
The Ffowcs Williams and Hawkings (FWH) equation is essentially an inhomogeneous wave equation that can be derived by manipulating the continuity equation and the NavierStokes equations. The FWH [ 43, 105] equation can be written as:
where  
=  fluid velocity component in the direction  
=  fluid velocity component normal to the surface  
=  surface velocity components in the direction  
=  surface velocity component normal to the surface  
=  Dirac delta function  
=  Heaviside function 
is the sound pressure at the far field ( ). denotes a mathematical surface introduced to "embed'' the exterior flow problem ( ) in an unbounded space, which facilitates the use of generalized function theory and the freespace Green function to obtain the solution. The surface ( ) corresponds to the source (emission) surface, and can be made coincident with a body (impermeable) surface or a permeable surface off the body surface. is the unit normal vector pointing toward the exterior region ( ), is the farfield sound speed, and is the Lighthill stress tensor, defined as
(21.22) 
is the compressive stress tensor. For a Stokesian fluid, this is given by
(21.23) 
The freestream quantities are denoted by the subscript .
The solution to Equation 21.21 is obtained using the freespace Green function ( ). The complete solution consists of surface integrals and volume integrals. The surface integrals represent the contributions from monopole and dipole acoustic sources and partially from quadrupole sources, whereas the volume integrals represent quadrupole (volume) sources in the region outside the source surface. The contribution of the volume integrals becomes small when the flow is low subsonic and the source surface encloses the source region. In FLUENT, the volume integrals are dropped. Thus, we have
where
where
(21.27)  
(21.28) 
When the integration surface coincides with an impenetrable wall, the two terms on the right in Equation 21.24, and , are often referred to as thickness and loading terms, respectively, in light of their physical meanings. The square brackets in Equations 21.25 and 21.26 denote that the kernels of the integrals are computed at the corresponding retarded times, , defined as follows, given the observer time, , and the distance to the observer, ,
(21.29) 
The various subscripted quantities appearing in Equations 21.25 and 21.26 are the inner products of a vector and a unit vector implied by the subscript. For instance, and , where and denote the unit vectors in the radiation and wallnormal directions, respectively. The dot over a variable denotes sourcetime differentiation of that variable.
Please note the following remarks regarding the applicability of this integral solution: