349 



If we use L and C defined in (1.1) and (1.2) as 

 units, equation (3.5) can be written in terms of non- 

 dimensional quantities as follows: 



a^[a^(l + aLpj_) - 2kha.^L^(p2 + ^ b^(l + a.V<p^)] 



^'•'^ + a^zz;l . ka-^(^-^) = 1, 



where ka" ^ ' is the internal energy of the gas ex- 

 pressed in non-dimensional quantities, k is defined in 

 equation (1.5) . 



It was shown in Report 37. IR that the period can be 

 found by assuming b is constant so that b = 0. Equation 



(3.6) reduces to 



(3.7) a^a^d + aL^) + a^ + ka"^^*^"^^ = 1. 



If we approximate (1 + aL^, ) ' by 1 + aLp-,/2, equation 



(3.7) can be solved for a and we get 



a^/^(l + aL9?T/2)da 



(3.8) . = dt. 



Let 



(3.9) 



^ ^ r^l a2/2 da 



1^1 - a^ - ka-^(*^-l)- 

 o 



n^l a^/2 da 



(3.10) I2 = / , t V =7T ' 



o 

 where a and a, are respectively the smallest and largest 

 zeros of the denominator. We find then that the time 

 from beginning to maximum is 



(3.11) h " ^1 ■*" ^ V^l' 



