Higher order interaction of elastic waves in weakly nonlinear hollow circular cylinders. I. Analytical foundation

Yang Liu, Clifford Jesse Lissenden, III, Joseph Lawrence Rose

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N+1)(N+2)/2-3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N≤K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.

Original language English (US) 214901 Journal of Applied Physics 115 21 https://doi.org/10.1063/1.4879459 Published - Jun 7 2014

Fingerprint

circular cylinders
elastic waves
hollow
harmonics
interactions
spherical coordinates
power series
S waves
synchronism
harmonic generations
parity
nonlinearity
energy

All Science Journal Classification (ASJC) codes

• Physics and Astronomy(all)

Cite this

title = "Higher order interaction of elastic waves in weakly nonlinear hollow circular cylinders. I. Analytical foundation",
abstract = "A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N+1)(N+2)/2-3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N≤K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.",
author = "Yang Liu and {Lissenden, III}, {Clifford Jesse} and Rose, {Joseph Lawrence}",
year = "2014",
month = "6",
day = "7",
doi = "10.1063/1.4879459",
language = "English (US)",
volume = "115",
journal = "Journal of Applied Physics",
issn = "0021-8979",
publisher = "American Institute of Physics Publising LLC",
number = "21",

}

In: Journal of Applied Physics, Vol. 115, No. 21, 214901, 07.06.2014.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Higher order interaction of elastic waves in weakly nonlinear hollow circular cylinders. I. Analytical foundation

AU - Liu, Yang

AU - Lissenden, III, Clifford Jesse

AU - Rose, Joseph Lawrence

PY - 2014/6/7

Y1 - 2014/6/7

N2 - A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N+1)(N+2)/2-3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N≤K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.

AB - A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N+1)(N+2)/2-3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N≤K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.

UR - http://www.scopus.com/inward/record.url?scp=84902596800&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902596800&partnerID=8YFLogxK

U2 - 10.1063/1.4879459

DO - 10.1063/1.4879459

M3 - Article

VL - 115

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 21

M1 - 214901

ER -