Editing Strouhal number (section)

== Derivation ==
Knowing [[Newton's laws of motion#Second|Newton's second law]] stating force is equivalent to mass times acceleration, or <math>F=ma</math>, and that acceleration is the derivative of velocity, or <math>\tfrac{U}{t}</math> (characteristic speed/time) in the case of fluid mechanics, we see

:<math> F=\dfrac{mU}{t}</math>,

Since characteristic speed can be represented as length per unit time, <math>\tfrac{L}{t}</math>, we get

:<math> F=\dfrac{mU^2}{L}</math>,

where,
: ''m'' = mass,
: ''U'' = characteristic speed,
: ''L'' = characteristic length.

Dividing both sides by <math>\tfrac{mU^2}{L}</math>, we get

:<math> \tfrac{FL}{mU^2}=1=\text{constant}</math> ⇒ <math>\tfrac{mU^2}{FL}=1=\text{constant}</math>,

where,
: ''m'' = mass,
: ''U'' = characteristic speed,
: ''F'' = net external forces,
: ''L'' = characteristic length.

This provides a dimensionless basis for a relationship between mass, characteristic speed, net external forces, and length (size) which can be used to analyze the effects of fluid mechanics on a body with mass.

If the net external forces are predominantly elastic, we can use [[Hooke's law]] to see

:<math> F=k\Delta L</math>,

where,
: ''k'' = spring constant (stiffness of elastic element),
: ''ΔL'' = deformation (change in length).

Assuming <math>\Delta L\propto L</math>, then <math>F\approx kL</math>. With the natural resonant frequency of the elastic system, <math>\omega_0^2</math>, being equal to <math>\tfrac{k}{m}</math>, we get

:<math> \dfrac{mU^2}{FL}=\dfrac{mU^2}{kL^2}=\dfrac{U^2}{\omega_0^2L^2}</math>,

where,
: ''m'' = mass,
: ''U'' = characteristic speed,
: ''<math>\omega_0</math>'' = natural resonant frequency,
: ''ΔL'' = deformation (change in length).

Given that cyclic motion frequency can be represented by <math>f=\tfrac{\omega_0^2L}{U}</math> we get,

:<math>\dfrac{U^2}{\omega_0^2L^2}=\dfrac{U}{fL}=\text{constant}=\dfrac{fL}{U}=\text{St (Strouhal Number)}</math>,

where,
: ''f'' = frequency,
: ''L'' = characteristic length,
: ''U'' = characteristic speed.