Theory

The end correction of an isolated hole is the mass of air set in motion beside it on both sides. When holes sit close together their near fields overlap, so each hole carries less extra mass. The end correction is multiplied by a factor $f_{int}(\sigma) \le 1$ :

Elnady: the short form $1 - 1.47\sqrt\sigma + 0.47\sqrt\sigma^3$ of (Elnady et al., 2010) eq. (11), equation (1); the Elnady impedance model always uses it.
Fok: the Fok function $\psi(\sqrt\sigma)$ , a series in $\xi = \sqrt\sigma$ , equation (2), with the coefficients as listed in Elnady's doctoral thesis (Elnady, 2004) (Paper III, eq. 39). It multiplies the end correction; the thesis notes that an older formulation divided by it.

Equations

f_{int} = 1 - 1.47\sqrt{\sigma} + 0.47\sqrt{\sigma}^{3}

(1)

\psi(\xi) = 1 - 1.4092\,\xi + 0.33818\,\xi^3 + 0.06793\,\xi^5 - 0.02287\,\xi^6 + 0.03015\,\xi^7 - 0.01641\,\xi^8, \qquad \xi = \sqrt{\sigma}

(2)

\zeta_{end} = \frac{i\omega\rho\,\epsilon\,f\,d}{\rho c}, \qquad f \in \{1,\ f_{int},\ \psi\}

(3)

where	meaning	unit
$\sigma$	Open-area ratio	-
$\epsilon$	End-correction factor (End-correction factor (thin plate))	-
$d$	Hole diameter	m

Default and rule

Default: none
Rule: None ( $f = 1$ , the isolated-hole end correction).
Source: (Elnady et al., 2010) eq. (11); the Fok series as listed in Elnady's doctoral thesis (Elnady, 2004), Paper III eq. 39.

Valid range none | Elnady short form | Fok function (thin-plate physics baseline only)

Effect

At σ = 0.1 the Elnady factor is 0.55 and the Fok function 0.57: the end correction roughly halves.

Applies to

The thin-plate physics baseline only.

Hole interaction (thin plate)

Theory

Equations

Default and rule

Valid range none | Elnady short form | Fok function (thin-plate physics baseline only)

References 2

Theory

Equations

Default and rule

Valid range none | Elnady short form | Fok function (thin-plate physics baseline only)

Related 5

References 2