The cut-off phenomenon of longitudinal waves propagating normal to the layers of composite is studied experimentally by measuring the attenuation of the ultrasonic waves. Present method is equivalent to measuring the imaginary part of the dispersion relation, while the conventional methods that treat the relation between phase velocity and frequency are equivalent to measuring the real part of the relation. Constituents of composite specimens are elastic or viscoelstic, and it is shown that the effect of constituents on cut-off phenomenon is in good agreement with that predicted theoretically. For viscoelastic composite, it is clarified that the extreme attenuation observed in a specific frequency is not due to its viscosity but mainly to cut-off phenomeon.