Building Acoustics

76 Building acoustics

Figure 3.10 Incident and reflected plane wave at a surface.

By using Equations (3.9) and (3.17) we may express the sound pressure and the particle
velocity in the incident wave as

j( )

ii

i j( )

i

00

(,) ˆ e

ˆ

and ( , ) e ,

tkx

tkx

pxt p

p

vxt

c

ω

ω

ρ

−

−

=⋅

=⋅

(3.64)

where ρ 0 c 0 is the characteristic impedance of the medium. For the reflected wave we get

j( ) j( )

rr i

i j( )

r

00

(,) ˆˆe e

ˆ

and ( , ) e.

tkx tkx

p

tkx

p

pxt p Rp

p

vxt R

c

ωω

ω

ρ

++

+

=⋅ = ⋅

=− ⋅

(3.65)

There will be a change in sign for the wave number due to the change of direction of the
wave. At the same time the particle velocity is changing sign as the gradient of the
pressure is changing sign along with the wave number. The total pressure at the
boundary surface (x = 0) will be

pt p t p t p R(0, )=+=+⋅ir i(0,) (0,) ˆ( (^1) p)ejȦt (3.66)
and the particle velocity:
(^) ir i ()j
00
(0, ) (0, ) (0, ) (^1) p e t.
p
vt v t v t R
c
ω
ρ

=+= −⋅ (3.67)

Inserting these expressions into Equation (3.63) we get

pi

pr Z

g

x = 0