Frequency-Dependent Friction

With a further generalization, we can consider losses that are dependent on frequency, so that the friction coefficient $\Upsilon$ is replaced by $\Upsilon(\omega_s)$. In such case, all the formulas up to (49) will be recomputed with this new $\Upsilon(\omega_s)$.

Quite often, losses are deduced from experimental data which give the value $V_R(\omega_s)$. In these cases, it is useful to calculate the value of $\Upsilon(\omega_s)$ so that the wave admittance can be computed. From (33) we find

$$\Theta = \frac{\vert V_R\vert}{\sqrt{\frac{\omega_s^2}{c^2} + {V_R}^2}}$$ (51)

and, therefore, from (46) we get

$$\Upsilon(\omega_s) = \frac{2 \vert V_R(\omega_s)\vert}{\omega_s} {\sqrt{\frac{\omega_s^2}{c^2} + {V_R(\omega_s)}^2}} \; .$$ (52)

For instance, in a radius-$a$ cylindrical tube, the visco-thermal losses can be approximated by the formula [45]

$$\vert V_R(\omega_s)\vert = \frac{3.0\times 10^{-5}}{a} \sqrt{\frac{\omega_s}{2 \pi}} \; ,$$ (53)

which can be directly replaced into (53).

In vibrating strings, the viscous friction with air determines a damping that can be represented by the formula [45]

$$\vert V_R(\omega_s)\vert = a_1 \sqrt{\frac{\omega_s}{2 \pi}} + a_2 \; ,$$ (54)

where $a_1$ and $a_2$ are coefficients that depend on radius and density of the string.

<1334>>