  This paper presents a simple Fourier-matching method to rigorously study
resonance frequencies of a sound-hard slab with a finite number of arbitrarily
shaped cylindrical holes of diameter ${\cal O}(h)$ for $h\ll1$. Outside the
holes, a sound field can be expressed in terms of its normal derivatives on the
apertures of holes. Inside each hole, since the vertical variable can be
separated, the field can be expressed in terms of a countable set of Fourier
basis functions. Matching the field on each aperture yields a linear system of
countable equations in terms of a countable set of unknown Fourier
coefficients. The linear system can be reduced to a finite-dimensional linear
system based on the invertibility of its principal submatrix, which is proved
by the well-posedness of a closely related boundary value problem for each hole
in the limiting case $h\to 0$, so that only the leading Fourier coefficient of
each hole is preserved in the finite-dimensional system. The resonance
frequencies are those making the resulting finite-dimensional linear system
rank deficient. By regular asymptotic analysis for $h \ll 1$, we get a
systematic asymptotic formula for characterizing the resonance frequencies by
the 3D subwavelength structure. The formula reveals an important fact that when
all holes are of the same shape, the Q-factor for any resonance frequency
asymptotically behaves as ${\cal O}(h^{-2})$ for $h\ll1$ with its prefactor
independent of shapes of holes.
