On the measurement of virtual height

A time dependent definition of the virtual height of a reflected wave train is suggested. This definition is such that its accuracy increases as the width of the incident pulse increases. Moreover, it is theoretically possible to obtain an estimate of the virtual height with as small an error as desired, no matter what the nature of the reflecting medium. Suppose an incident pulse having width
W
and a carrier frequency
\omega_{0}
produces a reflected wave which, when measured at a fixed point
x={0}
in space, is
R(\omega_{0}, w, t)
. The suggested definition of the virtual height
h'(\omega_{0})
is
h'(\omega_{0}) = lim \min{\omega\rightarrow\infty}c\{\int\min{-\infty}\max{\infty} t\|R(\omega_{0},w,t)\|^{2}dt/ \int\min{-\infty}\max{\infty}\|R(\omega_{0},w,t)\|^{2}dt -\int\min{-\infty}\max{\infty}\|I(\omega_{0},w,t)\|^{2}dt/\int\min{-\infty}\max{\infty}\|I(\omega_{0},w,t)\|^{2}dt},
where
c
is the free space velocity of light. This relation for
h'(\omega_{0})
holds for any physically reasonable incident wave train and is independent of its envelope shape. An example is given of a reflected wave whose virtual height cannot be determined by inspection in the usual manner. The expression given here for the virtual height provides the correct result.