412 BELL SYSTEM TECILMCAL JOIRSAL 



4. Solution- In-cludixc. Rkcombi.vatiox hut Neglectixg Diffusion 



It is plausible to expect by analogy with Fig. 2 that (24) can be solved, 

 neglecting the last term, by a similar construction in which the curve of /?/, 

 against .v at time / is derived from that at time l)y moving each point to 

 the right along a descending curve, instead of along a horizontal line as be- 

 fore. To show that this is indeed the case, and at the same time to show 

 that the diffusion term cannot so easily be taken into account, let (24) be 

 written, omitting its last term, as 



*: = -*w % - KM 



where * is just the translation into dimensionless variables of the velocity V 

 encountered in (4). This can ])e converted into a differential equation for ^ 

 l)y writing 



and multiplving through by ( — J 



\dv/ 



(dA\ 



A = R(^)(~) +*W (27) 



[dp 



ds dw 



a U + [ * </w j dU -\- j ^ chi'j 



ds dw 



whence the general solution is 



t ^ - j c[uhc -[- jXs + w) . (28) 



where/ is an arbitrary function. If the same transformation is tried on (24) 

 with the diffusion term retained, the equation corresponding to (27) has an 

 additional term on the right containing a quotient of second and first deriva- 

 tives of ^ with respect to v, and the simple explicit solution fails. 



To apply (28) to ex])licit calculation, or even to visualize it physically, it 

 is necessarv to determine the i)roi)er form of the arbitrary function/ to fit 



