Waves in fluid and solid media - Building Acoustics - page 85

⎛ ⎞ ⎛ ⎞

2 π f 2 π f

n n

cos ⋅ ⋅ x cos ⋅ x

⎜ ⎟ ⎜ ⎟ 0

∞

c c

∑

⎝ ⎠ ⎝ ⎠

0 0

pfxx (,, ) = C . (3.93)

0

2 2

f − f

n

n = 1

The constant C will contain the strength of the source having the frequency f . There are
several important comments to be made on this result. First, the pressure will be
dominated by the term having a natural frequency nearest to the driving frequency but
the response will contain contributions from many terms in the sum.
Second, as we have not introduced any form of energy losses in the tube, the
pressure will go to infinity when the driving frequency coincides with any of the natural
frequencies. To calculate on a more realistic situation we may formally add a loss term in
the denominator. This is carried out by calculating on a situation as depicted in Figure
3.16, where a small loudspeaker is placed at a distance x 0 from the wall in one end and
where the sound pressure is measured by a microphone placed at a distance x .

Loudspeaker Microphone

q p

x 0

x

L =1.7 m

Figure 3.16 A hard-walled tube with a sound source and a receiver (microphone).

The results of the calculations are shown in Figure 3.17 . Due to the 1.7 metre
length of the tube the first natural frequency will be 100 Hz and the following ones
multiples of this frequency. Using one of the source positions we get, as expected,
resonances at these frequencies. For the second source position, in the centre of the tube,
no resonances shows up at 100 Hz and 300 Hz. (Why is that?)
Last but not least a very important comment should be made on Equation (3.93)
when it comes to the symmetry in the expression: the source and the receiver may change
places without altering the results. This is an example of the aforementioned acoustical
reciprocity principle , a principle that is quite general and in many instances very useful
in practice. A generalization to three-dimensional sound fields is evident but the principle
of exchanging source and receiver does also apply when the system contains structural
components, albeit subject to some limitations. We shall return to this theme when
treating the subject of sound transmission in Chapter 6 where we deal with the general
subject of vibroacoustic reciprocity (see section 6.6.1).