Chapter 6: Impedance measurements

Acoustic impedance measurements
Fig. 6.19: Free field setup, oblique incidence (from [6]).
This assumption is only valid if the value kr 1 is sufficiently large [10]. The
wave which is travelling from that mirror source in the direction of the
probe is given by:
p
p R , e
0 ω + 2
( ω θ )
i( t kr )
out
= (6.28)
r2
The total pressure at the position (0,d) is given by:
ikr1 ikr2
⎛ e
e ⎞
ptot = pin + pout = p0
⎜ + R ( ω, θ ) ⎟e
⎝ r1 r2
⎠
iωt
(6.29)
The particle velocity, which is normal to the impedance plane, can now be
deduced:
u
tot
1
=
ρ
0
∫
∂ptot
dt
∂x
p ⎛ 1−
ikr 1−
ikr ⎞
= ⎜ e cos( θ ) − R ( ω, θ ) e cos( θ ) ⎟e
ρ0c ⎝ −ikr1 −ikr2
⎠
0 1 ikr1 2 ikr2
iω
t
(6.30)
The impedance at the position of the probe is given by:
ikr1 ikr2
e
e
+ R ( ω,
θ )
p
tot( d , ω ) r1 r2
Z( d , ω ) = = ρ0c
u
tot( d , ω ) ⎛ 1−
ikr ⎞
1 ikr
⎛ 1−
ikr ⎞
1 2 ikr2
⎜ ⎟e cosθ0
− R ( ω, θ ) ⎜ ⎟e cosθ
⎝ −ikr1 ⎠ ⎝ −ikr2
⎠
(6.31)
The reflection coefficient of the absorbing surface can be deduced:
6-22