The theory of the propagation, and in particular of the attenuation, of first and second sound in He II is considered in greater detail than has been done hitherto, using the two-fluid model as a basis. In addition to the waves of first and second sound there can occur near material boundaries two other kinds of waves: one which has an almost purely imaginary wave number and hence is aperiodic in space with exponential damping, reducing it to small proportions over a distance of 10 -7 m or less; the other of the character of the viscosity waves, also encountered in ordinary fluids and strongly damped over the distance of one wave length. All four types of waves can most conveniently be derived from velocity potentials and are in general necessary to fulfill the boundary conditions. The behaviour of these waves when the temperature passes the λ-point or approaches absolute zero is investigated. The experimental evidence is discussed in the light of the results obtained.