Bat Ne 4 Ag. 1999 Havelock, Note on the sinkage of a ship at low speeds 205 
If the length, beam and draft of the ship are L, B, D respectively, then L—2a, 
B=2b, D=c; for B>,<,=2Dwe use the expressions (12), (14) and (16) respectively. The 
numerical values shown in Table I have been calculated from these formulae. 
Table I. Values of gh/U’. 
B/D L|D=10 | LL|D = 16 
1 0.0253 0.0138 
2 0453 0231 
3 0612 .0318 
4 0735 0397 
6. The measured sinkage of ship models at low speeds has been analysed by Horn’), 
who has given an empirical formula derived as an average from available data for many 
different forms of model. His expression for the sinkage is, in the present notation 
h =55p (0.85 + P) {x (1125 — 3) +25} (18-55) apieyl pee Gay comet: LS): 
where L, B, D are length, beam and draft respectively, and P is the prismatic coefficient of 
the form; the formula is valid, as an average, for suitable ranges of these parameters. 
It should be noted that this formula is for actual measured sinkage, and is probably 
derived from velocities rather higher than those for which the preceding simple calculation 
is valid; moreover, the ellipsoid is not one of the ship forms included in the data. However 
we may use it to test the order of magnitude of the results. If we apply (18) to an 
ellipsoidal form with L/B=8 and B/D=2, we obtain h=0.0283 U*/g; this compares with 
the value 0.0231 U/g for this case given in Table I. 
Horn’) has suggested using the sinkage at low speeds to estimate the increased 
frictional drag for a model compared with a flat plate; his formula for the percentage increase 
in the resistance R is 
dL OOVARE/IE == 200 GTA TIM Perea Wes (lies | Aarne IRAs (9): 
For the prolate spheroid with L/B=8, the value of h in Table I gives, according to the 
formula (19), an increase of 4.6 per cent in the resistance. 
Amtsberg’) has recently determined the resistance of a submerged prolate spheroid 
experimentally; he gives two values for the increase, namely 5.2 per cent and 3.7 per cent, 
the smaller value being obtained after applying certain corrections. Amtsberg also 
investigated certain other surfaces of revolution, for which the velocity potential is given by 
a source distribution along the axis. He gives numerical values of the ordinates of the 
surface and of the theoretical distribution of velocity along the contour; from these, it is 
possible to evaluate numerically the integral we have denoted by Q in the preceding sections. 
Taking, for example, the values given by Amtsberg for his model #1257, we obtain 
approximately @ = 0.0284 o U? (area of section). This gives an equivalent sinkage of 0.0284 U?/g 
and, according to (19), an increase of resistance of about 5.7 per cent; the values deduced by 
Amtsberg from his experimental results are 7.3 and 4.9 per cent, the latter being the 
corrected value. 
It is well-known that in models of this type the measured distribution of pressure over 
the surface only differs appreciably from the theoretical value near the rear end of the model. 
Hence the effect of this divergence upon the resolved vertical pressure will only be a small 
correction; taking, for example, model #1257 and using Amtsberg’s measured values of 
the pressure instead of the theoretical values, a rough approximation gives a factor of 0.0288 
instead of 0.0284. 
7. Summary. The sinkage of a model at sufficiently low speeds is assumed to be due 
to stream line fluid motion round the submerged part of the model, neglecting the disturbance 
of the water surface. Taking an ellipsoidal form for the submerged part, exact expressions 
are found for the total defect of vertical pressure and hence for a certain equivalent sinkage. 
The results are compared numerically with available data and are found to be of the right 
order of magnitude. Further, reference is made to Horn’s approximate formula connecting 
the sinkage with the increase of resistance of the model compared with that of a flat plate. 
949 
2) Amtsberg, Jahrb. der Schiffsbautechn. Gesellsch., Bd. 38, 1937, S. 177. 
461 
