TY - JOUR

T1 - The complete infinite series solution of systems governed by the wave equation with boundary damping

AU - Sirota, Lea

AU - Halevi, Yoram

N1 - Funding Information:
This research was supported by The Israel Science Foundation (grant No. 1211/10 ). Lea Sirota was also supported by a Scholarship for Women in Science of Israel Ministry of Science & Technology .

PY - 2014/1

Y1 - 2014/1

N2 - A common method of solving initial boundary value problems is separation of variables, denoted as modal analysis in the field of flexible structures. For systems with undamped boundary conditions the method is well-established, but for systems with boundary damping it does not provide closed form solutions. In this paper the exact modal series solution for second order systems with damped boundaries is derived with explicit expressions for the series coefficients. Knowledge of these coefficients enables practical applications of the solution, such as finite dimension approximation. The key element of the derivation is a new orthogonality condition for the damped eigenfunctions. The modal series is also transformed into a traveling wave form. The solution, which is the extension of the classical D'Alembert formula, is represented by a single equivalent propagating wave. A component of the solution, denoted by "end waves", is identified to provide the continuity of the systems displacement response.

AB - A common method of solving initial boundary value problems is separation of variables, denoted as modal analysis in the field of flexible structures. For systems with undamped boundary conditions the method is well-established, but for systems with boundary damping it does not provide closed form solutions. In this paper the exact modal series solution for second order systems with damped boundaries is derived with explicit expressions for the series coefficients. Knowledge of these coefficients enables practical applications of the solution, such as finite dimension approximation. The key element of the derivation is a new orthogonality condition for the damped eigenfunctions. The modal series is also transformed into a traveling wave form. The solution, which is the extension of the classical D'Alembert formula, is represented by a single equivalent propagating wave. A component of the solution, denoted by "end waves", is identified to provide the continuity of the systems displacement response.

KW - Boundary damping

KW - Infinite series

KW - Modal analysis

KW - Separation of variables

KW - Vibration

KW - Waves

UR - http://www.scopus.com/inward/record.url?scp=84887623320&partnerID=8YFLogxK

U2 - 10.1016/j.wavemoti.2013.06.011

DO - 10.1016/j.wavemoti.2013.06.011

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AN - SCOPUS:84887623320

SN - 0165-2125

VL - 51

SP - 114

EP - 124

JO - Wave Motion

JF - Wave Motion

IS - 1

ER -