Waves in fluid and solid media - Building Acoustics - page 72

force input. In connection with sound sources it will be useful to introduce yet another
(mechanical) impedance concept, the radiation impedance . This represents the ratio of
the reaction force of the fluid medium on the source, i.e. caused by the motion of the
source, and the source velocity. Denoting this reaction force F r and the source velocity u
we write

F

r

Z ==+⋅ RX j . (3.53)

r r r

u

Using the piston source in the last section, assuming that it has mechanical impedance Z m
(in vacuum) and driven by a force F , the radiation impedance will be coupled in series
with the mechanical impedance. The velocity of the piston will therefore be

F

u = . (3.54)

Z + Z

mr

The real part R r of the radiation impedance will give us the power radiated by the source
so we may in general write

1 1

{ } { } { }

2 2

W = Re F ⋅ u * = Re Z ⋅ ⋅ u u * = u ⋅ Re Z = u ⋅ R . (3.55)

radiated r r r r

2 2

Using the monopole as an example we immediately get by using Equation (3.40) that

22 22 22

ρω S

kS kS

ka << 1

0

Rc = ρ ⎯⎯⎯→ ρ c = . (3.56)

r 0 0 0 0

22

4(1 π + ka ) 4 π 4 π c

0

The imaginary part of the radiation impedance will on the other hand represent a load on
the source, which, in many cases, may act as a contribution to the mechanical mass of the
source. The radiation impedance is therefore an important factor in a number of different
cases, not only when considering vibration of solid surfaces, but also generally when a
vibrating column of air brings about sound radiation. There is a diverse range of
examples one may mention here, ranging from sound radiation from musical instruments
to resonance sound absorbers; see Chapter 5 and further on to sound transmission
through holes and slits in wall or floors, see Chapter 8.
Finally, we shall consider the radiation impedance of a piston source where we
expect that the result for low frequencies will be of the same form as for a monopole.
However, in this case the impedance will attain quite another complexity. When
calculating Z r we shall have to use the general Equation (3.46) but in this case we must
calculate the pressure on the surface of the piston. Specifically, the pressure p on a
surface element d S´ is induced by the sum of the movements by all the other elements d S .
To arrive at the total pressure on the piston we therefore have to perform yet another
integration, namely over the elements d S´ . We shall not present this derivation, which
may be simplified by using the principle of reciprocity outlined in section 3.6, but the
end results are important and shall be commented upon. The radiation impedance for a
piston placed in a baffle may be written (for a derivation see e.g. Kinsler et al. (2000)).