Regularization in optimal trajectory problems.

Abstract

Fig. 3b Oscilloscope trace from the pressure gage in the surface of the 9° cone model. 20^5 Mach reflection theory. There is a kink in the reflected shock wave, downstream from the Mach stem. The kink suggests some further shock reflections in the flow behind the Mach stem. The pressure gage trace corresponding to the 41° indicident angle adds evidence that something more than classical Mach reflection is occurring. The trace, Fig. 2b, shows a step in the gage's initial response as if the gage had been struck by two successive shock waves, one close behind the other. The pressure rise associated with the first step is exactly what we would expect to get from the Mach stem alone. The additional pressure rise from the step to the peak brings the pressure up to a level very close to that attained by regular reflection at 39°. So the large drop in pressure from regular to Mach reflection predicted by simple theory does not really exist. The additional compression that we expected to follow the Mach stem does take place, and it appears as though the mechanism for this additional compression is a second reflected shock wave. Further proof of the existence of a second shock wave behind the Mach stem was found by looking at incident angles much larger than the limiting angle. Figure 3a shows a case where the incident angle is 54°. The second reflected shock wave is now clearly visible. The pressure gage trace, Fig. 3b, shows a well-defined step. The time between the two shock waves, as measured from the pressure gage trace, corresponds exactly to the separation distance between the Mach stem and the second shock wave. Again, the second shock wave compressed the gas behind the Mach stem to a level well above that predicted by simple Mach reflection theory. Conclusions