The paper discusses th e decay of a one-dim ensional periodic acoustic signal of m oderate am plitude after it has developed a saw tooth profile containing one th in shock in each period. A composite asym ptotic ex p an sion is constructed by inserting Taylor (1910) shock transitions a t regular intervals w ithin a piecewise linear profile. This representation, which form ally is valid to first order a t distances where shocks are thin, is shown to be an exact solution of B urgers equation a t

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... ll distances. I t is, in fact, th e profile due to F ay (1931) and is usually represented as a Fourier series. The F ay solution is th ereb y shown to have a simple in terp retatio n in term s of a periodic array of spreading shocks th a t appear n ot to in teract as th ey in terp en etrate. I t is also confirmed th a t this new representation corres ponds, under th e H opf-C ole transform ation, to the solution of the heat conduction equation which describes th e spreading of a periodic array of p o in t h eat sources. In an appendix, two identities involving double sums of hyperbolic functions are derived. The Burgers equation I ntroduction uT+ruus = 8uss (i.i) has been used by m an y au th o rs (Lighthill 1956 , Soluyan & K hokhlov 1961 , B lackstock 1964 to describe th e propagation of one-dim ensional acoustic signals of m oderate am plitude. H ere, -U --U ( Y) denotes the density, th e L agrangian coordinate Y m easures distance from th e driving piston, and, if t denotes tim e and c0 th e linearized sound speed, th en = . T hus S/c0 denotes th e tim e elapsed since th e passage of a reference w avelet while th e con sta n ts r and 8 q u an tify th e effects of am plitude dispersion and of diffusion re spectively, and th e subscripts S, Y denote p artial differentiation. I t is, of course, w idely know n th a t th e tran sfo rm atio n due to H opf (1950) and Cole (1951) reduces (1.1) to th e h eat conduction equation. Using this transform ation, Lighthill (1956) illum inates the com petition betw een nonlinearity and diffusion th ro u g h m any exam ples of shock form ation, interaction, spreading an d decay. U n fo rtu n ately , in nonlinear acoustics th e B urgers equation has only lim ited applif This paper is dedicated to the mem ory of A cadem ician R. V. K hokhlov. [ 409 ]