1060 Proceedings of Royal Society of Edinburgh. [suss. 
Deep Sea Ship-Waves. {Continued from Proc. R.S.E., 
January 23, 1905.) By Lord Kelvin. 
(Read July 17, 1905.) 
§ 65. Referring to § 63, we must, for the present, as time 
presses, leave detailed interpretation of the curves of fig. 17 : 
merely remarking that, according to § 44, if 3 = 0, (which means 
that J is an integer), the disturbance, d, is infinitely great; of 
which the dynamical meaning is clear in (70) of § 39. 
§ 66. Let us now find the depression of the water at distance x 
from the origin, when the disturbance is due to a single forcive, 
expressed by the formula * 
< 95 >’ 
travelling uniformly with any velocity v. If this forcive were 
applied steadily to the surface of water at rest it would produce a 
steady depression — Tl(x), as we are taking the density of the 
water, unity. Thus the forcive II (x) would shape the water to an 
infinitely long trough, of cross-section shown in fig. 25, representing 
z = Jcb 2 /(x 2 + b 2 ) on the scale of k = 10 cm. and 5 = 1 cm. 
1 f x 
Taking — / dx of (95) we find tan _1 (a;/&).5&. Hence the area 
9 J o 
of fig. 25 is 2 tan -1 S.bJc, or n rbk, and the total area of the 
diagram extended to infinity on each side is 7 rbk. Hence the area 
of fig. 25 is , or '92, of the total area. This total area, 7 rbk, I 
180 
call, for brevity, the forcive area ; and 7 rb, I call the mean breadth 
of the forcive area. The breadth of the forcive where z = *8 k (as 
shown by the dotted line B B in the diagram) is b. 
§ 67. Now let the forcive be suddenly set in motion, and kept 
moving uniformly with any velocity v in the rightward direction 
of our diagrams. This will produce a great commotion, settling 
* What is denoted by x in this and following expressions, is the (x - vt ) of 
§§ 36 . . . . 40 ; the origin of co-ordinates being now fixed relatively to the 
travelling forcive. 
