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Consider a string held fixed at $x=0$ and $x=L$. This string has a harmonic series associated with it with each harmonic time period, $T_n$, given by: $$T_n=\frac{2}{n}\int\frac{dx}{v(x)},$$ where $v(x)$ is the wave speed along the string. Do the harmonic time periods, $T_n$, correspond to resonant time periods? Let the resonant time periods be denoted with $T_{n}^{res}$, where these are defined to be the time periods such that when the string is driven at these time periods the maximum amplitude is produced. Is $$T_n=T_n^{res},$$ a true statement in general? My gut tells me no because I know that if a damping term is introduced by say placing the string in honey that this should probably change the resonant frequency while the harmonic time periods should always be given by: $$T_n=\frac{2}{n}\int\frac{dx}{v(x)}.$$ Do you know any sources books etc. which discuss driving strings and what determines the resonant frequency of the string?

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    $\begingroup$ The answer will depend strongly on the specific way that the string is driven, and the specific form of damping. Also, if the material of the string is non-linear in any significant degree then you may not have $v$ as just a function $v(x)$. That is, things may be very different for a larger amplitude if the string material does not follow Hooke's law to a significant degree. $\endgroup$
    – user93146
    Jul 3, 2018 at 14:18

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