A Note on the Secular Equation of the Product of Two Matrices

1961 ◽  
Vol 68 (9) ◽  
pp. 906 ◽  
Author(s):  
Harry Lass ◽  
Carleton B. Solloway
Keyword(s):  
Secular Equation

On a technique for deriving explicit transfer matrices of orthotropic layers

2015 ◽  
Vol 37 (4) ◽  
pp. 303-315 ◽  
Author(s):  
Pham Chi Vinh ◽  
Nguyen Thi Khanh Linh ◽  
Vu Thi Ngoc Anh
Keyword(s):  
Wave Propagation ◽  
Explicit Form ◽  
Half Space ◽  
Transfer Matrix ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Transfer Matrices ◽  
Orthotropic Layer ◽  
Wave Propagation Problem ◽  
Arbitrary Thickness

This paper presents  a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

Berechnung kinetischer Isotopeneffekte für die Reaktion von Methyljodid mit Cyanid-Ion

1966 ◽  
Vol 21 (9) ◽  
pp. 1377-1384
Author(s):  
A. V. Willi
Keyword(s):  
Transition State ◽  
Secular Equation ◽  
Isotope Effects ◽  
Force Constants ◽  
Bond Breaking ◽  
Bending Force ◽  
A Value ◽  
Order Of Magnitude ◽  
Deuterium Isotope Effects ◽  
The Difference

Kinetic carbon-13 and deuterium isotope effects are calculated for the SN2 reaction of CH3I with CN-. The normal vibrational frequencies of CH3I, the transition state I · · · CH3 · · · CN, and the corresponding isotope substituted reactants and transition states are evaluated from the force constants by solving the secular equation on an IBM 7094 computer.Values for 7 force constants of the planar CH3 moiety in the transition state (with an sp2 C atom) are obtained by comparison with suitable stable molecules. The stretching force constants related to the bonds being broken or newly formed (fCC, fCC and the interaction between these two stretches, /12) are chosen in such a way that either a zero or imaginary value for νʟ≠ will result. Agreement between calculated and experimental methyl-C13 isotope effects (k12/ k13) can be obtained only in sample calculations with sufficiently large values of f12 which lead to imaginary νʟ≠ values. Furthermore, the difference between fCI and fCC must be small (in the order of 1 mdyn/Å). The bending force constants, fHCI and fHCC, exert relatively little influence on k12/k13. They are important for the D isotope effect, however. As soon as experimental data on kH/kD are available it will be possible to derive a value for fHCC in the transition state if fHCI is kept constant at 0.205 mdynA, and if fCI, fCC and f12 are held in a reasonable order of magnitude. There is no agreement between experimental and calculated cyanide-C13 isotope effects. Possible explanations are discussed. — Since fCI and fCC cannot differ much it must be concluded that the transition state is relatively “symmetric”, with approximately equal amounts of bond making and bond breaking.

Rayleigh wave propagation at the boundary surface of dry sandy ($SiO_2$) thermoelastic solids

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Shishir Gupta ◽  
Rishi Dwivedi ◽  
Smita Smita ◽  
Rachaita Dutta
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Rayleigh Wave ◽  
Penetration Depth ◽  
Dispersion Equation ◽  
Half Space ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Rayleigh Surface Wave ◽  
Content Type

Purpose The purpose of study to this article is to analyze the Rayleigh wave propagation in an isotropic dry sandy thermoelastic half-space. Various wave characteristics, i.e wave velocity, penetration depth and temperature have been derived and represented graphically. The generalized secular equation and classical dispersion equation of Rayleigh wave is obtained in a compact form. Design/methodology/approach The present article deals with the propagation of Rayleigh surface wave in a homogeneous, dry sandy thermoelastic half-space. The dispersion equation for the proposed model is derived in closed form and computed analytically. The velocity of Rayleigh surface wave is discussed through graphs. Phase velocity and penetration depth of generated quasi P, quasi SH wave, and thermal mode wave is computed mathematically and analyzed graphically. To illustrate the analytical developments, some particular cases are deliberated, which agrees with the classical equation of Rayleigh waves. Findings The dispersion equation of Rayleigh waves in the presence of thermal conductivity for a dry sandy thermoelastic medium has been derived. The dry sandiness parameter plays an effective role in thermoelastic media, especially with respect to the reference temperature for η = 0.6,0.8,1. The significant difference in η changes a lot in thermal parameters that are obvious from graphs. The penetration depth and phase velocity for generated quasi-wave is deduced due to the propagation of Rayleigh wave. The generalized secular equation and classical dispersion equation of Rayleigh wave is obtained in a compact form. Originality/value Rayleigh surface wave propagation in dry sandy thermoelastic medium has not been attempted so far. In the present investigation, the propagation of Rayleigh waves in dry sandy thermoelastic half-space has been considered. This study will find its applications in the design of surface acoustic wave devices, earthquake engineering structural mechanics and damages in the characterization of materials.

Rayleigh-type surface wave on a rotating orthotropic elastic half-space with impedance boundary conditions

2020 ◽  
Vol 26 (21-22) ◽  
pp. 1980-1987
Author(s):  
Baljeet Singh ◽  
Baljinder Kaur
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Rayleigh Wave ◽  
Surface Waves ◽  
Half Space ◽  
Rotation Rate ◽  
Secular Equation ◽  
Elastic Half Space ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The propagation of Rayleigh type surface waves in a rotating elastic half-space of orthotropic type is studied under impedance boundary conditions. The secular equation is obtained explicitly using traditional methodology. A program in MATLAB software is developed to obtain the numerical values of the nondimensional speed of Rayleigh wave. The speed of Rayleigh wave is illustrated graphically against rotation rate, nondimensional material constants, and impedance boundary parameters.

To problem of reliable calculation of coefficients of secular equation by special functions

2013 ◽  
Author(s):  
Tamaz S. Vashakmadze ◽  
Rezo Chikashua ◽  
Gela Manelidze
Keyword(s):  
Secular Equation ◽  
Special Functions

Estimation of the Effect of a Parameter Change on the Roots of Stability Equations

1949 ◽  
Vol 1 (1) ◽  
pp. 39-58 ◽  
Author(s):  
K. Mitchell
Keyword(s):  
Secular Equation ◽  
Probable Error ◽  
Typical Case ◽  
Lateral Stability ◽  
Parameter Change ◽  
Longitudinal Stability ◽  
Quartic Equation ◽  
Spiral Motion ◽  
The Stability

SummaryA method is given for calculating approximately the changes in the roots of a stability secular equation caused by a change in any of the parameters involved. General formulas are given applicable to any quartic equation, and special formulae are also given applicable to the stability of an aeroplane: lateral stability in the text, and longitudinal stability in an appendix. The method of using the formulae is illustrated by applying them to a particular calculation of the lateral stability of an aeroplane, and a check of the results is made by comparing the predicted approximate changes with those calculated by solution of the modified period equations. It is shown that the formulae are reliable, for this typical case, for any reasonable changes in any parameter other than nv. If the changes in the derivatives are made equal to the probable error with which they can be measured, the formulae enable us to evaluate the probable errors of the roots. These are found to be considerable, and to arise mainly from uncertainties in yv, nv and nr: if these could be reduced to 0.03 in yv and 0.006 in the others, the uncertainties in the roots would be reduced to some ten per cent, of their values, except for a larger uncertainty in the root corresponding to the slow spiral motion.

Effects of Two-Temperature, Rotation, Initial Stressed Magnetic Field on Rayleigh Wave in Half-Plane

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Heena Sharma ◽  
Sangeeta Kumari
Keyword(s):  
Rayleigh Wave ◽  
Initial Stress ◽  
Secular Equation ◽  
Attenuation Factor ◽  
Radiation Condition ◽  
Amplitude Attenuation ◽  
Temperature Parameter ◽  
Velocity Of Propagation ◽  
Generalized Thermoelastic ◽  
Two Temperature

Abstract In the present paper, we consider the governing equation for generalized thermoelastic media under the effect of magnetic fleld, rotation, initial stress, and two-temperature parameter for Rayleigh wave in half-space. The secular equation of Rayleigh wave is also deduced using surface wave solution, which also satisfy the radiation condition for thermally insulated/isothermal surface. The velocity and amplitude attenuation factor of Rayleigh wave is also computed for a particular material. The effect of two-temperature, rotation, and initial stress parameters on velocity of propagation and amplitude attenuation factor is shown graphically.

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