  We consider the existence of periodic traveling waves in a bidirectional
Whitham equation, combining the full two-way dispersion relation from the
incompressible Euler equations with a canonical shallow water nonlinearity. Of
particular interest is the existence of a highest, cusped, traveling wave
solution, which we obtain as a limiting case at the end of the main bifurcation
branch of $2\pi$-periodic traveling wave solutions continuing from the zero
state. Unlike the unidirectional Whitham equation, containing only one branch
of the full Euler dispersion relation, where such a highest wave behaves like
$|x|^{1/2}$ near its crest, the cusped waves obtained here behave like
$|x\log|x||$. Although the linear operator involved in this equation can be
easily represented in terms of an integral operator, it maps continuous
functions out of the H\"older and Lipschitz scales of function spaces by
introducing logarithmic singularities. Since the nonlinearity is also of higher
order than that of the unidirectional Whitham equation, several parts of our
proofs and results deviate from those of the corresponding unidirectional
equation, with the analysis of the logarithmic singularity being the most
subtle component. This paper is part of a longer research programme for
understanding the interplay between nonlinearities and dispersion in the
formation of large-amplitude waves and their singularities.
