{'fraction_non_alphanumeric': 0.01344364012409514, 'fraction_numerical': 0.0, 'mean_word_length': 4.796407185628743, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nonlinear wave propagation on an arbitrary steady transonic flow

Here, we have studied the propagation of an arbitrary disturbance bounded in space on an arbitrary two- or three-dimensional transonic flow. First we have presented a general theory valid for an arbitrary system of n first-order quasi- linear partial differential equations and then used the theory for the special case of gasdynamic equations. If a disturbance is created in the neighbourhood of a sonic point, only a part of the disturbance stays in the transonic region and it is bounded by a wave front perpendicular to the streamlines. This part of the disturbance is governed by a very simple partial differential equation and the problem essentially reduces to the discussion of one-dimensional waves. The disturbance decays in the neighbourhood of the points where the flow acceler- ates from a subsonic state to a supersonic state and it attains a steady state where the flow is decelerating.