110 Building acoustics
In addition to the parameters given above, there are others related to our binaural
hearing, based on measurements using an artificial or dummy head. These so-called
inter-aural cross correlation measures are correlated to the subjective quality of “spatial
impression”.
4.4 Wave theoretical models
Obtaining analytical solutions to the wave equation (4.1) are difficult except in cases
where the room has a simple shape and simple boundary conditions. In section 3.6, we
arrived at a solution for the sound field in a simple one-dimensional case: a tube closed
in both ends and with stiff walls where we assumed that the particle velocity everywhere
was equal to zero. We may easily generalize these results to the three-dimensional case if
we assume that the room has a rectangular shape with dimensions Lx, Ly and Lz. We shall
use this as an example to illustrate some important properties of sound fields in rooms;
how the impulse response will depend on e.g. the room dimensions and furthermore, how
we may predict the impulse responses.
For a free wave field we shall have to solve the wave equation without the source
term. Assuming harmonic time dependence, we get the Helmholtz equation for the sound
pressure in three-dimensional form
∇^22 pkp+=0, (4.9)
where k is the wave number. Initially, we shall assume that all boundary surfaces are
infinitely stiff and there are no other energy losses in the room. The eigenfunctions for
the pressure will then be given by
(^) pnnnxyz(xyz C, , )=⋅cos(kxx)⋅cos(k yy)⋅cos(kzz), (4.10)
where C is a constant and where the eigenvalues for the wave number is given by
2 2 2
2222 x y z.
nxyz
xyz
n n n
kkkk
LLL
⎛⎞π ⎛⎞π ⎛⎞π
=++=⎜⎟+⎜⎟+⎜⎟
⎝⎠⎜⎟⎝⎠
⎝⎠
(4.11)
The corresponding eigenfrequencies are given by
1
2 2 2 2(^0).
xyz 2
x y z
nnn
xyz
cn n n
f
LLL
⎡⎛⎞⎛⎞⎛⎞⎤
=++⎢⎜⎟⎜⎟⎜⎟⎥
⎢⎝⎠⎜⎟⎝⎠⎥
⎣ ⎝⎠ ⎦
(4.12)
To each of these eigenfunctions or normal modes there is a set of numbers, a set of
indices. Equation (4.10) then represents a three-dimensional standing wave if we
multiply with the time-dependent factor exp(jω t). In the literature special names are used
for the wave forms associated with these sets of indices. We have an axial mode when
two of the indices are equal to zero, a tangential mode when just one of the indices is
zero, and finally, an oblique mode when all indices are different from zero. (Can you tell
the direction of the wave in the room in these three cases?)