579 



The use of the non-dimensional variables thus defined allows the 

 expression for the conservation of energy to be written in the form: 



i^(;2.i) .ka3(l-^) =1 



(9.6) 



in which the dot signifies differentiation with respect to non-dimensional 

 time. The parameter k is defined by the expression: 



(rQ)'*' (-r-l) (rQ)^ CV-l) 



(9.7) 



The following considerations give the theoretical relation between 

 the observed period and maximum radius and the value of rQ, under different 

 assumptions regarding the parameters k and 'Y in Sq. (9.6): At maximum or 

 minimum radius, a vanishes in Eq. (9.6) leading to an equation 



+ k a^^l- ^) = 1 (9.8) 



whose greatest and least roots should be the non-dimensional maximvim and 

 minimum radii; note that if k vanishes (internal energy negligible) the 

 maximiom radius becomes simply a^j = 1 or Ajj = L. The roots for various 

 values of TT and k appear in Table XX, where a^^^ is the non-dimensional 

 maximum radius and ajjj the non-dimensional minimum radius. 



The time for one oscillation is computed by obtaining twice the time 

 between minimum and maximum radii,* In the case where k is neglected. 

 Eq. (9.6) can be integrated in terms of the incomplete beta f\inction3) and 

 the numerical result is 1.492. On the other hand, when k is not neglected, 

 Shiffman and Friedman^l) have shown that the period can be approximated 

 very accurately by means of a quadratiire formula involving the Tchebycheff 

 polynomials. Such calculations were performed by Shiffman and Friedman 

 for a few values of V and k using fifth-order Tchebycheff polynomials; 

 those authors remark that the fifth-order approximation agrees within 1% 

 with the third-order approximation. In the present report Shiffman' s and 

 Triedman' s calculations have been extended to include a wider range of 

 values of the parameters 'J' and k for reasons described below. A sample 

 calculation was also performed comparing the fifth- and seventh-order 

 approximations which were found to agree within 0.025^. The results are 

 reported in Table XXI, and are also exhibited graphically in Fig. 2. 



* Examination of. experimental radius-time curves reveeils that they are 

 essentially symmetrical about the maximum radius. 



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