  Birch's law in high pressure physics postulates a linear relationship between
elastic wave speed and density and one of its most well known applications is
in investigations into the composition of the inner core of the Earth using the
Preliminary Reference Earth Model as the primary source of constraints.
However, it has never been subjected to high precision tests even at moderately
elevated temperatures. Here we carry out such a test by making use of the
Density Functional Theory of electronic structure calculation and the Density
Functional Perturbation Theory of calculating the phonon dispersion relation.
We show that a recently proposed modification to the Birch's law is
consistently satisfied more accurately than its original version. This modified
version states that it is the product of elastic wave speed and one-third power
of density that should be a linear function of density. We have studied the
cases of platinum, palladium, molybdenum and rhodium with cubic unit cell and
iron with hexagonal-close-packed unit cell with temperatures up to 1500K and
pressures up to about 360 GPa. We also examine the genericity of the validity
of a recently proposed extension of the Birch's law according to which elastic
wave speed is a linear function of temperature at a given density. Within the
error bars of our calculation, we find that this is consistent with our data
for the four cubic materials at temperatures up to 3300 K.
