  We study the effect of tidal forcing on gravitational wave signals from
tidally relaxed white dwarf pairs in the LISA, DECIGO and BBO frequency band
($0.1-100\,{\rm mHz}$). We show that for stars not in hydrostatic equilibrium
(in their own rotating frames), tidal forcing will result in energy and angular
momentum exchange between the orbit and the stars, thereby deforming the orbit
and producing gravitational wave power in harmonics not excited in perfectly
circular synchronous binaries. This effect is not present in the usual
orbit-averaged treatment of the equilibrium tide, and is analogous to transit
timing variations in multiplanet systems. It should be present for all LISA
white dwarf pairs since gravitational waves carry away angular momentum faster
than tidal torques can act to synchronize the spins, and when mass transfer
occurs as it does for at least eight LISA verification binaries. With the
strain amplitudes of the excited harmonics depending directly on the density
profiles of the stars, gravitational wave astronomy offers the possibility of
studying the internal structure of white dwarfs, complimenting information
obtained from asteroseismology of pulsating white dwarfs. Since the vast
majority of white-dwarf pairs in this frequency band are expected to be in the
quasi-circular state, we focus here on these binaries, providing general
analytic expressions for the dependence of the induced eccentricity and strain
amplitudes on the stellar apsidal motion constants and their radius and mass
ratios. Tidal dissipation and gravitation wave damping will affect the results
presented here and will be considered elsewhere.
