Can we hear beats with pulsar timing arrays?

An isolated supermassive black hole binary (SMBHB) produces an identical cross-correlation pattern of pulsar timings as an isotropic stochastic background gravitational waves (GWs) generated possibly by inflation. Can there remain the identical cross-correlation pattern in the presence of a secondary SMBHB? To address this issue, the present paper focuses on GWs with similar amplitudes but slightly different frequencies $f_1$ and $f_2$ coming from two different directions. Beats between the two GWs can modify angular correlation patterns. The beat-induced correlation patterns are not stationary but modulated with a beat frequency $f_{beat}\equiv \mathrm{\mid }{f}_1-f_2\mathrm{\mid }$. We obtain an analytic solution that allows us to infer $f_{beat}$ from the modulated angular correlations.