  We calculate the mean square amplitude of the shape fluctuation -- an
equal-time correlation -- of an almost planar fluid membrane immersed in a
near-critical binary fluid mixture. One fluid component is usually
preferentially attracted by the membrane, and becomes more concentrated around
it because of the near criticality. This generates osmotic pressure, which
influences the amplitude. The amplitude is also affected by the reversible
dynamics of the mixture, which moves with the membrane. By assuming the
Gaussian free-energy functional and weak preferential attraction, the author
previously showed that a new term is added to the restoring force of the
membrane and tends to suppress the amplitude. Not assuming both of them, but
still focusing on modes with wavelength longer than the correlation length, we
here calculate the amplitude of a tensionless membrane. First, within the
Gaussian model, we solve the governing equations to show that, for long
wavelength, the additional term becomes predominant, although decreased
hydrodynamic effects make its numerical factor approximately half that of the
previous result. The change in the term turns out not to be monotonic with the
wavelength, which is mainly caused by the change in the induced mass. Second,
assuming the critical composition far from the membrane, we calculate the
amplitude beyond the regime of the Gaussian model. The result coincides roughly
with the corresponding result in the Gaussian model if the correlation length
is interpreted as one close to the membrane.
