Waves in fluid and solid media 81
(^222)
stat
00
2()sincosd21Rp sincosd.
ππ
α ==−αφ φ φ φ ⎡⎤φ φ φ
∫∫⎢⎥⎣⎦
(3.79)
Inserting for Rp according to Equation (3.77) we get
(^) ( )
() ()
22
2
stat 22 21
81 ln12 Arctg.
1zz zz z
zz
zz zzzα′′⎡⎤′′′− ′′
=⋅ − ⋅ + + + ⋅⎢⎥′ ⋅
⎢⎥′′′+
⎣⎦
(3.80)
The symbol z is the surface impedance normalised by the characteristic impedance Z 0 of
the medium, i.e.
gg
00
jRe jIm.ZZ
zz z
ZZ⎧ ⎫⎧⎫
=+⋅=′′′⎨ ⎬⎨⎬+⋅
⎩⎭ ⎩⎭
(3.81)
Figure 3.14 shows the average value αstat as a function of the normalized surface
impedance. A comparison with Figure 3.11 generally shows that the statistical absorption
coefficient is higher than the normal incidence factor, but also that the absolute
maximum is slightly lower; (αstat)max ≈ 0.95 at z' ≈ 1.6.
Figure 3.14 Statistical absorption factor as a function of the normalized impedance components, (Z = Zg/Z 0 ).
3.5.3 Oblique sound incidence. Boundary between two media
A general treatment of the case of plane wave’s incident on a locally reacting surface was
given in the previous section. Implicitly, this means that we presuppose the impedance Zg
0 5 10 15 20-15-10-5051015Real ZImag Z0.864 0.605 0.519 0.432 0.3460.2590.0864 0.173