168 Building acoustics
5.4.1.3 Acoustical resistance
To calculate energy losses due to viscous forces, we have to modify the simple equation
of force, the Euler equation, to include the effects of such forces. The linearized Navier-
Stokes equation of force may be written as
(^0)
4
()
3
p
tρμ μ∂
∇ =− + ∇ ∇⋅ − ∇×∇×
∂
v
vv, (5.24)where we now have got two additional terms both containing the coefficient of viscosity
μ. This coefficient is approximately equal to 2⋅ 10 -5 kg/(m⋅s) for air.
To carry out calculations on absorbers where perforated plates are an element, we
shall have to predict the inherent viscous losses by sound propagation through
perforations such as holes or slits. For the former and also for calculation on the single
Helmholtz resonator in the next section, we shall start looking at thin tubes of circular
cross section and calculate the sound propagation in the axis direction, again assuming
harmonic time dependence. The equation of force may then be simplified to
j 0 x x
p v
vr.
x rr rμ
ρω∂∂⎛∂
=− + ⎜⋅
∂∂⎝⎠
⎞
∂⎟ (5.25)
The variable vx is the component of the particle velocity in the axial direction, and r the
radius vector (cylindrical coordinates). Assuming that the velocity is zero at the tube
walls, i.e. when r is equal to the radius a, the solution to the equation is (see Allard
(1993))
00
001 J( ) j
1,where
jJ()xp qr
vq
xqa.
ωρ
ωρ μ∂ ⎛⎞ −
=− ⎜⎟− =
∂ ⎝⎠
(5.26)
The symbol J 0 indicates a Bessel function of the first kind and zero order. From this
equation we may now calculate the mean particle velocity in a cross section of the tube.
We get
()
()
0 1
2(^00)
2d
12 Jj
1
j jJj
a
x
x
vrr
p s
v
asx s
π
π ωρ
⋅ ⎛⎞
∂ −
==−−⎜
∂ ⎜⎜⎟− −
⎝⎠
∫
⎟,
⎟
(5.27)
where
sa^0.
ωρ
μ= (5.28)
We may rewrite this expression as
jx ,
p
v
xωρ∂
=−
∂
(5.29)
where ρ is the effective density of the air in the tube, given by