Teaching wave propagation and the emergence of Viète's formula

ISSN： 0031-9120

Source： Physics Education, Vol.47, Iss.1, 2012-01, pp. : 87-91

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Abstract

The well-known result for the frequency of a simple spring–mass system may be combined with elementary concepts like speed = wavelength × frequency to obtain wave propagation speeds for an infinite chain of springs and masses (masses m held apart at equilibrium distance a by springs of stiffness γ). These propagation speeds are dependent on the wavelength of the wave. The dispersion is easily investigated by considering normal modes of increasing wavelength. This investigation also elegantly highlights how the dispersion physically arises in the form of effective spring constants due to the way in which neighbouring springs contribute to the propagation of each of the normal modes. The resulting propagation speeds v(λ) are given by an expression &v(lambda)=frac {p(lambda)}{pi }asqrt {frac {gamma }{m}} ;, which in the limit of large λ becomes &v=asqrt {frac {gamma }{m}} ;. This of course means that &lim_{lambda to infty } ({p(lambda)})= pi ;—the serendipitous emergence of what turns out to be Viète's formula for π in terms of nested roots of 2.