© The Author(s), Published by EDP Sciences, 2025

1 Introduction

To meet indoor noise regulations, sound abatement techniques that intervene at the transfer path are widely applied. An example of such techniques are sound absorbing panels, which lower the indoor sound pressure level by reducing the intensity of the reflected sound waves. These panels commonly consist of either porous layers (e.g. mineral wool) or resonance-based absorbers (e.g. Helmholtz-, λ/2- and λ/4-resonators). However, since the needed layer thickness or resonator length scales with the wavelength [1–4], conventional sound absorbing panels become bulky when targeting low-frequency sound, which poses issues in confined spaces such as vehicle interiors and airplane cabins.

In order to provide a high-performing and space-efficient solution to low-frequency sound problems, academic research has been focussing on acoustic metamaterials (AMMs), which exploit the same physical principles as conventional sound absorbing structures, but are characterized by a more complex geometry, which enables them to realise high absorption while maintaining a (deep) sub-wavelength thickness [5–36].

A commonly used type of AMMs are folded quarter-wavelength acoustic metamaterials (QW-AMMs), which can consist of solely folded λ/4-resonators (usually in a parallel assembly) or can be a combination of a folded λ/4-resonator and other absorber concepts [6–21].

The acoustic performance of such QW-AMMs can be modelled using either computationally extensive finite element simulations or an analytical model. This model requires several inputs, including the effective resonator length.

How this length must be calculated depends on the resonator's folding layout. For labyrinth folded λ/4-resonators, a distinction is made based on the assumed sound propagation path. If the sound waves are assumed to propagate along a zigzag path, as in a.o. [7–14, 19], the effective resonator length can be calculated using Pythagoras’ theorem. If the sound waves are assumed to propagate along a circular path near the fold, the effective length is calculated as the sum of the centreline length and a correction factor [20]. For λ/4-resonators with separate 90° folds, things are less straightforward: for this type of resonators, no consensus is found in literature on how the effective length must be calculated. For example, Zhang Z. et al. propose in [21] to assume the effective length equal to the centreline length. However, the accuracy of this approach is found to largely depend on the resonator layout. In hindsight, an adapted version of the formula presented in [20] could have been applied, but also the accuracy of this approach is expected to be highly dependent on the folding layout since this present formula does not account for the nearness of the fold to the resonator opening, although this parameter is found to be important [37]. In [18], on the other hand, Qinhao, L. and Guoqing, D. applied Pythagoras’ theorem to calculate the effective resonator length. However, it is unclear how the trajectory within each channel is determined. Additionally, the effective length of some channels is greater than the centreline length, which is not in line with the experimental findings in a.o. [21]. Lastly, one can also attempt to calculate the effective length of λ/4-resonators with separate 90° folds by combining the aforementioned formulas, as done in [17]. In their work, Wang Y. et al. propose a length formula which combines the Pythagoras’ theorem, the centreline assumption and a correction factor. Although good agreement between the analytical and numerical results is found, it is unclear on which physics the assumed sound propagation path and proposed length correction are based.

Due to the issues with determining the effective resonator lengths, QW-AMMs that consist of λ/4-resonators with separate 90° folds are barely studied in literature. This is unfortunate, since these structures show much potential, as shown in [21]. They also offer a larger freedom of design as compared with labyrinth folded resonators and, thus, lend themselves better for optimal use of the available space, which might result in a better acoustic performance. Hence, to facilitate the design and encourage further development of this type of QW-AMMs, this paper takes first steps towards the definition of a formula to calculate the effective length of a λ/4-resonator with separate 90° folds by (i) investigating the effect of a single 90° fold on the effective length of rectangle and slit-like λ/4-resonators and (ii) deriving a length correction factor δ to account for this effect into the analytical model, similar to as performed by Cambonie T. et al. for a bent resonator and by Catapane G. et al. for labyrinth resonators [20, 38]. The eventual expression for δ is verified numerically and experimentally validated.

The remainder of this paper is structured as follows. In Section 2, the basic analytical model used to predict the absorption coefficient of rectangular and slit-like λ/4-resonators is explained. Section 3 covers the stepwise derivation of δ and the numerical verification of δ against other single-fold resonator configurations and state-of-the-art length (correction) formulas. In Section 4, the experimental validation of δ by means of impedance tube measurements on 3D printed samples is discussed. Section 5 concludes this paper with an overview of the main results.

2 Analytical model

The effectiveness of a sound absorber is usually expressed in terms of the absorption coefficient α, which is defined as the ratio of the absorbed intensity to the incident intensity under normal incidence. This ratio can be rewritten in terms of impedances as given by equation (1):

$$\alpha =1-{\left|\frac{Z_{\mathrm{tot}}-{\varrho }_0{c}_0}{Z_{\mathrm{tot}}+{\varrho }_0{c}_0}\right|}^2,$$(1)

with ϱ₀ the ambient density, c₀ the ambient sonic speed and Z_tot the input impedance of the sound absorber [3]. The frequency at which α reaches its maximum value is called the resonance frequency f_R of the absorber. If the sound absorber consists of a single and rigid λ/4-resonator embedded in a finite baffle, Z_tot is given by equation (2):

$$Z_{\mathrm{tot}}=-j\frac{A_b}{A_r}\tilde{Z}\mathit{cot}\left(\tilde{k}L\prime \right)+j\frac{A_b}{A_r}{\varrho }_0\omega \varepsilon ,$$(2)

with j the imaginary unit, A_b the cross-section area of the baffle, A_r the cross-section area of the resonator, $\tilde{Z}$ the impedance of the medium inside the resonator, $\tilde{k}$ the wavenumber of the sound waves inside the resonator, L′ the effective resonator length, ω the angular frequency and ɛ the so-called end correction used to account for 3D effects occurring at the resonator opening [2, 3, 39, 40]. If the sound absorber consists of multiple structures in parallel, Z_tot can be calculated based on the electrical-acoustical analogy as given by equation (3):

$$Z_{\mathrm{tot}}={\left({\displaystyle \sum _{i=1}^i}\frac{1}{Z_{r,i}}\right)}^{-1},$$(3)

with Z_r,i the input impedance of the i-th structure. For a QW-AMM consisting of i λ/4-resonators in parallel, Z_r,i is given by equation (2).

2.1 Equivalent fluid parameters $\tilde{\varrho }$ and $\tilde{K}$

The parameters $\tilde{Z}$ and $\tilde{k}$ out of equation (2) can be calculated using equations (4) and (5), respectively:

$$\tilde{Z}=\sqrt{\tilde{K}\tilde{\varrho }},$$(4)

$$\tilde{k}=\omega \sqrt{\frac{\tilde{\varrho }}{\tilde{K}}},$$(5)

with $\tilde{K}$ and $\tilde{\varrho }$ the bulk modulus and density of the medium inside the resonator, respectively [1]. Since the resonators considered in this work are narrow, as classified by Weston [41], significant visco-thermal losses will occur inside the resonators. These losses are accounted for in the calculation of $\tilde{\varrho }$ and $\tilde{K}$.

For a rectangle resonator with cross-section 2w×2h, $\tilde{\varrho }$ and $\tilde{K}$ are calculated using equations (6) and (7), respectively:

$$\begin{array}{cc}\hfill \tilde{\varrho }& =\frac{\eta w^2{h}^2}{j4\omega }{\left[{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{A_m^2{B}_n^2\left(A_m^2+B_n^2+\frac{j\omega {\varrho }_0}{\eta }\right)}\right]}^{-1},\hfill \end{array}$$(6)

$$\begin{array}{cc}\hfill \tilde{K}& =P_0\gamma [\gamma -(\gamma -1)\frac{j4\omega {\varrho }_0\mathit{Pr}}{\eta w^2{h}^2}\hfill \\ \hfill & \times {{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{A_m^2{B}_n^2\left(A_m^2+B_n^2+\frac{j\omega {\varrho }_0\mathit{Pr}}{\eta }\right)}]}^{-1},\hfill \end{array}$$(7)

with A=(m+0.5)π/w, B=(n+0.5)π/h, η the dynamic viscosity of the medium, γ the specific heat ratio, Pr the Prandtl number and P₀ the ambient pressure [42].

For a slit-like resonator (i.e. a rectangle resonator for which 2h≫2w), $\tilde{\varrho }$ and $\tilde{K}$ are calculated using equations (8) and (9), respectively:

$$\tilde{\varrho }={\varrho }_0{\left[1-\frac{tanh(\beta \sqrt{j})}{\beta \sqrt{j}}\right]}^{-1},$$(8)

$$\tilde{K}=P_0\gamma {\left[1+(\gamma -1)\frac{tanh(\beta \sqrt{\mathit{jPr}})}{\beta \sqrt{\mathit{jPr}}}\right]}^{-1},$$(9)

with β = $w\sqrt{\omega {\varrho }_0/\eta }$ [42].

2.2 End correction ɛ

The end correction for a rectangle or slit-like resonator embedded in a finite rectangle baffle with dimensions 2w′×2h′ can be calculated using equation (10) or equation (11), respectively:

$$\begin{array}{cc}\hfill {\varepsilon }_{R\subset R}& =\frac{4\xi \chi }{\pi }{\displaystyle \sum _{{\mathit{mn}}^{*}}^{\infty }}{\nu }_{\mathit{mn}}{\left[\frac{sin\left(m\pi \xi \right)}{m\pi \xi }·\frac{sin(n\pi \chi )}{n\pi \chi }\right]}^2\hfill \\ \hfill & \times {\left[\sqrt{{\left(\frac{m}{w\prime }\right)}^2+{\left(\frac{n}{h\prime }\right)}^2}\right]}^{-1},\hfill \end{array}$$(10)

$${\varepsilon }_{S\subset R}={\displaystyle \sum _{m=1}^{\infty }}\frac{1}{m}{\left[\frac{sin\left(m\pi \xi \right)}{m\pi \xi }\right]}^2,$$(11)

with ξ=(2w)/(2w′) and χ=(2h)/(2h′) [40, 43]. Both equations (10) and (11) assume the resonator to be in the centre of the baffle.

2.3 Effective resonator length L′

The effective resonator length L′ is calculated differently depending on the resonator layout. For unfolded resonators, L′ equals the centreline length. For labyrinth folded λ/4-resonators, L′ is calculated using either Pythagoras’ theorem or the formula presented by Catapane G. et al. in [20]. For λ/4-resonators with separate 90° folds, however, no well-established approach to calculate the effective length is available in literature. In this work, steps are taken to fill this void. These steps are discussed in more detail in Section 3.

3 Derivation and verification of a length correction factor

In this paper, it is investigated if L′ of rectangle and slit-like λ/4-resonators with a single 90° fold can be expressed as L+δ with L the centreline length and δ a length correction factor, similar to as presented in [20]. To this end, three steps are taken:

1. The effect of a single 90° fold on the effective length of rectangle and slit-like λ/4-resonators is investigated by (i) modelling single-fold rectangle and slit-like λ/4-resonators numerically in COMSOL MultiPhysics^® , (ii) extracting the resonance frequency f_R of each resonator out of the simulation and (iii) calculating L′ from f_R and comparing this value with L;

2. A length correction factor δ is defined as L′−L and a dimensionless expression for δ is derived through curve fitting;

3. The accuracy of δ is verified against arbitrary single-fold resonator configurations and existing length (correction) formulas.

Each of these steps is further elaborated on in the following subsections.

3.1 Numerical simulations in COMSOL MultiPhysics^®

The numerical simulations are performed on a rudimentary resonator shape as illustrated in Figure 1. The resonator has a cross-section of 2w×2h and is centred in a baffle with dimensions 2w′×2h′. L is equal to S₁+S₂. The resonator is folded along 2w. The incident plane sound wave has an amplitude of 1 Pa and propagates through air in a direction normal to the resonator-baffle interface. On top of the baffle, a perfectly matched layer is added to mimic an unbounded domain and rigid boundary conditions are imposed on all outer walls. The resonator is set to be a narrow acoustics region, accounting for visco-thermal losses, while the air in the baffle is assumed to be lossless. The geometry is meshed with tetrahedral elements. The Helmholtz equation is solved in the frequency-domain for a given frequency range and for each of these frequencies, the absorption coefficient is calculated. The study is then re-run using a finer mesh. This process is repeated until the results have converged. Then, the resonance frequency of the folded resonator is extracted from the simulation. The resolution of the numerical simulations is 1 Hz. In total, 168 unique resonators are modelled, each having a resonance frequency between 250 and 2500 Hz. An overview of the dimensions of these resonators is given in Table 1. Four different resonator shapes are considered. The dimensions of the baffle are chosen such that a clear absorption peak should be noticeable. The dimensionless quantity S₁/L expresses the relative nearness of the centre of the fold to the resonator opening. An S₁/L-value of 1 corresponds to an unfolded resonator.

Figure 1. CAD design of a modelled λ/4-resonator. The design consists of three areas: a resonator (grey area), a baffle (white area) and a perfectly matched layer (yellow area).

For each of the 168 resonators, the length correction factor δ is calculated from the identified resonance frequency using equation (12):

$$\delta =L\prime -L=\frac{c}{4f_R}-\varepsilon -L,$$(12)

with c the sonic speed inside the resonator, equal to $|\sqrt{\tilde{K}/\tilde{\varrho }}|$.

3.2 Derivation of δ

Figure 2 shows δ in function of S₁/L for all 168 resonators. When looking at Figure 2, several conclusions can be drawn. Firstly, it can be seen that δ≤0; the effective length of a folded resonator is always smaller than its centreline length. This implies that folding leads to an increase in resonance frequency, which is in line with earlier experimental findings [21]. Secondly, it can be seen that |δ| decreases when S₁/L increases. This is expected, since δ must be zero (i.e. L′=L) when S₁/L=1. Lastly, it can be seen that, for a fixed value of L, |δ| increases when 2w increases.

Figure 2. Relation between S1/L and δ for all 168 modelled resonators, grouped per value of 2w. (a) 2w=1 mm. (b) 2w=3 mm. (c) 2w=4 mm. (d) 2w=6 mm.

Since δ∝2w and 1/δ∝S₁/L, the following non-dimensional relationship can be set out [20, 37, 38]:

$$\frac{\delta }{2w}=f\left(\frac{S_1}{L}\right).$$(13)

Figure 3 shows the relation between the left and right side of equation (13)¹ for all 168 resonators. When comparing Figures 3a–3f, it can be seen that the data points corresponding to L=0.2858 m do not follow the general trend. This is because the 1 Hz resolution of the numerical simulations is too coarse to get an accurate prediction of δ at very low frequencies. Hence, these 28 data points are excluded from further (regression) analyses. A bivariate regression analysis is now performed on the remaining 140 data points to derive the eventual expression for δ/(2w). Given the strong overlap between these points, a single regression line can be fit. This line is set to be a cubic polynomial, which has a general form given by a₁x³+a₂x²+a₃x+a₄. The coefficients of the polynomial are optimised using the least squares method (scipy.optimize.curve_fit), while meeting the following constraint: a₄=−(a₁+a₂+a₃). This constraint forces δ to be 0 when S₁/L=1. Eventually, equation (14) is obtained:

Figure 3. Relation between S1/L and δ/(2w) for all 168 resonators, grouped per value of L. (a) L=0.2858 m. (b) L=0.1319 m. (c) L=0.1072 m. (d) L=0.0572 m. (e) L=0.0463 m. (f) L=0.0407 m.

$$\begin{array}{cc}\hfill \frac{\delta }{2w}=& -0.624{\left(\frac{S_1}{L}\right)}^3+0.926{\left(\frac{S_1}{L}\right)}^2\hfill \\ \hfill & +0.190\left(\frac{S_1}{L}\right)-0.492.\hfill \end{array}$$(14)

The coefficients in equation (14) are rounded to increase readability. The accuracy of equation (14) is assessed for the 140 resonators using a three-step approach:

1. The absorption coefficient of each resonator is calculated analytically using the formulas presented in Section 2 with L′=L (i.e. without any length correction) and its resonance frequency is determined;

2. The relative error between the numerical and analytical resonance frequency is calculated;

3. Equation (14) is integrated into the analytical model so that L′=L+δ and the two previous steps are repeated.

The results are shown in Figure 4. When looking at Figure 4, it can be seen that the accuracy of the analytical model drastically increases when equation (14) is applied: the maximum absolute value of the relative error drops from 6.3% to 0.4%, or in terms of frequency, from 134 Hz to 8 Hz. It is seen that the gain in accuracy is largest for the smallest value of L. This is expected, as the relative impact of δ on L′ increases when L decreases. It is also noticed that, for a fixed value of L, the absolute value of the relative error generally increases with 2w. This implies an additional dependence of δ on 2w, which is not yet accounted for in the current equation. As it is expected that accounting for this additional dependence will further increase the model accuracy, equation (13) is adapted to equation (15):

$$\frac{\delta }{2w}=f{\left(\frac{S_1}{L}\right)}_{2w_0}f\left(\frac{2w}{2w_0}\right),$$(15)

with 2w₀ a reference width [44]. In this paper, the reference width is set to 6 mm, the largest 2w-value in the data pool. The relation between S₁/L and δ/(2w) for the 35 resonators with 2w=6 mm and between 2w/(2w₀) and δ/(2w) for the 140 data points is shown in Figure 5. Since δ/(2w) is now function of two dimensionless ratios, two bivariate regression analyses are performed to derive the eventual expression for δ/(2w). Firstly, the relation between S₁/L and δ/(2w) is analysed for the 35 6×5 mm²-resonators. Based on Figure 5a, this relation can be approximated by a cubic polynomial. The coefficients of this polynomial are again optimised using the least squares method (scipy.optimize.curve_fit), using the same constraint as when deriving equation (14). Since the fit is now based on 35 data points instead of 140, the coefficients of this cubic polynomial will be different compared with equation (14). Secondly, the relation between 2w/(2w₀) and δ/(2w) is analysed for all 140 data points. Based on Figure 5b, this relation can be approximated by a quadratic polynomial, which has a general form given by b₁x²+b₂x+b₃. The coefficients of this polynomial are also optimised using the least squares method (scipy.optimize.curve_fit), this time meeting the following constraint: b₃=1−(b₁+b₂). This constraint forces the fit to be optimal when 2w=2w₀. Eventually, equation (16) is obtained:

Figure 4. Overview of the relative error between the numerical and analytical resonance frequency for the 140 resonators, either or not applying equation (14). Each dot represents a different resonator. (a) No length correction. (b) Use of equation (14).

Figure 5. Relation (a) between S1/L and δ/(2w) for the 35 resonators with 2w=6 mm, and (b) between 2w/(2w0) and δ/(2w) for all 140 resonators. (a) δ/(2w)=f(S1/L). (b) δ/(2w)=f(2w/(2w0)).

$$\begin{array}{cc}\hfill \frac{\delta }{2w}& =\left(-0.722{\varsigma }^3+1.068{\varsigma }^2+0.132\varsigma -0.478\right)\hfill \\ \hfill & \times \left(0.181{\upsilon }^2-0.301\upsilon +1.120\right),\hfill \end{array}$$(16)

with ς and υ representing S₁/L and 2w/(2w₀), respectively [44]. The coefficients in equation (16) are rounded to increase readability. The accuracy of equation (16) is compared with the accuracy of equation (14) and the results of this comparison are visualised in Figure 6. When Figure 6 is looked at, it can be seen that accounting for the additional dependence of δ on 2w has indeed further increased the model accuracy: the maximum absolute value of the relative error drops from 0.4% to 0.35%, or in terms of frequency, from 8 Hz to 5 Hz. Additionally, the relative error has become less dependent on the values of L and 2w as compared with using equation (14).

Figure 6. Overview of the relative error between the numerical and analytical predicted resonance frequency for the 140 resonators, applying either equation (14) or equation (16). Each dot represents a different resonator. (a) Use of equation (14). (b) Use of equation (16).

3.3 Comparison with other length (correction) formulas

To support the earlier claim that the formula presented by Catapane G. et al. will not yield accurate results for resonators with separate 90° folds, said formula is applied to determine the resonance frequency of the remaining 140 data points. Since it now concerns λ/4-resonators with 90° folds instead of U-turns, this formula first has to be adapted, resulting in equation (17):

$$L\prime =L+(\pi -4)\frac{w}{2}·$$(17)

Then, equation (17) is implemented into the analytical model and, for each resonator, the relative error between the numerical and analytical resonance frequency is calculated. The results are visualised in Figure 7. When Figures 6b and 7 are compared, several conclusions can be drawn. Firstly, it can be seen that the dependence of the relative error on both L and 2w is much larger for equation (17) than for equation (16). Secondly, based on the distance between the whiskers, it can be concluded that the accuracy of equation (17) is generally more dependent on S₁/L as compared with equation (16). Lastly, it can be seen that equation (17) indeed yields significantly less accurate results than equation (16), especially at smaller centreline lengths. This was expected, since in equation (17), L′ solely depends on 2w, while in equation (16), L′ is function of both 2w and S₁. The use of Pythagoras’ theorem is not included in this comparison since it is unclear which zigzag sound propagation path should be assumed.

Figure 7. Relative error in predicted resonance frequency when applying equation (17). Each dot represents a different resonator.

3.4 Verification of δ for other single-fold resonators

Given the data used to derive δ, equation (16) should be valid for any resonator that meets the following requirements: (i) 0.1319 m≥L≥0.0407 m, (ii) 2w≤6 mm, and (iii) one 90° fold with its centre at a distance between 0.1L and 0.9L from the resonator opening. In this section, the validity of δ is verified for 160 unique and arbitrary resonator geometries that meet these requirements. These resonators are divided into three sets:

• Set A consists of resonators that have the same cross-section dimensions as those listed in Table 1, but have other values for L and S₁/L;

• Set B consists of resonators that have different cross-section dimensions than those listed in Table 1, but have the same values for L and S₁/L;

• Set C consists of resonators that have the same cross-section dimensions as the resonators in Set B and the same values for L and S₁/L as the resonators in Set A.

An overview of the resonator geometries used for model verification is given in Table 2. Each resonator is modelled in COMSOL MultiPhysics^® as described in Section 3.1 and the accuracy of equation (16) is determined as described in Section 3.2. The results are shown in Figure 8.

Figure 8. Overview of the relative error between the numerical and analytical predicted resonance frequency for the resonators in Set A (top), Set B (middle) and Set C (bottom), either or not applying equation (16). Each dot represents a different resonator. (a) No length correction. (b) Use of equation (16).

When looking at Figure 8, it can be seen that applying equation (16) leads to an improved model accuracy. The maximum absolute value of the relative error when applying equation (16) is now 0.20%, or in terms of frequency, 4 Hz. As these values are in line with the error values in Section 3.2, it can be concluded that the use of equation (16) is justified within the proposed ranges for 2w, L and S₁/L.

4 Experimental validation

Now that equation (16) is successfully verified, its use is also experimentally validated by means of impedance tube measurements on 3D printed samples.

An overview of the samples used for model validation is given in Table 3. To ensure sufficiently high absorption, each sample consists of either eight identical 3×3 mm² or four identical 5×3 mm² λ/4-resonators in parallel. The dimensions of the resonators fall within the validity range of equation (16) and are chosen so that the effect of folding should be clearly noticeable. All samples are printed with an Ultimaker S3 printer using the intent Engineering-profile [45]. As this profile is associated with a high dimensional accuracy, the dimensions of the printed resonators are expected to be close to the CAD dimensions, which should lead to representative experimental results. The samples are made out of Makerfill red PLA, as shown in Figure 9. The printer itself is placed in a climate-conditioned room to avoid additional temperature variability.

Figure 9. Pictures of two 3D printed samples used for model validation.

Once printed, each sample is mounted on an in-house impedance tube (∅=40 mm) to determine its maximum absorption coefficient and resonance frequency. The impedance tube test setup is illustrated in Figure 10. The measurements are performed according to the ISO 10534-2:1998 standard [46].

Figure 10. Sketch of the measurement set-up.

4.1 Manufacturing inaccuracies in 3D printed parts

Due to manufacturing inaccuracies, the geometry of 3D printed resonators will be different from the CAD design, which might translate into an off-design performance. Since the aim of this section is to validate the accuracy of equation (16), it is important to know to which extent manufacturing inaccuracies contribute to the discrepancy between experimental and analytical results so that conclusive statements can be made with regard to the model accuracy. To this end, sample I (see Tab. 3) is printed four consecutive times using the exact same G-code file and without recalibrating the printer in between prints. The theoretical and experimental absorption curves of samples I(1) to I(4) are shown in Figure 11.

Figure 11. Analytical and experimental absorption curves of samples I(1) to I(4) [44].

When looking at Figure 11, it can be seen that the difference in maximum experimental absorption coefficient α_max (exp) between the four samples is limited, while the difference in experimental resonance frequency f_R (exp) between the samples is significant. It can also be seen that the four experimental absorption curves deviate from the analytical result. The discrepancies in both α_max (exp) and f_R (exp) between the samples can be attributed to manufacturing inaccuracies and tolerances.

More specifically, the discrepancies in α_max (exp) can be attributed to the non-ideality of the resonator cross-section, as shown in Figure 12: neither resonator has a perfectly rectangular cross-section, all resonators measure under 5 and 3 mm at their widest points and the rounding radius is not constant within and between samples, even though all resonators are printed with the same nozzle. These non-idealities will affect the visco-thermal losses that occur inside the resonator and, hence, the value of α_max (exp). There are also clear differences in surface quality between the four samples, which might also affect the absorption coefficient. The discrepancies in f_R (exp) between the samples can be attributed to geometric tolerances in the Z-axis. Each resonator of the reference sample is designed with an outer length of 0.042 m, but since a Z-dimension of 42 mm requires a non-integer number of layers (279.7), the slicer software rounds the number of layers to the nearest integer value (280), which results in a targeted Z-dimension of 42.05 mm instead. However, the outer lengths of the printed resonators are found to vary between 41.9 mm and 42.25 mm, as shown in Table 4. Hence, it is assumed that L will also differ from its targeted value of 41 mm, resulting in an off-design resonance frequency.

Figure 12. Microscopic images of the resonator cross-sections of the four reference samples, grouped per row. The scale of the microscopic images is 0.25 mm/division [44].

In sum, it can be stated that (i) there can be substantial differences in dimensional accuracy between 3D printed samples, even when the same printer settings are used and (ii) the influence of the overall print quality (i.e. manufacturing inaccuracies, tolerances and surface quality) on the acoustic performance of a λ/4-resonator cannot be neglected. Hence, it is suggested to use measured resonator dimensions as model input instead of CAD dimensions such that the shift in α and f_R due to dimensional inaccuracies is already accounted for in the analytical model and the remaining error between model and experiments can mainly be attributed to model inaccuracies.

4.2 Repeatability of impedance tube measurements

Prior to the experimental validation, the repeatability of the impedance tube measurements was determined. To this end, a sample out of Table 3 was selected and repeatedly (re)mounted to the impedance tube. Each time, its absorption coefficient was measured and the resonance frequency of the sample was determined. In this work, sample G was selected since it has average values for L and S₁/L. In total, eight test runs were performed and the repeatability of the test procedure was evaluated using the repeatability coefficient κ, which is defined as the value under which the absolute difference between two repeated measurements will lie with a 95% probability and can be calculated using equation (18):

$$\kappa =1.96\sqrt{2{\sigma }^2},$$(18)

with σ² the variance on the measured quantity [47]. In this work, the measured quantity is the resonance frequency and κ was found to be ≊4 Hz.

4.3 Accuracy of δ

To evaluate the accuracy of equation (16), impedance tube measurements are performed on all samples out of Table 3. Once a sample is tested, the test data is saved and the dimensions of each resonator of that sample are measured as follows:

• The average cross-section dimensions $\overline{2w}$ and $\overline{2h}$ of each resonator are determined using a microscope. It should be noted that the values of $\overline{2w}$ and $\overline{2h}$ are assumed to be constant over depth, which might not be the case in reality.

• The actual centreline length $\overline{L}$ of each resonator is measured using a digital caliper. For samples A–D, $\overline{L}$ of each resonator is estimated out of the outer dimensions as the cross-section is too small to fit the depth rod of the caliper. For samples E–G, the resonators are cut open and direct depth measurements are performed.

Then, for each sample, four analytical absorption curves are calculated: α (C,−), α (C,δ), α (M,−), and α (M,δ). The letters “C” and “M” indicate whether the CAD or measured resonator dimensions were used as model input, while “–” and “δ” specify whether L′ in equation (2) equalled L or L+δ, respectively, with δ calculated using equation (16). The properties of air are calculated using the same ambient air temperature and pressure as measured during the tests. These four curves are then compared to the experimental results (α (exp)). An overview of the results for samples A–D and samples E–H is given in Figures 13 and 14, respectively.

Figure 13. Experimental and analytical results for samples A to D. The grey area indicates the area in which the experimental resonance frequency is situated, taking into account the repeatability of the test procedure.

Figure 14. Experimental and analytical results for samples E to H. The grey area indicates the area in which the experimental resonance frequency is situated, taking into account the repeatability of the test procedure.

The accuracy of equation (16) is now assessed in terms of the absolute error ψ as defined by equations (19)–(22):

$$\psi (C,\delta )=f_R(\exp )-f_R(C,\delta ),$$(19)

$$\psi (C,-)=f_R(\exp )-f_R(C,-),$$(20)

$$\psi (M,\delta )=f_R(\exp )-f_R(M,\delta ),$$(21)

$$\psi (M,-)=f_R(\exp )-f_R(M,-).$$(22)

An overview of the measured and calculated resonance frequencies and absolute errors for samples A to D and samples E to H is given in Tables 5 and 6, respectively. When looking at Tables 5 and 6, several conclusions can be drawn. Firstly, it can be concluded that, in general, $\left|\psi (M,-)\right|<\left|\psi (C,-)\right|$ and $\left|\psi (M,\delta )\right|<\left|\psi (C,\delta )\right|$, which implies that using the measured resonator dimensions as model input indeed allows for a more accurate prediction of the resonance frequency compared with using the CAD dimensions. Secondly, when looking closer into the values of $\left|\psi (M,\delta )\right|$, it is seen that, on average, $\left|\psi (M,\delta )\right|$ is larger for samples A to D than for samples E to H. This is attributed to L′ of the resonators of samples A to D being estimated instead of directly measured, but further research is needed to establish this conclusion. Lastly, it can be seen that implementing equation (16) does increase the accuracy of the model as |ψ(M,δ)|≤|ψ(M,−)| for each sample. The maximum value for |ψ(M,δ)| is 5 Hz, while the maximum value for |ψ(M,−)| is 49 Hz. When accounting for the repeatability of the test procedure, these values increase to 9 and 53 Hz, respectively.

5 Conclusions

In this paper, steps are taken towards the definition of a formula to calculate the effective length of a λ/4-resonator with separate 90° folds. More specifically, the effect of a 90° fold on the effective length of rectangle and slit-like λ/4-resonators is investigated numerically and a length correction factor δ is derived to account for this effect in the analytical model. It is found that δ is directly proportional to the resonator width and inversely proportional to the nearness of the fold to the resonator opening. The expression for δ is then verified numerically using COMSOL MultiPhysics^® and validated experimentally by means of impedance tube measurements on 3D printed samples. Out of the experimental validation, it follows that δ grasps the effect of folding correctly and that the implementation of δ leads to a significant increase in the model accuracy: the maximum error between model and experiments drops from 49 Hz to 5 Hz. When the results of the experimental validation are studied more closely, it is noticed that using the measured resonator dimensions generally results in a higher model accuracy compared with using the CAD dimensions. Further steps include investigating (i) if and how δ can be applied to model rectangle and slit-like λ/4-resonators with multiple separate 90° folds, (ii) if and how δ can be applied to model rectangle and slit-like λ/4-resonators with U-turns (cf. labyrinth layout) and (iii) if the presented approach can also be applied to model λ/4-resonators with non-90° folds and λ/4-resonators with other cross-section shapes.

Funding

The research of F. De Bie (fellowship no. 1S17424N) is funded by a personal grant from the Research Foundation Flanders (FWO).

Conflicts of interest

The authors declare to have no known competing financial interests or personal relationships that could have influenced the work described in this paper.

Data availability statement

The data will be made available upon request.

Author contribution statement

Femke De Bie: Conceptualization, Formal analysis, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Hervé Denayer: Conceptualization, Methodology, Supervision, Writing – review & editing. Elke Deckers: Conceptualization, Methodology, Supervision, Writing – review & editing.

References

All Tables

All Figures

	Figure 1. CAD design of a modelled λ/4-resonator. The design consists of three areas: a resonator (grey area), a baffle (white area) and a perfectly matched layer (yellow area).
In the text

	Figure 2. Relation between S1/L and δ for all 168 modelled resonators, grouped per value of 2w. (a) 2w=1 mm. (b) 2w=3 mm. (c) 2w=4 mm. (d) 2w=6 mm.
In the text

	Figure 3. Relation between S1/L and δ/(2w) for all 168 resonators, grouped per value of L. (a) L=0.2858 m. (b) L=0.1319 m. (c) L=0.1072 m. (d) L=0.0572 m. (e) L=0.0463 m. (f) L=0.0407 m.
In the text

	Figure 4. Overview of the relative error between the numerical and analytical resonance frequency for the 140 resonators, either or not applying equation (14). Each dot represents a different resonator. (a) No length correction. (b) Use of equation (14).
In the text

	Figure 5. Relation (a) between S1/L and δ/(2w) for the 35 resonators with 2w=6 mm, and (b) between 2w/(2w0) and δ/(2w) for all 140 resonators. (a) δ/(2w)=f(S1/L). (b) δ/(2w)=f(2w/(2w0)).
In the text

	Figure 6. Overview of the relative error between the numerical and analytical predicted resonance frequency for the 140 resonators, applying either equation (14) or equation (16). Each dot represents a different resonator. (a) Use of equation (14). (b) Use of equation (16).
In the text

	Figure 7. Relative error in predicted resonance frequency when applying equation (17). Each dot represents a different resonator.
In the text

	Figure 8. Overview of the relative error between the numerical and analytical predicted resonance frequency for the resonators in Set A (top), Set B (middle) and Set C (bottom), either or not applying equation (16). Each dot represents a different resonator. (a) No length correction. (b) Use of equation (16).
In the text

	Figure 9. Pictures of two 3D printed samples used for model validation.
In the text

	Figure 10. Sketch of the measurement set-up.
In the text

	Figure 11. Analytical and experimental absorption curves of samples I(1) to I(4) [44].
In the text

	Figure 12. Microscopic images of the resonator cross-sections of the four reference samples, grouped per row. The scale of the microscopic images is 0.25 mm/division [44].
In the text

	Figure 13. Experimental and analytical results for samples A to D. The grey area indicates the area in which the experimental resonance frequency is situated, taking into account the repeatability of the test procedure.
In the text

	Figure 14. Experimental and analytical results for samples E to H. The grey area indicates the area in which the experimental resonance frequency is situated, taking into account the repeatability of the test procedure.
In the text