Abstract.The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.Introduction and survey of literature. The quasilinear parabolic equation now known as the "one-dimensional Burgers equation,"first appeared in a paper by Bateman [4], who derived two of the essentially steady solutions (1.3 and 1.5 of our Table). It is a special case of some mathematical models of turbulence introduced about thirty years ago by J. M. Burgers [10], [11]. The distinctive feature of (1) is that it is the simplest mathematical formulation of the competition between convection and diffusion. It thus offers a relatively convenient means of studying not only turbulence but also the distortion caused by laminar transport of momentum in an otherwise symmetric disturbance and the decay of dissipation layers formed thereby. Moreover, the transformation u = -(2v/6)(dd/dx)relates u{x, t) and d(x, t) so that if 6 is a solution of the linear diffusion equationthen u is a solution of the quasilinear Burgers equation (1). Conversely, if u is a solution of (1) then 9 from (2) is a solution of (3), apart from an arbitrary time-dependent multiplicative factor which is irrelevant in (2).In connection with the Burgers equation, transformation (2) appears first in a technical report by Lagerstrom, Cole, and Trilling [38, especially Appendix B], and was published by Cole [21]. At about the same time it was discovered independently by Hopf [30] and also-in the context of the similarity solution u = t~l/2S(z), z = (4vt)~1/2x-by Burgers [14, p. 250]. The similarity form of the Burgers equation-the quasilinear ordinary differential equation for S(z)-is a Riccati equation [51], and can thus be regarded as a basis for motivating transformation (2) inasmuch as (2) is a standard means of linearizing the Riccati equation. More general hydrodynamical applications of this transformation have been discussed by Ames [1, chapter 2], Chu [20], and Shvets and Meleshko [55].

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Dissipation of tidal energy is expressed here as an integral on the surface of a sphere that encloses the mass of the planet. When developed in constituent form, this surface integral depends linearly on the secondary potential that arises from the tidal disturbance; it can therefore be expressed as the sum of one part due to the body tide and another due to the fluid tides. The body tide part depends only on the anelastic response of the solid earth to the primary potential. The fluid tide part depends mainly on the elastic solid earth response to tidal loading and on the sum of the ocean tide and atmospheric tide mass disturbances. The atmosphere's contribution can be evaluated reliably from published analyses of the observed barometric tide. Tide gage data, on the other hand, do not suffice for a comparably reliable analysis of the observed ocean tide and must therefore be supplemented by dynamical interpolation through numerical integration of Laplace's tidal equations. Dissipation obtained from ocean tide models published during the past 5 years has a range of at least ±15% from the average value. The main cause of this uncertainty is the difficulty of modeling dissipation, but there are other uncertain aspects of existing models, notably the lack of provision for absorption of energy into baroclinic motions and for the anelastic response of the solid earth to tidal loading.

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