Mean Values Connected with the Sawtooth Curve of Figure 5

Abstract

For large values of t the solution of the nonlinear Equation (1.1), with v→0, asymptotically approaches to a ‘sawtooth’ curve, consisting of upward sloping segments, having a slope equal to 1/t, separated by steep vertical descents or ‘shocks’, as indicated in Figure 5 and in more detail in the lower part of Figure 13 . The location of a ‘shock’ is determined by the position of the axis of the doubly contacting parabola at the instant t under consideration and is denoted by x _k . The endpoints of the parabolic arc are at ${x_k} + {x'_k}$ and ${x_k} + {x'''_k}$ so that x ^′_k and x ^″_k are relative coordinates with reference to the axis of the parabola They correspond to x ₁, x ₂, respectively, occurring in the integrals of the preceding chapter.