10 



Evaluating to the first two terms only 



Q (t)-Q (16) = 



V^4 



VF 



■2]fW- 



dt 



-H"(f-i)«< 



[8] 



= log t + — 5- - log 16 



s 2t 2 512 



Q (u) was evaluated for 16 < t < 30 using Equation [8]. A tabulation of these values is given 



in Table 1. 



The values of Q,(u) = / Q n (t) dt were obtained from these by integration; see Table 



1 Q u 



2. 



The P-functions are not defined for negative values of t. P(-\t\) = was used. Thus, 

 it can be seen from Equation [6] for Z w that for < <f < 1 - p only the terms P Q ( y f ) and 

 P -1 (y£) are defined. For 1 - p < f <1 + p the term P Q _1 {y( f - 1 - p) } is defined. For 

 1 + p<£<2 the term P _1 {y(£ - 1 + p)\ is also defined and for £>2 all are defined. It 

 would appear then that the first two terms mentioned would give the ordinates of the bow wave 

 system, the next the ordinates for the waves proceeding from the foreshoulder, the fourth the 

 ordinates of the waves proceeding from the aftershoulder, and the last two the ordinates of 

 the stern wave system. 



In the example given in the present paper (Figure 6), these four waves are plotted 

 separately to show the components of the total wave system contributed by each of the terms 

 considered above. 



The ^-functions appearing in the equation for the symmetrical disturbance are defined 

 for negative values of t as: 



Q (-O = Q (t) and Q i (-t)= -QAt) 



Thus all the terms of Equation [7] are defined for all values of £, . 



The evaluation of Equations [6] and [7] give the wave profiles in terms of dimension- 

 less coefficients containing Z w and Z,. The multiplication of these values by the constants 



tz v- and - — -,- r respectively, transforms them into the dimensions of 6. In 



Try(l-p) Try(i-p) ^ J ' 



Figure 7 the calculated and observed profiles are plotted to an inch scale. In Figure 6, how- 

 ever, the components are plotted in dimensionless form. 



The comparison was made as outlined above for five different speeds. At the lower 

 speeds (4.5 and 5 knots) the agreement between the observed and calculated profiles is good; 

 for higher speeds the agreement of the crests is good but the troughs of the calculated pro- 

 files are deeper than those observed (Figure 7). This is at least partly accounted for by the 

 assumption of infinite draft made throughout the calculation. The influence of a finite draft 

 will be more marked at higher speeds. This will be evident from a consideration of the 



