W hen a sound source generates an acoustic field the resulting pressure fluctuations can be thought o f as a sum o f an infinite number o f modes (Morse and Ingard, 1968). This observation reflects that, generally, there is more than one, non-zero, solution o f the wave equation which satisfies a set o f boundary conditions. Each solution is termed an eigenfunction^ and has an associated eigenfrequency.
Consider the case o f harmonic stimulation o f air within a hard walled circular tube o f radius a, driven by a piston at the z = 0 boundary. Referring to the usual cylindrical co ordinates, r, 0 and z, the radial, azimuth and axial co-ordinates respectively, the eigenfunction, i.e., the solution for sound pressure, is given by (Kinsler et al, 1982):-
Pm« = A^„^[A:,„r]cos[m0]exp[7(co/ - k^z)]
(4.6)
where m and n are natural numbers and refer to the mode number; , is the m'* order Bessel function o f the first kind; j = (-1 )’^^ ; co = 2tc/ w h e r e /is the driving frequency; is the axial wavenumber, which determines the nature o f propagation, and is given by:-
k l = k ^ - k l
(4.7)
where the propagating wavenumber, A: = co / c , and c is the speed o f sound, k^^ is the modal wavenumber given by:-
(4.8)
where is the (n+1)"' zero o f dJ„[x] / d x, i.e., inflection point o f / „ [ ; r ] .
^ Prefix from the German eigen, signifying own, characteristic.
Chapter 4: Probe assem bly calibration: M ethods
Table 4.3: Table o f inflection points o f the order Bessel function o f the fir s t kind.
i^ inflection point
m*^’ order 1 2 3
0 0 3.83 7.02
1 1.84 5.33 8.54
2 3.05 6.71 9.97
For the (0,0) mode, Pog = 0 , therefore kQQ=0, leading to k^=d) / c . Since k^ is real, the (0,0) mode propagates with speed equal to c, for co > 0 . The (0,0) mode is therefore an example o f a propagating mode. In addition, because the (0,0) mode is completely described by one spatial variable, the axial co-ordinate, z, this propagating mode is an example o f a plane wave. Here, the phase o f neighbouring wavefronts relative to the stimulus varies with distance, but is related through A: in a linear manner
However, for the (1,0) mode, k^Q % 1.84, such that k^ is imaginary for low frequencies but real for frequencies above a point termed the modal cut-off f r e q u e n c y T h e cut o ff frequency for the (1,0) mode, which is the lowest f„„, is given by f ^ « 1 0 1 / a (Kinsler et al, 1982). For a=4.5 mm the lowest cut-off frequency is then 22.4 kHz.
When / > f„„, a propagating mode exists, resulting in a non-planar wave due to the dependence upon more than one spatial co-ordinate.
When the wavenumber k^ is purely imaginary, that is when f < f^n then the mode is termed (non-propagating) evanescent^. The consequence o f an evanescent mode is an acoustic wave in the medium which has two distinct properties from the plane wave case. Firstly, points in close proximity to the source vibrate in synchrony, in other words the displacement o f the medium is in phase with the displacement o f the sound source. Under such a condition, the wave velocity is in quadrature to the displacement.
^ Such m odes turn up in optics and quantum m echanics and give rise to the phenom enon o f tunnelling.
Chapter 4: Probe assem bly calibration: Methods
This leads to the observation that the disturbance has zero capacity for energy dissipation. Rather than being dissipated, the energy is stored, as if the medium acted as a purely reactive impedance. Secondly, a snapshot in time o f the amplitude against z is not sinusoidal, as for plane waves, but rather decays exponentially.
Table 4.4 lists the, customary. He distance, which is the axial length in which the evanescent mode amplitude falls by a factor o f approximately 0.37. Rearranging equation 4.1 gives / a){\ - { k a I . When k a « \ x ^ ^ , termed the low frequency approximation, the wavenumber approximates to k^ I a) and is independent o f frequency. Therefore, is approximately equal to {a! \x^^) (Keefe and Benade, 1981). For the worst case, that is in the (1,0) mode, = 1.84, and for /= 1 0 kHz and a=A.5 mm, the largest radius in the calibration tube set, the error in for
the low frequency approximation is -11%.
Table 4,4: Me distances fo r circular tube, a=4.5 mm. Low frequency approximation used. Distance in mm. n^’’ mode m“’ mode 0 1 2 0 N/A 1.17 0.64 1 2.25 0.84 0.53 2 1.48 0.67 0.45
The discussion so far has been valid for the origin o f excitation to be an oscillatory boundary, equivalent to a piston source. Consider the case where the source is taken to act at a p o i n t , A microphone situated at (r^,Qy,z^) samples the pressure disturbance. The preceding treatment o f propagating and evanescent modes equally applies to this case. Additionally, the following discussion is valid for one dimensional acoustic wave propagation only, in that all eigenfunctions are evanescent modes, except for the (0,0) mode which is a propagating mode, planar in dispersion.
Chapter 4: Probe assem bly calibration: Methods
The sum o f the evanescent modes is given by a single real valued non-dimensional parameter, termed the evanescent mode factor, s, which as Keefe and Benade (1981) note is a function o f the configuration o f the sound source relative to the measurement point.
We define an evanescent wave as that disturbance which is a result o f all contributing non-propagating modes.
The presence o f the evanescent pressure wave is modelled as a result o f an additional impedance lumped element, in series with the input impedance o f the acoustic load, Z .„, which is dependent upon the plane wave only, as illustrated in figure 4.2.
Figure 4.2: Network analogue illustrating the effect o f the additional evanescent term as an additional impedance, which is purely reactive, assuming one dimensional wave propagation.
Under harmonic stimulation, , the evanescent dependant impedance element, Z^^, is then given by , 0 , , z j (Keefe and Benade, 1981; Brass and Locke, 1997), where is the lossless characteristic impedance given by p c ln a f . is, by virtue o f the 90° operator, y, purely reactive and exhibits an argument which is governed by the sign o f the evanescent mode factor; +90° for s > 0 , and -90°, for s < 0 . As stated previously, the pressure disturbance due to the evanescent wave dissipates zero energy, compatible with the inclusion o f a reactive impedance. Also, the frequency behaviour
Chapter 4: Probe assem bly calibration: Methods
o f Zgy is mass-like due to the proportional dependence upon co , so that | | increases at 6 dB/Oct.
Consider the situation in which in cylindrical co-ordinates, the loudspeaker port is situated at ( r ^ = R ) and the microphone located at = 7^,8, =7t,z, = 0 ), which roughly corresponds to the geometry o f the BS probe used in this study. From Brass and Locke (1997) this configuration approximately corresponds to the z = 0.025a curve in panel D in their figure 5. When the radius o f the acoustic load is a = 4.5 mm and
R = 1.5 mm then s « -0.3.