An Alternative Example of the Method of Multiple Scales

Edwards, David A.

doi:10.1137/s0036144598348454

Cited by 5 publications

(2 citation statements)

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“…Equation ( 11) is a linear wave equation with slowly varying coefficients, which makes it an ideal candidate for a multiple-scales asymptotic expansion [16,17,18,19,46,47], the generalization of Cole's two-variable expansion procedure [48,Chap. 3].…”

Section: Solution By a Multiple-scales Expansionmentioning

confidence: 99%

On mechanical waves and Doppler shifts from moving boundaries

Christov

2017

Math Methods in App Sciences

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We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, $X_\mathrm{s}(t) = vt$ is the location of the source with constant velocity $v$. In the present work, however, we consider a source co-located with a moving boundary $x=X_\mathrm{s}(t)$, where $X_\mathrm{s}(t)$ can have an arbitrary functional form. For "slowly moving" boundaries (\textit{i.e.}, ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale asymptotic analysis of the moving-boundary problem for the linear wave equation. We obtain a closed-form leading-order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extending the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (a) $\dot X_\mathrm{s}$ for non-uniform boundary motion can be inserted in place of the constant velocity $v$ in the classical Doppler formula and (b) that the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated.Comment: 14 pages, 4 figures, Wiley journal style; accepted for publication in Mathematical Methods in the Applied Sciences; v2 corrects typographical error

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Section: Solution By a Multiple-scales Expansionmentioning

confidence: 99%

On mechanical waves and Doppler shifts from moving boundaries

Christov

2017

Math Methods in App Sciences

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“…However, note that the secularity arises in a vastly different way from that in a standard multiple-scale problem. (A further discussion of this type of multiple-scale problem may be found in [26].) Once the multiple-scale expansion has been postulated, the correction due to transport may be calculated using a Fourier series.…”

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Biochemical Reactions on Helical Structures

Edwards¹

2000

SIAM J. Appl. Math.

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Abstract. Analyses of biochemical surface-volume reactions often focus on the reaction-dominated case, where the free-floating analyte is well-mixed and the binding kinetics are unaffected by transport. In actuality, transport effects often play a role. A mathematical model is formulated for a cylinder with a helical reacting strip on its surface, which is a good model for a helical biopolymer such as DNA that interacts with molecules that bind to periodic structures along its backbone. Perturbation techniques are used to analyze the concentration of the reacting species for association and dissociation kinetics when the cylinder is immersed in a quiescent medium. In the case of an insulating boundary for the analyte region, a multiple-scale expansion must be used. The secularity which necessitates such an expansion is significantly different from canonical examples. The expressions for the bound state provide a direct way to estimate the rate constants from raw data. Remarks on the calculation of effective rate constants for general transport-affected systems are presented.

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Advanced Asymptotic Methods

Kuehn

2014

Applied Mathematical Sciences

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An Alternative Example of the Method of Multiple Scales

Cited by 5 publications

References 10 publications

On mechanical waves and Doppler shifts from moving boundaries

On mechanical waves and Doppler shifts from moving boundaries

Biochemical Reactions on Helical Structures

Advanced Asymptotic Methods

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