. On Deep-Sea Water Waves Caused by a Local Disturbance on or beneath the Surface. gt; CO cos * Jo(Aot)^, (79) f = — f Wff sin <r*. e™6* Jo (few) & (80) Or, performing the operation —37 upon (52) and (53), we obtain gp at *=%t!„-t2 efdr, (84) o in which, for simplicity, it is put v = (pb/A)^/(gb)U r = t^ifffib). (85) For large values of t, using (57), we can take for 77 its asymptotic expansionof the form 1 2t3 2V 2V 24t9 (86) From these expressions we can make out the general feature of the move-ment of the point under consideration. By the aid of I we can
. On Deep-Sea Water Waves Caused by a Local Disturbance on or beneath the Surface. gt; CO cos * Jo(Aot)^, (79) f = — f Wff sin <r*. e™6* Jo (few) & (80) Or, performing the operation —37 upon (52) and (53), we obtain gp at *=%t!„-t2 efdr, (84) o in which, for simplicity, it is put v = (pb/A)^/(gb)U r = t^ifffib). (85) For large values of t, using (57), we can take for 77 its asymptotic expansionof the form 1 2t3 2V 2V 24t9 (86) From these expressions we can make out the general feature of the move-ment of the point under consideration. By the aid of I we can draw the approximate diagram whichrepresents the displacement at the origin, the units of scales being modifiedby (85).. For t = 0, we have rj = 0 and drj/dr = 4. Therefore as soon as the impulseis applied on the free surface the point at the origin gains suddenly a finitevelocity and begins to move downwards. It will be seen, from the abovefigure, that at about r = 0*6 the depression becomes maximum, at aboutt-— 1*5 it takes the value zero and then begins the elevation of the surface,and it takes the maximum value at about t = 2, after that it decreasesslowly until it tends asymptotically to its limiting value zero, and the point comes to rest. If we assign another assumption to the initial impulse like that expressed Waves caused by a Local Disturbance. 73 in (59), it will probably take place by a similar mode of motion, but in asomewhat complicated form. III. § 7. Now we come to the problem of surface waves caused by an explosionwhich takes place at a finite depth from the free surface.* In this case, wemight take the value of the impulse, that of p(j>, given by the explosion asthe initial
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