Waves in fluid and solid media 75
basis of the intensity in the two waves. As the intensity both in a plane and a spherical
wave is proportional to the sound pressure squared this reflection factor will be equal to
|Rp|^2. The part of the incident energy being lost in the reflection process, i.e. 1 - |Rp|^2 , is
called the absorption factor having the symbol α
2
α=−1.Rp (3.62)Another characteristic quantity to characterise a boundary surface is what we shall
denote surface impedance Zg defined as
(^) g
n boundary
ˆ
ˆ
p
Z
v⎛⎞
=⎜⎟
⎝⎠
(3.63)
The quantity vn is the component of the particle velocity normal to the boundary surface.
In the example mentioned above, in connection with boundary surfaces, the velocity vn
would be equal to the velocity of the boundary surface. The surface impedance may be
considered as a variant of the general quantity specific impedance defined in Equation
(3.18). A similar quantity is denoted transmission impedance or more commonly wall
impedance, as one normally will use it as a characteristic for the wall surfaces in a room.
However, in this case the quantitypˆis the pressure difference between the two sides, not
only the total pressure on one side.
In the following sections, we shall derive expressions for the reflection and
absorption factors assuming that the boundary surface is characterized by the surface
impedance Zg. We shall restrict our derivation to plane waves and, in the first place,
assume that the wave is incident normally on the surface. At oblique incident there will
be an important distinction whether the surface impedance will be a function of the angle
of incidence or not. In the latter case, the surface is called locally reacting, which means
that we need not consider in-plane wave propagation. This implies that the normal
component of the particle velocity at a given point on the surface depends on the sound
pressure at this point only. In other words, pressure on the surface at a certain point
causes no movement elsewhere on the surface. In practice, this is a reasonable
assumption for many types of porous absorber, at least in the lower frequency range but
in general it may be difficult to decide whether an absorber may be treated as locally
reacting or not. One may of course prevent sound propagation along the surface by
subdividing the absorber using a lattice of some kind, e.g. a honeycomb core structure
but such solutions may not be desirable due to other requirements.
3.5.1 Sound incidence normal to a boundary surface
We shall assume that a plane wave is incident normally on a boundary surface, which is
coincident with the plane having the coordinate x = 0 (see Figure 3.10).