Plasma waves in a Fermi-degenerate quantum plasma are studied in the framework of the Vlasov-Poisson self-consistent-field theory. A complete time-dependent analytical solution of the initial-value problem is obtained for a multistream model both by stationary-wave and Laplace-transform methods. In the continuum limit, the excitation spectrum can be expressed by the imaginary part of the response function to the initial perturbations. The relaxation of plasma waves is discussed for one-dimensional systems with both Fermi and Maxwellian statistics. Apart from the usual exponential Landau damping, regimes of sub- and superexponential damping can be identified due to the phase relaxation of single-particle excitations. In addition, beat waves and echoes are discussed.