On Melde's String

Recently, Kidachi and Onogi 1 ) considered Melde's string and derived a nonlinear Mathieu equation for the Fourier component of the transverse displacement. They announced to analyze experimental data in terms of the temporal behavior they found in the solution of the nonlinear Mathieu equation. Here we point out that the master equation for Melde's string that they use is incorrect. Their equation reads, {Pu {Pu { (au) 2}-~ p at 2 - T ax2 1 + ax = 0, with T = To{l- Ecoswt), (1) where u is the displacement of the string, p is the mass density of the string per unit length, and To is the mean strength of the tension applied to the string; E and ware the magnitude and the angular frequency of the periodic variation of the tension, respectively. They assume that the tension is independent of the displacement and that non linearity appears where one calculates the component of the tension along the dis placement using au/ax. We note that the string changes its length in vibration, so the tension clearly depends on the displacement. One might think this effect is so small that it should be negligible, but the simple estimate given below shows it is much larger than the effect considered in Ref. 1). Let the tension T be the sum of the equilibrium value Teq , the oscillating part Tosc and the increment due to the displacement L1T. Then the small displacement approximation gives the coefficient of U xx in (1) as