By J.M. Burgers

Since the 'Introduction' to the most textual content provides an account of how within which the issues taken care of within the following pages originated, this 'Preface' will be restricted to an acknowledgement of the aid the paintings has obtained. It begun throughout the pe riod whilst i used to be professor of aero- and hydrodynamics on the Technical collage in Delft, Netherlands, and lots of discussions with colleagues ha ve in:fluenced its devel opment. Oftheir names I point out the following basically that ofH. A. Kramers. Papers No. 1-13 ofthe record given on the finish ofthe textual content have been written in the course of that interval. numerous! ofthese have been makes an attempt to discover rules which later needed to be deserted, yet progressively a line of inspiration emerged which promised extra sure effects. This line started to come to the foreground in pa in line with No. three (1939}, whereas a initial formula ofthe effects used to be given in paper No. 12 (1954}. at the moment, in spite of the fact that, there nonetheless was once lacking a practica! strategy for manipulating a undeniable distribution functionality of significant curiosity. A six months remain on the Hydrodynamics Laboratories ofthe California Institute of expertise, Pasadena, California (1950-1951}, was once supported via a freelance with the dep. of the Air F orce, N o. AF 33(038}-17207. A process lectures was once given in this interval, that have been released in typescript below the name 'On Turbulent Fluid Motion', as file No. E-34. 1, July 1951, of the Hydrodynamics Laboratory.

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_ 2jnv(t-O) 2v 2 t-e a(O') de'}]. 2) This condition can be satisfied only if a is positive. We again make use of the geometrica! 3) to * A solution ofthis nature has been discussed in paper No. 10, pp. 48-51. 1 ). 4) S(e)=-(-)+C, 2t-e both curves now referring to a e-scale. For fixed x, t the S-curve is a hyperbola, which for small x approaches to a rectangular corner, formed by its two asymptotes. From Figure 11 it will be seen that no contact point can be found in a 'valley' of the s-curve, and the solution of the contact problem will not change if an s-curve with 'valleys' hyperbola for small x o ~ e-a~s Fig.

14, pp. 128-134 (1964). 3) is taken from a more complicated case treated by S. Go1dstein, 'On Diffusion by Discontinuous Movements, and on the Te1egraph Equation', Quart. J. Mech. Appl. Math. 4, 129-156 (1951). 8) -oo and the average added totallength will be: L 00 + 00 O -oo Jd~ Jdn p(~, n) ~. 9) Evidently this quantity must be equal to L. 10} -00 This is a normalization condition for the function p(~, n), which will be used later on. 11. Transformation of the Statistica) Problem The statistica!

As bas been indicated before, with large values of t, the contact points are very close to the actual maxima and in the following we neglect the small remaining distances. Hence we assume that xk + x~ and xk + x~ are determined by the maxima themselves. The series of parabolic arcs then presents the aspect of a festoon without gaps; each arc is suppoşed to follow immediately upon its predecessor. 2). 1) (using ~k here in a new notation); and (2) the difference in height of its endpoints 11k = [(x~)l- (x~) 2 ]/2t.