The dust lattice waves in a one-dimensional Debye binary dust chain consisting of two distinct dust particle species with different charges and masses are investigated. It is found that there are two branches for both longitudinal and transverse modes, namely the optical mode of high frequency and the acoustic mode of low frequency, which will be merged into one ordinary longitudinal (transverse) mode of single dust chain. The influence of the parameters, i.e., the dimensionless lattice parameter α, the mass ratio σ, and the charge ratio ε of the two particles, on the dispersion relation of longitudinal and transverse waves is discussed. Furthermore, the branching and the merging of longitudinal and transverse waves are discussed in detail.