airframe
========

.. automodule:: airframe
	:members:
	:undoc-members:
	:show-inheritance:
	:exclude-members:

theory
------

The Fink method is used to assess airframe noise. The general equation for the mean-square acoustic pressure for the aircraft noise components is:

.. math::

    <p^2>^*_{af,comp} = 10\log{10} \left[ \frac{\Pi^*_{comp}}{4\pi(r_s^*)^2p_{\textrm{ref}}^2} \frac{D_{comp}\theta, \phi) F_{comp}(S_{comp})}{(1 - M_0 cos(\theta))^4} \right]

This equation is applied to the individual noise source contributions (denoted by subscript comp), namely wing trailing edge (including main wing, horizontal and vertical tail), leading edge slat, trailing edge flap and landing gear noise (including main and nose gear). 

Wing trailing edge noise
""""""""""""""""""""""""

The equations in this section are applicable to the main wing, horizontal tail or vertical tail surface (denoted by the subscript x). The acoustic power for trailing edge noise is given by:

.. math::

    \Pi^*_{xte} = K_{xte} M_0^5 \delta_{xte}^*G_x \left(\frac{b_x}{b_w}\right)^2

In this equation, the constant :math:`K_{xte} = 7.075\cdot 10^{-6}` for an aerodynamically clean wing configuration and :math:`K_{xte} = 4.464\cdot 10^{-5}` for non-clean configuration. The boundary layer thickness at the wing trailing edge in this equation is given by:

.. math::

    \delta^*_{xte} = 0.37 \frac{A_x}{b_x^2}\left(\frac{\rho_0 M_0 c_0 A_x}{\mu_0 b_x} \right)^{-0.2}

where :math:`A_w` and :math:`b_w` are the wing area and span of the respective wing element. The spectral distribution function, :math:`F_{xte}(S_{xte})`, is given by:

.. math::

    F_{xte}(S_{xte}) = 
    \begin{cases}
    0.613 (10 S_{xte})^4 [(10S_{xte})^{1.35} + 0.5]^{-4} & \textrm{if delta wing} \\
    0.485 (10 S_{xte})^4 [(10S_{xte})^{1.5} + 0.5]^{-4} & \textrm{if rectangular wing} \\
    \end{cases}

The directivity function, :math:`D_{xte}`, the Strouhal number, :math:`S_{xte}`, and the spectral distribution function, :math:`F_{xte}(S_{xte})`, are given by:

.. math::

    D_{xte} = 4cos^2(\phi) cos^2\left(\frac{\theta}{2}\right)
    
.. math::

    S_{xte}  = \frac{f \delta^*_{xte}b_x}{M_0 c_0} (1 - M_0 cos\theta)


Leading edge slat
"""""""""""""""""

The noise from the leading edge slats has 2 components, because of 2 different noise generating mechanisms. Firstly, an increase in the main wing trailing edge noise, because of the change in boundary layer thickness (*component 1*), and secondly, the noise generated by the slat itself (*component 2*). The noise power of both components is equal to :math:`\Pi^*_{les,1} = \Pi^*_{les,2} = 4.464\cdot 10^{-5} M_0^5 \delta_{wte}^*`, where the boundary layer thickness of the main wing, :math:`\delta_{wte}^*`, is given by the equation in the wing section. The directivity function, :math:`D_{les}`, of both noise components is equal and given by the same equation from the wing section. The spectral distribution function, :math:`F_{les}`, is given by: 

.. math::

    \begin{cases}
    F_{les,1}(S_{les,1}) = 0.613 (10 S_{les}) ^ 4 \cdot [(10S_{les,1}) ^ 1.5 + 0.5] ^{-4} & \textrm{if component 1}\\
    F_{les,1}(S_{les,1}) = 0.613 (2.19 S_{les}) ^ 4 \cdot [(2.19S_{les,2}) ^ 1.5 + 0.5] ^{-4} & \textrm{if component 2}\\
    \end{cases}

where the spectrum functions :math:`S_{les,1} = S_{les,2}` are given by the same equation from the wing section.

Trailing edge flaps
"""""""""""""""""""

The noise power of the trailing edge flaps depends on the number of slots and is given by:

.. math::

    \Pi^*_{af,tef} = 
    \begin{cases}
    2.787e-4  M_0 ^ 6 \frac{A_{flap} }{ b_w ^ 2} sin^2(\delta_{flap}) & \textrm{if $n_{slots}<$  3}  \\
    3.509e-4  M_0 ^ 6 \frac{A_{flap} }{ b_w ^ 2} sin^2(\delta_{flap}) & \textrm{if $n_{slots}$ = 3}            
    \end{cases}

The spectral distribution function, :math:`F_{tef}`, for single, double and triple slotted flaps is given by:

.. math::

	\begin{array}{ll}
	    F_{tef, 1-2 slot} = \begin{cases}
	    0.0480 S_{tef} & \textrm{if $S_{tef} <2$}\\
	    0.1406S_{tef}^{-0.55}& \textrm{if $2\leq S_{tef}\leq20$}\\
	    216.49S_{tef}^{-3}& \textrm{if $20<S_{tef}$}\\
	    \end{cases} 
	    & 
	    F_{tef, 3 slot} = \begin{cases}
	    0.0257S_{tef}& \textrm{if $S_{tef}<2$}\\
	    0.0536S_{tef}^{-0.06525}& \textrm{if $2\leq S_{tef}\leq75$}\\
	    17078S_{tef}^3& \textrm{if $75<S_{tef}$}\\
	    \end{cases}
	\end{array}


The directivity function, :math:`D_{tef}`, and the Strouhal number, :math:`S_{tef}`, are given by: 

.. math::

    D_{tef}(\theta, \phi) = 3(\sin\delta_{flap}\ \cos\theta + \cos\delta_f \ \sin \theta \cos \phi)^2

.. math::

    S_{tef} = \frac{f A_{flap}}{M_0 b_f c_0} (1 - M_0 cos\theta)

Landing gear
""""""""""""

If the landing gear is extracted, the noise for each landing gear (i.e. main gear, nose gear) consists of wheel-noise and strut-noise. The noise power for the landing gear noise is given by:

.. math::

    \begin{cases}
    \Pi^*_{lg,wheel} = 4.349e-4 M_0^6 n_{wheel} \left(\frac{d_{tire}}{b_w}\right)^2 & \textrm{for wheel noise; one/two-wheel landing gear} \\
    \Pi^*_{lg,wheel} = 3.414e-4 M_0^6 n_{wheel} \left(\frac{d_{tire}}{b_w}\right)^2 & \textrm{for wheel noise; one/two-wheel landing gear} \\
    \Pi^*_{lg,strut} = 2.735e-4 M_0^6 \left(\frac{d_{tire}}{b_w}\right)^2\left(\frac{l_{strut}}{d_{tire}}\right) & \textrm{for wheel strut noise} \\
    \end{cases}

The spectral distribution function, :math:`F_{lg}`, for the landing gear noise is given by:

.. math::

    \begin{cases}
    F_{lg, wheel} = 13.59S_{lg}^2 (12.5 + S_{lg}^2)^{-2.25} & \textrm{for wheel-noise; 2-wheel gear} \\
    F_{lg, wheel} = 0.0577S_{lg}^2 (1 + 0.25S_{lg}^2)^{-1.5} & \textrm{for wheel-noise; 4-wheel gear} \\
    F_{lg, strut} = 5.325S_{lg}^2(30+S_{lg}^8){-1} & \textrm{for strut-noise; 2-wheel gear} \\
    F_{lg, strut} = 1.280S_{lg}^3(1.06+S_{lg}^2){-3} & \textrm{for strut-noise; 2-wheel gear} \\
    \end{cases}
    
The directivity function, :math:`D_{lg}`, and the Strouhal number, :math:`S_{lg}`, for the landing gear noise are given by:

.. math::

    \begin{cases}
    D_{lg, wheel} = \frac{3}{2}sin^2 \theta & \textrm{for wheel-noise} \\
    D_{lg, strut} = 3 sin^2\theta sin^2\phi & \textrm{for strut-noise} \\
    \end{cases}

.. math::

    S_{lg} = \frac{f d_{\textrm{tire}}}{M_0 c_0}(1 - M_0 cos\theta)

Combining the aircraft noise components
""""""""""""""""""""""""""""""""""""""""""""

The total mean-square acoustic pressure of the aircraft noise is given by the sum of the individual components:.

.. math::

    <p^2>^*_{af,total} = \sum <p^2>^*_{af,wte} + <p^2>^*_{af,les} + <p^2>^*_{af,tef} + \sum <p^2>^*_{af,lg}

HSR suppression
"""""""""""""""

An airframe suppression factor, :math:`\sigma`, as function of frequency, :math:`f`, and polar directivity angle, :math:`\theta` is applied to the total mean-square acoustic pressure:

.. math::

    <p^2>^*_{af,total, s} = \sigma_{supp}(f, \theta)<p^2>^*_{af,total}