  We study first order fluctuations of a relativistic membrane in the curved
background of a black hole. The zeroth-order solution corresponds to a
spherical membrane tightly covering the event horizon. We obtain a massive
Klein-Gordon equation for the fluctuations of the membrane's radial coordinate
on the 2+1 dimensional world-volume. We finally suggest that quantization of
the fluctuations can be related to black hole's mass quantization and the
corresponding entropy is computed. This entropy is proportional to the membrane
area and is related to the one-loop correction to the thermodynamical entropy
$A_H/4$. With regards to the membrane model for describing effectively a
quantum black hole, we connect these results with previous work on critical
phenomena in black hole thermodynamics.
