A general scheme for the derivation of evolution equations describing mixed nonlinearity

Cramer, M. S.; Sen, Rahul

doi:10.1016/0165-2125(92)90053-5

Cited by 24 publications

(28 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When the medium ahead is uniform and the source term is absent, the evolution equation reduces to that of Kluwick and Cox [5] with the coefficients being absolute constants. The linear term vanishes too, if we consider one-dimensional wave propagation in the uniform medium and we get an evolution equation similar to that of Cramer and Sen [4].…”

Section: Singular Ray Expansionsmentioning

confidence: 91%

“…When the medium ahead of the wave is uniform and the source term is absent, Eq. (16) reduces to the form obtained by Kluwick and Cox [5]; further, the case of purely one-dimensional wave propagation is recovered on setting χ = 0, and yields an equation derived by Cramer and Sen [4].…”

Section: Small Amplitude High Frequency Wavesmentioning

confidence: 96%

“…In this context, we refer to the work of Cramer and Sen [4] and Kluwick and Cox [5], who examine the propagation of high frequency pulses when the nonlinear effects are noticeable over times of order O(ε −2 ) rather than those of O(ε −1 ); their main results focus on the case when the quadratic nonlinearity parameter Γ defined as…”

Section: Small Amplitude High Frequency Wavesmentioning

confidence: 99%

“…The effectiveness of the geometrical optics method prompted many workers to extend the underlying ideas to wave motion (see, for example, Choquet-Bruhat [1], Hunter and Keller [2], Majda and Rosales [3], Cramer and Sen [4], Kluwick and Cox [5], and Srinivasan and Sharma [6]); the technique involves the introduction of slow and fast variables and the phase functions. These multiple scale expansions have been successfully employed in the context of ODEs [7], where these were introduced by Krylov-Bogoliubov as a variant of the methods employed earlier by Poincaré and Lindstedt to eliminate secular terms in the perturbation expansions of Celestial Mechanics.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear wave propagation through a stratified atmosphere

Sharma

Madhumita

2005

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.  2005 Elsevier Inc. All rights reserved.

show abstract

Section: Singular Ray Expansionsmentioning

confidence: 91%

Section: Small Amplitude High Frequency Wavesmentioning

confidence: 96%

Section: Small Amplitude High Frequency Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear wave propagation through a stratified atmosphere

Sharma

Madhumita

2005

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…In Sec. III, we apply an extension of the perturbation technique of Taniuti and Wei (1968), as developed by Cramer and Sen ( 1991 ), to derive the evolution equation governing the nonlinear distortion not only at, but in the neighborhood of, each critical point. 'The resultant evolution equation is a modification of Burgers' equation containing both quadratic and cubic nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Nonlinearity parameters for pulse propagation in isotropic elastomers

Carman

Cramer

1992

The Journal of the Acoustical Society of America

View full text Add to dashboard Cite

The propagation of one-dimensional shear waves in isotropic hyperelastic materials is examined. Both compressible and incompressible materials subject to arbitrarily large shear prestrains are considered. The nonlinear evolution equation governing small disturbances on prestrained undisturbed states is derived. The prestrain is seen to change the nonlinearity from the cubic form found in the unstrained case to the stronger quadratic form governed by the conventional Burgers' equation. Additional results include explicit and general expressions for the quadratic and cubic nonlinearity parameters. Numerical estimates are also provided for natural rubber and foamed polyurethane. It is demonstrated that the quadratic nonlinearity parameter does not increase monotonically with prestrain and may actually vanish at nonzero values of the prestrain.

show abstract

Undercompressive shock waves in Hall‐magnetohydrodynamics

Shukla

Sharma

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

Funding informationUniversity Grants Commission (UGC), India JEL Classification: 35C20; 35G25; 35L67; 76N15; 76W05We study the propagation of nonlinear waves in a Hall-magnetohydrodynamic model. An asymptotic method is used to derive the Gardner-Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second-and third-order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splitting-a phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave-fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.

show abstract

A general scheme for the derivation of evolution equations describing mixed nonlinearity

Cited by 24 publications

References 17 publications

Nonlinear wave propagation through a stratified atmosphere

Nonlinear wave propagation through a stratified atmosphere

Nonlinearity parameters for pulse propagation in isotropic elastomers

Undercompressive shock waves in Hall‐magnetohydrodynamics

Contact Info

Product

Resources

About