## Inverse eigenvalue problem for a vibration of a string with viscous drag.(English)Zbl 0717.73046

A pair of eigenvalue problems is considered describing eigenvibrations of a string with viscous drag: $d/dx[T(x)(dv(x)/dx)]-\lambda k(x)v(x)=\lambda^ 2\rho (x)v(x)\quad (0\leq x\leq 1),$ and $v(0)=v(1)=0\quad or\quad v(0)=dv/dx(1)=0.$ It is shown that all the eigenvalues of these two problems uniquely determine both a tension T and a linear mass density $$\rho$$, provided that the coefficient of viscous drag k(x) is a given nonzero constant function.
Reviewer: E.Syrkin

### MSC:

 74H45 Vibrations in dynamical problems in solid mechanics 74B99 Elastic materials 74H99 Dynamical problems in solid mechanics 34A55 Inverse problems involving ordinary differential equations 74J25 Inverse problems for waves in solid mechanics 74K05 Strings 34L05 General spectral theory of ordinary differential operators
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### References:

 [1] Borg, G, Eine umkehrung der Sturm-liouvilleschen eigenwertfrage, Acta math., 78, 1-96, (1946) · Zbl 0063.00523 [2] Hochstadt, H, The inverse Sturm-Liouville problem, Comm. pure appl. math., 26, 715-729, (1973) · Zbl 0281.34015 [3] Levinson, N, The inverse Sturm-Liouville problem, Mat. tidsskr., B, 25-30, (1949) · Zbl 0041.42310 [4] Russell, D.L, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. math. anal. appl., 40, 336-368, (1972) · Zbl 0244.93025 [5] Russell, D.L, Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems, J. math. anal. appl., 62, 186-225, (1978) · Zbl 0371.93010 [6] Timoshenko, S, Vibration problems in engineering, (1955), Van Nostrand New York · JFM 63.1305.03 [7] Yamamoto, M, Inverse spectral problem for systems of ordinary differential equations of first order, I, J. fac. sci. univ. Tokyo sect. IA math., 35, 519-546, (1988) · Zbl 0667.34018
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