular cylinder about the z-axis of large radius R, then with (R, 0, z) as polar coordinates, it follows that
a 5) dS (14)
aS = R dz de 0/on = —0/OR and thus 2a 09; Oe X,; = twpe™ a (« a 9; oe) a dzd@ (15)
Exciting Forces on a Submerged Ellipsoid
For the submerged ellipsoid with semi-axes (a), de, as), defined by the equation
(16)
the asymptotic representations of the radiation potentials ¢, were derived in reference [2], for the case of sinusoidal oscillations with constant forward velocity. These re- sults are an approximation based on the assumption of a moderately large depth of submergence. In order to study the exciting forces on a fixed ellipsoid we use the potentials derived in [2], setting the forward velocity equal to zero. Thus, from equation (21) therein, we obtain
¢; = —2 (+5)" P;(4 + 6) exp[K(z — h — iR)
argh + 1/4] (17)
where
P,(u) = —2r7iwK a,a,a;D, cos wi
P2(u) = —2miwK a,a2a;D. sin u H(@)
P3(u) = 22K a\a203 D; ja)
P,(u) = —2miwk 2a,a20;(a2? — a3?)D, sin 29)
P,(u) = 2miwK? aya2a3(a,2 — a3”)Ds cos wil
Pe(u) = —2rwK?a,a,03(a12 — ap)De¢ cos u sin we and
q = K[(a2 — a3?) cos? w + (a2? — a3?) sin? u]'”
Here ai, ae, and a; are the semi-axes of the ellipsoid, with 2a, the length, 2a, the beam, and 2a; the depth, h = depth of submergence of the centroid, 7,,(q) is the spheri- cal Bessel function, and the coefficients D, are related to the virtual-mass coefficients of the ellipsoid in an in- finite fluid, and are defined by
D; = (2 a a;)~! @) = 1, 2, 3)
Q;+17 — A;+2" ii Dies = | 2 tga) + aes — as] G = 1,2,3) (19)
and
(18)
a dy
ens i} (a? + A)[@? + (a? + Aa? +H]? (20) For regular incident waves of amplitude A, progressing in a direction which makes an angle 6 with the z-axis,
the velocity potential is
A
eo = — exp[Kz — iKR cos(@— 8)] (21) Substituting equations (17) and (21) in (15), we obtain
1/2 2% 0 X, = —pAKe' @ ok) I f {1 — cos (@ —8)]
-exp{2Kz — Kh — iKR[1 + cos(@ — £)] + ri/4}P,(m + 0)dz do
s 2 fe 5 ; — —ipA € a) e—Khttat+ xi/4 2a it [1 — cos(@ — 6)]-exp{—7KR 0
[1 + cos(@ — 6)]}P,(x + 0)d8
20 = —ipA ( of) eta eal (1 + cos u) -exp{ —iKR(1 — cos u)}P,;(u + B)du (22)
where in the last integral we have replaced the variable @ by uw = x + 6 — 8. Since the radius of the control surface is large, KR > 1, and we may evaluate the integral over u by the method of stationary phase:
2a Qn
f(uje*KRA—cosu) dy, = (zs). {e-™/47(0) + eri/4—2iKRF(e)} + O(1/R) Thus,
X, = —Atpg 4 e~ KhtiotP,(g) (23) Substitution of the appropriate expressions for P,(8) gives the six exciting forces, as functions of the angle of incidence B.
In the special case of a spheroid at zero speed, the fore- going results are in agreement with Havelock’s expres- sions, obtained directly by finding the diffraction po- tential and integrating the pressure over the surface of the spheroid.
We now restrict our attention to the roll moment X,:
X, = —82pgA K2a,a203(a2?
— a3?)D, sinBe~ ** ++! (24)
2
0) 4
where
q = K[(a:? — a;*) cos? 6 + (a? — a”) sin? B]?/? In the special case of beam waves, 8 = 7/2, and thus X, = —8mpgAaara3Die~ ** + '47.[K (a2? — as*)'/?] (25)
We note that this moment depends on the length 2a, through the factor a,D,. The spherical Bessel function je oscillates about zero for real values of its argument, and thus for a2? — a3? > 0, or a beam-depth ratio greater than one, the roll moment coefficient will oscillate about zero.2 This holds for all values of 8. On the other hand, for a2? — a3?-< 0, the parameter gq will be either real or imaginary, according as a, — a32 2 ——— tan? B S Reiners For the angles of incidence between head (or following) waves and the critical angle, where a? => a3?
tan? B = 5 32 = a2”
q will be real and the roll-moment coefficient will oscillate. For angles between the critical angle and beam waves, ¢ will be imaginary and j2(q)/q? will be a (real) monotonic increasing function of g. Thus for these angles the co-
3 It is assumed in this discussion that a; > a3.
0.007
0.006
0.005
0.003
0.002
0.001
-.0005, ° ' 2 3 4
Fig. 1 Coefficient of roll-exciting moment for ellipsoid a2/a, = 1/7, a3/a, = 1/14, b/a, = 2/7, for various angles of incidence
efficient of the moment X, will be positive and non- oscillatory, rising from zero at zero frequency to a maxi- mum, and then decreasing to zero at large frequencies, due to the exponential factor e~**. However for fairly slender ellipsoids, with a,? >> a;? > a,?, this sector of angles will be quite narrow. In all cases the roll moment is 90 deg out of phase with the wave height at the centroid.
Computations of the roll moment X, are easily per- formed from equation (24). The inertia coefficient D, can be computed directly using tabulated values of the inte- grals a; in Zahm [6], or in terms of the entrained inertia coefficient
a (a2? — as”) (a2 ie a3) 2(a24 — a3*) + (a2 — as) (a2? + a”)?
which is plotted in Kochin, Kibel’, and Rose, [7], and tabulated (with the notation ya = 4,’) by Zahm [6]. In both references, the semi-axes are denoted (a, b, c) rather than (a, a2, a3), with the restriction a > b > c. Thus (a, b, c) should be replaced by (ai, a2, a3) if a, > a2 = a; or by (ai, a3, @2) if a, $ G3; S a2: The spherical Bessel function can be evaluated from various tables, or from the relation
M44 =
0.007
0.006
0.006
0.004
ot 0.003
0.002
0.001
Fig. 2 Coefficient of roll-exciting moment for ellipsoid 22/4, = 1/14, a3/a, = 1/7, b/a, = 2/7, for various angles of incidence
(oy Canes Jn2) = a IN WS
2
For illustration we shall compute the roll-moment coefficient
— Xs ~ 8rpgAaia2azD, cos wt
= —K(a,? — a;?) sin Be ** 0) q
C (26)
in the case of the two ellipsoids for which damping com- putations were given in [2]; the first of these has a beam- length ratio a2/a,; = 1/7, a beam-depth, ratio a2/a; = 2, and a depth of submergence h = 2 ap, or equal to the beam. The second ellipsoid has the beam and the depth interchanged, or a2/a; = 1/14, a3/a: = 1/7, h = 4ax. Curves of the coefficient C, for various angles of inci- dence @ are shown in Figs. 1 and 2, as functions of
K (\az? es a3?|)*/2
In view of the definition of C,, the curves for 8 = 90 deg or beam waves can be considered as independent of the length 2a;, for any ellipsoid with the given beam-depth
ratios and depth of submergence. Fig. 2 also shows the critical angle 8 = 83 deg where the coefficient C, ceases to oscillate about zero.
Relation Between Damping and Exciting Forces
The fact that the exciting forces can be determined from the far-field asymptotic behavior of the radiation potentials ¢, implies a relation between the damping forces and the exciting forces, since it is well known that the damping forces can be found from energy radiation at infinity. In fact, for an arbitrary three-dimensional body at zero speed, the six principal damping coefficients B,, are given by the integrals [2]
4p oo 2x Bay Ss = Ik ii |P(u)|2du (27) TW 0 where for the particular body considered, the functions P, characterize the far-field potential, in accordance with equation (17).
The functions P, can be replaced by the exciting forces X ,, from equation (23). It is convenient for this purpose to define the exciting-force amplitudes X,, where
X; = X Met (28)
Then, from (23),
tw IO) = Tam where X,(8) denotes the exciting force amplitude for waves at an angle of incidence 8. With this notation, it follows from (27) that
wk 2a By = cael |X, (8)|748
0
ef" X (8) (29)
(30)
Thus the damping coefficients B,, are proportional to the integrals of the squares of the corresponding exciting- force amplitudes, integrated over all angles of incidence. This relation is valid for an arbitrary three-dimensional floating or submerged body, since its derivation only requires that the far-field potentials y, be of the form (17), with the functions P, corresponding to the particular body under consideration.
Equation (30) allows us to compute the damping coefficients, if we know the exciting forces for waves of all angles of incidence. However in practice it is more likely that one may desire the inverse, i.e., given the damping coefficients, can we find the exciting forces? In general this is not possible, for the damping coefficients are constants while the exciting forces depend on the angle of incidence. One exception is for bodies of revolu- tion with a vertical axis of symmetry, such as a sphere or spar buoy. Then clearly the heave exciting force is in- dependent of the angle of incidence, 8, while the remain- ing nonzero exciting forces will depend linearly on cos 8 or sin 6. Thus, for example, for surge,
X1(8) = X10) cos 6 and from (30) it follows that
2» |
— (0) 2 2 are (0)| Je cos? BdB
= Fagen KO)
In this manner we can find a relation for each of the excit- ing forces in terms of the corresponding damping co- efficient. Without loss of generality we can set 6 = 0, and we thus obtain the expressions
4 /2
X,(@0)| = A (“= Bu) (31) 2paq3 Vy
[Xs| = 4 ‘Ce Bu) (32) 4 3 Vy
[X.(0)| = A (22 Bu) (33)
Thus for bodies with a vertical axis of symmetry, the damping coefficients are sufficient to determine the amplitudes of the exciting forces (although not their phases), and vice versa.
Two-Dimensional Case
We consider now the case of plane two-dimensional motion, such as beam waves incident on an infinitely long cylinder. Then equation (14) is replaced by a line integral at infinity, and if we take as the control contour a large rectangle consisting of the free surface, the bottom
at 2 = — o, and two vertical lines —~ Z z € 0 at x = +o, ‘ve then obtain, in place of equation (15), cane 0 de Pay z=to GS iwpeiat [ [ <a — 9 a dz (84)
For the two-dimensional incident-wave system, pro- gressing in the positive x-direction, A go = cr exp (Kz — iKz) (35) while the asymptotic radiation potentials, analogaus to equation (17), are
¢; = P*exp(Kz —iK2x) forr—~> +0 (36)
Here the functions P,+ depend only on the wave number K and the particular body under consideration, and the superscript + corresponds to the case r > +o. In general the two functions P;+ and P,~ will be unequal, but in the practical cases of importance, involving bodies symmetrical about the y-axis, the magnitudes of these two functions will be the same, or more precisely,
P\+t = —P,~ (surge) P;+ = P3;- (heave) Pst = —Ps;- (pitch)
(It is-more conventional to apply these concepts in the two-dimensional x — y plane, and j = 1 may be thought of as either surge or sway, while 7 = 5 corresponds to either pitch or roll.)
We now proceed as before to find the exciting forces and damping coefficients as functions of P;. Substituting (35) and (36) in (34) we obtain
0 X, = iwpe'" aA (f eXa2) [P,+(—71K + 7K)
— P,-(iK + iK)] (87)
pqgA cP;
which is the two-dimensional analog of equation (23). Thus the exciting force is proportional to the amplitude of the radiation potential at infinity, in the direction from which the waves are incident.
The damping coefficients are given in terms of energy radiation, by the expressions
By; = 4pe[|P,+]? + |\P;-|?] or, for a symmetrical body, By = po|P;al2 (38) These are the two-dimensional analogs of equation (27).
Comparing (38) with (37) it follows that for an arbitrary two-dimensional body with transverse symmetry,
B;; = OMA: or
X,% = (39) which is the desired relation for the amplitude of the exciting force in each mode, in terms of the corresponding damping coefficient. This equation is to be compared with equations (31-33) for a three-dimensional body of rotation with vertical axis. We emphasize again that within the framework of linearized water-wave theory, equation (39) is an exact expression which holds for any two-dimensional body with transverse symmetry, in each of the three degrees of freedom.
‘In the two-dimensional theory one frequently uses the “wave-height ratio” rather than the damping coefficient, especially for heave, where the wave-height ratio A is de- fined as the wave amplitude at infinity, per unit ampli- tude of heave displacement. Thus in the present nota- tion, where the velocity potential is the potential per unit heave velocity, and the wave height is given by the expression
tw ¢ 5 e(z, 0) it follows that the wave-height ratio for heaving oscilla- tions will be
= K|PA (40) where we delete the superscript (+) since for a sym-
metric body |P3+| = |P3-|. Substituting (40) in (37), it follows that (41)
[x20] = 2 Aa
4 $ke
Fig. 3 Coefficient of heave-exciting force for a floating ellipse
or the amplitude of the heave exciting force, per unit amplitude of the incident wave, is x, AP PEK: This relation is of practical importance since the wave- height ratio A has been computed [8-12] for various floating cyliadrical forms, including the circular cylinder, ellipse, flat plate, and the so-called Lewis-forms and their generalizations. Figs. 3 and 4 show the nondimensional heave-excit- ing force coefficient
(42)
where B = beam, for various elliptic cylindrical sections. In all cases the coefficient A was obtained from the calcu- lations of Porter [11]. In Fig. 3 the abscissa is the con- ventional parameter, w2B/2g, while in Fig. 4 the abscissa is w?T/g, with T = draft. Also shown in the latter figure is the thin-ship theory curve for an ellipse, which may be regarded as the limiting curve for an ellipse of small beam-draft ratio, or as the thin-ship approximation to an arbitrary ellipse. This thin-ship result correspoads to a source distribution on the centerline of the ellipse, of strength proportional to the normal velocity on the sur- face of the ellipse. One obtains in this manner the ex- pression
1 Tedroft 07 HeB/2T
03 02
0.1
t) ! 2 3 4 5 6 7 8 KT
Fig. 4 Coefficient of heave-exciting force for a floating ellipse
C3 = 2n|L_,(KT) — lh(KT)]
where L_, and J, are the modified Struve and Bessel functions, respectively, defined by the series
_-< (x/2)? Liz) = 2 ta +) rm +h) = (0/2)
AG) = 2, aes doy
It is evident in Fig: 4 that the dependence upon beam is relatively small, and thus that the thin-ship result is a fairly good approximation, at least for small or moderate frequencies and moderate values of the beam-draft ratio.
The coefficient C3 is nondimensionalized with the force pgAB, or the hydrostatic buoyancy force due to the wave amplitude A. Thus in the limit of low frequency, or long waves, Cz; = 1.0. However Figs. 3 and 4 show that in practice, this limit is a poor approximation since the values of C; fall off very rapidly for finite frequencies. In the limit of high frequencies, it can be shown that
= Yel Aw~ RT (1 + H), for the ellipse of beam B and draft T, and H = B/2T. Thus for large frequencies,
KT>1
H(i + H)
(KT)? , KT >1
oe
Acknowledgments
Unpublished data for the wave-height ratio of the floating elliptic cylinder were furnished by Cdr. W. R. Porter, and the author is very grateful to Dr. Porter for supplying these data. The curves of the roll-exciting moment for the submerged ellipsoid were constructed by Mrs. Helen W. Henderson.
he author also takes this opportunity to acknowledge two constant sources of assistance: Dr. T. F. Ogilvie for his supervision and innumerable stimulating discussions, and Miss Evelyn I. Giesler for her patience in typing.
References
1 M. D. Haskind, ‘‘The Exciting Forces and Wet- ting of Ships in Waves,” (in Russian), [zvestia Akademii Nauk S.S.S.R., Otdelenie Tekhnicheskikh Nauk, No. 7, 1957, pp. 65-79. (English translation available as David Taylor Model Basin Translation No. 307, March 1962.)
2 J. N. Newman, ‘The Damping of an Oscillating Ellipsoid Near a Free Surface,” JourRNAL oF SHIP RE- SEARCH, vol. 5, no. 3, December 1961, reprinted as David Taylor Model Basin Report 1500.
3 TT. H. Havelock, “The Forces on a Submerged Body Moving Under Waves,” Transactions, Institution of Naval Architects, vol. 95, 1954, pp. 77-88.
4 R.G. Dean and F. Ursell, ‘Interaction of a Fixed, Semi-Immersed Circular Cylinder with a Train of Surface Waves,” M. I. T. Hydrodynamics Laboratory Technical Report No. 37, September 1959.
~ 5 O. Grim, “Die Schwingungen von schwimmen- den, zweidimensionalen Kérpern,’’ Hamburgische Schiff- bau-Versuchsanstalt, Report No. 1171, 1959.
6 A. F. Zahm, “Flow and Force Equations for a Body Revolving in a Fluid,” NACA Report 323.
7 N. E. Kochin, I. A. Kibel’, and N. V. Rose, “Theoretical Hydrodynamics,” (in Russian), Gostekhiz- dat, 1948. (German translation, Akademie-Verlag, Berlin, Germany,.1954.)
8 F. Ursell, “On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid,” Quarterly Journal of Mechanics and Applied Mathematics, vol. 2, 1949, pp. 218-231.
9 O. Grim, “Berechnung der durch Schwingungen eines Schiffskérpers erzeugten hydrodynamischen Kriafte,’ Jahrbuch der Schiffbautechnischen Gesell- schaft, vol. 47, 1953, pp. 277-299.
10 F. Tasai, “On the Damping Force and Added Mass of Ships Heaving and Pitching,” Journal of Zosen Kiokat, vol. 105, 1959, pp. 47-56. (English translations: Reports of Research Institute for Applied Mechanics, vol. 7, no. 26, 1959 and University of California, Insti- tute of Engineering Research, Series No. 82, Issue No. 15, 1960.)
11 W. R. Porter, ‘‘Pressure Distributions, Added Mass, and Damping Coefficients for Cylinders Oscillating in a Free Surface,’’ University of California, Institute of Engineering Research, Series No. 82, Issue No. 16, 1960.
12 R. C. MacCamy, ‘On the Heaving Motion of Cylinders of Shallow Draft,” JourNnaL or Suip ReE- SEARCH, vol. 5, no. 3, 1961, pp. 34-43.
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DIR, Scripps Inst of Oceanography, Univ of California
DIR, ORL Penn State
DIR, WHOI
0 in C, PGSCOL, Webb
DIR, lowa Inst of Hydrau Res DIR, St Anthony Falls Hydrau Lab
Head, NAME, MIT 1 Prof Abkowitz 1 Prof Kerwin
Inst of Mathematical Sciences, NYU, New York
Dept of Engin, Inst of Eng Res, Univ of California 1 Dr. J. Wehausen
Hydronautics, Inc., Pindell School Road, Laurel, Maryland
Pres, Oceanics, Inc., 114 E 40 St, New York, 16
Dr. Willard J. Pierson, Jr., Coll of Engin, NYU, New York
Dr. Finn Michelsen, Dept of Nav Arch, Univ of Michigan, Ann Arbor
Prof Richard MacCamy, Carnegie Tech, Pittsburgh 13
Dr. T.Y. Wu, Hydro Lab, CIT, Pasadena
Dr. Hartley Pond, 4 Constitution Road, Lexington 73, Massachusetts
Dr. J. Kotik, TRG, 2 Aerial Way, Syosset, New York
Prof Byrne Perry, Dept of Civil Eng, Stanford Univ, Palo Alto, California
Prof B.V. Korvin-Kroukovsky, East Randolph, Vermont
Prof L.N. Howard, Dept of Math, MIT, Cambridge 39, Massachusetts
INITIAL DISTRIBUTION (continued) Copies
1 Prof M. Landahl, Dept of Aero & Astro, MIT, Cambridge 39, Massachusetts
1 Mr. Richard Barakat, Itek, 700 Commonwealth Ave, Boston 15, Massachusetts
10
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