The Initial Wave Resistance of a Moving Surface Pressure. 248 
It should be noted that for a pressure system which leaves regular waves 
in its rear, we cannot take (26) as an approximation for the limiting value 
of (13) when «0, except under certain further limitations. For the 
present this difficulty does not arise, as we are considering a waveless system 
with @ («) of the form («—x») w(x); we have seen that in this case the 
integrals (12) and (13) remain finite and determinate with be zero. 
In particular, for the forms (16) and (18), the group formula (26) 
reduces to 
—T?9pR = hq gto Wt V2e—*/N cos (gt/4c+ 4), (27) 
which, from (22), agrees with the first term in the asymptotic expansion of 
the exact solution for this case. 
Instead of expressing R as a function of the time ¢, we can use the 
distance travelled, or again the number n of free wave-lengths X» through 
which the system has moved ; in the last case the circular function in (27) 
becomes cos }(2n+1) 7. The form of (27) agrees with the definition of the 
wave resistance as the resolved total pressure. For after a sufficient time, 
the surface in the neighbourhood of the moving origin consists chiefly of the 
simple waves whose group velocity is the velocity ¢ of the pressure system ; 
thus the wave-length there is 4X9. 
7. Consider now two numerical examples of the exact solution (21) with 
different values of the ratio 7/No. 
In the first place, we shall adopt units used by Kelvin, for comparison and 
for simplicity of calculation. 
Case i: 9=4, r=1; M=2; mem; C= BBr/r. 
From (18) the pressure system F (z) can be obtained by graphing 
4m cos */4 8 cos § 8—5 cos %* 6 cos 2 6, 
where 6=tan~1(a/r). The graph is shown in curve (1) of fig. 1; the 
curve has maxima nearz = + 0-2, though they are almost inappreciable on 
the diagram. 
It is convenient to graph the resistance curve upon a base &=ct/2r; in 
this particular case £ is also the number of wave-lengths A» through which 
the system has moved. The angles of the formula (21) are now 
6 = tan; b = 7&3/2(1+ &). 
It is unnecessary to repeat the expression (21) with these values ; each of 
the 14 terms can be easily calculated for any given value of &. The results 
are shown in curve (1) of fig. 2; toobtain the curve 15 points were calculated 
by the formula (21). 
The wave resistance decreases ultimately to zero, as it should for a waveless 
113 
