  There have been many studies of the instability of a flexible plate or flag
to flapping motions, and of large-amplitude flapping. Here we use inviscid
simulations and a linearized model to study more generally how key quantities
-- mode number (or wavenumber), frequency, and amplitude -- depend on the two
dimensionless parameters, flag mass and bending stiffness. In the limit of
small flag mass, flags perform traveling wave motions that move at nearly the
speed of the oncoming flow. The flag mode number scales as the -1/4 power of
bending stiffness. The flapping frequency has the same scaling, with an
additional slight increase with flag mass in the small-mass regime. The
flapping amplitude scales approximately as flag mass to the 1/2 power. For
large flag mass, the dominant mode number is low (0 or 1), the flapping
frequency tends to zero, and the amplitude saturates in the neighborhood of its
upper limit (the flag length). In a linearized model, the fastest growing modes
have somewhat different power law scalings for wavenumber and frequency. We
discuss how the numerical scalings are consistent with a weakly nonlinear
model.
