To propagate both longitudinal and transverse waves, a material must have

In transverse waves, the particle motion is normal to the direction of the wave. Therefore, as the wave propagates each element of the medium undergoes shearing strain. In longitudinal waves, the constituents of the medium oscillate about their mean position along the direction of wave propagation like propagation of sound wave. This involves compressive stress (pressure). Thus, transverse waves be propagated in medium which can sustain shearing stress. Similarly, longitudinal waves can be propagated in medium which can sustain bulk modulus.