We next turn to a famous example worked out by the eminent British fluid
dynamicist G. I. Taylor.^{1} In a nuclear explosion there is an essentially instantaneous
release of energy *E* in a small region of space. This produces a spherical
shock wave, with the pressure inside the shock wave thousands of times
greater that the initial air pressure, which may be neglected. How does the
radius *R* of this shock wave grow with time *t*? The relevant governing
variables are *E*, *t*, and the initial air density , with
dimensions , [*t*]=*T*, and . This set
of variables has independent dimensions, so *n*=3, *k*=3. We next determine
the exponents:

so that

with the solution *a*=1/5, *b*=-1/5, *c*=2/5. Therefore

with *C* an undetermined constant. If we plot the radius *R* of the shock as
a function of time *t* on a log-log plot, the slope of the line should be 2/5. The intercept of the graph would provide information about the energy *E* released in the explosion, if the constant *C* could be determined. By
solving a model shock-wave problem Taylor estimated *C* to be about 1; he
was able to take de-classified movies of nuclear tests, and using his model,
infer the yield of the bombs [6]. This data, of course, was
strictly classified; it came as a surprise to the American intelligence
community that this data was essentially publicly available
to those well versed in dimensional analysis.

Footnote:

^{1}
Taylor's name is associated with many phenomena in fluid mechanics: the Rayleigh-Taylor instability, Saffman-Taylor fingering, Taylor cells, Taylor columns, etc.

Thu Jul 10 16:27:59 EDT 1997