# Transmission of Low-Frequency Acoustic Waves in Seawater Piping Systems with Periodical and Adjustable Helmholtz Resonator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Acoustic Equations and Calculation Method

_{H}= ρ

_{w}S

_{n}l

_{n}

_{e}and spring stiffness K

_{H}= κ

_{w}S

_{n}

^{2}/V

_{c}, or could be analogous to an inductor-capacitor circuit, whose acoustic impedance L

_{H}and capacitance C

_{H}are L

_{H}= ρ

_{w}l

_{n}

_{e}/S

_{n}and C

_{H}= V

_{c}/κ

_{w}, respectively, for the HR and the pipe, which are filled with liquid (e.g., water) only [29]. Symbol l

_{n}

_{e}is the effective length of the neck, which can be calculated by l

_{n}

_{e}= l

_{n}+ 1.4r

_{n}[31,32]. l

_{n}, r

_{n}, and S

_{n}represent the length, the radius, and the cross-sectional area of HR neck, respectively. ρ

_{w}and κ

_{w}are, respectively, the fluid density and the bulk modulus. Subscripts “w” and “a” indicate the fluid “water” and “air,” respectively. Such an equivalent physical model is sufficiently accurate if the frequency range concerned was low enough. The dimension of HR will be smaller than the corresponding wavelength. In this sense, the HR can be defined as the local resonator according to the AMs theory. When the HR cavity is composed of a liquid chamber and a gas chamber, e.g., a water and air hybrid chamber, the acoustic impedance L

_{H}and capacitance C

_{H}should be changed to L

_{H}= ρ

_{w}l

_{n}/S

_{n}and C

_{H}= (V

_{cw}/κ

_{w}

^{2}+ V

_{ca}/κ

_{a}

^{2}), respectively. Mounting this HR periodically into the seawater piping system with a fixed lattice space l

_{a}that is much smaller than the acoustic wavelength λ (l

_{a}~ λ/5), the periodic system will behave as a homogenized effective medium where the acoustic band gaps may be expected [32]. Numerical validation will be addressed in next section. Construction of the periodic seawater piping system is sketched in Figure 1. Wave equation of the acoustic medium inside the seawater pipe can be given by the following formula [4,17,28]:

_{cw}and V

_{ca}; S

_{p}is the cross-sectional area of the pipe; bulk modulus κ is calculated by ρc

^{2}, in which ρ and c denote the density and the acoustic speed of fluid inside the pipe and the HR, respectively. Based on the time-harmonic assumption, the above acoustic equations can be simplified to be the following form:

_{t}and A

_{r}indicate the amplitude coefficients of transmitted and reflected waves, respectively. Acoustic speed v is derived from pressure p through the relation between the sound pressure and the acoustic velocity: v = −ρ

_{0}

^{−1}∫∂p/∂xdt. Volume speed Q is the product of acoustic speed v and cross-sectional area of pipe S

_{p}. Thus, acoustic states at the two ends of a uniform pipe section with length of l

_{a}(e.g., the uniform pipe section between the (n−1)th and the nth periodic pipe cells as sketched in Figure 1) has the following transfer matrix relation [1,17]:

_{H}is the acoustic impedance of HR. It can be formulated by Z

_{H}= jωL

_{H}+ (jωC

_{H})

^{−1}. Introducing the state vector Γ = {p, Q}’ into the above equation, Equation (6) can be rewritten into the following abbreviated form:

_{c}is the corresponding transfer matrix relating to the state vectors at the two ends of a periodic cell. Moreover, the two state vectors at the left and the right sides of periodic cell should also satisfy the following restriction due to the periodic boundary condition [33]:

_{p}= exp (±jkl

_{a}). Accordingly, the sound transmission loss TL can be given by 10lg(t

_{p}

^{−1}). Up to now, propagation characteristics of acoustic waves in the infinite and the finite periodic pipes can be examined by the calculation of TL and band structure, respectively, upon which relative analysis for the periodic system could be carried out.

## 3. Results and Discussion

_{p}= 5 cm, r

_{c}= 2r

_{p}, and r

_{n}= 0.8r

_{p}; lengths for the periodic cell, HR chamber, and neck are chosen to be l

_{a}= 0.96 m, l

_{cw}= 4.5r

_{p}, and l

_{n}= 1.5r

_{p}, respectively.

^{3}. Figure 2 exhibits the corresponding numerical results for a seawater pipe system with four HRs installed equidistantly, as shown by the solid line; the dashed line corresponds to the simulation of Comsol commercial software for the same periodic system. The good agreement of solid line with dashed line validates the accuracy of the numerical algorithm developed in this paper. Comparing the sound transmission loss of the periodic pipe system with that of a same pipe system with a single HR mounted, as illustrated by the dash-dotted line, one can see that there are two attenuation ranges of sound transmission in the former case. They are 356.5–645.5 Hz and 781–1094 Hz. Within these two frequency zones, acoustic wave propagation in the periodic system is attenuated evidently; in contrast, sound transmitting in the latter one, i.e., the pipe system with one HR attached, is damped only in a very narrow frequency range near the resonant frequency peak f

_{H}of HR that determined by f

_{H}= (2π)

^{−1}(C

_{H}L

_{H})

^{−1/2}. With regard to the infinite periodic pipe system, the acoustic propagation characteristics are captured by the band structure, as shown in Figure 3, of which the lattice constant l

_{a}and other geometric parameters involved in the calculation are the same as those in Figure 2. On examination of Figure 3, it can be seen that the location and bandwidths of these two BGs are exactly the same as that of the sound suppression zones in the finite periodic structure, as well as the attenuation effects, as revealed by the imaginary part of μ in Figure 3 and the TL in Figure 2. In fact, both the sound transmission loss and the band structure are equivalent in describing acoustic characteristics for periodic systems. The difference of these two approaches lies in that the former is used for finite periodic systems and the latter is for ideal periodic structures, i.e., the infinite periodic systems. The damping in the band gap has nothing to do with energy loss transformed into heat, so it can occur in the lossless cases. Introduction of visco-thermal losses into the metamaterial-type seawater pipe system may strengthen the damping effects in both the stop and the pass bands, yet it tends to smooth out the sharply cusped features that occur at the boundaries of the Bragg gaps and at scatterer resonance gaps [27]. Moreover, the diffusive effect of sound in a plausible case is very small except for the stopping bands, thus the dissipation is not taken into account in the current work.

_{H}will be changed from C

_{H}= V

_{cw}/κ

_{w}

^{2}to C

_{H}= (V

_{cw}/κ

_{w}

^{2}+ V

_{ca}/κ

_{a}

^{2}), such that the location of the first BG will be notably lowered, as the resonant frequency of HR f

_{H}is decreased with the increase of C

_{H}. Figure 4 validates such a conclusion. Lengths for the water and the air chambers employed in the calculation are l

_{cw}= 0.4r

_{p}and l

_{ca}= 0.5r

_{p}, respectively. Other parameters are kept the same with those in Figure 3. Density and sound speed of air are 1.225 kg/m

^{3}and 342 m/s, respectively.

_{H}of HR, i.e., 13.3 Hz under the aforementioned parameters. Consequently, this band gap is categorized as resonant gap (RG). As for the second band gap in Figure 4, namely the frequency range 781–1036 Hz, one can see that it is almost unchanged as compared to that in Figure 3. Tracing it to its causation, we know that the behavior of the second band gap is dominated by the Bragg scattering mechanism. In other words, the generation of the second gap could be ascribed to the effects of interference between the incident, reflected, and transmitted acoustic waves in the system cells. Thereupon, one of the band edge f

_{B}, is determined by mc/2l

_{a}, i.e., the Bragg condition; m = 1, 2, 3, …, denotes the mth band gap induced by this Bragg scattering mechanism. In this sense, the second band gap is defined as Bragg-type gap (BG). From the Bragg condition, it can be known that the adoption of air-water hybrid chamber in the HR medium in fact has little influence on the Bragg condition, hence the BG would experience no change in its band gap features, such as bandwidth, location, etc.

_{a}. Material and geometric parameters involved in the calculation are kept the same as those applied in Figure 4. The solid and the dashed lines correspond to the simulation of TM method and Comsol commercial software, respectively. The TL curves predicted by these two different numerical methods agree in the low frequency range, while only small differences can be observed near the Bragg gap between the TM method and the Comsol simulation. This may due to a nonlinear conversion of propagative waves toward evanescent waves and due to the hypothesis used in the TM algorithm that the HR neck connected to pipe wall is viewed as a point such that its neck geometric dimension is neglected. The attenuation zones in this plot agree well with the two gaps shown in Figure 4, thus the effectiveness of low-frequency noise suppression capacity by changing the cavity of a water-filled HR by a mix between the liquid and gas cavity (water and air) is validated once again.

_{H}, therein resulting in a low-frequency and broad RG. Hence, it should be noted that the filling ratio of air chamber plays an important part in generating this low-frequency and wide RG. RG with broader width and lower central frequency may be achieved if the volume of HR air cavity is increased. Figure 7 presents a complete surface of imaginary parts of μ, as functions of frequency and volume of air chamber (i.e., length of the air chamber l

_{ca}), through which detailed information on the behavior of the band gap location, width, and attenuation coefficient can be roundly known. According to the planform view of this attenuation constant surface, a conclusion can be reached as follows: a little bit of air chamber added to the water chamber of HR will lead to a sharply decrease in the central frequency of the resonant gap, as well as an extension in the bandwidth; however, a further increase of air volume cannot bring in such a remarkable change in the RG as that in the beginning. Still, it does broaden the bandwidth and lower the band gap location to some extent. Anyway, revelation of the RG and BG behavior against these key parameters will eventually prove to be useful in obtaining a broad, low-frequency band gap for noise transmitting suppression, mainly to form experience for reference.

_{f}/ρ)

^{1/2}, in which the bulk modulus K

_{f}is simply the gas pressure p multiplied by the dimensionless adiabatic index γ, which is about 1.4 for air. Effects of external pressure on the sound transmission properties of the seawater pipe system with air-water chamber HRs attached periodically are illustrated in Figure 8. Geometric parameters are chosen to be the same as those in Figure 4. Temperature of the surroundings and the internal media of the pipe system is assumed to be 25 degrees Celsius. The dash-dotted, the dashed, and the solid lines correspond to pressure values of 1 bar, 5 bars, and 15 bars, respectively.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Sound transmission losses for the seawater pipe system with a single HR and a HR array mounted, respectively.

**Figure 5.**Sound transmission loss of a finite length pipe with four liquid-air HRs attached equidistantly.

**Figure 6.**Von misses stress and acoustic pressure field for the seawater pipe system: (

**a**,

**c**,

**e**) are respectively corresponding to the von mises stress fields at 235 Hz, 340 Hz and 500 Hz; (

**b**,

**d**,

**f**) correspond to the acoustic pressure fields at 235 Hz, 340 Hz and 500 Hz, respectively.

**Figure 7.**Imaginary parts of μ, as functions of l

_{ca}, for the periodic pipe of which its HR chambers are filled with air and water.

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## Share and Cite

**MDPI and ACS Style**

Liu, B.; Yang, L.
Transmission of Low-Frequency Acoustic Waves in Seawater Piping Systems with Periodical and Adjustable Helmholtz Resonator. *J. Mar. Sci. Eng.* **2017**, *5*, 56.
https://doi.org/10.3390/jmse5040056

**AMA Style**

Liu B, Yang L.
Transmission of Low-Frequency Acoustic Waves in Seawater Piping Systems with Periodical and Adjustable Helmholtz Resonator. *Journal of Marine Science and Engineering*. 2017; 5(4):56.
https://doi.org/10.3390/jmse5040056

**Chicago/Turabian Style**

Liu, Boyun, and Liang Yang.
2017. "Transmission of Low-Frequency Acoustic Waves in Seawater Piping Systems with Periodical and Adjustable Helmholtz Resonator" *Journal of Marine Science and Engineering* 5, no. 4: 56.
https://doi.org/10.3390/jmse5040056