Editing Dimensional analysis (section)

=== A more complex example: energy of a vibrating wire ===
Consider the case of a vibrating wire of [[length]] {{math|''ℓ''}} (L) vibrating with an [[amplitude]] {{math|''A''}} (L).  The wire has a [[linear density]] {{math|''ρ''}} (M/L) and is under [[Tension (physics)|tension]] {{math|''s''}} (LM/T<sup>2</sup>), and we want to know the energy {{math|''E''}} (L<sup>2</sup>M/T<sup>2</sup>) in the wire.  Let {{math|''π''<sub>1</sub>}} and {{math|''π''<sub>2</sub>}} be two dimensionless products of [[Power (mathematics)|powers]] of the variables chosen, given by
: <math>\begin{align}
  \pi_1 &= \frac{E}{As} \\
  \pi_2 &= \frac{\ell}{A}.
\end{align}</math>

The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
: <math>F\left(\frac{E}{As}, \frac{\ell}{A}\right) = 0 ,</math>
where {{math|''F''}} is some unknown function, or, equivalently as
: <math>E = As f\left(\frac{\ell}{A}\right) ,</math>
where {{math|''f''}} is some other unknown function.  Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension.  Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function&nbsp;{{math|''f''}}.  But our experiments are simpler than in the absence of dimensional analysis.  We'd perform none to verify that the energy is proportional to the tension.  Or perhaps we might guess that the energy is proportional to&nbsp;{{math|''ℓ''}}, and so infer that {{math|1=''E'' = ''ℓs''}}.  The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.

The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex.  Consider, for example, a small pebble sitting on the bed of a river.  If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water.  At what critical velocity will this occur?  Sorting out the guessed variables is not so easy as before.  But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a [[dimensionless number]] such as the [[Reynolds number]], which may be interpreted by dimensional analysis.