Every plume-rise and dispersion equation in this chapter divides by $u$, the wind speed at stack height. Every anemometer in the country stands at 10 m. That gap is a whole calculation, and skipping it is one of the quiet ways a dispersion assessment goes wrong.

The power law is $u_2 = u_1\left(\dfrac{z_2}{z_1}\right)^{p}$ — read aloud u-two equals u-one, z-two over z-one, to the p. The subscript convention, said once: 1 is where you MEASURED, 2 is where you WANT. So $u_1$ is the measured wind in m/s, $z_1$ the height it was measured at (10 m, essentially always), $z_2$ the height wanted, $u_2$ the answer in m/s, and $p$ the profile exponent, a bare number with no units.

$p$ is the ground's fingerprint. About 0.10 over open water, 0.15 over open country, 0.25 in the suburbs and 0.35 in a city centre — rougher ground drags the low air harder, so the profile is steeper and there is more to gain by climbing. Stable air raises it further still. Reading the wrong row of that table is not a rounding error: over 70 m it moves the answer by half again.

Two things fall out of the shape. First, the ratio $z_2/z_1$ is a bare number, so both heights may be in any unit as long as it is the SAME unit. Second, the exponent is well under 1, which means the profile bends over: the first ten metres cost the most and the rest is gained slowly. An 80 m stack over farmland sees about 1.37 times the gatehouse wind, not eight times it.

And the law runs backwards. With two anemometers at two heights, $p = \dfrac{\ln(u_2/u_1)}{\ln(z_2/z_1)}$ fits the site's own exponent instead of borrowing one from a table — which is what a serious assessment does. Either logarithm base works, natural or base ten, provided it is the same base above and below.