We present a new formula for the total statistical weight of all Rydberg levels G^{ion} (n_{l}, n_{h} ) for which the principal quantum number n is between higher, n_{h} , and a lower, n_{l} , limits. This formula can be used for all atoms in the periodic table and for all corresponding ions.
Wave propagation in a near-bottom layer containing gas bubbles is analyzed. Evolution equations are derived for intense acoustic waves and wave beams in a medium with an inhomogeneous bubble distribution. The field of an intense beam along the axis of the focusing sound channel is calculated. The coefficients of reflection and passage of sound from a homogeneous medium into a bubble layer and back again are calculated. It is shown that the near-bottom layer can effectively trap rays incident on it and ensure a waveguide propagation character. The presence of bubbles increases both the interval of angles at which the wave penetrates the layer and the interval of angles at which rays undergo total internal reflection and do not depart the layer. The acoustic field in the layer from a point source is calculated.
One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like Burgers’ equation stick together as absolutely inelastic particles when colliding. The interactions of single pulses in media with modular nonlinearity considered here reveal new physical features that are still poorly understood. There is an analogy between the single pulses collision and the interaction of clots of chemical reactants, such as fuel and oxidant, where the smaller component disappears and the larger one decreases after a reaction. At equal “masses” both clots can be annihilated. In this work various interactions of two and three pulses are considered. The conditions for which a complete annihilation of the pulses occurs are indicated. © 2017 The Author(s)
We study experimentally the behavior of a nonlinear element, a light plate pressed to the opening in the cavity of an acoustic resonator. Measurements of field oscillations inside and outside the cavity have shown that for large amplitudes, they become essentially anharmonic. The time dependences of displacement of the plate with increasing amplitude of the exciting voltage demonstrates a gradual change in the shape of vibrations from harmonic to half-period oscillation. A constant component appears in the cavity: rarefaction or outflow of the medium through the orifice. We construct a theory for nonlinear oscillations of a plate taking into account its different elastic reactions to compression and rarefaction with allowance for monopole radiation by the small-wave-size plate or radiation of a plane wave by the plate. We calculate the amplitudes of the harmonics and solve the problem of low-frequency stationary noise acting on the plate. We obtain expressions for the correlation function and mean power at the output given a normal random process at the input.
The subsurface exploration of other planetary bodies can be used to unravel their geological history and assess their habitability. On Mars in particular, present-day habitable conditions may be restricted to the subsurface. Using a deep subsurface mine, we carried out a program of extraterrestrial analog research â MINe Analog Research (MINAR). MINAR aims to carry out the scientific study of the deep subsurface and test instrumentation designed for planetary surface exploration by investigating deep subsurface geology, whilst establishing the potential this technology has to be transferred into the mining industry. An integrated multi-instrument suite was used to investigate samples of representative evaporite minerals from a subsurface Permian evaporite sequence, in particular to assess mineral and elemental variations which provide small-scale regions of enhanced habitability. The instruments used were the Panoramic Camera emulator, Close-Up Imager, Raman spectrometer, Small Planetary Linear Impulse Tool, Ultrasonic drill and handheld X-ray diffraction (XRD). We present science results from the analog research and show that these instruments can be used to investigate in situ the geological context and mineralogical variations of a deep subsurface environment, and thus habitability, from millimetre to metre scales. We also show that these instruments are complementary. For example, the identification of primary evaporite minerals such as NaCl and KCl, which are difficult to detect by portable Raman spectrometers, can be accomplished with XRD. By contrast, Raman is highly effective at locating and detecting mineral inclusions in primary evaporite minerals. MINAR demonstrates the effective use of a deep subsurface environment for planetary instrument development, understanding the habitability of extreme deep subsurface environments on Earth and other planetary bodies, and advancing the use of space technology in economic mining. Copyright Â© Cambridge University Press 2016
A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types—modular, quadratic, and quadratically cubic—which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves.
Solutions to a partial differential equation of the third order containing the modular nonlinearity are studied. The model describes, in particular, elastic waves in media with weak high-frequency dispersion and with different response to tensile and compressive stresses. This equation is linear for solutions preserving their sign. Nonlinear phenomena only manifest themselves to alternating solutions. Stationary solutions in the form of solitary waves or solitons are found. It is shown how the linear periodic wave becomes nonlinear after exceeding a certain critical value of the amplitude, and how it transforms into a soliton with further increase in the amplitude.
Two models of an anharmonic oscillator that have exact solutions are considered. The equationsdescribe motion in a “modulus” potential well with a singularity at the minimum and in a double symmetricwell with a singularity at the vertex of the potential barrier. The forms and spectra of the oscillations are computed. Forced oscillations caused by a random force are analyzed on the basis of equations with Langevinsources. Nonstationary solutions of the corresponding Fokker–Planck equations are constructed. Thesesolutions describe