Building Acoustics

Waves in fluid and solid media 83

ir t

irt ir

11 2 2

cos cos cos or

cos cos cos cos.

vv v

ppp pp

ZZ Z Z

φφψ

φ φψ ψ

+=

+

−= =

(3.86)

Figure 3.15 Incident wave on a boundary surface between two media having different characteristic impedance.
Specular reflection and transmission into medium two.

In this case, the pressure reflection factor will be

21

21

cos cos

cos cos

p

ZZ

R

ZZ

φ ψ

φ ψ

−

=

+

(3.87)

Comparing with the former Equation (3.77) we find that the surface impedance Zg now is
replaced by Z 2 /cosψ. The requirement that the boundary surface is locally reacting, i.e. ψ
≈ 0, presupposes that the sound speed in medium two must be much lower than in
medium one, which follows from Equation (3.85). Alternatively, we may envisage that
the energy losses in medium two are very large; the attenuation of sound waves along the
boundary surface is so large that there is, practically speaking, a local reaction only.

3.6 Standing waves. Resonance

In section 3.5.1.1, we calculated the total pressure in front of a surface as being the sum
of the pressures in the incident plane wave and the reflected (plane) wave, respectively.
Letting the reflected wave now hit a second surface, a standing wave may appear
between these two surfaces. This may be realized in practice by using a tube with stiff
walls and closed at both ends by some kind of lid. The cross sectional dimensions must
be much smaller than the wavelength but we shall not put any restrictions on the length L
of the tube. For simplicity, we shall assume that the lids closing off the tube are totally

pi

pr

x = 0

y

x

φ

φ

ψ

pt

12

ρ 1 c 1 ρ 2 c 2