We consider the problem of emission of sound by an airplane in the stage of its take-off run along the runway of an airdrome under the conditions of wind action. The aircraft engines play the role of the sources of sound. In the problem, they are modeled by point sources moving in air with a constant acceleration parallel to the plane interface between the acoustic and solid elastic half spaces. The solution of the problem is obtained by using the Fourier integral transformation with respect to time and space coordinates and the method of stationary phase. We also perform the numerical analysis of the spacetime distributions of sound pressure and the levels of sound pressure for the case of take-off of a sixengine An-225 "Mriya" aircraft.
Formulation of the ProblemAirdromes used for the take-of and landing of the aircrafts are the sites of concentration of strong noise and, hence, the sources of the propagation of acoustic waves to the neighboring areas. Therefore, at present, the problem of acoustic climate near airdromes attracts significant attention of the researchers [8,12,17], in particular, in the fields of measurements of the level of acoustic pressure [7], empirical methods of its evaluation [19,20], in the problems of optimization [5,11], reduction of the aircraft noise [13], and protection against this type of noise [18]. However, the mathematical modeling of acoustic wave processes capable of large-scale evaluation of this kind of hazard to the human health [10] is still poorly developed.In the present work, we make an attempt to construct a mathematical model of this kind. Due to the complexity of the problem, we restrict ourselves to the study of the process of the emission of acoustic energy by aircrafts only in the stage of take-off run.Consider the problem of emission of sound by a starting aircraft. The trajectory of motion of this object consists of four parts. The first of them represents the accelerated motion of the aircraft along the airdrome runway beginning from the initial point of its start. The second part is the accelerated motion in air at certain relatively low heights parallel to the runway. The third part can be described as the motion along a curvilinear path with passing to the fourth part, i.e., to the upward motion with a constant velocity at a certain angle to the runway.In the present work, we study the process of noise generation in the course of the take-off run, i.e., in the first part of the trajectory, because this stage has the largest duration and, as a rule, is realized near the inhabited areas.We replace the engines by point sources of sound with a certain intensity and angular frequency of harmonic vibrations Ω. First, we consider a single source of sound moving in the plane Oxz at a height z = z 0 over
