WAVE PATTERNS AND WAVE RESISTANCE. 3 
and this will represent a free wave pattern of some form travelling steadily parallel to Ox 
with velocity c. 
We may again obtain a rough picture of the result by simple graphical methods. 
Suppose we represent a component plane wave of (5) by parallel straight lines showing the 
crests and troughs at, say, the instant t= 0, in the manner shown in Fig. 1, the full lines 
representing crests and the broken lines troughs. 
Now draw similar lines on the same diagram for a large number of values of 6 in the 
range from —90° to + 90°. It is instructive to take, for instance, intervals of 10° and to 
Fig. 1. 
draw 19 sets of lines as in Fig. 1. Such a diagram is not given here, as there is too much 
detail for reproduction on a small scale; but it is interesting to see the picture of a familiar 
wave pattern emerging from such a diagram. The curves which we see in process of 
formation are shown in Fig. 2. 
These curves are, of course, the envelopes of the lines of constant phase of the com- 
ponent waves, and their mathematical equations are most easily obtained by expressing 
Fig. 2. 
that fact. When we look into the formation of the curves we see that they represent places 
where component crests, or troughs as the case may be, combine together to give prominent 
features of the pattern; on the other hand, we may say that at points at some distance 
outside the region covered by these curves the component crests and troughs tend to cancel 
each other out on the average. We arrive in this way at the picture of a wave pattern of 
transverse and diverging waves, with a focus point O, and extending in advance of this point 
as well as to the rear; the whole forms a freely moving pattern travelling forward with 
steady velocity. It need hardly be said that this description of the pattern represented 
379 
