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Acoustic bandpass filters employing shaped resonators
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Acoustic bandpass filters employing shaped resonators M. Červenka n, M. Bednařík Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Czech Republic
a r t i c l e i n f o
abstract
Article history: Received 13 January 2016 Received in revised form 9 June 2016 Accepted 29 June 2016 Handling Editor: Y. Auregan
This work deals with acoustic bandpass filters realized by shaped waveguide-elements inserted between two parts of an acoustic transmission line with generally different characteristic impedance. It is shown that the formation of a wide passband is connected with the eigenfrequency spectrum of the filter element which acts as an acoustic resonator and that the required filter shape substantially depends on whether the filter characteristic impedance is higher or lower than the characteristic impedance of the waveguide. It is further shown that this class of filters can be realized even without the need of different characteristic impedance. A heuristic technique is proposed to design filter shapes with required transmission properties; it is employed for optimization of low-frequency bandpass filters as well as for design of bandpass filters with wide passband surrounded by wide stopbands as it is typical for phononic crystals, however, in this case the arrangement is much simpler as it consists of only one simple-shaped homogeneous element. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Acoustic bandpass filter Shaped resonator Heuristic optimization
1. Introduction It is well known that noise pollution is a serious problem in contemporary urban environment. A substantial part of this pollution can be attributed to the exhaust noise of internal combustion engines which is one of many reasons why a good understanding of wave propagation in ducts is important in order to be able to address this and related issues. Noise propagating in ducts is usually treated by means of mufflers (silencers), see e.g. [1] and references therein, which are essentially filters which either dissipate acoustic energy into heat, reflect it back to the source or do partially both. In some applications, it may be desirable to employ acoustic filters with wide passband, which is the subject of this work. The article elaborates the idea by Rudenko and Shvartsburg [2] who have noticed that if a suitably shaped element (filter) is inserted between two parts of a waveguide filled with a medium with different characteristic impedance, wide passband at low frequencies can be formed, which is a behaviour similar to the one observed in phononic crystals, see e.g. [3–6], however, in this case, the arrangement is much simpler as it consists of only one simple-shaped homogeneous element. Within this work, it is shown that the appearance of the wide passband is caused by mutual approaching of eigenfrequencies or lowering the first eigenfrequency of the inserted element which essentially forms an acoustic resonator. In Section 2, it is demonstrated on two analytically treatable examples that the element shape giving rise to a wide passband formation depends on whether the characteristic impedance of the filter element is higher or lower than the characteristic impedance of the waveguide and, that the formation of a wide passband can be achieved even without the need of different
n
Corresponding author. E-mail address: [email protected] (M. Červenka).
http://dx.doi.org/10.1016/j.jsv.2016.06.045 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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characteristic impedances. In this case, the filter element forms a shaped expansion chamber (studied e.g. by [7]), but with more complex behaviour. Section 3 presents an algorithm which allows designing filter-element shapes with prescribed transmission properties; two examples are provided for a lowpass filter with wide passband and extended adjacent stopband and a filter with a wide passband surrounded by wide stopbands. Section 4 of the article draws some concluding remarks.
2. Simple filters Let us assume a circular acoustic transmission line (waveguide) with constant radius r a filled with medium with ambient density ρ0a and speed of sound c0a divided into two parts between which an element (filter) of length d is inserted with generally variable radius r b ¼ rðxÞ and different medium (ρ0b , c0b ), see Fig. 1. Let us assume further that there are only plane waves propagating in the waveguide. In front (upstream) of the element, there is an incoming wave p0i ðx; tÞ and wave p0r ðx; tÞ which reflects on the element. As the region behind (downstream) the element is assumed to be semi-infinite, there is only the transmitted outgoing wave p0t ðx; tÞ. The incoming wave is supposed to be harmonic with angular frequency ω so that phasor representation is employed iωt ^ . Hence, we can write the following formulas describing the incoming, reflected and transmitted as p0 ðx; tÞ ¼ ℜ½pðxÞe waves: p^ i ðxÞ ¼ P 0i e ika x ;
p^ r ðxÞ ¼ P 0r eika x ;
p^ t ðxÞ ¼ P 0t e ika ðx dÞ ;
(1)
where P 0i , P 0r and P 0t are (complex) constants (amplitudes) and ka ¼ ω=c0a is the wavenumber. As it is shown below, the frequency characteristics of the transmission coefficient of the filter strongly depend on its geometry and characteristic impedance. 2.1. Constant-cross-sectioned filter element Let us assume the case of r b ¼ r b0 ¼ const: In this case, the acoustic pressure phasor inside the element has the form ^ pðxÞ ¼ A1 e ikb x þ A2 eikb x ;
(2)
where A1 and A2 are integration constants and kb ¼ ω=c0b is the wavenumber. The transmission coefficient through the element can be calculated from the continuity of acoustic pressure [given by Eqs. (1) and (2)] and acoustic volume velocity at x ¼0 and x ¼d as T ¼
P 0t 4 ¼ pffiffiffi pffiffiffiffiffiffiffiffi ; pffiffiffi pffiffiffiffiffiffiffiffi P 0i ð α þ 1=αÞ2 eiπΩ ð α 1=αÞ2 e iπΩ
(3)
ρ0a c0a r 2b0 ρ0b c0b r 2a
(4)
where α¼
which is the product of characteristic impedances ratio with the ratio of waveguides' cross-sections and Ω ¼ kb d=π is the dimensionless frequency.
Fig. 1. Geometry of the problem.
Fig. 2. Transmission coefficient modulus for r b ¼ r b0 ¼ const: and individual values of α.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Analysis of formula (3) shows that if α ¼ 1, the acoustic impedances of individual waveguides are matched and perfect transmission occurs ðjT j ¼ 1Þ for all frequencies, see e.g. [8]. Otherwise, if α a1, perfect transmission occurs only if Ω ¼ n; with n being an integer. The perfect transmission for non-zero frequencies is caused by resonance of waves within the element. On the contrary, if Ω ¼ ð2n 1Þ=2, the transmission is minimal and it holds jT jmin ¼
2α : 1 þ α2
(5)
Frequency characteristics of the transmission coefficient modulus for various values of α can be seen in Fig. 2. It can be observed that the resonance Q-factor increases with increasing value of α (or 1=α). For α⪢1, the frequency bandwidth of resonance peaks can be calculated as ΔΩ 4π=α. The case of r b0 4 r a and za ¼ zb corresponds to a simple expansion-chamber filter, see e.g. [1]. 2.2. Cosine-shaped filter element Let us assume the filter element with shape described by function d ; r b ðxÞ ¼ r m cos γ 0 x 2
0 rx rd;
(6)
where r m is its maximum radius. The coefficient γ 0 is defined as γ0 ¼
2 r arccos b0 ; d rm
γ 0 d A 〈0; πÞ:
(7)
If r b0 ¼ r a , then r b ð0Þ ¼ r b ðdÞ ¼ r a and the connection to the external waveguides is continuous. For this shape function, acoustic pressure within the element can be calculated, see Appendix A.1, as 0
p^ ðxÞ ¼
0
A1 eiμ x þA2 e iμ x ; cos γ 0 x d=2
(8)
2
where μ02 ¼ kb þ γ 02 . Applying the continuity of acoustic pressure [given by Eqs. (1) and (8)] and volume velocity at x ¼0 and x¼d results in the transmission coefficient T ¼
P 0t 4αμ0 kb ¼ ;
2 0
2 P 0i 0 0 iμ αa þ iðαμ þ kb Þ e d αa0 iðαμ0 kb Þ e iμ0 d
where a0 ¼
0 1 dr γd : ¼ γ 0 tan r dx x ¼ 0 2
(9)
(10)
The central waveguide is effectively an acoustic resonator with acoustically soft end-walls if α o1 or with acoustically hard end-walls if α 41. As it is shown in Appendix A.1, resonance frequencies of cosine-shaped resonator with acoustically soft end-walls decrease with increasing parameter γ 0 d, see Eq. (A.5). It is thus possible, in case of αo 1, to decrease the resonator first eigenfrequency to such a low value that the two first peaks of perfect transmission get overlapped and make a low-frequency passband. This situation can be seen in Fig. 3 in the case of α ¼ 1=10 and γ 0 d ¼ 2:7. In the case of α 41, the eigenfrequencies increase with increasing parameter γ 0 d and the low-frequency passband cannot appear.
Fig. 3. Left: transmission coefficient moduli of cosine-shaped filter elements for α ¼ 1=10 and individual values of γ 0 d; right: the corresponding filter element shapes for r b0 ¼ r a .
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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2.3. Hyperbolic-cosine-shaped filter element The case described in this section corresponds to the one studied in [2]. Let us assume the filter element with shape described by function d ; 0 rx r d: (11) r b ðxÞ ¼ r m cosh γ x 2 where r m is its minimum radius. The coefficient γ is defined as γ¼
2 r arccosh b0 : d rm
(12)
If r b0 ¼ r a , then r b ð0Þ ¼ r b ðdÞ ¼ r a and the connection to the external waveguides is continuous. For this shape function, acoustic pressure within the filter element can be calculated, see Appendix A.2, as p^ ðxÞ ¼
A1 eμx þ A2 e μx ; cosh γ x d=2
(13)
2
where μ2 ¼ γ 2 kb . Applying the continuity of acoustic pressure [given by Eqs. (1) and (13)] and volume velocity at x¼0 and x ¼d results in the transmission coefficient T ¼
P 0t 4iαμkb ¼ ; P 0i ðαμ αa þikb Þ2 eμd ðαμ þ αa ikb Þ2 e μd
(14)
1 dr γd : ¼ γtanh r dx x ¼ d 2
(15)
where a¼
As it is shown in Appendix A.2, the first resonance frequency of hyperbolic-cosine-shaped resonator with acoustically hard end-walls decreases with increasing parameter γd, see Fig. A2. It is thus possible, in case of α 4 1 (contrary to the case of cosine-shaped resonator), to decrease the resonator first eigenfrequency to such a low value that the two first peaks of perfect transmission get overlapped and make a low-frequency passband. This situation can be seen in Fig. 4 in the case of α ¼ 10 and γd ¼ 5. It can be observed in the figure that for this value of γd the higher resonance peaks are shifted towards the higher frequencies extending the bandstop frequency range. It is also noteworthy to mention that for kb oγ the perfect transmission occurs even if the waves in the filter element are evanescent. In the case of α o1, the eigenfrequencies increase with increasing parameter γd and the low-frequency passband cannot appear. 2.4. Numerical validation As it is obvious from formula (4), the impedance mismatch (α a1) can be achieved either by different characteristic impedances (ρ0a c0a a ρ0b c0b ), by cross-section discontinuity (r b0 a r a ) or by the combination of both. Applicability of formulas (9) and (14) for calculation of transmission coefficients in case of homogeneous media and cross-section discontinuities is demonstrated by comparison of predicted analytical and numerical results. The numerical results were calculated in 2D axisymmetric geometry by finite element method in COMSOL Multiphysics (Acoustics Module, Pressure Acoustics, Frequency Domain) for cosine-shaped and hyperbolic-cosine-shaped filter elements. In both the cases, d ¼500 mm and r a ¼ 10 mm and air at normal conditions as a medium in the entire geometry were assumed. In the case of cosine-shaped element, α ¼ 1=10 and γ 0 d ¼ 2:7, see Fig. 3, which results in r b0 ¼ 3:16 mm and r m ¼ 14:44 mm. In the case of hyperbolic-cosine-shaped resonator, α ¼ 10 and γd ¼ 5, see Fig. 4, which results in r b0 ¼ 31:62 mm and r m ¼ 5:16 mm. The geometry is depicted for illustration in Fig. 5. The comparison of numerical and analytical results can be seen in Fig. 6. It can be observed that the agreement is very good. In both the cases the agreement of numerical and analytical results decrease with increasing frequency as the validity
Fig. 4. Left: transmission coefficient moduli of hyperbolic-cosine-shaped filter elements for α ¼ 10 and individual values of γd; right: the corresponding filter element shapes for r b0 ¼ r a .
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 5. Cosine-shaped and hyperbolic-cosine-shaped resonators used as lowpass filters.
Fig. 6. Comparison of analytical and numerical results.
of Webster's equation used here for description of plane waves in shaped waveguides is limited by relation krjdr=dxj=2⪡1, see [9].
3. Optimized resonant filters The above-mentioned analytical examples have shown that the shape of the resonant filter element can strongly influence the resulting transmission coefficient characteristics. Depending on the value of α, resonance frequencies can be shifted upwards or downwards; in the case of the first resonance frequency shifted downwards, the low-frequency passband can appear. Shaped resonators are known to have non-trivial eigenfrequency spectra, which is the property utilized for generation of high-amplitude acoustic fields, see e.g. [10]. Utilizing the perturbation theory, Hamilton et al. [11] have shown that each eigenfrequency can be independently shifted via appropriate modulation of the resonator wall radius. It is thus in principle possible to find (optimize) the resonator shape in such a way that the first resonant frequency is shifted downwards, the second one upwards so that the low-frequency bandpass region is followed by adjacent wide bandstop region. For this purpose, the approach used in work [12] for resonant cavities shaping could be employed in a straightforward way. Within this work, a more direct approach was adopted which is based on search for resonant-filter-element shape approximating prescribed transmission coefficient spectrum. As it is demonstrated below, this approach allows for designing a wider variety of acoustic filters. 3.1. Riccati equation An effective way of calculating the reflection/transmission of acoustic waves impinging upon a non-uniform region utilizes the Riccati equation, see e.g. [13,14]. The Riccati equation for variable-cross-sectioned waveguide can be derived from linearized fluid-dynamics equations written in the form ∂ρ0 1 ∂ 2 r ρ0b v ¼ 0; þ ∂t r 2 ∂x
ρ0b
∂v ∂p0 ¼ ; ∂t ∂x
p0 ¼ c20b ρ0 ;
(16)
where ρ0 is the acoustic density and v is the acoustic velocity. Introducing the acoustic volume velocity w ¼ πr 2 v, Eq. (16) can be for harmonic waves rewritten as ^ dw πr 2 ^ ¼ iω p; dx ρ0b c20b
dp^ ρ ^ ¼ iω 0b2 w: dx πr
^ w ^ can be written as Using these equations, Riccati equation for acoustic impedance Z ¼ p= ^ p^ dw dZ d p^ 1 dp^ ρ πr 2 2 ¼ ¼ 2 ¼ iω 0b2 þiω Z : ^ ^ dx w dx dx w w πr ρ0b c20b ^ dx
(17)
(18)
Introducing dimensionless variables x X¼ ; d
R¼
r r b0
;
Z¼
πr 2a Z; ρ0a c0a
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Eq. (18) can be non-dimensionalized into the form dZ iπΩ ¼ 2 þ iπΩαR2 Z 2 ; dX αR
(19)
which is more suitable for numerical integration. The coefficient α can describe, see Eq. (4), the ratio of the characteristic impedances, the ratio of cross-sections at the cross-section discontinuities or the combination of both. The normalized Riccati equation (19) is employed similarly as in [14] for the calculation of the filter-element transmission coefficient spectrum as follows. Behind the filter element, there is only an outgoing transmitted wave, so that for the normalized acoustic impedance, it holds Z ¼ 1 for X 41. The value of Z ¼ 1 is used as an initial condition for the Riccati equation (19) integrated from X¼1 to X ¼0, which provides the acoustic impedance Zð0Þ “seen” by the wave impinging upon the filter element. The corresponding reflection coefficient is then calculated as R
P 0r Zð0Þ 1 : ¼ P 0i Zð0Þ þ 1
(20)
As dissipation of acoustic energy is neglected within this model, the following relation holds between the reflection coefficient R and the transmission coefficient through the filter element T : jRj2 þ jT j2 ¼ 1; resulting in jT j ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ℜ½Zð0Þ : jZð0Þ þ 1j
(21)
3.2. Optimization procedure The optimization procedure for designing the resonant filter elements works as follows. The filter-element properties are parametrized using an M-dimensional parameter vector q which defines the element shape and the parameter α, where M is supposed to be odd. The filter-element shape is described using a set of ðM 1Þ=2 þ 2 control points ½X i ; Ri ¼ ½0; 1; ½X 1 ; R1 ; ½X 2 ; R2 ,..., ½X K ; RK ; ½1; 1;
(22)
where K ¼ ðM 1Þ=2, which are interconnected using the cubic splines with zero derivatives at X¼0 and X¼1. The value of Rð0Þ ¼ Rð1Þ ¼ 1 is kept fixed [see Eq. (22)] in order to control the impedance mismatch only by the parameter α, whose value is also the subject of the optimization. The parameter vector q thus reads q ¼ ½q1 ; q2 ; q3 ; …; qM ¼ ½X 1 ; X 2 ; …; X K ; R1 ; R2 ; …; RK ; α:
(23)
The required transmission coefficient spectrum is sampled at N frequencies Ω1 ; Ω2 ; …; ΩN providing the values jT req ðΩi Þj for i ¼ 1; …; N. The parameter vector q is sought in the sense of the least squares method by minimization the M-dimensional objective function Q ðqÞ ¼
N
2 1X w jT req ðΩi Þj jT ðq; Ωi Þj ; Ni¼1 i
(24)
where wi are weighting factors for individual frequencies and jT ðq; Ωi Þj is calculated employing the Riccati equation (19) and relation (21). The objective function (24) was minimized heuristically employing a self-adaptation variant ðμ; λÞ-ES of Evolution Strategies, see e.g. [15], as the method has been successfully used earlier for solution of related problems [12,14]. Eq. (19) was integrated numerically using adaptive step-size Dormand–Prince method [16], implemented in Cþ þ. In brief, the ðμ; λÞ-ES was implemented as follows: Initialization: The population of λ individuals is generated each of whom has a genome represented by the parameter vector q [Eq. (23)]. The individual values of the parameter vector are generated randomly in intervals: X i A 〈0; 1〉, Ri A 〈Rmin ; Rmax 〉, α A 〈αmin ; αmax 〉. The control points' abscissas are generated in order to have ascending values. Then, each
Fig. 7. Example of evolutions of the objective function; Q min stands for the minimum value of Q within the given generation.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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individual is checked to meet RðxÞ A 〈Rmin ; Rmax 〉 for X A 〈0; 1〉 as the cubic splines can cause crossing the pre-defined limits. If this is the case, the corresponding individual is rejected and re-generated. All the individuals are assessed, the μ-ones with the lowest value of the objective function Q ðqÞ [Eq. (24)] are selected as parents for the evolution. The evolution: From the μ parents, pairs are randomly drawn to produce a new generation of λ offspring. The genome of a new individual is created by cross-breeding (individual components of genome qi are randomly taken from the parent's genomes) and mutation (qi -qi þ si , where si is a random number with zero mean value and Gaussian distribution). The new individual is checked whether RðxÞ A 〈Rmin ; Rmax 〉 for X A 〈0; 1〉 and α A 〈αmin ; αmax 〉, if not, it is rejected. After all the generation has been produced, the individuals are assessed, the μ of them with the lowest objective function are chosen as the parents for next generation and the evolution continues for the pre-defined number of generations. Fig. 7 shows an example of evolution lasting for 500 generations repeated 100 (the case studied in Section 3.3.1 for α o1) with different (randomly generated) initial conditions. It can be seen that there are three minima; the algorithm converged several times to the global one. More on the heuristic optimization can be found e.g. in [15]. 3.3. Numerical results The numerical results presented within this section were obtained using the above-described procedure. The optimization was performed for M ¼11 (7 control points), for both αo 1 and α 4 1. As it can be expected, the filter-element spectrum depends on the range in which its radius can vary. For this reason, some reasonable Rmin and Rmax were defined delimiting the allowed resonator dimensions. In each case studied here, the optimization was repeated 1000 each population consisted of λ ¼ 105 individuals with μ ¼ 15 parents, each evolution lasted for 500 generations. 3.3.1. Optimized lowpass filter For the lowpass filter, the required transmission coefficient characteristics was prescribed as ( 1: Ωi rΩcut ; where Ωi ¼ 0:1; 0:2; 0:3; …; Ωmax : jT req ðΩi Þj ¼ 0: Ωcut oΩi rΩmax ;
(25)
As an example, the results of optimization for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α o 1 are shown in Fig. 8. In this case, the filterelement radius was restricted to RðXÞ A 〈1:0; 5:0〉 The resulting shape for optimum αopt ¼ 0:265 is seen in the left panel; the positions of the control points are marked by the crosses. It can be observed that the shape has two maxima, it is rather simple and more-or-less symmetric about its centre. As the optimization employs 1D model, the numerical results are validated by comparison with the ones calculated using 2D axisymmetric model implemented in COMSOL Multiphysics (Acoustics Module, Pressure Acoustics, Frequency Domain). Air at normal conditions was used as medium in entire geometry, cross-section discontinuities were introduced to implement non-unitary α. These parameters were used in the 2D model: d¼500 mm and r a ¼ 10 mm. The right panel of Fig. 8 shows the comparison of required and obtained transmission coefficient spectrum together with the spectrum calculated employing 2D model for the optimized shape. It can be observed that the numerical results correspond to each other very well and that the optimization provides much (approximately two-times) wider bandstop region compared to the non-optimized cosine-shaped resonator (see Fig. 6a). As it was mentioned earlier, the value of αopt is together with the control points the result of the optimization procedure. The value of α, similarly as it was discussed in Section 2.1, influences the quality factor of individual resonance peaks (for given filter-element geometry). This behaviour is shown in Fig. 9. For α oαopt , the quality factor is higher, which means that the roll-off is steeper, however, the passband is not as flat as in the optimum case. On the other hand, if α 4αopt , the passband is flatter than in the optimum case, however, the roll-off is less steep. It is thus obvious that generally, the wider the prescribed dimensionless bandwidth, the less steep the filter roll-off1. The ability of the filter-element shape to control its individual resonance frequencies, as it follows from the analysis given in the Appendix, depends on the value of Rmax =Rmin ratio. This fact is demonstrated in Fig. 10. Here, the prescribed transmission characteristic (25) is approximated by filters with values of Rmax ¼ 2:5; 3:0; 3:5; 4:0; 4:5; 5:0 for Ωcut ¼ 0:5 and
Fig. 8. Lowpass filter optimized for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α o 1, RðXÞ A 〈1:0; 5:0〉.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 9. Transmission coefficient spectrum of the filter from Fig. 8 for various values of α; dashed lines: α o αopt , solid line: α ¼ αopt (corresponds to Fig. 8), dash-dot lines: α 4 αopt .
Fig. 10. Transmission coefficient spectrum of the filter optimized for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α o 1 with individual values of Rmax . In all the cases, Rmin ¼ 1.
Fig. 11. Transmission coefficient spectra and corresponding filter-element shapes for Ωcut ¼ 0:5 and different values of Ωmax – optimization for α o 1.
Fig. 12. Lowpass filter optimized for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α 4 1, RðXÞ A 〈0:2; 1:0〉.
Ωmax ¼ 3:0. In all the cases Rmin ¼ 1. It can be observed that for low values of Rmax there is a resonance peak at Ω 2 present which is suppressed (shifted behind Ωmax ) for higher values of Rmax . The increasing value of Rmax can be used for further increasing the filter-element stopband width, see Fig. 11. The transmission coefficient spectrum from Fig. 8 (Ωmax ¼ 3:0, Rmax ¼ 5:0, dashed line) is compared here with the one for prescribed stopband extended to Ωmax ¼ 3:5 (Rmax ¼6.0, dash-dot line) and the one for prescribed stopband extended to Ωmax ¼ 4:0 (Rmax ¼ 8:0, solid line). In all the cases, Rmin ¼ 1:0 and Ωcut ¼ 0:5. It can be observed that for the highest value of Rmax ¼ 8:0, the first as well as the second resonance peak is shifted leftwards which results in a flat and wide passband with steep roll-off (αopt ¼ 0:0706). The corresponding filter-element shapes are depicted in the right panel of Fig. 11. It can be seen that all the shapes are very similar. In the previous paragraphs, all the filter-element shapes were optimized for αo 1, however, similar results can be obtained for α 4 1. Fig. 12 shows the filter-element shape and its transmission coefficient spectrum for prescribed characteristics (25), Ωcut ¼ 0:5, Ωmax ¼ 3:0 and RðXÞ A 〈0:2; 1:0〉. The calculated transmission coefficient spectrum is compared with the one obtained using the 2D model with the same parameters as before. Again, the correspondence of the results is very good. The resulting shape for optimum α ¼ 6:85 has two minima and contrary to its counterpart depicted in Fig. 12, it is noticeably non-symmetric about its centre. Compared to the hyperbolic-cosine resonator, see Fig. 6b, the bandstop region is substantially wider, the numerical results have shown, analogously as in the previous case, that the stopband can be further extended by lowering the constraint Rmin . Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 13. Bandpass filter optimized for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α o 1; RðXÞ A 〈0:2; 1:0〉.
Fig. 14. Bandpass filter optimized for various values of ΔΩ ¼ Ωl Ωh , Ωmax ¼ 3:5 and α o 1; RðXÞ A 〈0:2; 1:0〉.
3.3.2. Optimized bandpass filter Within this paragraph, it is demonstrated that the presented approach can be employed for designing a bandpass filter. For this purpose, the required transmission coefficient characteristics can be prescribed as 8 > < 0: Ωi o Ωl ; where Ωi ¼ 0:1; 0:2; 0:3; …; Ωmax : (26) jT req ðΩi Þj ¼ 1: Ωl r Ωi rΩh ; > : 0: Ω oΩ rΩ max ; i h As an example, the results of optimization for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α o 1 are shown in Fig. 13. In this case, the filter-element radius was restricted to RðXÞ A 〈0:2; 1:0〉. The resulting shape for optimum αopt ¼ 0:110 is seen in the left panel; the positions of the control points are marked by the crosses. It can be observed that the shape has one minimum, it is rather simple and more-or-less symmetric about its centre. Contrary to the case of lowpass filter optimized for α o1, see Fig. 8, the filter element is contracted. The right panel of the figure shows the comparison of required and obtained transmission coefficient spectra together with the spectrum calculated employing 2D model for the optimized shape. Air at normal conditions was used as medium in the external waveguides. To implement non-unitary α, a medium with c0b ¼ c0a and ρ0b ¼ ρ0a =α was assumed to fill the filter element. These parameters were used in the 2D model: d¼500 mm and r a ¼ 10 mm. It can be observed that the numerical results correspond to each other very well. In this case, the transmission band appears because of the fact that the first and second resonant frequencies are shifted towards each other and the corresponding resonance peaks mutually overlap. Fig. 14 demonstrates the possibility of shifting the passband; the parameters of the optimization are the same as in the previous case except for the frequencies Ωl , Ωh , which are varied as Ωl ¼ 1:1; …; 1:6 and Ωh ¼ 1:4; …; 1:9 (the width of the passband is kept the same Ωh Ωl ¼ 0:3). The left panel of the figure shows the obtained transmission characteristics, the right panel of the figure shows the corresponding filter-element shapes. It can be observed that the shapes are rather symmetric with the exception of the case of the highest frequency range; the numerical results show that the shape gets symmetric if the ratio Rmax =Rmin is increased. In the previous paragraphs, the filter-element shapes were optimized for α o 1, again, similar results can be obtained for α 41. Fig. 15 shows the filter-element shape and its transmission coefficient spectrum for prescribed characteristics (26), for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α 41. In this case, the filter-element radius was restricted to RðXÞ A 〈1:0; 5:0〉. The calculated transmission coefficient spectrum is compared with the one obtained using the 2D model with the same parameters as before. Again, the correspondence of the results is very good. The resulting shape for optimum αopt ¼ 7:70 resembles an inversion of the shape obtained for α o 1 – see Fig. 13.
4. Conclusions It has been shown by means of analytical results that the formation of wide passband in non-uniform resonant acoustic filters is caused by mutual approaching of individual eigenfrequencies and that the corresponding shapes depend on Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 15. Bandpass filter optimized for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α 4 1; RðXÞ A 〈1:0; 5:0〉.
whether the filter-element characteristic impedance is higher or lower than the characteristic impedance of the external waveguide. It has also been demonstrated that this class of filters can be realized even without the necessity of different characteristic impedances. An evolutionary algorithm based on numerical integration of the Riccati equation has been proposed to demonstrate that the acoustic filters can be optimized and designed to approximate prescribed transmission characteristics. The proposed algorithm was used to optimize low-frequency bandpass filters; it has been shown that suitable filter shaping results in substantial extending of the bandstop region adjacent to the passband. The algorithm was also used to design bandpass filters with wide passband surrounded by wide stopbands as it is typical for phononic crystals, however, the proposed arrangement is much simpler as it consists of only one simple-shaped homogeneous element. The techniques employed within this work can be further elaborated to include inhomogeneous medium, waveguides with different characteristic impedances or cross-sections where optimized filter elements could serve for broadband impedance matching. This is the subject of our future work.
Acknowledgements This work was supported by GACR grant 15-23079S. The authors would also like to acknowledge the contribution of the COST Action CA15125. Some numerical calculations were performed at Computing and Information Centre of CTU.
Appendix A. Eigenfrequencies of cosine- and hyperbolic-cosine-shaped resonators Webster's equation describing planar harmonic waves in a waveguide with variable cross-section SðxÞ ¼ πr 2 ðxÞ can be written in the form 1 d 2 dp^ 2 r þ k p^ ¼ 0; (A.1) 2 dx r dx see e.g. [9], where p^ is the acoustic pressure phasor and k ¼ ω=c0 is the wavenumber. If a new variable F^ is introduced such that p^ ¼ F^ =r and substituted into Eq. (A.1) it becomes ! 2 2 d F^ 1d r ^ 2 F ¼ 0: (A.2) þ k r dx2 dx2 Eq. (A.2) has a simple analytical solution if 2
1d r ¼ γ 2 ¼ const: r dx2
(A.3)
The relation (A.3) is fulfilled for γ 2 40, if rðxÞ ¼ A1 eγx þ A2 e γx ;
rðxÞ ¼ A coshðγxÞ;
rðxÞ ¼ A sinhðγxÞ;
where A; A1 ; A2 are constants; for γ o 0, if 2
rðxÞ ¼ A1 sin ðγ 0 xÞ þ A2 cos ðγ 0 xÞ; where γ 0 ¼ iγ; and for γ ¼ 0, if rðxÞ ¼ A1 þ A2 x:
A.1. Cosine-shaped resonator Let us assume a resonant cavity with shape described by relation (6). For this shape, Eq. (A.2) reads d F^ þμ02 F^ ¼ 0; dx2 2
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2
where μ02 ¼ k þ γ 02 and it has the solution 0 0 F^ ðxÞ ¼ A1 eiμ x þA2 e iμ x ;
and thus the formula for acoustic pressure phasor has the form of Eq. (8). From here, acoustic velocity phasor can be calculated using Euler's equation v^ ðxÞ ¼
i dp^ ; kz0 dx
(A.4)
where z0 ¼ ρ0 c0 is the characteristic impedance. A.1.1. Acoustically soft boundaries ^ ^ At acoustically soft boundaries, it holds pð0Þ ¼ pðdÞ ¼ 0. Substitution into Eq. (8) results in A1 þA2 ¼ 0; 0
0
eiμ d A1 þe iμ d A2 ¼ 0; This set of equations has a non-trivial solution if 0
e2iμ d ¼ 1 ) μ0 d ¼ nπ; with n being an integer. From here, we can write for eigenfrequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωd 2 2 ¼ n2 π 2 γ 02 d ) Ω ¼ n2 γ 02 d =π 2 : c0
(A.5)
Comparing the relation (A.5) with the one for a constant-cross-sectioned resonator ðΩ ¼ nÞ shows that in the case of cosineshaped resonator with acoustically soft boundaries, the values of eigenfrequencies decrease with increasing value of γ 0 d, see Fig. A1a. A.1.2. Acoustically hard boundaries ^ ^ At acoustically hard boundaries, it holds vð0Þ ¼ vðdÞ ¼ 0. Substitution into Eqs. (8) and (A.4) results in ðiμ0 a0 ÞA1 ðiμ0 þa0 ÞA2 ¼ 0; 0
0
0
ðiμ þ a0 Þeiμ d A1 ðiμ0 a0 Þe iμ d A2 ¼ 0; where a0 is defined by relation (10). This set of equations has a non-trivial solution if 0 iμ a0 2 0 ¼ e2iμ d : iμ0 þa0 As the modulus of both the sides of this equation is unitary, the equation is satisfied if 2μ0 a0 ¼ μ0 d nπ: arctan 02 a μ02
(A.6)
Numerical solution of Eq. (A.6) shows that in the case of cosine-shaped resonator with acoustically hard boundaries, the values of eigenfrequencies increase with increasing value of γ 0 d, see Fig. A1b. A.1.3. Duality According to dualities of horns, see [17], eigenfrequencies of a resonator with the shape function r b ðxÞ ¼
r m ; d 0 cos γ x 2
0 rx r d;
(A.7)
are in the case of acoustically hard boundary conditions given by Eq. (A.5) and in the case of acoustically soft boundary conditions by solution of Eq. (A.6).
Fig. A1. Normalized eigenfrequencies of a cosine-shaped resonator with acoustically soft and hard boundaries; m is the numerical order of the corresponding mode.
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A.2. Hyperbolic-cosine-shaped resonator Let us assume a resonant cavity with shape described by relation (11). For this shape function, Eq. (A.2) reads d F^ μ2 F^ ¼ 0; dx2 2
2
where μ2 ¼ γ 2 k and it has the solution F^ ðxÞ ¼ A1 eμx þ A2 e μx : and thus the formula for acoustic pressure phasor has the form of Eq. (13). It is evident that for low frequencies, when k o γ, the two waves forming solution (13) are evanescent. Acoustic velocity phasor can be calculated from Eq. (13) using Euler's equation (A.4). A.2.1. Acoustically soft boundaries ^ ^ At acoustically soft boundaries, it holds pð0Þ ¼ pðdÞ ¼ 0. Substitution into Eq. (13) results in A1 þ A2 ¼ 0; eμd A1 þ e μd A2 ¼ 0; This set of equations has a non-trivial solution if e2μd ¼ 1: This condition can be fulfilled for non-zero μ only if μ is pure-imaginary, i.e. if k 4 γ. From here, we can write for eigenfrequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωd 2 2 ¼ n2 π 2 þ γ 2 d ) Ω ¼ n2 þγ 2 d =π 2 : (A.8) c0 Comparing the relation (A.8) with the one for a constant-cross-sectioned resonator ðΩ ¼ nÞ shows that in the case of hyperbolic-cosine-shaped resonator with acoustically soft boundaries, the values of eigenfrequencies increase with increasing value of γd, see Fig. A2a. A.2.2. Acoustically hard boundaries ^ ^ At acoustically hard boundaries, it holds vð0Þ ¼ vðdÞ ¼ 0. Substitution into Eqs. (13) and (A.4) results in ðμ þ aÞA1 ðμ aÞA2 ¼ 0; ðμ aÞeμd A1 ðμ þ aÞe μd A2 ¼ 0; where a is defined by relation (15). This set of equations has a non-trivial solution if aþμ 2 ¼ e2μd : aμ
(A.9)
Eq. (A.9) can be solved for real (positive) μ [γ 4 k – low-frequency evanescent waves; the numerical results show that the qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 real solution exists for γd≳2:39936 (r a =r m ≳1:810)] as well as for pure-imaginary μ ¼ iμ00 where μ00 ¼ k γ 2 (k 4γ – highfrequency waves) in this case, both sides of Eq. (A.9) have unitary modulus and the solution can be found as 2μ00 a ¼ μ00 d nπ: arctan 2 a μ002
(A.10)
Numerical solution of Eqs. (A.9) and (A.10) shows that in the case of hyperbolic-cosine-shaped resonator with acoustically hard boundaries, the eigenfrequency of the first mode decreases with increasing value of γd, the eigenfrequencies of the higher modes first slightly decrease and then increase with increasing value of γd, see Fig. A2b.
Fig. A2. Normalized eigenfrequencies of a hyperbolic-cosine-shaped resonator with acoustically soft and hard boundaries; m is the numerical order of the corresponding mode.
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A.2.3. Duality According to dualities of horns, see [17], eigenfrequencies of a resonator with the shape function r b ðxÞ ¼
r m ; d cosh γ x 2
0 r x r d:
(A.11)
are in the case of acoustically hard boundary conditions given by Eq. (A.8) and in the case of acoustically soft boundary conditions by solution of Eqs. (A.9) and (A.10).
References [1] M.L. Munjal, Acoustics of Ducts and Mufflers, 2nd edition, Wiley, Chichester, 2014. [2] O.V. Rudenko, A.B. Shvartsburg, Nonlinear and linear wave phenomena in narrow pipes, Acoustical Physics 56 (2010) 429–434, http://dx.doi.org/ 10.1134/S1063771010040044. [3] M. Shen, W. Cao, Acoustic band-gap engineering using finite-size layered structures of multiple periodicity, Applied Physics Letters 75 (1999) 3713–3715, http://dx.doi.org/10.1063/1.125438. [4] I.K. Lee, Y.J. Kim, J.H. Oh, Y.Y. Kim, One-dimensional broadband phononic crystal filter with unit cells made of two non-uniform impedance-mirrored elements, AIP Advances 3 (2013) 022105, http://dx.doi.org/10.1063/1.4790638. [5] H.J. Lee, J.K. Lee, Y.Y. Kim, Elastic metamaterial-based impedance-varying phononic bandgap structures for bandpass filters, Journal of Sound and Vibration 353 (2015) 58–74, http://dx.doi.org/10.1016/j.jsv.2015.05.012. [6] I.K. Lee, H.M. Seung, Y.Y. Kim, Realization of high-performance bandpass filter by impedance-mirroring, Journal of Sound and Vibration 355 (2015) 86–92, http://dx.doi.org/10.1016/j.jsv.2015.06.006. [7] Y. Kagawa, T. Omote, Finite-element simulation of acoustic filters of arbitrary profile with circular cross section, Journal of the Acoustical Society of America 60 (1976) 1003–1013, http://dx.doi.org/10.1121/1.381199. [8] R. Fleury, A. Alù, Extraordinary sound transmission through density-near-zero ultranarrow channels, Physical Review Letters 111 (2013) 055501, http: //dx.doi.org/10.1103/PhysRevLett.111.055501. [9] A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, Melville, 1994. [10] Y.A. Ilinskii, B. Lipkens, T.S. Lucas, T.W.V. Doren, E.A. Zabolotskaya, Nonlinear standing waves in an acoustical resonator, Journal of the Acoustical Society of America 104 (1998) 2664–2674, http://dx.doi.org/10.1121/1.423850. [11] M.F. Hamilton, Y.A. Ilinskii, E.A. Zabolotskaya, Linear and nonlinear frequency shifts in acoustical resonators with varying cross sections, Journal of the Acoustical Society of America 110 (2001) 109–119, http://dx.doi.org/10.1121/1.1379080. [12] M. Červenka, M. Bednařík, On the optimization of an acoustic resonator shape with respect to acoustic pressure amplitude, Acta Acustica United with Acustica 99 (2013) 183–191, http://dx.doi.org/10.3813/AAA.918601. [13] K. Walther, Reflection factor of gradual-transition absorbers for electromagnetic and acoustic waves, IRE Transactions of Antennas and Propagation 8 (1960) 608–621. [14] M. Červenka, M. Bednařík, Calculation of an axial temperature distribution using the reflection coefficient of an acoustic wave, Journal of the Acoustical Society of America 138 (2015) EL359–EL364, http://dx.doi.org/10.1121/1.4931830. [15] A.E. Eiben, J.E. Smith, Introduction to Evolutionary Computing, Springer, Berlin, 2010. [16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, Cambridge, 2007. [17] Y.M. Ram, S. Elhay, Dualities in vibrating rods and beams: continuous and discrete models, Journal of Sound and Vibration 184 (1995) 759–766, http://dx. doi.org/10.1006/jsvi.1995.0345.
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Acoustic bandpass filters employing shaped resonators M. Červenka n, M. Bednařík Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Czech Republic
a r t i c l e i n f o
abstract
Article history: Received 13 January 2016 Received in revised form 9 June 2016 Accepted 29 June 2016 Handling Editor: Y. Auregan
This work deals with acoustic bandpass filters realized by shaped waveguide-elements inserted between two parts of an acoustic transmission line with generally different characteristic impedance. It is shown that the formation of a wide passband is connected with the eigenfrequency spectrum of the filter element which acts as an acoustic resonator and that the required filter shape substantially depends on whether the filter characteristic impedance is higher or lower than the characteristic impedance of the waveguide. It is further shown that this class of filters can be realized even without the need of different characteristic impedance. A heuristic technique is proposed to design filter shapes with required transmission properties; it is employed for optimization of low-frequency bandpass filters as well as for design of bandpass filters with wide passband surrounded by wide stopbands as it is typical for phononic crystals, however, in this case the arrangement is much simpler as it consists of only one simple-shaped homogeneous element. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Acoustic bandpass filter Shaped resonator Heuristic optimization
1. Introduction It is well known that noise pollution is a serious problem in contemporary urban environment. A substantial part of this pollution can be attributed to the exhaust noise of internal combustion engines which is one of many reasons why a good understanding of wave propagation in ducts is important in order to be able to address this and related issues. Noise propagating in ducts is usually treated by means of mufflers (silencers), see e.g. [1] and references therein, which are essentially filters which either dissipate acoustic energy into heat, reflect it back to the source or do partially both. In some applications, it may be desirable to employ acoustic filters with wide passband, which is the subject of this work. The article elaborates the idea by Rudenko and Shvartsburg [2] who have noticed that if a suitably shaped element (filter) is inserted between two parts of a waveguide filled with a medium with different characteristic impedance, wide passband at low frequencies can be formed, which is a behaviour similar to the one observed in phononic crystals, see e.g. [3–6], however, in this case, the arrangement is much simpler as it consists of only one simple-shaped homogeneous element. Within this work, it is shown that the appearance of the wide passband is caused by mutual approaching of eigenfrequencies or lowering the first eigenfrequency of the inserted element which essentially forms an acoustic resonator. In Section 2, it is demonstrated on two analytically treatable examples that the element shape giving rise to a wide passband formation depends on whether the characteristic impedance of the filter element is higher or lower than the characteristic impedance of the waveguide and, that the formation of a wide passband can be achieved even without the need of different
n
Corresponding author. E-mail address: [email protected] (M. Červenka).
http://dx.doi.org/10.1016/j.jsv.2016.06.045 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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characteristic impedances. In this case, the filter element forms a shaped expansion chamber (studied e.g. by [7]), but with more complex behaviour. Section 3 presents an algorithm which allows designing filter-element shapes with prescribed transmission properties; two examples are provided for a lowpass filter with wide passband and extended adjacent stopband and a filter with a wide passband surrounded by wide stopbands. Section 4 of the article draws some concluding remarks.
2. Simple filters Let us assume a circular acoustic transmission line (waveguide) with constant radius r a filled with medium with ambient density ρ0a and speed of sound c0a divided into two parts between which an element (filter) of length d is inserted with generally variable radius r b ¼ rðxÞ and different medium (ρ0b , c0b ), see Fig. 1. Let us assume further that there are only plane waves propagating in the waveguide. In front (upstream) of the element, there is an incoming wave p0i ðx; tÞ and wave p0r ðx; tÞ which reflects on the element. As the region behind (downstream) the element is assumed to be semi-infinite, there is only the transmitted outgoing wave p0t ðx; tÞ. The incoming wave is supposed to be harmonic with angular frequency ω so that phasor representation is employed iωt ^ . Hence, we can write the following formulas describing the incoming, reflected and transmitted as p0 ðx; tÞ ¼ ℜ½pðxÞe waves: p^ i ðxÞ ¼ P 0i e ika x ;
p^ r ðxÞ ¼ P 0r eika x ;
p^ t ðxÞ ¼ P 0t e ika ðx dÞ ;
(1)
where P 0i , P 0r and P 0t are (complex) constants (amplitudes) and ka ¼ ω=c0a is the wavenumber. As it is shown below, the frequency characteristics of the transmission coefficient of the filter strongly depend on its geometry and characteristic impedance. 2.1. Constant-cross-sectioned filter element Let us assume the case of r b ¼ r b0 ¼ const: In this case, the acoustic pressure phasor inside the element has the form ^ pðxÞ ¼ A1 e ikb x þ A2 eikb x ;
(2)
where A1 and A2 are integration constants and kb ¼ ω=c0b is the wavenumber. The transmission coefficient through the element can be calculated from the continuity of acoustic pressure [given by Eqs. (1) and (2)] and acoustic volume velocity at x ¼0 and x ¼d as T ¼
P 0t 4 ¼ pffiffiffi pffiffiffiffiffiffiffiffi ; pffiffiffi pffiffiffiffiffiffiffiffi P 0i ð α þ 1=αÞ2 eiπΩ ð α 1=αÞ2 e iπΩ
(3)
ρ0a c0a r 2b0 ρ0b c0b r 2a
(4)
where α¼
which is the product of characteristic impedances ratio with the ratio of waveguides' cross-sections and Ω ¼ kb d=π is the dimensionless frequency.
Fig. 1. Geometry of the problem.
Fig. 2. Transmission coefficient modulus for r b ¼ r b0 ¼ const: and individual values of α.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Analysis of formula (3) shows that if α ¼ 1, the acoustic impedances of individual waveguides are matched and perfect transmission occurs ðjT j ¼ 1Þ for all frequencies, see e.g. [8]. Otherwise, if α a1, perfect transmission occurs only if Ω ¼ n; with n being an integer. The perfect transmission for non-zero frequencies is caused by resonance of waves within the element. On the contrary, if Ω ¼ ð2n 1Þ=2, the transmission is minimal and it holds jT jmin ¼
2α : 1 þ α2
(5)
Frequency characteristics of the transmission coefficient modulus for various values of α can be seen in Fig. 2. It can be observed that the resonance Q-factor increases with increasing value of α (or 1=α). For α⪢1, the frequency bandwidth of resonance peaks can be calculated as ΔΩ 4π=α. The case of r b0 4 r a and za ¼ zb corresponds to a simple expansion-chamber filter, see e.g. [1]. 2.2. Cosine-shaped filter element Let us assume the filter element with shape described by function d ; r b ðxÞ ¼ r m cos γ 0 x 2
0 rx rd;
(6)
where r m is its maximum radius. The coefficient γ 0 is defined as γ0 ¼
2 r arccos b0 ; d rm
γ 0 d A 〈0; πÞ:
(7)
If r b0 ¼ r a , then r b ð0Þ ¼ r b ðdÞ ¼ r a and the connection to the external waveguides is continuous. For this shape function, acoustic pressure within the element can be calculated, see Appendix A.1, as 0
p^ ðxÞ ¼
0
A1 eiμ x þA2 e iμ x ; cos γ 0 x d=2
(8)
2
where μ02 ¼ kb þ γ 02 . Applying the continuity of acoustic pressure [given by Eqs. (1) and (8)] and volume velocity at x ¼0 and x¼d results in the transmission coefficient T ¼
P 0t 4αμ0 kb ¼ ;
2 0
2 P 0i 0 0 iμ αa þ iðαμ þ kb Þ e d αa0 iðαμ0 kb Þ e iμ0 d
where a0 ¼
0 1 dr γd : ¼ γ 0 tan r dx x ¼ 0 2
(9)
(10)
The central waveguide is effectively an acoustic resonator with acoustically soft end-walls if α o1 or with acoustically hard end-walls if α 41. As it is shown in Appendix A.1, resonance frequencies of cosine-shaped resonator with acoustically soft end-walls decrease with increasing parameter γ 0 d, see Eq. (A.5). It is thus possible, in case of αo 1, to decrease the resonator first eigenfrequency to such a low value that the two first peaks of perfect transmission get overlapped and make a low-frequency passband. This situation can be seen in Fig. 3 in the case of α ¼ 1=10 and γ 0 d ¼ 2:7. In the case of α 41, the eigenfrequencies increase with increasing parameter γ 0 d and the low-frequency passband cannot appear.
Fig. 3. Left: transmission coefficient moduli of cosine-shaped filter elements for α ¼ 1=10 and individual values of γ 0 d; right: the corresponding filter element shapes for r b0 ¼ r a .
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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2.3. Hyperbolic-cosine-shaped filter element The case described in this section corresponds to the one studied in [2]. Let us assume the filter element with shape described by function d ; 0 rx r d: (11) r b ðxÞ ¼ r m cosh γ x 2 where r m is its minimum radius. The coefficient γ is defined as γ¼
2 r arccosh b0 : d rm
(12)
If r b0 ¼ r a , then r b ð0Þ ¼ r b ðdÞ ¼ r a and the connection to the external waveguides is continuous. For this shape function, acoustic pressure within the filter element can be calculated, see Appendix A.2, as p^ ðxÞ ¼
A1 eμx þ A2 e μx ; cosh γ x d=2
(13)
2
where μ2 ¼ γ 2 kb . Applying the continuity of acoustic pressure [given by Eqs. (1) and (13)] and volume velocity at x¼0 and x ¼d results in the transmission coefficient T ¼
P 0t 4iαμkb ¼ ; P 0i ðαμ αa þikb Þ2 eμd ðαμ þ αa ikb Þ2 e μd
(14)
1 dr γd : ¼ γtanh r dx x ¼ d 2
(15)
where a¼
As it is shown in Appendix A.2, the first resonance frequency of hyperbolic-cosine-shaped resonator with acoustically hard end-walls decreases with increasing parameter γd, see Fig. A2. It is thus possible, in case of α 4 1 (contrary to the case of cosine-shaped resonator), to decrease the resonator first eigenfrequency to such a low value that the two first peaks of perfect transmission get overlapped and make a low-frequency passband. This situation can be seen in Fig. 4 in the case of α ¼ 10 and γd ¼ 5. It can be observed in the figure that for this value of γd the higher resonance peaks are shifted towards the higher frequencies extending the bandstop frequency range. It is also noteworthy to mention that for kb oγ the perfect transmission occurs even if the waves in the filter element are evanescent. In the case of α o1, the eigenfrequencies increase with increasing parameter γd and the low-frequency passband cannot appear. 2.4. Numerical validation As it is obvious from formula (4), the impedance mismatch (α a1) can be achieved either by different characteristic impedances (ρ0a c0a a ρ0b c0b ), by cross-section discontinuity (r b0 a r a ) or by the combination of both. Applicability of formulas (9) and (14) for calculation of transmission coefficients in case of homogeneous media and cross-section discontinuities is demonstrated by comparison of predicted analytical and numerical results. The numerical results were calculated in 2D axisymmetric geometry by finite element method in COMSOL Multiphysics (Acoustics Module, Pressure Acoustics, Frequency Domain) for cosine-shaped and hyperbolic-cosine-shaped filter elements. In both the cases, d ¼500 mm and r a ¼ 10 mm and air at normal conditions as a medium in the entire geometry were assumed. In the case of cosine-shaped element, α ¼ 1=10 and γ 0 d ¼ 2:7, see Fig. 3, which results in r b0 ¼ 3:16 mm and r m ¼ 14:44 mm. In the case of hyperbolic-cosine-shaped resonator, α ¼ 10 and γd ¼ 5, see Fig. 4, which results in r b0 ¼ 31:62 mm and r m ¼ 5:16 mm. The geometry is depicted for illustration in Fig. 5. The comparison of numerical and analytical results can be seen in Fig. 6. It can be observed that the agreement is very good. In both the cases the agreement of numerical and analytical results decrease with increasing frequency as the validity
Fig. 4. Left: transmission coefficient moduli of hyperbolic-cosine-shaped filter elements for α ¼ 10 and individual values of γd; right: the corresponding filter element shapes for r b0 ¼ r a .
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 5. Cosine-shaped and hyperbolic-cosine-shaped resonators used as lowpass filters.
Fig. 6. Comparison of analytical and numerical results.
of Webster's equation used here for description of plane waves in shaped waveguides is limited by relation krjdr=dxj=2⪡1, see [9].
3. Optimized resonant filters The above-mentioned analytical examples have shown that the shape of the resonant filter element can strongly influence the resulting transmission coefficient characteristics. Depending on the value of α, resonance frequencies can be shifted upwards or downwards; in the case of the first resonance frequency shifted downwards, the low-frequency passband can appear. Shaped resonators are known to have non-trivial eigenfrequency spectra, which is the property utilized for generation of high-amplitude acoustic fields, see e.g. [10]. Utilizing the perturbation theory, Hamilton et al. [11] have shown that each eigenfrequency can be independently shifted via appropriate modulation of the resonator wall radius. It is thus in principle possible to find (optimize) the resonator shape in such a way that the first resonant frequency is shifted downwards, the second one upwards so that the low-frequency bandpass region is followed by adjacent wide bandstop region. For this purpose, the approach used in work [12] for resonant cavities shaping could be employed in a straightforward way. Within this work, a more direct approach was adopted which is based on search for resonant-filter-element shape approximating prescribed transmission coefficient spectrum. As it is demonstrated below, this approach allows for designing a wider variety of acoustic filters. 3.1. Riccati equation An effective way of calculating the reflection/transmission of acoustic waves impinging upon a non-uniform region utilizes the Riccati equation, see e.g. [13,14]. The Riccati equation for variable-cross-sectioned waveguide can be derived from linearized fluid-dynamics equations written in the form ∂ρ0 1 ∂ 2 r ρ0b v ¼ 0; þ ∂t r 2 ∂x
ρ0b
∂v ∂p0 ¼ ; ∂t ∂x
p0 ¼ c20b ρ0 ;
(16)
where ρ0 is the acoustic density and v is the acoustic velocity. Introducing the acoustic volume velocity w ¼ πr 2 v, Eq. (16) can be for harmonic waves rewritten as ^ dw πr 2 ^ ¼ iω p; dx ρ0b c20b
dp^ ρ ^ ¼ iω 0b2 w: dx πr
^ w ^ can be written as Using these equations, Riccati equation for acoustic impedance Z ¼ p= ^ p^ dw dZ d p^ 1 dp^ ρ πr 2 2 ¼ ¼ 2 ¼ iω 0b2 þiω Z : ^ ^ dx w dx dx w w πr ρ0b c20b ^ dx
(17)
(18)
Introducing dimensionless variables x X¼ ; d
R¼
r r b0
;
Z¼
πr 2a Z; ρ0a c0a
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Eq. (18) can be non-dimensionalized into the form dZ iπΩ ¼ 2 þ iπΩαR2 Z 2 ; dX αR
(19)
which is more suitable for numerical integration. The coefficient α can describe, see Eq. (4), the ratio of the characteristic impedances, the ratio of cross-sections at the cross-section discontinuities or the combination of both. The normalized Riccati equation (19) is employed similarly as in [14] for the calculation of the filter-element transmission coefficient spectrum as follows. Behind the filter element, there is only an outgoing transmitted wave, so that for the normalized acoustic impedance, it holds Z ¼ 1 for X 41. The value of Z ¼ 1 is used as an initial condition for the Riccati equation (19) integrated from X¼1 to X ¼0, which provides the acoustic impedance Zð0Þ “seen” by the wave impinging upon the filter element. The corresponding reflection coefficient is then calculated as R
P 0r Zð0Þ 1 : ¼ P 0i Zð0Þ þ 1
(20)
As dissipation of acoustic energy is neglected within this model, the following relation holds between the reflection coefficient R and the transmission coefficient through the filter element T : jRj2 þ jT j2 ¼ 1; resulting in jT j ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ℜ½Zð0Þ : jZð0Þ þ 1j
(21)
3.2. Optimization procedure The optimization procedure for designing the resonant filter elements works as follows. The filter-element properties are parametrized using an M-dimensional parameter vector q which defines the element shape and the parameter α, where M is supposed to be odd. The filter-element shape is described using a set of ðM 1Þ=2 þ 2 control points ½X i ; Ri ¼ ½0; 1; ½X 1 ; R1 ; ½X 2 ; R2 ,..., ½X K ; RK ; ½1; 1;
(22)
where K ¼ ðM 1Þ=2, which are interconnected using the cubic splines with zero derivatives at X¼0 and X¼1. The value of Rð0Þ ¼ Rð1Þ ¼ 1 is kept fixed [see Eq. (22)] in order to control the impedance mismatch only by the parameter α, whose value is also the subject of the optimization. The parameter vector q thus reads q ¼ ½q1 ; q2 ; q3 ; …; qM ¼ ½X 1 ; X 2 ; …; X K ; R1 ; R2 ; …; RK ; α:
(23)
The required transmission coefficient spectrum is sampled at N frequencies Ω1 ; Ω2 ; …; ΩN providing the values jT req ðΩi Þj for i ¼ 1; …; N. The parameter vector q is sought in the sense of the least squares method by minimization the M-dimensional objective function Q ðqÞ ¼
N
2 1X w jT req ðΩi Þj jT ðq; Ωi Þj ; Ni¼1 i
(24)
where wi are weighting factors for individual frequencies and jT ðq; Ωi Þj is calculated employing the Riccati equation (19) and relation (21). The objective function (24) was minimized heuristically employing a self-adaptation variant ðμ; λÞ-ES of Evolution Strategies, see e.g. [15], as the method has been successfully used earlier for solution of related problems [12,14]. Eq. (19) was integrated numerically using adaptive step-size Dormand–Prince method [16], implemented in Cþ þ. In brief, the ðμ; λÞ-ES was implemented as follows: Initialization: The population of λ individuals is generated each of whom has a genome represented by the parameter vector q [Eq. (23)]. The individual values of the parameter vector are generated randomly in intervals: X i A 〈0; 1〉, Ri A 〈Rmin ; Rmax 〉, α A 〈αmin ; αmax 〉. The control points' abscissas are generated in order to have ascending values. Then, each
Fig. 7. Example of evolutions of the objective function; Q min stands for the minimum value of Q within the given generation.
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individual is checked to meet RðxÞ A 〈Rmin ; Rmax 〉 for X A 〈0; 1〉 as the cubic splines can cause crossing the pre-defined limits. If this is the case, the corresponding individual is rejected and re-generated. All the individuals are assessed, the μ-ones with the lowest value of the objective function Q ðqÞ [Eq. (24)] are selected as parents for the evolution. The evolution: From the μ parents, pairs are randomly drawn to produce a new generation of λ offspring. The genome of a new individual is created by cross-breeding (individual components of genome qi are randomly taken from the parent's genomes) and mutation (qi -qi þ si , where si is a random number with zero mean value and Gaussian distribution). The new individual is checked whether RðxÞ A 〈Rmin ; Rmax 〉 for X A 〈0; 1〉 and α A 〈αmin ; αmax 〉, if not, it is rejected. After all the generation has been produced, the individuals are assessed, the μ of them with the lowest objective function are chosen as the parents for next generation and the evolution continues for the pre-defined number of generations. Fig. 7 shows an example of evolution lasting for 500 generations repeated 100 (the case studied in Section 3.3.1 for α o1) with different (randomly generated) initial conditions. It can be seen that there are three minima; the algorithm converged several times to the global one. More on the heuristic optimization can be found e.g. in [15]. 3.3. Numerical results The numerical results presented within this section were obtained using the above-described procedure. The optimization was performed for M ¼11 (7 control points), for both αo 1 and α 4 1. As it can be expected, the filter-element spectrum depends on the range in which its radius can vary. For this reason, some reasonable Rmin and Rmax were defined delimiting the allowed resonator dimensions. In each case studied here, the optimization was repeated 1000 each population consisted of λ ¼ 105 individuals with μ ¼ 15 parents, each evolution lasted for 500 generations. 3.3.1. Optimized lowpass filter For the lowpass filter, the required transmission coefficient characteristics was prescribed as ( 1: Ωi rΩcut ; where Ωi ¼ 0:1; 0:2; 0:3; …; Ωmax : jT req ðΩi Þj ¼ 0: Ωcut oΩi rΩmax ;
(25)
As an example, the results of optimization for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α o 1 are shown in Fig. 8. In this case, the filterelement radius was restricted to RðXÞ A 〈1:0; 5:0〉 The resulting shape for optimum αopt ¼ 0:265 is seen in the left panel; the positions of the control points are marked by the crosses. It can be observed that the shape has two maxima, it is rather simple and more-or-less symmetric about its centre. As the optimization employs 1D model, the numerical results are validated by comparison with the ones calculated using 2D axisymmetric model implemented in COMSOL Multiphysics (Acoustics Module, Pressure Acoustics, Frequency Domain). Air at normal conditions was used as medium in entire geometry, cross-section discontinuities were introduced to implement non-unitary α. These parameters were used in the 2D model: d¼500 mm and r a ¼ 10 mm. The right panel of Fig. 8 shows the comparison of required and obtained transmission coefficient spectrum together with the spectrum calculated employing 2D model for the optimized shape. It can be observed that the numerical results correspond to each other very well and that the optimization provides much (approximately two-times) wider bandstop region compared to the non-optimized cosine-shaped resonator (see Fig. 6a). As it was mentioned earlier, the value of αopt is together with the control points the result of the optimization procedure. The value of α, similarly as it was discussed in Section 2.1, influences the quality factor of individual resonance peaks (for given filter-element geometry). This behaviour is shown in Fig. 9. For α oαopt , the quality factor is higher, which means that the roll-off is steeper, however, the passband is not as flat as in the optimum case. On the other hand, if α 4αopt , the passband is flatter than in the optimum case, however, the roll-off is less steep. It is thus obvious that generally, the wider the prescribed dimensionless bandwidth, the less steep the filter roll-off1. The ability of the filter-element shape to control its individual resonance frequencies, as it follows from the analysis given in the Appendix, depends on the value of Rmax =Rmin ratio. This fact is demonstrated in Fig. 10. Here, the prescribed transmission characteristic (25) is approximated by filters with values of Rmax ¼ 2:5; 3:0; 3:5; 4:0; 4:5; 5:0 for Ωcut ¼ 0:5 and
Fig. 8. Lowpass filter optimized for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α o 1, RðXÞ A 〈1:0; 5:0〉.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 9. Transmission coefficient spectrum of the filter from Fig. 8 for various values of α; dashed lines: α o αopt , solid line: α ¼ αopt (corresponds to Fig. 8), dash-dot lines: α 4 αopt .
Fig. 10. Transmission coefficient spectrum of the filter optimized for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α o 1 with individual values of Rmax . In all the cases, Rmin ¼ 1.
Fig. 11. Transmission coefficient spectra and corresponding filter-element shapes for Ωcut ¼ 0:5 and different values of Ωmax – optimization for α o 1.
Fig. 12. Lowpass filter optimized for Ωcut ¼ 0:5, Ωmax ¼ 3:0 and α 4 1, RðXÞ A 〈0:2; 1:0〉.
Ωmax ¼ 3:0. In all the cases Rmin ¼ 1. It can be observed that for low values of Rmax there is a resonance peak at Ω 2 present which is suppressed (shifted behind Ωmax ) for higher values of Rmax . The increasing value of Rmax can be used for further increasing the filter-element stopband width, see Fig. 11. The transmission coefficient spectrum from Fig. 8 (Ωmax ¼ 3:0, Rmax ¼ 5:0, dashed line) is compared here with the one for prescribed stopband extended to Ωmax ¼ 3:5 (Rmax ¼6.0, dash-dot line) and the one for prescribed stopband extended to Ωmax ¼ 4:0 (Rmax ¼ 8:0, solid line). In all the cases, Rmin ¼ 1:0 and Ωcut ¼ 0:5. It can be observed that for the highest value of Rmax ¼ 8:0, the first as well as the second resonance peak is shifted leftwards which results in a flat and wide passband with steep roll-off (αopt ¼ 0:0706). The corresponding filter-element shapes are depicted in the right panel of Fig. 11. It can be seen that all the shapes are very similar. In the previous paragraphs, all the filter-element shapes were optimized for αo 1, however, similar results can be obtained for α 4 1. Fig. 12 shows the filter-element shape and its transmission coefficient spectrum for prescribed characteristics (25), Ωcut ¼ 0:5, Ωmax ¼ 3:0 and RðXÞ A 〈0:2; 1:0〉. The calculated transmission coefficient spectrum is compared with the one obtained using the 2D model with the same parameters as before. Again, the correspondence of the results is very good. The resulting shape for optimum α ¼ 6:85 has two minima and contrary to its counterpart depicted in Fig. 12, it is noticeably non-symmetric about its centre. Compared to the hyperbolic-cosine resonator, see Fig. 6b, the bandstop region is substantially wider, the numerical results have shown, analogously as in the previous case, that the stopband can be further extended by lowering the constraint Rmin . Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 13. Bandpass filter optimized for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α o 1; RðXÞ A 〈0:2; 1:0〉.
Fig. 14. Bandpass filter optimized for various values of ΔΩ ¼ Ωl Ωh , Ωmax ¼ 3:5 and α o 1; RðXÞ A 〈0:2; 1:0〉.
3.3.2. Optimized bandpass filter Within this paragraph, it is demonstrated that the presented approach can be employed for designing a bandpass filter. For this purpose, the required transmission coefficient characteristics can be prescribed as 8 > < 0: Ωi o Ωl ; where Ωi ¼ 0:1; 0:2; 0:3; …; Ωmax : (26) jT req ðΩi Þj ¼ 1: Ωl r Ωi rΩh ; > : 0: Ω oΩ rΩ max ; i h As an example, the results of optimization for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α o 1 are shown in Fig. 13. In this case, the filter-element radius was restricted to RðXÞ A 〈0:2; 1:0〉. The resulting shape for optimum αopt ¼ 0:110 is seen in the left panel; the positions of the control points are marked by the crosses. It can be observed that the shape has one minimum, it is rather simple and more-or-less symmetric about its centre. Contrary to the case of lowpass filter optimized for α o1, see Fig. 8, the filter element is contracted. The right panel of the figure shows the comparison of required and obtained transmission coefficient spectra together with the spectrum calculated employing 2D model for the optimized shape. Air at normal conditions was used as medium in the external waveguides. To implement non-unitary α, a medium with c0b ¼ c0a and ρ0b ¼ ρ0a =α was assumed to fill the filter element. These parameters were used in the 2D model: d¼500 mm and r a ¼ 10 mm. It can be observed that the numerical results correspond to each other very well. In this case, the transmission band appears because of the fact that the first and second resonant frequencies are shifted towards each other and the corresponding resonance peaks mutually overlap. Fig. 14 demonstrates the possibility of shifting the passband; the parameters of the optimization are the same as in the previous case except for the frequencies Ωl , Ωh , which are varied as Ωl ¼ 1:1; …; 1:6 and Ωh ¼ 1:4; …; 1:9 (the width of the passband is kept the same Ωh Ωl ¼ 0:3). The left panel of the figure shows the obtained transmission characteristics, the right panel of the figure shows the corresponding filter-element shapes. It can be observed that the shapes are rather symmetric with the exception of the case of the highest frequency range; the numerical results show that the shape gets symmetric if the ratio Rmax =Rmin is increased. In the previous paragraphs, the filter-element shapes were optimized for α o 1, again, similar results can be obtained for α 41. Fig. 15 shows the filter-element shape and its transmission coefficient spectrum for prescribed characteristics (26), for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α 41. In this case, the filter-element radius was restricted to RðXÞ A 〈1:0; 5:0〉. The calculated transmission coefficient spectrum is compared with the one obtained using the 2D model with the same parameters as before. Again, the correspondence of the results is very good. The resulting shape for optimum αopt ¼ 7:70 resembles an inversion of the shape obtained for α o 1 – see Fig. 13.
4. Conclusions It has been shown by means of analytical results that the formation of wide passband in non-uniform resonant acoustic filters is caused by mutual approaching of individual eigenfrequencies and that the corresponding shapes depend on Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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Fig. 15. Bandpass filter optimized for Ωl ¼ 1:5, Ωh ¼ 1:8, Ωmax ¼ 3:5 and α 4 1; RðXÞ A 〈1:0; 5:0〉.
whether the filter-element characteristic impedance is higher or lower than the characteristic impedance of the external waveguide. It has also been demonstrated that this class of filters can be realized even without the necessity of different characteristic impedances. An evolutionary algorithm based on numerical integration of the Riccati equation has been proposed to demonstrate that the acoustic filters can be optimized and designed to approximate prescribed transmission characteristics. The proposed algorithm was used to optimize low-frequency bandpass filters; it has been shown that suitable filter shaping results in substantial extending of the bandstop region adjacent to the passband. The algorithm was also used to design bandpass filters with wide passband surrounded by wide stopbands as it is typical for phononic crystals, however, the proposed arrangement is much simpler as it consists of only one simple-shaped homogeneous element. The techniques employed within this work can be further elaborated to include inhomogeneous medium, waveguides with different characteristic impedances or cross-sections where optimized filter elements could serve for broadband impedance matching. This is the subject of our future work.
Acknowledgements This work was supported by GACR grant 15-23079S. The authors would also like to acknowledge the contribution of the COST Action CA15125. Some numerical calculations were performed at Computing and Information Centre of CTU.
Appendix A. Eigenfrequencies of cosine- and hyperbolic-cosine-shaped resonators Webster's equation describing planar harmonic waves in a waveguide with variable cross-section SðxÞ ¼ πr 2 ðxÞ can be written in the form 1 d 2 dp^ 2 r þ k p^ ¼ 0; (A.1) 2 dx r dx see e.g. [9], where p^ is the acoustic pressure phasor and k ¼ ω=c0 is the wavenumber. If a new variable F^ is introduced such that p^ ¼ F^ =r and substituted into Eq. (A.1) it becomes ! 2 2 d F^ 1d r ^ 2 F ¼ 0: (A.2) þ k r dx2 dx2 Eq. (A.2) has a simple analytical solution if 2
1d r ¼ γ 2 ¼ const: r dx2
(A.3)
The relation (A.3) is fulfilled for γ 2 40, if rðxÞ ¼ A1 eγx þ A2 e γx ;
rðxÞ ¼ A coshðγxÞ;
rðxÞ ¼ A sinhðγxÞ;
where A; A1 ; A2 are constants; for γ o 0, if 2
rðxÞ ¼ A1 sin ðγ 0 xÞ þ A2 cos ðγ 0 xÞ; where γ 0 ¼ iγ; and for γ ¼ 0, if rðxÞ ¼ A1 þ A2 x:
A.1. Cosine-shaped resonator Let us assume a resonant cavity with shape described by relation (6). For this shape, Eq. (A.2) reads d F^ þμ02 F^ ¼ 0; dx2 2
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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2
where μ02 ¼ k þ γ 02 and it has the solution 0 0 F^ ðxÞ ¼ A1 eiμ x þA2 e iμ x ;
and thus the formula for acoustic pressure phasor has the form of Eq. (8). From here, acoustic velocity phasor can be calculated using Euler's equation v^ ðxÞ ¼
i dp^ ; kz0 dx
(A.4)
where z0 ¼ ρ0 c0 is the characteristic impedance. A.1.1. Acoustically soft boundaries ^ ^ At acoustically soft boundaries, it holds pð0Þ ¼ pðdÞ ¼ 0. Substitution into Eq. (8) results in A1 þA2 ¼ 0; 0
0
eiμ d A1 þe iμ d A2 ¼ 0; This set of equations has a non-trivial solution if 0
e2iμ d ¼ 1 ) μ0 d ¼ nπ; with n being an integer. From here, we can write for eigenfrequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωd 2 2 ¼ n2 π 2 γ 02 d ) Ω ¼ n2 γ 02 d =π 2 : c0
(A.5)
Comparing the relation (A.5) with the one for a constant-cross-sectioned resonator ðΩ ¼ nÞ shows that in the case of cosineshaped resonator with acoustically soft boundaries, the values of eigenfrequencies decrease with increasing value of γ 0 d, see Fig. A1a. A.1.2. Acoustically hard boundaries ^ ^ At acoustically hard boundaries, it holds vð0Þ ¼ vðdÞ ¼ 0. Substitution into Eqs. (8) and (A.4) results in ðiμ0 a0 ÞA1 ðiμ0 þa0 ÞA2 ¼ 0; 0
0
0
ðiμ þ a0 Þeiμ d A1 ðiμ0 a0 Þe iμ d A2 ¼ 0; where a0 is defined by relation (10). This set of equations has a non-trivial solution if 0 iμ a0 2 0 ¼ e2iμ d : iμ0 þa0 As the modulus of both the sides of this equation is unitary, the equation is satisfied if 2μ0 a0 ¼ μ0 d nπ: arctan 02 a μ02
(A.6)
Numerical solution of Eq. (A.6) shows that in the case of cosine-shaped resonator with acoustically hard boundaries, the values of eigenfrequencies increase with increasing value of γ 0 d, see Fig. A1b. A.1.3. Duality According to dualities of horns, see [17], eigenfrequencies of a resonator with the shape function r b ðxÞ ¼
r m ; d 0 cos γ x 2
0 rx r d;
(A.7)
are in the case of acoustically hard boundary conditions given by Eq. (A.5) and in the case of acoustically soft boundary conditions by solution of Eq. (A.6).
Fig. A1. Normalized eigenfrequencies of a cosine-shaped resonator with acoustically soft and hard boundaries; m is the numerical order of the corresponding mode.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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A.2. Hyperbolic-cosine-shaped resonator Let us assume a resonant cavity with shape described by relation (11). For this shape function, Eq. (A.2) reads d F^ μ2 F^ ¼ 0; dx2 2
2
where μ2 ¼ γ 2 k and it has the solution F^ ðxÞ ¼ A1 eμx þ A2 e μx : and thus the formula for acoustic pressure phasor has the form of Eq. (13). It is evident that for low frequencies, when k o γ, the two waves forming solution (13) are evanescent. Acoustic velocity phasor can be calculated from Eq. (13) using Euler's equation (A.4). A.2.1. Acoustically soft boundaries ^ ^ At acoustically soft boundaries, it holds pð0Þ ¼ pðdÞ ¼ 0. Substitution into Eq. (13) results in A1 þ A2 ¼ 0; eμd A1 þ e μd A2 ¼ 0; This set of equations has a non-trivial solution if e2μd ¼ 1: This condition can be fulfilled for non-zero μ only if μ is pure-imaginary, i.e. if k 4 γ. From here, we can write for eigenfrequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωd 2 2 ¼ n2 π 2 þ γ 2 d ) Ω ¼ n2 þγ 2 d =π 2 : (A.8) c0 Comparing the relation (A.8) with the one for a constant-cross-sectioned resonator ðΩ ¼ nÞ shows that in the case of hyperbolic-cosine-shaped resonator with acoustically soft boundaries, the values of eigenfrequencies increase with increasing value of γd, see Fig. A2a. A.2.2. Acoustically hard boundaries ^ ^ At acoustically hard boundaries, it holds vð0Þ ¼ vðdÞ ¼ 0. Substitution into Eqs. (13) and (A.4) results in ðμ þ aÞA1 ðμ aÞA2 ¼ 0; ðμ aÞeμd A1 ðμ þ aÞe μd A2 ¼ 0; where a is defined by relation (15). This set of equations has a non-trivial solution if aþμ 2 ¼ e2μd : aμ
(A.9)
Eq. (A.9) can be solved for real (positive) μ [γ 4 k – low-frequency evanescent waves; the numerical results show that the qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 real solution exists for γd≳2:39936 (r a =r m ≳1:810)] as well as for pure-imaginary μ ¼ iμ00 where μ00 ¼ k γ 2 (k 4γ – highfrequency waves) in this case, both sides of Eq. (A.9) have unitary modulus and the solution can be found as 2μ00 a ¼ μ00 d nπ: arctan 2 a μ002
(A.10)
Numerical solution of Eqs. (A.9) and (A.10) shows that in the case of hyperbolic-cosine-shaped resonator with acoustically hard boundaries, the eigenfrequency of the first mode decreases with increasing value of γd, the eigenfrequencies of the higher modes first slightly decrease and then increase with increasing value of γd, see Fig. A2b.
Fig. A2. Normalized eigenfrequencies of a hyperbolic-cosine-shaped resonator with acoustically soft and hard boundaries; m is the numerical order of the corresponding mode.
Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i
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13
A.2.3. Duality According to dualities of horns, see [17], eigenfrequencies of a resonator with the shape function r b ðxÞ ¼
r m ; d cosh γ x 2
0 r x r d:
(A.11)
are in the case of acoustically hard boundary conditions given by Eq. (A.8) and in the case of acoustically soft boundary conditions by solution of Eqs. (A.9) and (A.10).
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Please cite this article as: M. Červenka, & M. Bednařík, Acoustic bandpass filters employing shaped resonators, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.06.045i