  The Whitham equation was proposed as a model for surface water waves that
combines the quadratic flux nonlinearity $f(u) = \tfrac{1}{2}u^2$ of the
Korteweg-de Vries equation and the full linear dispersion relation $\Omega(k) =
\sqrt{k\tanh k}$ of uni-directional gravity water waves in suitably scaled
variables. This paper proposes and analyzes a generalization of Whitham's model
to unidirectional nonlinear wave equations consisting of a general nonlinear
flux function $f(u)$ and a general linear dispersion relation $\Omega(k)$.
Assuming the existence of periodic traveling wave solutions to this generalized
Whitham equation, their slow modulations are studied in the context of Whitham
modulation theory. A multiple scales calculation yields the modulation
equations, a system of three conservation laws that describe the slow evolution
of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly
nonlinear limit, explicit, simple criteria in terms of general $f(u)$ and
$\Omega(k)$ establishing the strict hyperbolicity and genuine nonlinearity of
the modulation equations are determined. This result is interpreted as a
generalized Lighthill-Whitham criterion for modulational instability.
