The bulk waves in double-porosity media and the reflection of P1 wave at the free permeable and impermeable surfaces are theoretically derived and solved. The variations of the amplitude reflection coefficients of the reflecting waves with incident angle and frequency under different surface boundaries

as well as the dispersion and attenuation characteristics of the four kinds of bulk waves

are numerically studied. The results show that: (1) P1

P2

P3 and S waves all show dispersion and attenuation characteristics. Therein

P1 wave propagates fastest with lowest attenuation

and P3 wave attenuates fastest with lowest propagation velocity. (2) As the incident angle of P1 wave increases

the amplitude reflection coefficients of the reflecting P2

P3 and S waves increase at first then decrease

but the case of the reflecting P1 wave is just opposite. Moreover

the permeability of the surface has distinctive effect on the amplitude reflection coefficients of P2 and P3 waves and has slight effect on P1 and S waves. Nevertheless

whether the surface is permeable or impermeable

the changing tendency of the amplitude reflection coefficients is consistent. (3) The amplitude reflection coefficients of the reflecting P2 and P3 waves grow as frequency increases. In contrast with the reflecting P3 wave

for the reflecting P2 wave

the amplitude reflection coefficient in the case of free impermeable surface is larger than that in the case of free permeable surface. (4) For the reflecting P1 wave

with the increase of frequency

the amplitude reflection coefficient in the case of free permeable boundary increases firstly and then decreases

but fluctuates (decrease-increase-decrease) in the case of free impermeable surface. The amplitude reflection coefficients of the reflecting S wave just vary oppositely with the reflecting P1 wave.