. On Deep-Sea Water Waves Caused by a Local Disturbance on or beneath the Surface. (2» + l)lUrThis series can be put in the form 5> = - -^ «-** fW-r, (98) in which r is defined by (93). o * Since this work was finished my attention has been called to a paper by Prof. Lambin the l Annali di Matematica, vol. 21, p. 237 (1913), where the surface waves thataccompany a cylinder travelling uniformly transverse to its length under water areinvestigated. t H. Weber, Die part. ,, vol. 1, p. 43. Gr A 76 Mr. K. lerazawa. On Deep-Sea Water For large values of t, if we use the asymptotic exp
. On Deep-Sea Water Waves Caused by a Local Disturbance on or beneath the Surface. (2» + l)lUrThis series can be put in the form 5> = - -^ «-** fW-r, (98) in which r is defined by (93). o * Since this work was finished my attention has been called to a paper by Prof. Lambin the l Annali di Matematica, vol. 21, p. 237 (1913), where the surface waves thataccompany a cylinder travelling uniformly transverse to its length under water areinvestigated. t H. Weber, Die part. ,, vol. 1, p. 43. Gr A 76 Mr. K. lerazawa. On Deep-Sea Water For large values of t, if we use the asymptotic expansion (57), then ^-!Hi+i(f)+i-3(iy+i-3-6 @),+- Making use of Table I we can trace the curve for £ In this figure the units of length and time are modified by fi> = 4Ai7o/vW> ^a/4* = **- From this figure it will be seen that the line which was initially situatedalong the axis of y is suddenly put in motion and there begins an elevation ofthe free surface, and that after a certain time (about t = 0*9) there is the oneonly maximum of the elevation, and then it decreases more and more untilthe surface takes the original plane form. If we take r = 0*9 for the value of r which gives the maximum of theelevation, then the time at which it occurs is given by t = 045^(1^), in which t is reckoned from the time of explosion in seconds and h is measured by feet. Lastly, at a point very far from the source of disturbance, if we neglect hcompared with x, we may put q = x and ^ = tt/2 in (97). Then we have 2Atf 1 (gfi\ 1 /gg\» - 1 _/ggY„ ^« x + ll\2a? (99) * This formula and those that follow are applicable only when gt2/2a; is moderatelysmall, this being the
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