The pressure of water waves upon a fixed obstacle 419 
where the asterisk denotes the conjugate complex. Putting in the value 
of b,, and using properties of the Bessel functions, these expressions may be 
reduced to a simple form; we obtain finally 
4 27 2 2 
= oe eee) 
7°Ka3 0 Ka? j (War ar 14) (Jnat Yen) 
the argument of the Bessel functions being xa. 
The series in (39) occurs also in an expression given by Nicholson (1912) 
in a similar problem for electromagnetic waves. Some values had been 
calculated before this reference was known, and with Nicholson’s values 
for the series we have the results in table 3. 
TABLE 3 
Ka 0-5 1:0 2:0 3:0 5:0 
R/gpha 0-429 0-998 0-940 0-950 0-965 
It was shown by Nicholson that when xa is large, the series approximates 
to the value §7?x%a’; hence when the wave-length is small compared with 
the diameter of the cylinder, we have approximately 
P = 29ph?a. (40) 
This agrees with the limiting value if we assume total reflexion over the front 
half of the cylinder and a complete shadow over the rear half, and apply 
to each element the expression (32) for total oblique reflexion from a plane; 
for we then have 
t 
P=| igph?acos6d0 = 2gph2a. (41) 
=i7 
Although this limiting valué is obtained theoretically as an extreme case 
for short waves, it is interesting to note from the preceding table that it is 
practically attained for comparatively long waves of wave-length even 
larger than the diameter. This consideration suggests using the method to 
give an upper limit for cylinders whose section is more like that of a ship. 
8. Consider a cylinder with vertical sides, the horizontal section being of 
ship form and symmetrical about Ox. We assume total reflexion by the 
sides of the ship from the bow back to where the sides become parallel to 
Ox, and we assume a complete shadow aft of that point. 
For the model of § 3, in which the bow is a wedge of semi-angle «, we obtain 
for the total resistance 
R = tgph?B sin* a, (42) 
where B = 2b = beam. 
480 
