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.
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