It is well known that Poisson's
ratio for all types of structures increases with pressure, so
the shear velocity must increase more slowly with pressure than the
longitudinal velocity. This phenomenon can be explained
in terms of both lattice dynamics and finite strain [ Anderson, 1995,
Chapter 9]. Stacey [1995] has quantified
this idea for iron with an empirical
relationship between the shear modulus, 
,
and the bulk modulus, 
, given by 
,
where 
and 
are positive constants determined by
the second derivative of the potential function. This formula
has physical significance when transformed into
where 
is the value
of 
at infinite pressure (the parameter in the Keane EoS (see
Anderson, [1995], p. 175). Stacey pointed out that the PREM data
on the inner core fit this equation quite well, and that 
is
defined as the limit when 
; that is, when
Poisson's ratio is 0.5. The Keane equation suggests that 
and
asymptotically approach the limit 
. Thus 
must
monotonically decrease with P, a feature outside the resolution
of 
in PREM. W.W. Anderson and Ahrens [1989]
noted that 
for pure iron is 10% higher
than for PREM, in spite of good agreement between 
of
pure iron and 
of PREM. For the outer core, PREM yields
values of 
near 
, but the analysis of Stacey [1995] shows that this
is a numerical artifact---a kind of average value, and that 
really has to obey a linear law in P, which descends
from about 5 at 
to
3.3 as 
.