- 13 



157 



When P/8 is much greater than 10', the compressibility of the nater will be important during 

 the (short) time thit a is near to 1, and will give rise to a radiation of acoustic energy and a 

 consequent decay of the pulsations. This effect is Ignored in the present calculations. 

 Nevertheless the result may be expected to indicate what happens during the greater part of a 

 pulsation and particularly when a > 2, for when a has reached Z, the pressur will have fallen to 

 a value of orJer lO^Q or less. 



The method adopted In the calculations was to determine first two fundamental solutions of 

 equation (k?) f or /£L , as a function of a, using the expansions (50) and (5U). These solutions, 

 denoted respect ively'by g, and/la- l) j , were chosen to satisfy the conditions, 



dg, 

 9, = 1, — i = at a = 1 (67) 



d a 



/( 



a- 1) j J = 0, 



''j, 



(68) 



and thus correspond to the expansions given in (SOl when a, = 1, and * « i, i . = i. Their 

 values were determined from (50) for the range a = 1 to a = J. 5. These solutions were then 

 extended by analytic continuation over the range a = 1.5 to a = 10 by means of Taylor expansions 

 of the form (51) relative to the points k = 2 and 6. For the remainder of the range, a = lO to 

 a = a . the solutions were Continued by means of the expansions (50) with z-a-a. The various 



■ J ^ . ■ . m 



expansions used have certain common regions, where their convergence is sufficiently rapid for 



calculation, so that a number of checks on the results could be made. The values of g and 

 v(a- i) j found in this way are given in Table I. 



Apart from the variable a, the functions 9 and j depend on two parameters, CT (=^'P) and n. As 

 far as cr is concerned, the values of g^ and j^ given In the table are unaffected up to a = 2 by taking 

 any value of a not greater than 10"*, :ind they are not affected by more than about ijS up to a = 6. 

 increasing cr to 10"^ does not change the values up to a = 3 by more than about l%. Consequently 

 the e^rly part of T^ble I will give a fair approximation to the solution of equation (49) for 

 values cf a from 1 to 3 (and possibly as far as 6), for any value of PfQ likely to arise in connection 

 with explosions. On the jther hand, altering n has a great effect on the values of 3 and j , as 

 is oovious from the aparoximate expressions (58) for /3 near a = 1. ' ^ 



The determination of ,3 during a contracting phase, following the known variation during 

 the preceding expanding phase, is detemilned, as shown above, by expressing /5 in terms of the two 

 fundamental solutions 9^ and /(a^ -a) j^. Hence the values of these solutions from a = 9 to 

 a = a^^ were alsc calculated and are shown in Table I. 



The relation between the two pairs of fundamental solutions in Table I is found to be 



