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The scaling of earthquake source displacement spectra is analytically studied based on the continuous form of one-dimensional dynamical spring-slider model in the presence of either linearly slip-weakening friction or linearly velocity-weakening friction. The main parameters of the model are the natural angular frequency, wo, and the (dimensionless) decreasing rate, D, of friction with slip (or the characteristic displacement) for slip-weakening friction as well as the (dimensionless) decreasing rate, u, of friction with velocity (or the characteristic velocity) for velocity-weakening friction. The analytic solution includes the complementary and particular parts. The former shows the travelling wave and the latter denotes vibrations at a site. The complementary solution exhibits w-1 scaling in the whole range of w for both friction laws. For the particular solution, slip-weakening friction results in spectral amplitudes only at three values of w. For velocity-weakening friction with u>0.5, the log-log plot of spectral amplitude versus w exhibits almost w0 scaling when w is lower than the corner angular frequency, wc, which is independent on u and increases with wo. When w>wc, the spectral amplitude monotonically decreases with w following a line with a slope value of -1, which is the scaling exponent

A dynamical study of frictional effect on scaling of earthquake source displacement spectra