5 - 



211 



The aisplacement o' the sphere is then, from equation (16), given Dy, 

 s _ B k X (2k - 2 - q) ■> k - 1 - kT 



6 (k-l) Sjj 2 k m X 



e Sin m T 



* I (e- "^ Cos m T - e" '^'^) 



» q "- 1- q (i . 3- kT ^^ ^ Tj 

 2kqX 



+ _L_ (") 



2 k X 



s = 2 = -US (28) 



qc /3 qc 



The unbalanced impulse A, defined by 



rt 



p at (29) 







can be simply obtained from equation (25) by virtue of the relation 



where 



Ar = - 



cq 



(30) 



(31) 



The total impulse A. directly incident on the sphere can be obtained from equations (2), 

 (lO) and (29) which give, 



X^ + (x^+ 2)/ x^- 1 I (32) 



-1 = -i O - X 

 Aj 3X 



For a plane Incident wave (X-*co) we note that A = A., while p , v and s are the 



1 I 



(iBximum pressure, velocity and displacement in the Incident wave. 



The preceding equations enable the motion of the sphere to be calculated for any special 

 case. For the purpose of obtaining qualitative conclusions the solutions for the limiting cases 

 of incident pulses long and snort relative to the diameter of the sphere wl)I now be given. 



(l) Short pulses . 



For short pulses, i.e. pulses in which the pressure becomes negligible in a time small 

 compared with the time taken by the pulse to travel past the sphere, the value of q is large and 

 retaining only the predominant items the preceding equations become, 



(2) Long pulses . 



For long pulses where q Is small we shall give the simplified asymptotic forms on the 

 additional assumption that x is large since, as discussed later, the present analysis Is not of 

 practical apfJication to underwater explosions when qX is smaller than order unity. On these 



assumptions ..... 



