2 THE EFFECT OF SPEED OF ADVANCE UPON THE DAMPING OF HEAVE AND PITCH 
form to a long plank, of length L and beam B, submerged to a 
draught d, moving forward with velocity c and making small 
pitching oscillations with angular velocity Q sin p t. 
The theoretical work is given in the Appendix. It begins with 
a different derivation of the fluid motion due to a moving 
pulsating source. Then by integration of an assumed source 
distribution we obtain the velocity potential for the plank. The 
fluid pressure is obtained for any point of the base and hence the 
moment of this pressure. Dealing only with the moment due to 
the pitching motion, the periodic part will be of the form 
M, sinpt-+M,cospt. The second term is in phase with the 
angular acceleration and can be considered as giving a ‘Virtual 
addition to the moment of inertia. The first term is in phase 
with the angular velocity and gives the corresponding damping 
coefficient; this is the only term which is examined here, and 
expressions for M, are given in equations (13), (14), and (15). 
For numerical computation we have taken L/B = 20 and 
d/B = 2. These ratios do not allow any direct comparison with 
the usual models; they were chosen partly to lessen computation 
and partly so as to bring out certain points. Fig. 1 shows curves 
lO'nM, 
P giBALn 
O2 O3 O04 OSs 06 O7 O8 
Fic. 1.—DAMPING MOMENT FOR PITCHING ON A BASE OF FREQUENCY 
FOR FROUDE NUMBERS 0, 0:07, 0:14, 0:18 FOR SPEED OF ADVANCE 
for the variation of 10?>7M,/pg*B*L*Q with the usual 
parameter p (B/g)+ for certain values of the Froude number F. 
The curves do not need any detailed discussion, but one or 
two remarks may be made. The curve F = 0 is for zero speed 
of advance and is of the usual type. It may be noted that the 
integrals in equation (13) include an oscillating factor and we 
might expect humps and hollows on the curve; but they occur at 
higher values of the frequency where the value of the moment is 
small. These possible oscillations are not interference effects 
connected with the beam, such as have given rise to discussion 
in two-dimensional problems: the latter have been ruled out of 
the present calculation by the assumption of small enough beam. 
The interference effects here are in length, between bow and 
stern; no doubt the rectangular form of the base would tend to 
exaggerate any such effects. 
Comparing the curve for zero speed with the other curves, a 
general effect is like moving the curve towards the lower fre- 
quencies with increasing speed, and we can see the interference 
effects coming into evidence. The other main point is the 
infinity at the critical value with a steep fall after this point 
followed by a small gradual rise. The critical point for F = 0-07 
is at p (B/g)* = 0:8; it is not shown in the diagram as the infinity 
is highly localized and computation would be tedious. The 
critical points for Froude numbers 0-14 and 0-18 are at 0-4 and 
0-31 respectively. It should be stated that there are certain 
speeds for which M, does not become infinite at the critical 
point, though there are still peak values; these speeds are such 
that the Bessel Function in the integrals (13) has zero value for 
6 = 0 when p e/g = 
Naturally, in any experimental results the mathematical 
infinities would be smoothed down as in other resonance effects ; 
and also they are likely to be highly localized and sensitive to 
small disturbances. Nevertheless they have their effect upon the 
rest of the curve; and with suitably devised experiments one 
might expect peaks on the damping curves in the region of the 
critical value of the parameter. 
References 
(1) Haskinp, M. D.: Priklad. Mat. i Mekh., 10, p. 33 (1946). 
(2) BRARD, R.: Assoc. Tech. Mar. Aero., 47, p. 455 (1949). 
(3) Hanaoka, T.: Journ. Zosen Kyokai, 93, p. 1 (1953). 
(4) GoLovaTo, P.: T.M.B. Report 1074 (Washington) (1957). 
(5) HAvetock, T. H.: Quart. Jour. Mech. App. Math., 11, p. 332 
(1949). 
Appendix 
A point source of strength m sin p ¢ is moving with velocity c 
at a depth d below the free surface of the water. We take moving 
axes with the origin O in the free surface immediately above the 
travelling source, O x in the direction of motion, O y transversely, 
and Oz vertically upwards. We suppose the motion to have 
been started from rest and the solution we require is that to which 
the motion approximates at a sufficiently long time after the 
start. The result can be obtained by integrating the effect of 
infinitesimal steps in the motion from the start up to the time f. 
Suppose the motion started at a time u before the present instant, 
that is at a time t — uw; then, using a general result for a variable 
source,©) we have for the velocity potential 
1 WN 
p=m(———)sinpt + $1 (1) 
where 
4megt ". 
= lim ——..] sinp(t—u)du 
by = im —t I p(t—w) 
7/2 
j a0 cos [« (x +c u) cos 6] cos (« y sin ) 
x sin(ugt K)Kte-K4-Ddxk (2) 
with r?=x?+y?+(z+d)?; R=x?+y+(2—d)? 
Carrying out the integration with respect to u we obtain 
1/2. 
fs in Gof 
u>a 7 
+ e—k(4—2) cos (K y sin 8) 
cos [k ccos 0 —p — (gk)4]u 
x (oe +po{ 
«ccos@ — p —(gx)t 
_ cos[kecos#@—p+(gk)t]u _ 2(g Kt 
«ccos@ —p+(gk)t (« ccos 0 — p)? —gK 
sin [k ccos 8 — p —(gk)*]u 
kccos@ — p —(g kt 
+ cos (« x cos 0 +po{ 
__ sin [k ccos 8 —p + (g cel) 
x ccos 6 — p + (g k)t 
— two similar terms with — p trp | Ke (3) 
Considering the integration with respect to « and the limiting 
value as u —> 00, we require the positive values of « for which 
the various denominators in equation (3) are zero, and the corre- 
618 
