Porous frame excitation by point sources in air and by stress circular and line sources –modes of air saturated porous frames - Propagation of Sound in Porous Media - page 168

(Melon et al . 1998, Park 2005, Pritz 1986, 1994; Sfaoui 1995a, b, Langlois et al . 2001,
Tarnow 2005), but the rigidity coefficients can present noticeable variations between the
quasi-static regime and the audible frequency range.

8.2 Prediction of the frame displacement

8.2.1 Excitation with a given wave number component parallel
to the faces
In Figure 8.1(a), a plane acoustic field in air impinges upon a porous layer of finite
thickness l at an angle of incidence θ . The layer is bonded onto a rigid impervious
backing. The geometry of the problem is two-dimensional in the incident plane. The
total pressure field p (incident plus reflected) at the surface of the layer is given by
p p e exp (

k 0 sin θ .
The first representation of the Biot theory is used in this chapter. Three Biot waves
can propagate in the porous layer, toward the backing and toward the air - porous mate-
rial boundary. In Figure 8.1(b), the incident plane wave is replaced by a normal stress
field τ zz = exp ( − jξx) , with the same dependence on x ,appliedtothesurface.Ina
first step, the Biot theory is used to predict the radial and the vertical displacement
components u s x and u s z of the porous frame created by a plane wave in air with a hor-
izontal (trace) wave number ξ .Let k 1 and k 2 be the wave numbers of the two Biot
compressional waves, and k 3 the wave number of the shear wave. Each plane wave is
related to a displacement u

= − jk x x) , the time dependence exp( jωt ) is discarded and k x = ξ =

s .Let N be the
shear modulus and ν the Poisson ratio of the frame. Let P,Q ,and R be the Biot rigid-
ity coefficients. The quantities k 1 , k 2 , k 3 , µ 1 , µ 2 , µ 3 , P , Q ,and R , are described in
Chapter 6.
The excitation by the pressure field or the stress field is related to Biot waves prop-
agating up and down the layer with the same horizontal wave number component. The
superscript + will be used for waves propagating from the upper to the lower face
and the superscript - for waves propagating in the opposite direction. Let ϕ i ,i = 1 , 2
be the velocity potentials of the compressional waves, with the related displacement
u

s f of air, with u f

of the frame and u = µ u

s

± = n ϕ 3

=− j ∇ ϕ/ω ,andlet

be the vector potentials, n being the unit vector on

the y axis and u s =− j ∇ ∧ /ω the related frame displacement. The scalar functions

q

s
zz

t

x

porous layer porous layer

z

Figure 8.1 Two sources inducing Biot wave in a porous medium: (a) incident plane
field, (b) normal stress field.