Annotation: The purpose of this study is the enhancement of Euler equation possibilities in order to solve the brachistochrone problem that is the determination of a curve of fastest descent. There are two circumstances: 1) the first integral of an Euler equation does not contain a partial derivative of integrand with respect to y in an explicit form; 2) when the classical Euler equation is derived, only the second term of integrand is integrated by parts. This allowed formulating a problem of determination of new conditions of functional extremality. It is assumed that the integrand of the first variation of a functional is equal to zero. Taking into account this pro vision and some other assumptions, the procedures have been determined for simultaneous application of the Euler equation and its analogue being non-invariant in relation to the coordinate system. The brachistochrone problem was solved using these equations: the curves that satisfy the conditions of weak minimum optimality were plotted. The time of a material point’s descent along the suggested curves and the classic extremals was numerically compared. It is shown that the application of suggested curves ensures short descent time as compared to the classic extremals.

Annotation: Due to an increase of power of rocket engines, the high intensity sound field generated by the exhaust jets have become an important factor, which determines the success rate of a rocket launch. Ensuring a successful launch of a rocket system became harder due to new engineering problems. Identification and definition of acoustic sources structure within a complex supersonic jet, being a one of the most important scientific problems, which have to be solved to find the ways to control accoustic radiation. A three components of acoustic sources can be defined here – broadband signals from large and small components of of turbulent jet and tonal signals which usually being overlooked during the estimation of overall sound pressure level. The paper considers various aspects of acoustics of the launch of rocket systems, which includes characteristics of acoustic sources in supersonic jets, possibilities and physical limitation factors, under which it is possible to control the sound radiation. Among the possible ways to control the process of sound generation by a jet, a method of water injection in a jet is being studied. While saving the general thrust of the engine this method can not greatly reduce the sound radiation by a jet. It is recommended to use big amounts of water-air mix to protect the launch pad from damage. Significant progress on the topic of understanding the process of sound generation by supersonic jets can be achieved via mathematical modeling of sound radiation. The latest achievements of mathematical modeling of sound generation by supersonic jets being presented.

Annotation: The semi-empirical technique of evaluation of external acoustic loads during lift-off is a modification of the known NASA SP-8072 technique based on empirically determined universal dependencies of jet section acoustic power on normalized coordinate and frequency (Strouhal number). The proposed technique was updated in respect of evaluation of mechano-acoustic effectiveness of supersonic jet, determination of spatial scale of sound generating zone in jet, and formation of sound radiation direction towards rocket’s head module. A detailed presentation of the sequence of solving the task set is provided. A comparative analysis of the NASA SP-8072 technique and the proposed modification is made.

Annotation: The article considers the procedure for evaluation of acoustic stressing parameters at the observation point nearby the launch vehicle nose cone when passing the sectors with maximum velocity heads and close to 1 Mach numbers. And the problem is set to determine the overall sound pressure level and the corresponding levels in octave and 1/3-octave frequency bands. Procedure under consideration is based on the semi-empirical dependency of characteristics of the wideband aerodynamic noise, which occurs during the launch vehicle flight at high velocities due to the turbulent pressure fluctuations and dimensionless aerodynamic parameters of the main stream. General idea of this approach is to establish relation of the velocity heads with wall pressure fluctuations in the boundary layer, calculating shear stress (friction) on the shell surface based on relationships applicable in the boundary layer theory and engineering experience. Attempts of development of similar calculation models go back to the early efforts, dedicated to the study of the aeroacoustics of the launch vehicle in flight. Main advantages of the procedure are its simplicity and versatility since it can be used to determine the acoustic loads around the payload fairings of launch vehicles of different sizes and shapes within the wide range of flight velocities and altitudes.