Van Wijngaarden 



— + ^— V p = p-p (3.5) 



In the linear case Ap p^ is a small quantity, c say. The pressure Pg is of order 



Pq(1+ 6), 9pg/Bpg is of order 1, and ^g'^y is of order e. 



We now assume that the operator working on p in the left side of (3.5) may 

 be approximated by 3^/By^. Then we may write instead of (3.5) 



—. (P-Pg) - (P-Pg) = 0- (3.6) 



Solving this equation yields using the boundary conditions (2.10), 



cosh y 



P-Pg= (^P-Po-Pg)^^;^. (3.7) 



Substituting (3.7) in the integrand in (3.3), we obtain 



cosh y 



Ap 1 



cosh A 



(3.8) 



Inserting (3.7) and (3.8) in (3.5) shows that reducing the differential operator in 

 (3.4) to 3^ 'dy^ is a reasonable approximation for small values of t and inde- 

 pendently for small values of y . 



If we take t << i, it follows that at y = for small t 



Pg-Po ;^ Apt^e-^ (3.9) 



where from the hyperbolic function in (3.8) only the first term in the expansion 

 in terms of k has been taken into account. This expression is identical with 

 (2.31), confirming what is said above about the validity of the used approxima- 

 tion for small t. 



For other values of t, (3.8) yields at y = (cosh X."' being replaced by 2e"'^) 



p„ - p„ ^ Ap n - cos 



■1 -'0 



1, 2^^-X 2 



(3.10) 



This expression shows just like the results of the foregoing section a slow vari- 

 ation of the gas pressure at y= 0. Comparison of (3.10) with (2.34) shows that 

 due to the approximation used in this section the variation of the gas pressure 

 as given by (3.10) is considerably too slow. 



Qualitatively, however, the results of this section agree with those of sec- 

 tion 2. Equation (3.7) indicates that for > ^^ 1, the difference between p andp 

 is small at y o, whereas Eq. (3.10) expresses that as time proceeds Pg in- 

 creases at y = slowly (compared with o-^^) from p^ at t = to the value 



Po + 2Ap. 



124 



