Acoustical Modeling and Auralization 217

In the absence of acoustical absorption, the relative
size of the wavelength of a sound wave to the size of an
object in the path of the wave dictates the primary phys-
ical phenomenon that takes place when the sound waves
reach the object. If the object can be considered acousti-
cally hard (i.e., having a low absorption coefficient) so
that the energy of the wave is not significantly reduced
by absorption and if the characteristic dimension of the
object (its largest dimension in the path of the sound
wave) is denoted by a then the product ka can be used to
predict how the sound waves will be affected by the
presence of the object.

Table 9-1 shows a range of products ka and the
resulting primary effect that the object will have on the
sound waves.

In our example if an acoustically hard object of
dimension a= 0.5 m is placed inside a full-size room,
then the sound waves will be reflected by the object
since ka= 9.1, a value clearly above the lower limit of 5
at which reflections become the primary phenomenon.

If a 1:10 scale model of the room is now created and
the object is scaled by the same amount its dimension
has now become ac= 0.05 m. Under the earlier condi-
tions where f= 1000 Hz the product kacnow has a value
of kac= 0.91 and the conclusion that must be drawn
from the guidelines presented in Table 9-1 is that the
sound waves diffract around the object: in other words,
the model has failed at predicting accurately the primary
physical phenomenon of sound reflection that takes
place when a 1000 Hz sound wave strikes an acousti-
cally hard object of 0.5 m in size.

In order for the model to yield the proper conclu-
sion—i.e., the sound waves are reflected by the
object—the wavelength of the sound waves has to be
scaled down by the same amount as the physical dimen-

sions of the room, or looking at Eq. 9-1—and keeping

the speed of sound a constant—the frequencies of the

sound waves have to be scaled up by an amount equal to

the inverse of the model’s physical scale. In our

example, a 10 kHz frequency needs to be used in the

model in order to assess the conditions that exist inside

the full-size room under 1 kHz excitation. If data

pertaining to the room needs to be available from 50 Hz

to 20 kHz, the use of a 1:10 scale physical model will

require that frequencies from 500 Hz to 200 kHz be

used during the investigation.

9.2.1.2 Time and Distance Considerations in Physical

Models

A sound wave traveling at a velocity c will take a time t

to cover a distance x according to the relation

(9-3)

In a physical model the dimensions are scaled down

and as a consequence the time of travel of the waves

inside the model is reduced by the same factor. If

time-domain information with a specific resolution is

required from the model, then the required resolution in

the time data must increase by a factor equal to the

inverse of the scale in order to yield the desired accuracy.

As an example, if a sound source is placed at the end

of a room with a length x=30 m and c=344 m/s, the

application of relation, Eq. 9-3, shows that the sound

waves can be expected to reach the other end of the

room in 87.2 ms. If an accuracy of ±10 cm is desired in

the distance information, then a time resolution of

±291μs is required for the time measurements.

In a 1:10 scale situation—and under the same condi-

tions of sound velocity—the sound waves will now take

8.72 ms to travel the length of the model and a resolu-

tion of ±29.1μs will be required in the time-measuring

apparatus to yield the required distance resolution.

9.2.1.3 Medium Considerations in Physical Models

As a sound wave travels through a gaseous medium like

air it loses energy because of interaction with the mole-

cules of the media in a phenomenon known as thermal

relaxation. Energy is also lost via spreading of the wave

as it travels away from the source. The absorption in the

medium as a function of distance of travel x and other

parameters such as temperature and humidity can be

represented by a loss factor K that is given by:

(9-4)

Table 9-1. Effect of Wave Number and Object
Dimension on the Propagation of Sound Waves

Value of kaPrimary Phenomenon Taking Place When the

Sound Waves Reach the Object (Not Including

Absorption)

Diffraction: the sound waves travel around the

object without being affected by its presence. The

object can be considered invisible to the waves.

Scattering: the sound waves are partially reflected

by the object in many directions and in a compli-

cated fashion. This scattering phenomenon is asso-

ciated with the notion of acoustical diffusion.

Reflection: the sound waves are deflected by the

object in one or more specific direction(s) that can

be predicted from application of basic geometry

laws.

kad 1

1 ka 5

ka! 5

xct=

Ke

• 
  

  mx