The spreading of the position and momentum probability distributions for the stable free oscillations of a circular membrane of radius l is analyzed by means of the associated Boltzmann-Shannon information entropies in the correspondence principle limit (n → ∞, m fixed), where the numbers (n,m), n ∈ N and m ∈ Z, uniquely characterize an oscillation of this two-dimensional system. This is done by solving the short-wave asymptotics of the physical entropies in the two complementary spaces, which

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... oils down to the calculation of the asymptotic behaviour of certain entropic integrals of Bessel functions. It is rigorously shown that the position and momentum entropies behave as 2 ln(l) + ln(4π) − 2 and ln(n) − 2 ln(l) + ln(2π 3 ) when n → ∞, respectively. So the total entropy sum has a logarithmic dependence on n and it does not depend on the membrane radius. The former indicates that the ordering of short-wavelength oscillations is exactly identical for the entropic sum and the single-particle energy. The latter holds for all oscillations of the membrane because of the uniform scaling invariance of the entropy sum.