Alexander E. Dubinovand

A simple construction of a multi-pendulum in the form of a flexible chain of beads is described, together with a straightforward experimental methodology for studying a complex phenomenon—the oscillations of this multi-pendulum initially excited by a bending disturbance. The equipment and experimental procedure are accessible to school students and to educational institutions at the introductory level of physics education. The equipment and experiments are safe and can be rapidly implemented in the educational process. Importantly, the study of oscillations of a multi-pendulum allows for unlimited variability in laboratory results, which is essential for developing students’ independent experimental skills. Any student can construct their own multi-pendulum differing in parameters—such as length, number of elements, method of suspension, etc—from that of a neighbouring student. As a result, it is possible, for example, to measure the damping of oscillations and bending in a suspended multi-pendulum.

A non-ideal dusty plasma is considered, consisting of a mixture of three gases: inertia less electrons and ions, and a gas of negatively charged granules. Electrons and ions in the plasma are assumed to obey the Boltzmann distribution, while the granular gas follows the equation of state of a non-ideal gas proposed by C. Dieterici. For the first time, stationary granular-acoustic waves (GAWs) in such a plasma are investigated within the framework of a gasdynamic approach. A linear theory of GAWs is developed, from which the dispersion relation is derived and an exact expression for the linear speed of granular sound is obtained. Furthermore, a nonlinear theory of large-amplitude GAWs in a non-ideal granular plasma is developed for the first time, based on the Bernoulli pseudopotential method. Analysis of the pseudopotential shows that stationary GAWs in the considered plasma can take the form either of nonlinear periodic waves or of granular-acoustic (GA)-solitons. The corresponding ranges of wave velocities for both forms are determined. The dependences of the maximum soliton velocity and the maximum Mach number on the relative electron concentration in the plasma are calculated. It is found that, with increasing relative electron concentration, the maximum velocity of the GA-soliton increases slightly and almost linearly. The developed theory can be applied to describe GAWs arising in dense non-ideal granular plasmas occurring in volcanic ejecta of planets or in soil ejection during meteorite impacts on planetary surfaces and their satellites.

A concept and design of a relativistic two-cavity reflex klystron with a magnetic mirror has been proposed. A 2.5D particle-in-cell (PIC) computer model of a relativistic two-cavity reflex klystron with a magnetic mirror has been created on the basis of the KARAT PIC code. The model takes into account the supply of an external microwave from the master oscillator and allows self-consistent calculation of the electron dynamics and the output characteristics of the klystron. Comparative modeling of a relativistic two-cavity reflex klystron with a magnetic mirror and a relativistic two-cavity transit klystron without a magnetic mirror has been performed. It has been shown that the spectral characteristics of both klystrons are close to each other. The output power of the reflex and transit klystrons has been compared. It has been found that the output power of the reflex klystron is approximately 1.4 times greater than that of the transit klystron. The effect of the magnetic mirror position on the value of the generated microwave power of the reflection klystron has been studied.

This study explores a physical system in which particle motion is characterized by linearly increasing acceleration and constant jerk over time. A vacuum planar electron diode operating under constant voltage in the space-charge-limited regime is analyzed. The derivation of the Child–Langmuir law is presented in detail. It is demonstrated that in such diode, all electrons travel from the cathode to the anode with constant jerk and the linearly increasing acceleration. Higher-order time derivatives of motion—snap (or jounce), crackle, and pop are all zero. An exact formula for the constant jerk is derived for the first time.

Представлены некоторые свойства новой трансцендентной W-функции Ламберта: определение функции, ее график, координаты характерных точек, простейшие тождества. Даны несколько примеров, показывающих, как с помощью W-функции можно решать аналитически трансцендентные уравнения, содержащие степенные, логарифмические и экспоненциальные члены. Представлено полученное недавно решение сравнения двух функциональных выражений степенного типа и , которое возникает благодаря обобщению сравнения чисел и . Впервые получено точное решение нового обобщенного сравнения более сложных степенных выражений и на множествах положительных действительных чисел и для положительных значений показателя . Решение представлено как в виде точной формулы с помощью W-функции Ламберта, так и с помощью графиков.

The authors consider a nonideal plasma. The plasma consists of a mixture of two gases—a gas of inertial electrons and a gas of positive single-charge ions. Electrons in plasma are considered to be Boltzmann distributed, and an ion gas obeyed the Van der Waals equation of state. Ion-acoustic waves (IAWs) in such a plasma are studied. A linear theory of IAWs is developed. Within the framework of this theory, a dispersion equation is derived and an exact expression is obtained for the linear velocity of an ion sound. A non-linear theory of high-amplitude IAWs is developed; it is based on the Bernoulli pseudopotential method. It has been shown that stationary IAWs in the studied plasma can have the form of either a nonlinear periodic wave or the form of solitons. The wave velocity ranges in these two forms are defined. Dependence of the maximum velocity of solitons on the value of plasma thermodynamic nonequilibrium parameter is calculated.

The results of the 1D study of a multifluid model of nonlinear ion-acoustic solitons in a bi-ion plasma are presented. The model is based on gasdynamic equations for each plasma component. For the first time, the effect of taking into account Dalton's law on the characteristics of ion-acoustic solitons was assessed. The dispersion equation of ion-acoustic waves is derived both taking into account Dalton's law and without it. In both cases, the dispersion equation provides two branches—an optical branch and an acoustic branch. It is shown that taking into account Dalton's law leads to a change in the values of phase velocities for both branches in a short-wave part of the spectrum. The velocity of an ion sound is 2 times higher when Dalton's law is taken into account than without it. Profiles of physical quantities in ion-acoustic solitons were calculated in the framework of a nonlinear theory. It was found that taking into account Dalton's law leads to an amplitude increase in an electrostatic potential in a soliton and to an amplitude decrease in both concentrations of components in plasma; a concentration burst length in a soliton when taking into account Dalton's law is noticeably greater than without taking into account this law; some soliton profiles may take the form of solitons of a W-type at large Mach numbers.