  This manuscript puts forward and verifies an analytical approach to design
phononic crystals that feature a bandgap at the lowest possible frequencies.
This new approach is verified against numerical optimization. It rests on the
exact form of the trace of the cumulative transfer matrix. This matrix arises
from the product of N elementary transfer matrices, where N represents the
number of layers in the unit cell of the crystal. The paper presents first a
new proof of the harmonic decomposition of the trace (which, unlike the
original derivation, does not resort to group-theoretical concepts), and then
goes to demonstrate that the long-wavelength asymptotics of the function that
governs the dispersion relation can be described in closed form for any
layering, plus that it possesses a simple and explicit form that facilitates
its study. Using this asymptotic result for low frequencies, we then propose an
analytical curvature-minimization approach to devise layerings that yield an
opening of the first frequency bandgap at the lowest possible frequency value.
Finally, we also show that minimizing the frequency at which the first gap
opens can be at odds with obtaining broadband attenuation, i.e., maximizing the
width of the first frequency bandgap.
