300 T. H. Havelock 
terms would be of a smaller order of magnitude, and of a similar order to 
those which have already been neglected in obtaining an expression for ¢, . 
on theassumption that the angle between the tangent plane and theza-plane 
is always small. The pressure equation is now 
Op: OP 
pip = const. —gz—c~"—¢ mat (3) 
and the wave resistance is given by 
le = -2| [p stdeds, (4) 
taken over the longitudinal section of the ship. 
The term in ¢, in (8) gives from (4) the expression for the wave resistance 
for the ship advancing into still water; we shall denote this by R,. We give, 
for reference later, the expression for R, in terms of the equivalent surface 
distribution o,, namely, 
ta 
R= Lén§p| (P? + Q?) sec? 0d8, 
0 
where IP, +70, =| Io eko28ec? O+ixor sec 8 dy dz, (5) 
The term in ¢’ in (3) gives from (4) the additional resistance R’ due to the 
standing waves; we have 
rt Op’ oy 
f =? — — i 
R pe Aa a ae de 
=— 2aph| | = exo? sin(K yx — f) da dz. (6) 
3—Consider a simple form of model, of uniform draft d and length 21, 
whose surface for y > 0 is given by 
y = b(L —22/d2) (1— 22/2). (7) 
From (4) we obtain, after carrying out the integrations, 
, _ 8gpbhl { 2 ( 1 realise . 
ly = PAE ie R ae) | (sin gl — Kol e08 ey) cos (8) 
The factor cos # in (8) shows how R’ varies with the position of the ship 
among the waves; for # = 0 or # = 7, the surface elevation is anti-sym- 
metrical with respect to the mid-section of the ship. Further, the factor 
(sin Kol — Kol cos Kgl) /(Kol)3 (9) 
