In a uniform distribution of gas, the shocks resulting from an explosion decay as they propagate outward; the object of the present investigation is to discover what may occur in a star, for which very different possibilities are presented by the decrease of the equilibrium values of pressure, density and sound speed with increasing distance from the center.
First, the linearized theory is discussed, since although it is inadequate in certain respects, even for the weak disturbances considered here, it can be corrected to give a true description of the flow. A complete solution of the linear equations is not known in general, but a solution is found in the form of an expansion, which is valid in a certain region of the flow provided that the sound speed A(r) in the equilibrium configuration satisfies A'(r) = O ( A ( r ) / r ) (r is the distance from the center of the star). This solution is improved, following the method used in [4], to give the correct (non-linear) theory of the propagation of weak disturbances, by replacing the linearized characterized variable which appears in it by a better and sufficiently accurate approximation to the exact characteristic variable. The region of validity of the corrected solution includes the head shock; thus the shock may be determined. Near the surface of a finite polytropic star the condition on A(r) is not satisfied, but an approximate solution of the linearized equation is obtained by a separate investigation, and again the behaviour of the shock is studied.
It is always found that the jump of pressure, Ap, a t the shock tends to zero as A -+ 0, but that the particle velocity behind the shock and the shock strength (defined as the ratio of A p to the undisturbed pressure ahead of the shock) may increase or decrease depending upon the equilibrium configuration of the star.