Building Acoustics

58 Building acoustics

adiabatically. This implies that the changes happen so fast that the temperature exchange
with the surrounding medium is negligible. This is opposed to wave propagation in
capillary tubes or generally in porous media, a theme we will treat in Chapter 5.
Starting from the general adiabatic gas equation

PV⋅=γ constant, (3.12)

where P and V are the pressure and volume of the gas, respectively and where γ is the
adiabatic constant (≈ 1.4 for air), we may show that Equation (3.4) gives

2 0

0

.

P

pc

γ

ρ ρ

ρ

= ⋅= ⋅ (3.13)

This also implies that the phase speed is proportional to the absolute temperature T (°K)
because we have

0 0

0

.

P

cT

γ

ρ

=∝ (3.14)

Due to our application here we have here added an index zero to the phase speed. In the
literature several approximate expressions may be found. A more accurate one is:

ct 0 (air)=20.05⋅+273.2 , (3.15)

where the temperature t is given in degree Celsius (°C). When it comes to the particle
velocity v, dealing with linear acoustics, it is implicitly assumed that its absolute value is
much less than the phase speed. That the assumption is fulfilled for the sound pressures
normally experienced in our daily life is illustrated in the example below. (We disregard
sound pressure levels that may even briefly damage our hearing). For the one-
dimensional case we may write Equation (3.3) as

0 x,

p v

x t

ρ

∂ ∂

=−

∂ ∂

(3.16)

which for a harmonic time dependency gives

0

00

j j and thereby

.

x

x

kp v

pcv

ωρ

ρ

−⋅=− ⋅

=⋅

(3.17)

The quantity ρ 0 c 0 is the characteristic impedance of the medium, and it is a special case
of the specific acoustic impedance defined by

(^) s
Pa s
.
m
p
Z
v

⎛⎞⋅

= ⎜⎟

⎝⎠

(3.18)

Example What is the magnitude of the particle velocity at a sound pressure of 1.0 Pa,
being equivalent to a sound pressure level of ≈ 94 dB?