For over a century, side-branch resonators have served as effective acoustic filters, yet the explanation for their sound reduction capability has varied. This paper introduces a novel theory applicable to all types of side-branch resonators from an energy perspective and explains sound reduction as a consequence of acoustic energy redistribution. Our theory posits that a standing wave inside the resonator induces air vibration at the opening, which then acts as a secondary sound source, emitting acoustic energy predominantly in the form of kinetic energy. Due to the formation process of the standing wave, the sound wave generated by the resonator undergoes a phase shift relative to the original sound wave in the main pipe. Consequently, this generated sound wave, while matching the amplitude, possesses an opposite phase compared to the original noise wave within the main pipe. This antiphase relationship results in the cancellation of sound waves when they interact post-resonator in the main pipe. Our theory, grounded in an energy perspective, is derived from the principles of standing wave vibration and energy conservation.

Side-branch resonators, including Helmholtz resonators1 and quarter-wavelength resonators,2 are widely utilized as acoustic filters to reduce low-frequency noise. However, explanations for their sound reduction mechanism vary. Mainstream theories explaining the sound reduction mechanism in resonators include acoustic impedance,3–11 describing the medium’s resistance to sound wave propagation; friction damping during resonance, in particular in muffler-integrated resonators;12–20 and the cancellation of sound waves through destructive interference.21–27 Each of these explanations contributes to our understanding of how these resonators effectively reduce unwanted noise.

For side-branch Helmholtz resonators, the most widely accepted explanation is rooted in the concept of acoustic impedance, initially proposed by Webster using an analogy with electrical circuit systems.3 This theory has since been expanded by various researchers. Stewart formulated a theory on acoustic transmission in pipes with a Helmholtz resonator acting as the branch cavity, calculating acoustic transmission based on acoustic impedance.4,5 Canac applied electrical filter impedance formulas to elucidate and design acoustic filters.6 Building on the work of Stewart and Canac, Irons developed a theory for acoustic filters that includes phase changes in the main pipe.7 Karal introduced the end correction theory to the analogous acoustical impedance model.8 Ingard presented a Helmholtz resonator design theory based on the analogy electric circuit system.9 Keefe et al. developed a frequency‐domain based system to measure acoustic impedance and reflection coefficient.10 Kinsler summarized and explained the sound reduction of resonators based on the acoustic impedance in his book.11 He presented that the volume velocity reaches the maximum at the natural frequency of the side-branch resonator, causing the impedance beneath the neck to become infinitesimal. This area, effectively acting as a soft boundary, reflects incident sound waves back toward their source. As a result, the transmission loss of the incident sound wave reaches the maximum at the natural frequency of the resonator. Many studies16,28–48 about the acoustic performance of the side-branch resonator sound reduction are based on the above-described acoustic impedance theories. In addition, when Helmholtz resonators are employed in muffler design, the sound reduction working principle is further interpreted through the concept of friction damping.12–20 In resonance, these resonators capture sound energy and convert it into kinetic energy of air vibrations inside their cavity and neck. This energy is subsequently dissipated as heat due to the damping effect, thereby lowering the sound intensity at targeted frequencies. In systems with higher damping, a greater amount of energy is needed to sustain these vibrations, leading to an increased transmission loss.

In quarter-wavelength resonators, the more commonly accepted explanation for the sound reduction working principle is sound wave destructive interference.49–58 It posits that the out-of-phase reflected wave interacts with the incoming sound wave, leading to a cancellation effect at the open end of the resonator. This cancellation reduces the amplitude of the sound wave at the target frequency, thereby reducing the noise.

In those discussed theories, acoustic impedance explains the propagation characteristics of sound within the resonator, friction damping provides a view of energy loss, and destructive interference elucidates the interaction and cancellation of sound waves. However, regardless of the perspective used to explain sound reduction in side-branch resonators, the principle of energy conservation remains paramount. Consequently, there should be a unified theory at the energy level that coherently explains the sound reduction working principle. Furthermore, this theory should be applicable at the energy level to explain the sound reduction working principles of all types of side-branch resonators, whether the focus is on Helmholtz resonators or quarter-wavelength resonators. In this paper, we present an innovative theory that advances our understanding of the sound reduction working principle of side-branch resonators. This theory offers a unified explanation for the mechanism of diverse types of resonators, unified through the fundamental concept of energy conservation. It elucidates how standing wave-induced acoustic energy redistribution leads to a phase shift in sound waves, resulting in destructive interference after sound waves pass through a side-branch resonator.

The working principle of sound reduction in side-branch resonators fundamentally relies on the properties of standing waves. Our discussion begins with an examination of sound propagation and the formation of standing waves. Following this, we delve into the sound reduction mechanism of side-branch resonators, underpinned by the principle of energy conservation.

Sound propagation within a medium generates distinct zones of compression and rarefaction, influenced by the phase relationships of particle motion.59 In areas of compression, particles congregate closely, creating high-pressure zones along the sound wave’s direction of travel. As particles are closer to the center of compression, they experience greater positive pressure but exhibit smaller displacements. Conversely, rarefaction zones are characterized by particles spreading farther apart, leading to areas of lower pressures. In these zones, particles nearer to the center of rarefaction experience greater negative pressures but demonstrate smaller displacements. The compression/rarefaction center shows maximum positive/negative pressures but exhibits minimum displacements. At the centers of compression and rarefaction, the pressure peaks positively and negatively, respectively, while particle displacement remains minimal in both situations.

First, let us explore the concept of standing waves in a straight branch pipe. After establishing this foundation, we will then extend our analysis to encompass branch cavities with random shapes. As a sound wave enters a straight branch pipe, it undergoes reflection, interacting with the subsequent sound wave entering the pipe, and forms a standing wave inside the branch pipe. Specifically, when the wavelength of the incident sound wave is four times the length of the branch pipe,2 unique phenomena occur at both ends of the pipe. At the open end of the branch pipe, the compression area of the incoming wave consistently aligns with the rarefaction center of the reflected wave. This consistent alignment results in the formation of a pressure node, which is simultaneously a displacement antinode, at the open end. Conversely, at the closed end, there is always an alignment of either the compression or rarefaction areas of the incoming and reflected waves. Such an alignment leads to the creation of a pressure antinode, which is also a displacement node, at the closed end. Consequently, when the frequency of the incident sound matches the natural frequency of the branch pipe, a maximal pressure and minimal air displacement are observed at the closed end. The opposite is true for the open end, where pressure is minimal and air displacement is maximal. This condition results in maximum air vibrations at the open end of the branch pipe. The vibrating air at this open end then acts as a secondary sound source, emitting sound that propagates back into the main pipe.

Utilizing a finite element analogous method,60 we model the air within the standing wave inside a branch pipe as a series of infinitesimal air layers, each characterized by its pressure distribution. In this model, each air layer of the branch cavity is represented as a mass–spring unit. When end correction factors are disregarded, the air displacement profile in the branch pipe exhibits a sinusoidal shape, typical of standing waves. Given that pressure and air displacement in a standing wave display a quarter-wavelength phase difference, their inverse relationship is illustrated in Fig. 1.

FIG. 1.

Standing wave air particle displacement and pressure in a straight branch pipe.

FIG. 1.

Standing wave air particle displacement and pressure in a straight branch pipe.

Close modal

In a branch cavity with a random shape, the entry of a sound wave initiates a complex propagation process. The wave travels from the neck into the cavity, diverging in multiple directions and reflecting off the rigid boundaries. These reflections lead to the superposition of incident and reflected waves, resulting in the formation of standing waves. These standing waves, while maintaining the original frequency, manifest in various directions within the cavity. A key feature of these standing waves is the location of the pressure nodes, which are consistently situated beneath the neck, and the pressure antinodes found at the rigid boundaries.

Employing the same analogous method as utilized in the straight branch pipe, the air within the branch cavity can be conceptualized as numerous infinitesimal air layers. These layers are determined by the distribution of pressure and are each represented as individual mass–spring units. The formation of pressure iso-surfaces, which define the boundaries of these air layers, is inherently influenced by the geometry of the rigid boundary. This analogy mass–spring system is ideally suited for a three-dimensional model due to the multidirectional standing waves from the neck into the cavity. Given the uniform frequency of the standing waves, we simplify these air layers into planar sections while adhering to the principle of mass conservation. This simplification enables the modeling of the three-dimensional air vibrations in these layers as uniformly oscillating mass–spring units, each moving in a single direction. The air near the neck, undergoing vibration, acts as a secondary sound source, reintroducing sound back into the main pipe. Figure 2 provides a visual representation of this phenomenon, illustrating the air displacement profile for a randomly shaped branch cavity, in conjunction with the corresponding pressure and air displacement profiles.

FIG. 2.

Standing wave air particle displacement and pressure in a random shaped branch cavity.

FIG. 2.

Standing wave air particle displacement and pressure in a random shaped branch cavity.

Close modal

As we discussed before, when a sound wave enters a branch cavity and undergoes reflection, a standing wave is formed within the cavity. The standing wave forms immediately after the reflected sound wave reaches the opening and interacts with the incoming wave. As the incident sound wave aligns with the natural frequency of the cavity, the reflected wave inside the cavity travels half a wavelength more than the external wave, leading to a phase delay. Consequently, the sound wave generated by the cavity is delayed by half a wavelength relative to the incident noise wave in the main pipe and is in the opposite phase. When the sound wave entering the resonator is from a distant source, we can model it as plane wave radiation.

As the incident sound wave interacts with the reflected sound wave, assuming no energy loss during the reflection and interaction, both the incident and reflected sound waves contain identical acoustic energy. Upon their interaction, they form a standing wave. Drawing from the principle of energy conservation in traveling and standing waves,61–63 it is posited that the energy of the standing wave equals the cumulative acoustic energy of these two component waves during their superposition. The energy of the standing wave is then radiated as kinetic energy, acting as a new sound source at its pressure node, the resonator’s open end. Consequently, a sound wave with double the energy of the incident wave is emitted back into the main pipe, manifesting as a new sound wave driven by the standing wave’s action. This phenomenon is an energy transformation facilitated by the standing wave. Given the main pipe’s unchanged structure after the branch cavity’s introduction, the newly generated sound wave by the branch cavity retains an identical waveguide while propagating in both directions along the main pipe. The relationship between waveguide and sound wave energy distribution64–66 suggests that the new sound’s acoustic energy is uniformly dispersed at both ends of the main pipe. Given our initial assumption that the standing wave’s energy is double that of the incident wave, and the new sound wave arises from the standing wave’s energy conversion, it is logical to deduce that the acoustic energy of the sound wave emanating from the branch cavity in each direction of the main pipe is equivalent to the incident wave’s energy. Consequently, the amplitude of the sound wave generated by the branch cavity67 is deemed equal to that of the incident wave. Therefore, the generated sound wave, traveling in the same direction as the noise in the main pipe, cancels out due to having equal amplitude but opposite phase. Conversely, the generated wave traveling in the opposite direction forms a standing wave with the noise in the main pipe. The proposed mechanism for sound reduction in side-branch resonators primarily revolves around energy redistribution, which fundamentally leads to destructive interference, as illustrated in Fig. 3.

FIG. 3.

Standing wave-induced sound reduction working principle.

FIG. 3.

Standing wave-induced sound reduction working principle.

Close modal

This conclusion is rooted in the premise that the standing wave’s energy equals the sum of the energies of the incident and reflected waves. Therefore, we only need to substantiate that the standing wave’s energy is the sum of the energies of the incident and reflected waves, corroborating the assumption that the generated sound wave’s amplitude matches that of the incident wave, to substantiate our theory.

Here, we substantiate the new explanation for the sound reduction phenomenon in side-branch resonators, grounding our analysis in the principle of energy conservation. By demonstrating that the acoustic energy of a standing wave is twice the incident acoustic energy, we can substantiate that the acoustic energy emitted from the standing wave is double that of the traveling wave in the main pipe. Given that the sound generated by the standing wave is evenly distributed across the two open ends of the main pipe due to its consistent geometry, it follows that the amplitude of the sound wave emanating from the branch cavity is equivalent to that of the incident sound wave within the main pipe.

We approach the explanation of the sound reduction phenomenon from an energy perspective, assuming no loss of energy during the propagation and superposition of the sound waves. All our derivations adhere to the isothermal model. We begin by considering a traveling wave in the air,
u=Asinkx2πft.
(1)
Here, u is the air displacement, A is the amplitude, k is the wavenumber, and f is the frequency. Let us begin by deriving the energy in the traveling wave, focusing on an air unit within the sound propagation path. The mass of this air unit is
dm=ρdV.
(2)
The velocity of the air particle in the air unit is
v=ut=2πfAcoskx2πft.
(3)
Therefore, the kinetic energy of the air unit can be expressed as
Ek=12dmv2=2ρπ2f2A2cos2kx2πftdV.
(4)
For the potential energy, the force exerted by the pressure change Fp on the air unit is equivalent to the force applied by the air unit’s elastic force, which is characterized by the air unit’s elastic coefficient s,
Fp=dadp,
(5)
Fs=sdu.
(6)
Here, da represents cross-sectional area of the air unit, oriented perpendicular to the displacement direction of the air particle and dp denotes the acoustic pressure resulting from the sound wave propagation. From the force balance, we deduce that
Fp=Fs,
(7)
s=dadpdu.
(8)
To determine the air unit elastic coefficient s, the acoustic pressure dp needs to be derived. From thermodynamic theory,68 the speed of sound c is expressed as
dp=c2dρ.
(9)
Next, the expression for density changes needs to be derived. Based on the conservation of mass principle, the mass of the air unit remains constant after deformation,
ρV=ρV=ρ+dρVdV=ρVρdV+dρVdρdV.
(10)
We neglect the second-order term in Eq. (10), and the density change becomes
dρ=ρdVV=ρdadudadx=ρdudx.
(11)
Therefore, the air unit elastic coefficient s can be expressed as follows:
s=c2ρdadx.
(12)
Then, the potential energy in the air unit is
Ep=12sdu2=12sdudx2dx2=12ρc2A2k2cos2kx2πftdadx.
(13)
From the wave equation property,
ck=2πf.
(14)
Rearrangement and simplification yield
Ep=2ρπ2f2A2cos2kx2πftdV.
(15)
Drawing from our earlier derivations, it becomes evident that the kinetic and potential energies within a traveling wave of an air medium unit are invariably equal. This balance underscores that the propagation of a traveling wave involves simultaneous transmission of both motion and energy through the medium. Accordingly, the total energy of the traveling wave in an air medium unit can be reformulated as follows:
Ett=Ek+Ep=4ρπ2f2A2cos2kx2πftdV.
(16)
The energy density of the traveling wave is
Wt=dEttdV=4ρπ2f2A2cos2kx2πft.
(17)
Our next derivation centers on the acoustic energy within a standing wave. We begin by considering two traveling waves that possess identical amplitude and frequency, yet propagate in opposite directions,
u1=Asinkx2πft,
(18)
u2=Asinkx+2πft.
(19)
The standing wave is formed due to the superposition of that two traveling waves,
us=u1+u2=2Asinkxcos2πft.
(20)
The derivation continues with an infinitesimal air unit. Utilizing Eqs. (2) and (12), the mass and elastic coefficient of the air unit are directly applied to calculate the kinetic and potential energies within this unit. The velocity of the air particles in the air unit is determined as follows:
v=ust=4πfAsinkxsin2πft.
(21)
Therefore, the kinetic energy of the air unit can be expressed as
Ek=12mv2=8ρπ2f2A2sin2kxsin22πftdV
(22)
and the potential energy becomes
Ep=12sdus2=12sdusdx2dx2=2ρc2A2k2cos2kxcos22πftdadx.
(23)
Rearrangement and simplification by plugging Eq. (14) in Eq. (23) yield
Ep=8ρπ2f2A2cos2kxcos22πftdV.
(24)
It is important to note that in a standing wave, the kinetic energy in a unit of air is not equivalent to its potential energy. Rather, these forms of energy undergo a continuous transformation into one another during the dynamic process. The total energy in a standing wave of an air unit can be rewritten as
Est=Ek+Ep=8ρπ2f2A2sin2kxsin22πft+cos2kxcos22πftdV.
(25)
The energy density of this standing wave is
Ws=dEstdV=8ρπ2f2A2sin2kxsin22πft+cos2kxcos22πft.
(26)
To validate the conservation of energy during wave superposition, comparing energy in a singular, arbitrary air unit is not suitable, as the energy distribution in traveling waves varies across different locations due to their energy transfer mechanisms. Similarly, in standing waves, energy is continuously converted between kinetic and potential forms. Therefore, to effectively demonstrate energy conservation in wave superposition, we should analyze the total energy within a wavelength range of a standing wave and juxtapose it with the cumulative energies of the two interacting traveling waves. The total energy within a wavelength λ of a traveling wave is calculated as follows:
Etλ=0λWtdx=0λ4ρπ2f2A2cos2kx2πftdx.
(27)
From the above equation,
Etλ=2ρπ2f2A2λ.
(28)
The total energy in a standing wave within a wavelength λ is
Esλ=0λWsdx=0λ8ρπ2f2A2sin2kxsin22πft+cos2kxcos22πftdx.
(29)
Simplifying the above equation and comparing with Eq. (28) gives
Esλ=4ρπ2f2A2λ=2Etλ.
(30)
Equation (30) confirms that energy conservation is maintained in wave interactions, indicating that the total energy of a standing wave is equal to the sum of the energies of the two interacting traveling waves. In a standing wave, potential energy is concentrated at the pressure antinodes, while kinetic energy is predominant at the displacement antinodes. Notably, when a displacement antinode of the standing wave aligns with the opening neck of a branch cavity, it effectively functions as a sound source. This source radiates the standing wave’s energy, entirely in kinetic form, into the main pipe. Our derived energy equations demonstrate that the acoustic energy emitted from the branch cavity is double the incident acoustic energy. Half of this acoustic energy generates a sound wave in the main pipe with the same amplitude and direction as the incident wave, but in the opposite phase, leading to cancellation post-neck interaction. The remaining half forms a sound wave with an amplitude identical to the incident wave but in the reverse direction. This wave propagates back toward the main pipe’s inlet, creating a standing wave in conjunction with the existing sound wave in the main pipe. Consequently, the energy in the main pipe undergoes redistribution due to the standing wave in the branch cavity. As a result, the acoustic energy dissipates beyond the branch cavity’s opening neck, effectively eliminating noise.

The proposed theory provides significant insights into energy conservation during the sound reduction process in resonators. Commonly, side-branch resonators are integral to silencing mechanisms across various applications. Notably, Helmholtz resonators and quarter-wavelength resonators are pivotal in designing vehicle exhaust systems and ventilation pipes.2,12,17,18 This theoretical framework validates the principle of energy conservation in side-branch resonators when employed as silencers. It elucidates the mechanism of destructive interference within these resonators, offering a perspective grounded in energy dynamics to explain noise cancellation.

Beyond their traditional use as silencers, side-branch resonators find application in enhancing acoustic performance, such as in architectural acoustics to mitigate unwanted noise or in musical instruments to enrich sound quality.69–72 In industrial settings, they are utilized to reduce noise in HVAC systems and machinery,73–75 contributing to a safer and more pleasant working environment. When resonators function as loudspeakers,76,77 especially with proximate sound sources, the impact of acoustic material damping becomes crucial in determining resonance amplitude and energy dissipation within the resonator. This aspect underscores the versatility of side-branch resonators, highlighting their utility in both amplifying and attenuating sound, depending on the application context. The forthcoming section will delve deeper into the influence of different sound source types on resonators.

In our previous discussion, when the sound wave entering the resonator originates from a distant source, the incident wave and the reflected wave interfere and form a standing wave, preventing the incident sound wave from acting as a constant pressure sound source with a continuous and stable excitation capability to induce resonance. Thus, this type of plane wave input typically only forms standing waves within the resonator and usually does not excite resonance. However, if the entrance of the side branch resonator is near a stable vibrating sound pressure source, then at the resonance frequency, the sound energy from this constant source will be effectively absorbed by the side branch resonator, resulting in resonance. Figure 4 shows that when a pipe with one open end and one closed end is subjected to 5 Pa constant pressure and plane wave radiation at the open end, resonance occurs with the constant pressure input but not with the plane wave radiation input.

FIG. 4.

Resonator’s reaction to different sound source input types.

FIG. 4.

Resonator’s reaction to different sound source input types.

Close modal

In Fig. 4, for the far-field sound source represented by a plane wave radiation input, the acoustic pressure at the pipe’s closed end remains constant, serving as the pressure antinode of the standing wave. The pressure node, which is also the displacement antinode, is situated at the pipe’s open end, corresponding to the pipe’s natural frequency. The standing wave’s acoustic energy is then emitted through air vibrations at the open end of the pipe. Regarding the near-field sound source, characterized by the constant pressure input, when the sound source’s vibration frequency aligns with the resonator’s resonant frequency, the resonator system maximally absorbs the energy emitted by each sound source vibration, converting it into the resonator’s resonant motion. Without damping, this energy absorption at the resonant frequency would continuously increase due to the persistent pressure excitation at the pipe’s open end, significantly elevating the acoustic pressure at the pipe’s closed end.

Without damping to moderate these vibrations, the resonator’s amplitude could theoretically increase indefinitely, potentially affecting nearby frequencies as well. In this scenario, the sound produced by the resonator’s resonance, even after interference cancellation, could be significantly greater than the original sound in the main pipe because the resonator’s resonance accumulates and converts the original sound source’s energy into another consistent source. Therefore, when designing practical applications with near-field sound sources, damping can be strategically employed to achieve noise reduction.78–80 

Acoustic materials help stabilize the resonator’s amplitude, ensuring that the absorbed sound energy counteracts the damping’s frictional forces, maintaining the resonator’s dynamic equilibrium.81–85 With damping, part of the sound energy from the constant source is continuously absorbed by the acoustic materials to sustain the resonant oscillations. During this process, a portion of the sound energy is transformed into heat through intermolecular friction, aiding in noise attenuation. The higher the damping at the same amplitude, the more sound energy is required to maintain the resonance vibration. Effective damping not only reduces the resonator’s amplitude at its resonant frequency but also smooths out the resonant peak, broadening the resonator’s noise reduction impact across a wider frequency spectrum, rather than confining it to a narrow band. Thus, under the influence of damping, the resonator’s resonance amplitude and resonant frequency range are effectively controlled and smoothed. In addition, the sound waves produced by the controlled resonance, along with another part of the sound that continues to propagate along the main pipe without being dissipated by the damping effect, create destructive interference, further reducing noise. This process is same as the earlier discussed far-field sound source scenario.

In conclusion, for a far-field sound source, the incident sound wave and the sound wave generated by the resonator interfere and cancel each other out in the main tube after passing through the side-branch resonator. For a near-field sound source, part of the sound energy is converted into heat energy under the action of damping. The remaining sound energy in the incident sound wave interferes destructively with the sound wave generated by the resonator in the main pipe after passing through the resonator. In applications, resonators are typically employed to reduce noise in the context of far-field sound sources. For near-field sound sources, higher damping is favored to dissipate energy and mitigate noise, while lower damping is advantageous for sound scattering, preserving sound energy and maintaining resonance.86,87

The proposed theory encompasses energy conservation in the context of sound reduction through destructive interference in side-branch resonators, regardless of the sound source type or the presence of damping. However, it exhibits limitations in accounting for the energy conservation throughout the entire noise reduction process in resonators affected by damping. The theory is capable of explaining the noise reduction that occurs due to the destructive interference of sound energy remaining after frictional heat losses. Yet, it does not provide a method to quantify the portion of sound energy that is converted to thermal energy due to damping prior to interference. Therefore, while the theory can depict the reduction of sound through destructive interference in side-branch resonators, it lacks the ability to calculate the thermal energy loss, thereby not fully elucidating the principle of energy conservation during the noise reduction process in such scenarios.

In this study, we introduce a comprehensive theory to elucidate the sound reduction phenomenon in side-branch resonators, approached from an energy perspective. This theory meticulously analyzes the distribution of kinetic and potential energy within traveling and standing sound waves. It reveals that in a unit of air medium, the kinetic and potential energies of a traveling wave are consistently equal, underscoring the dynamic transfer of motion and energy during wave propagation, which results in varied energy distribution along the wave path. In standing waves, an ongoing interconversion between kinetic and potential energy is observed. Notably, when an incident sound wave’s frequency matches the natural frequency of a branch cavity, the standing wave’s displacement antinode aligns with the cavity’s opening neck, transforming the acoustic energy into kinetic form. This kinetic energy then serves as a sound source, effectively radiating energy into the main pipe. Our derivations establish that the total acoustic energy of the standing wave within a wavelength in the branch cavity is double the incident wave’s acoustic energy within the same distance. In addition, the phase delay induced by the travel distance inside the branch cavity causes the energy redistribution and leads to distinct interactions: sound waves with the same propagation direction as the main pipe noise cancel each other out due to opposite phases and identical amplitudes, while waves in the opposite direction form a standing wave with the main pipe noise. This theoretical framework not only elucidates the sound reduction mechanism in side-branch resonators as a form of destructive interference but also interprets the destructive interference as energy redistribution from an energy perspective, while covering the influence of sound source type and damping in this process. In essence, our theory offers a universal explanation applicable across all types of side-branch resonators, detailing the underlying principles governing the sound reduction phenomenon.

In conclusion, this theoretical framework offers a comprehensive method to understand noise reduction in resonators from an energy perspective, aiding researchers in grasping the energy conservation aspect of destructive interference caused by resonator standing waves. This theory is helpful for designers in creating resonator-based silencers, enabling the design of more energy-efficient noise reduction solutions. However, the theory, predicated on the assumption of no energy loss, falls short in calculating sound energy loss due to damping, especially with near-field sound sources. Future research could integrate this theory with thermodynamic principles, allowing for a more nuanced understanding of energy conservation in resonators by accounting for damping effects. This integration could enhance damper design within resonators, elevating the effectiveness of noise reduction. Expanding this theory to encompass energy transfer, including damping, could also enrich research in the field of acoustic metamaterials88,89 and energy harvesting,90–92 providing a deeper understanding of energy dynamics in resonators.

The authors have no conflicts to disclose.

Jiaming Li: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Hae Chang Gea: Supervision (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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