  A homogeneous isotropic compressible Hadamard material has the property that
an infinitesimal longitudinal homogeneous plane wave may propagate in every
direction when the material is maintained in a state of arbitrary finite static
homogeneous deformation. Here, as regards the wave, 'homogeneous' means that
the direction of propagation of the wave is parallel to the direction of
eventual attenuation and 'longitudinal' means that the wave is linearly
polarized in a direction parallel to the direction of propagation. In other
words, the displacement is of the form u = n cos k(n.x - ct), where n is a real
vector.
  It is seen that the Hadamard material is the most general one for which a
'longitudinal' inhomogeneous plane wave may also propagate in any direction of
a predeformed body. Here, 'inhomogeneous' means that the wave is attenuated in
a direction distinct from the direction of propagation and 'longitudinal' means
that the wave is elliptically polarized in the plane containing these two
directions. In other words, the displacement is of the form u = Re {S exp[i
(omega S.x - ct)]}, where S is a complex vector (or bivector).
  Then a Generalized Hadamard material is introduced. It is the most general
homogeneous isotropic compressible material which allows the propagation of
infinitesimal 'longitudinal' inhomogeneous plane circularly polarized waves for
all choices of the isotropic directional bivector.
  Finally, the most general forms of response functions are found for
homogeneously deformed isotropic elastic materials in which 'longitudinal'
inhomogeneous plane waves may propagate with a circular polarization in each of
the two planes of central circular section of the B^n-ellipsoid, where B is the
left Cauchy-Green strain tensor corresponding to the primary pure homogeneous
deformation.
