Usage of an array of Helmholtz resonators to equalize
underwater transmission of an exponential horn
Tamar Vis
August 31, 2020
Abstract
As for transmission in air, for the transmission of sound waves under water a horn may be used to match the high driver impedance to the low impedance of empty space. The impedance of an exponential horn changes periodically with the frequency of the driven sig-nal due to reflections of the sound wave from the mouth back into the horn. This results in a transmission loss changing periodically with the driven signal frequency. For most practical applications, it is favorable to have a horn with constant transmission loss over a range of frequencies. In order to design a horn with constant transmission loss, an array of Helmholtz resonators has been added to the horn as to compensate a peak in transmission. This has not resulted in a continuous transmission over a band of frequencies. However, it has been shown that the array of Helmholtz resonators works comparably for an exponential horn as for a cylindrical waveguide with continuous cross-section. Further research that may yield a continuous transmission loss over a band of frequencies has been suggested.
Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between 01-04-2020 and 01-08-2020
Author Tamar Vis 11317248
Study B`eta-gamma, mj. Physics
Supervisor Rudolf Sprik
Examiner Corentin Coulais
Institute Van der Waals-Zeeman Institute
Faculty Faculty of Science, University of Amsterdam Date of submission August 31, 2020
1
Introduction
Underwater transmission of sound has a broad range of applications, from SONAR-like technolo-gies used in different aspects of navigation to academic research to the ocean floor and maintenance of offshore material. An example of this is shown in figure 1.
Part of the technical assembly used for transmitting sound waves is the acoustical horn. An acoustical horn is a flaring waveguide used for both restricting the angle of transmission [13], increasing the so-called gain of the transmitter [21] as well as imposing a load impedance on the driver of the system, as will be clearer from next section. Any horn, however, transmits some frequencies better than other frequencies as a result of reflections from the mouth of the horn back into the horn and the subsequent formation of standing waves in the horn. In any practical application, such as the use of SONAR in military submarines[22] or underwater communication [23] sound of multiple frequencies has to be transmitted. In practice, equal transmission of sound for a broad band of frequencies (that is, equalized[6]) can be achieved by electronically attenu-ating the signal or by using multiple driver-horn combinations. On the other hand, a horn with equal transmission may be designed using Acoustic Metamaterials. Acoustic Metamaterials as a phenomenon are described by Zangeneh-Nejad and Fleury [4] as ”rationally designed composites of composed of mesoscopic resonant inclusions placed in a host medium”[4] making it clear that metamaterials are to be designed specifically for a technical application. Furthermore, they are periodic structures made of ordinary materials [16]. In this thesis, an acoustic metamaterial as described by Fang et al. [16] is used to equalize te transmission of sound through an acoustic horn for a band of frequencies in a simulation of an acoustical horn filled with water. This is done by simulating an acoustical metameterial integrated into an acoustic horn, as described in section 3. However, first the workings of acoustical horns as well as the chosen metameterial are explained in section 2. After treating the design of the horn and the simulation of the horn in COMSOL Multiphysics R (COMSOL), in section 4 the horn is evaluated using the graph of the Transmission
Loss as a function of the transmitted frequency returned by COMSOL. Although this graph is not, as is ideally the case, flat, it is shown that the use of an array of Helmholtz Resonators can be used to equalize the transmission of an acoustical horn in water. Multiple suggestions for further research, which may lead to a better equalization of the signal, are given in section 5.
2
Theory
In order to design an acoustical horn capable broadcasting a signal with constant transmission over a broad frequency range, some basic knowledge of the field of physics concerning the prop-agation of sound is necessary. This section therefore begins by introducing some basic acoustics. The treatment of the acoustical horn follows. The section ends with the treatment of the used metamaterial, the array of Helmholtz resonators.
Figure 1: An example of the usage of underwater communication through sound waves. From: Arnon [1].
2.1
Acoustics
Sound, independent of the propagation medium, is a travelling, longitudinal pressure wave. The driver, that is, an oscillating element, causes an inhomogeniety in the pressure of the surrounding, formerly homogeneous, medium. This inhomogeneity is in the form of medium being pressed together due to its mass intertia, increasing the pressure and density at the point. This pressure makes the medium expand in all directions where the pressure is smaller. This compresses the next ”layer” of medium, giving this a higher-than-normal pressure. This process is the propagation of a sound wave: the repeated oscillation of a driver creates a repeated compression of medium and consequent repeated increase in pressure. This way, sound consists of a travelling wave of pressure. This propagation of sound waves in a fluid medium is mathematically described by the universal wave equation[5], which goes
∇2p = 1
c2
∂2p
∂t2, (1)
where p is the pressure at a given point in the propagation medium and c the velocity of the sound wave. For air, this velocity is 343 m s−1 whereas the sound velocity in water is 1440 m s−1 [5].
Equation 1 is a partial differential equation that relates the time- and spatial derivatives of physically possible sound waves. In any practical application, a stationary solution for the pressure
amplitude is of much greater use. The wave equation is solvable by separation of variables. This means solutions can be found by writing a solution in the form of
p(~r, t) = ψ(~r) · φ(t)[18]. (2)
Substituting this solution into equation 1, one acquires
∇2ψ(~r) ψ(~r) = 1 c2φ(t) ∂2φ(t) ∂t2 , (3)
with a left-hand side depending only on ~r and the right-hand side only depending on t. This means both sides may be set to an arbitrary constant. Here, the position-dependent part (left) is set to the value −k2 whereas the time-dependent part (right) is set to −ω2. Considering the
dispersion relation for a linear acoustic medium c = ωk with ω the phase velocity and k the wave number this is still a valid solution to equation 1. Then,
∇2ψ(~r) = −k2ψ(~r), (4)
∂2φ(t) ∂t2 = −ω
2φ(t), (5)
which implies the solutions
ψ(~r) = Aeik·~r+ Be−ik·~r (6)
and
φ(t) = eiωt, (7)
where the solution e−iωt has been omitted: where equation 6 deals with waves going in two directions (”forward” and ”backward”) along any arbitrary axis along ~r, in the time domain the waves only oscillate forward. The parameters A and B of the solution 6 can be complex and together determine the amplitude profile of the space in which the forward- and backwardgoing waves propagate. Equation 4 can be written into the form
∇2ψ(~r) + k2ψ(~r) = 0 (8)
which is known as the Helmholtz equation. This Helmholtz equation is the basis for the simulation of acoustical waves by COMSOL.
2.2
Acoustics within waveguides
Within waveguides, the Helmholtz Equation 8 spatially governs the propagation of pressure waves. A waveguide imposes some more boundary conditions on the this relation [3].
2.2.1 Hard wall boundary
A waveguide usually consists of a hard wall surrounding the medium through which a pressure wave propagates. This means that no normal component of the pressure wave is let through the waveguide wall. This is described mathematically by
∂p
∂~n = 0 (9)
directly at the surface of the wall. This means that in every situation where a hard wall boundary is present, the pressure wave should propagate along the wall with the pressure-front directly at the location of the wall perpendicular to it [3].
2.2.2 Soft wall boundary
The soft wall boundary is a boundary of a waveguide where the acoustic pressure vanishes. A real-life example of this would be a nonreflective wall, such as a perforation in a hard wall or a liquid-gas interface. It should be noted that contradictilly this boundary condition may not be used as a nonreflective ending of a waveguide: this is because the soft wall boundary forces the pressure locally to zero [3]. The soft wall boundary would be valid, however, as a waveguide side-wall boundary, effecting not the axial but only the radial distribution of pressure within the waveguide. As this boundary has not been used in the models and has no further value in explaining the working of waveguides, it will not be explored any further.
2.3
Flaring waveguides: horns
Horns in general are a subgroup of waveguides, characterized by the increasing cross-section as a direct function of the distance along the horn axis: the horn has a flaring shape. Horns are classified by the relation between the radius and the distance along the horn: for example, for an exponential horn, the horn radius goes exponential with the distance. Examples of other horn types are the conical horn (r ∼ x) and parabolic horn (r ∼ x2) with r the radius of the horn and x the distance along its axis[13].
On the end of the horn with a smaller cross-section, the driver is usually installed. The end of the horn with a larger cross-section is open to admit for radiation of pressure waves.
Acoustical horns have two purposes: adding directivity to the system and providing the driver with a load impedance. As this thesis does not cover the directivity of the horn, it will be suffici¨ent to note that the directivity of a horn is a measure for how much of the transmitted signal is transmitted along the axis of the horn as opposed to towards the sides of the horn.
However, the impedance of a horn is much more important in this thesis. Acoustic impedance is defined as the ratio between the pressure and the volume velocity U at any given point
Z ≡ p
with the volume velocity given as
U ≡ v · S (11)
with v the maximum velocity the particles in the acoustic medium reach due to the travelling pressure wave and S the cross-section of the wave. In the case of plane waves in a waveguide, this is the cross-section of the waveguide. In the absence of a waveguide, the sound wave propagates spherically in all directions and this cross-section is the area of a growing sphere [19].
As the impedance generally increases with decreasing waveguide cross-section, the waveguide imposes a boundary for the pressure and volume velocity at any point. This is necessary to match the impedance of the driver, which is relatively high, with that of the air, which is relatively low: whereas the driver is usually small compared to the wavelength of the sound it produces, thus being able to exert high pressure to the surrounding air but only move a small volume, the surrounding air is light and easilly moved. This way most power is lost internally in the driver.
To solve this problem, a cylinder with continuous cross-section would do. However, as the driver is usually of subwavelength dimensions, the mouth of the cylinder would also be subwavelength. This would cause most of the signal coming from the driver to be reflected back into the horn, considerably restricting transmission.
The acoustical horn is a solution for this: it starts narrow, with a small cross-section to impose a large acoustic impedance on the driver. This part is called the throat of the horn. On the other end, called the mouth, the horn is wide. The ideal radius of the mouth, that is, the radius of the mouth with a minimum of reflection of waves back into the horn, would be equal to the wavelength of the sound wave. Between the throat and the mouth of the horn, the horn flares smoothly as any abrubt change in cross-section also creates a source of reflection.
It should be noted that even at a mouth radius equal to the wavelenth of the transmitted wave, reflections are minimized but not diminished. Kemp [12] shows that reflections from the mouth do occur. This is treated more specifically in the next section.
2.3.1 Reflection and transmission within the horn
Given the above description of a horn, it can be useful to describe a horn as an array of discrete concentric cylinders as Kemp [12] shows. Although this approximation has not been used in this thesis, it is worth mentioning for two reasons: first off it illustrates the way a sudden change in cross-section of a waveguide creates a reflected wave; secondly, it considers the propagation of a plane wave along a cylindrical waveguide. In the system described are two mayor elements: the cylinder of continuous cross-section and the surface between two cylinders, being a change of cross-section in the waveguide. The description of a travelling pressure wave in a cylinder of continuous cross-section is given in Kemp [12]:
p0= cos(kd)p1+ i sin(kd)ZcU1 (12)
with p0 the pressure at a certain location along the waveguide, p1 the pressure at another
d the distance between the points (positive for a distance along the direction of the pressure wave, negative for the opposite direction) and Zc the characteristic impedance Zc = ρcS. U1 is here
the volume velocity at the location where the pressure amplitude equals p1. This result is used
in section 2.6 where the array of Helmholtz resonators is discussed, as the propagation along a straight waveguide between two Helmholtz resonators is used.
Equation 12 describes the propagation along a waveguide with continuous cross-section. How-ever, at any point where the cross-section changes, a part of the wave may be reflected back. In fact, Kemp [12] gives the amplitude reflection- and transmission coefficients
R = S1 S2 − 1 S1 S2 + 1 (13) T = 2S1 S2 S1 S2 + 1 (14)
Note that these coefficients do not add up to one, although their absolutes do: the reflected wave is transmitted backward from the change in cross-section with a π phase difference, which is shown in a −1 factor integrated into the reflection coefficient.
2.4
Horn impedance
As the impedance of a waveguide at a certain point within the waveguide is found by equation 10 and the pressure at any point can be found by equation 12, only a term for the volume velocity at any point is necessary to calculate the impedance of the waveguide. Similar to equation 12, the volume velocity may be described by
U0= i sin(kd)Zc−1p1+ cos(kd)U1 (15)
which, using equation 10, the impedance of a waveguide can be projected by
Z0= cos(kd)Z1+ i sin(kd)Zc i sin(kd)Zc−1Z1+ cos(kd)
(16) In the case of only a forward-propagating wave, Z1 = Zc, which reduces Z0 to Zc [12]. This
shows that the impedance of the horn depends on the geometry of the horn, but not on the wavelength. If this would be the entirety of it, the transmission loss would be constant for any frequency.
On the open end of the horn, another source of acoustical impedance is present: where, at the mouth of the horn, the horn ends. Here, it is found that the mouth in itself has a finite, nonzero, impedance. As stated before a sudden change in impedance is a source of reflection. As Kemp [12] states, it is possible to see the end of the mouth as a change of cross-section of the horn from a finite value (the cross-section of the mouth itself) to infinity. However, this reduces the reflection coefficient from equation 13 to become −1 - the incoming wave would be perfectly reflected with a phase difference of exactly π, causing standing waves with an amplitude of 0 at
the mouth. Consequently, no sound at all is transmitted. Modelling the mouth of a horn as a change in cross-section from the mouth surface to inf, acts as an impedance change from a finite Zend = ρc/Send to Zmouth = 0, which is nonphysical - a volume velocity wave would be present
without any corresponding pressure wave. Taking this into account, Fletcher and Rossing [19] model the mouth of a horn as a rigid piston taken to be placed in an infinite baffle. Then, as Fletcher and Rossing [19] show, the impedance of a flanged opening can be shown to be
Zmouth(F ) = R + iX (17) with R = Z0[ (ka)2 2 − (ka)4 12 + (ka)6 144 − ...] (18) X = Z0 πk2a2[ (2ka)3 3 − (2ka)5 45 + (2ka)7 1575 − ...] (19)
with k the wavenumber of the transmitted wave, a the radius of the mouth horn and Z0 the
standard impedance of the horn at the mouth. From equations 17-18, it is clear that the mouth impedance, and thus the reflection and transmission of waves at the mouth, depent explicitly on the wavenumber and so also on the wavelength. Note that for ka < 1, the mouth impedance is smaller than Z0 which means that most of the incoming sound wave is reflected back into the
horn. For ka > 2, most (although not all) of the sound wave is transmitted. For an unflanged mouth, similar terms described the mouth impedance of a horn, the difference being a factor 0.5 reduction of R and a factor 0.7 reduction of X [19].
2.4.1 Impedance of an exponential horn
Until here, everything has been valid for all acoustic horns. As in the simulations an exponential horn has been used, the mouth impedance of exponential horns will be discussed. By projecting the mouth impedance 17 along a consequent series of cylinders, the impedance of a horn at the throat can be found. In fact, Fletcher and Rossing [19] shows that
Zin= ρc S1 " Zmcos(bL + θ) + i(Sρc2sin(bL)) iZmsin(bL) + Sρc 2cos(bL − θ) # (20)
with S1 the throat- and S2 the mouth cross-section, b the factor
√
k2− m2 with m defined by
the horn geometry. θ is here the arctangent of the factor mb. Equation 20 shows that the input impedance of a horn fluctuates periodically in the complex plain with an angular frequency in the order of kL if m is taken to be very small compared to k. Note that if the mouth impedance Zm takes the value of Sρc
2, that is, the standard impedance of the last section of horn, the input
impedance of the entire horn is simply Sρc
1, the standard impedance of the throat. The condition
where Zm=Sρc2 is valid is when the radius of the mouth is roughly the same as the wavelength of
2.4.2 Transmission loss
The impedance of a horn, as described above, should for a horn be maximized as the power effectively transmitted instead of lost internally in the driver increases with the horn impedance. However, in simulations this impedance is difficult to measure. The transmission loss (TL) of a horn is easier to measure and found by
T L = 10 · log10(
Pin
Pout
) (21)
with Pin the acoustic power entering the waveguide (that is, effectively radiated from the
driver) and Pout the power leaving the waveguide. This can be directly found in a COMSOL
simulation. In order to equalize the transmission of an acoustical horn, the transmission loss has to be equalized. As equation 20 shows, the impedance fluctuates, so so does the transmission. A band-stop filter may be used to attenuate the peaks in transmission. A band stop filter, or reject filter, is a filter that attenuates the transduction of an acoustical wave for a limited range of frequencies.
2.5
Helmholtz resonator
One example of an acoustic reject filter uses the Helmholtz resonator. The Helmholtz resonator is a structure comprising a narrow neck (relative to the wavelength) and a resonance chamber of a given volume and shape. Although the shape of the resonance chamber influences the resonance frequency calculated later on, Chanaud [2] shows that this calculation is most valid for cubical resonance chambers, as are used in the All dimensions of the Helmholtz resonator are subwave-length. This guarantees that no standing waves can form within and that no degenerate modes may be excited. As the neck of the Helmholtz resonator is narrow, the fluid inside the neck can be approximated by being incrompressible. This means it stops acting as a fluid or acoustic medium and acts as a solid rod of mass m = ρSLneckinstead. On the other hand, the cavity is many times
larger than the neck and the fluid inside the cavity can be compressed. By the universal gas law, neglecting viscous effects, the pressure and volume of a section of fluid are inversely related. This means that if when the ”rod” moves a distance dx into the chamber due to a positive pressure change at the open end of the neck, the volume of the air which formerly filled the entire chamber is compressed into a volume of V − Sdx. The new volume of this air may be written as a fraction of the cavity itself:
Vair=
Vcavity− Sdx
Vcavity
· Vcavity (22)
which implies that the internal pressure of this mass of air is increased by this fraction:
Pair =
Vcavity
Vcavity− Sdx
· P0 (23)
P (dx) = [1 + S Vcavity
· dx + ...] · P0 (24)
where the cavity volume is much larger than the cross-section of the neck. For any small displacement dx, higher orders may be ignored and the air in the cavity exerts a pressure p0+
S
Vcavity · dx to the solid rod in the neck. This way, the system may be viewed as a mass-spring
system, in which the air in the cavity acts as a spring and the air in the neck, being incompressible, acts as a mass.
The mass-spring system is driven by the pressure at the beginning of the neck. This driving pressure can be transcribed into a driving force using the cross-section of the neck. This yields the force-balance
P0· S · eiωt= m¨x + k · x (25)
with x the deviation of the rod from its ”normal” position and eiωt the periodic term for the
driving oscillation. Note that for simplicity, friction has been ignored. More on this is written in section 5.3. Here, the spring constant k is equal to V 1
chamber, as
F0= P (dx)S =
1 Vchamber
dx (26)
The solution to the differential equation 25 is
x(t) = Aeiωt (27)
with A the positive real amplitude of the oscillation. The angular frequency ω is given by the driving acoustic wave at the entrance of the resonator, whereas the resonance frequency ω0is given
byqmk. This relation gives us the periodic displacement of the rod due to a given periodically changing pressure at the end of the neck. Substituting the solution 27 into the equation 25 gives the frequency-dependence of the amplitude of the periodic displacement of the ”rod” of the resonator:
A = P0S m 1 ω2 0− ω2 (28) with which the movement of the air at the entrance of the neck due to a pressure wave can now be described[20].
2.6
Helmholtz Resonator as a band-stop filter
Whereas in the previous section, the Helmholtz resonator was viewed on its own, in most practical applications the neck of the resonator is attached to another waveguide, as shown in figure 2. This way, an incoming acoustical wave encounters a junction of the waveguide. On one hand, the power of the oscillation is dispersed into moving the column of air in the resonator’s neck, whereas on the other hand the acoustical wave continues propagating through the waveguide itself. The fraction
of acoustical power lost to moving the Helmholtz resonator’s neck varies with the frequency of the signal.
As the pressure at the neck of the helmholtz resonator and the continued horn is the same, in both directions, the pressure amplitude of the oscillation is the same. However, the volume velocity in both directions is different depending on the impedance of the continued waveguide and the impedance of the Helmholtz resonator. In order to explicitly relate the acoustical power led through to the signal frequency, as necessary to design a correctly working band-stop filter, a relation for the impedance of a Helmholtz resonator should be found.
As stated, the acoustic impedance Z is defined by the ratio of the pressure amplitude to the volume velocity amplitude. This can be exploited as the pressure amplitude at the start of a Helmholtz resonator is known: this is the driving pressure of the rod and given by the incoming acoustic wave. The volume velocity amplitude can then be deduced from equation 27: equation 27 is an explicit definition of the displacement x of the rod, for a certain frequency, given a certain pressure amplitude at the start of the rod. The volume velocity is then found
U = vmaxS (29) = iAωS (30) = iωSω 2 0− ω2 p m (31)
With p the amplitude of the driving pressure wave. As the pressure and volume velocity are known, the impedance of the section of a waveguide in direct contact with the Helmholtz resonator can be found:
Z = iρL ωS(ω
2− ω2
0) (32)
This result for the impedance of a Helmholtz Resonator is the same as given in Wang and Mak [25].
Using the impedance of the Helmholtz resonator, and given the known impedance of a waveg-uide of continuous cross-section Z0= ρcS, it follows from the continuity of mass flux and pressure
that for a waveguide with a resonator connected to it, the in- and outgoing waves are found using the matrix [16][26][17] p+ p− ! x=D = (1 − ζ)e
−ikD −ζeikD
ζe−ikD (1 + ζ)eikD
! · p+ p− ! x=0 (33)
with ζ the fraction Zresonator
2Zwaveguide. p+ and p− are the amplitudes of the pressure waves going
in the positive and negative ˆx-direction respectively. The use of this matrix notation becomes clear in the next section, where a system of multiple Helmholtz Resonators spaced periodically is treated.
2.7
Periodic effect
In the previous section, the single Helmholtz resonator attached to a waveguide is treated mathe-matically. An array of periodically spaced Helmholtz resonators may improve the band-stopping effect of the resonator. In fact, Wang and Mak [26] shows that the periodic spacing of multiple Helmholtz arrays increases the quality of the band-stop filter. Furthermore, Wang and Mak [25] show that there is a secondary effect when multiple resonators are placed on equal distances. First, the single equation governing the propagation of a sound wave through an array of Helmholtz res-onators as if modulated by a Bloch wave is deduced. After that, the first of these two effects is explained. As in the design for the simulation the secondary effect has not been used, only some more of this is explained in section 5.4. Firstly, the propagation of a pressure wave has been shown to be described by the matrix given in equation 33. In a system of N Helmholtz Resonators, being an array, the matrix has to be applied repeatedly to a known vector respresenting the state at the driven end of a waveguide, usually p0
0 !
to find the consequent state-vektor at the other end. This repeated multiplication can be simplified by assuming the matrix has a (possibly complex) eigenvalue [17]. Both Sugimoto and Horioka [17] and Wang and Mak [26] argue that there is no reason for this eigenvalue not to be writable in the form of λ = eiqdwith q a complex mathematical
wavenumber and d the constant distance between either two adjacent Helmholtz resonators. This way, the matrix relation 33 can be written as
p+ p− ! x=d = eiqd· p+ p− ! x=0 (34) with again p+ and p− the pressure amplitudes of the left- and right-going waves. A sound wave
passing multiple Helmholtz resonators of the same dimensions is described by the repeated mul-tiplication of the matrix from equation 33. If a sound wave encounters n resonators this way,
p+ p− ! x=nd = An· p+ p− ! x=0 = eiqnd p+ p− ! x=0 (35)
which shows that repeated application of the matrix to a known state p+ p−
!
shifts the con-sequent resulting state by a complex value. This way, the amplitude of the left- and right-going waves are modulated by a periodic function as if by a wave. This mathematical wave is called a Bloch wave [17][25]. By calculating the eigenvalue of matrix A directly and equating this to the eigenvalue eiqd, an implicit relation is found for the complex wavenumber q:
cos(qD) = cos(kD) + iζsin(kD) (36)
This equation can easilly be solved for two cases: the first being that ω is very close to ω0,
that is, the Helmholtz resonators are or are almost in resonating mode; the second being when kD = nπ with n ∈ {1, 2, 3...}. This last case has to do with so-called Bragg-reflection as described in section 5.4.
2.7.1 High resonator activity
The first case, in which the signal attenuation stems from resonator activity, is in the case that ω is very close to ω0. This can be rewritten as
ω ω0
= 1 + ∆ (37)
with k∆k ≤ 1
As a stop band occurs when the Bloch wavenumber q has a real part, the signal is attenuated when the RHS of equation 36 has a modulus smaller than or equal to unity. The boundaries of such stop band are then [25]
∆1= Vck0 4Sd cot k0D 2 (1 + ∆1) (38) ∆2= − Vck0 4Sdtan k0D 2 (1 + ∆1) (39) However, the resonance (and so the highest attenuation) occurs when ∆ vanishes (supposing kD 6= nπ, in which case sin(kD) = 0) so when the angular frequency of the signal is the same as the resonant frequency described in section 2.5. Although it is difficult to match the stop band exactly with a peak in transmission from a horn, the resonance frequency of the resonators can be be matched to the top of the peak. A way to further match the attenuation from the resonators to the peak in transmission is given in section 5.3.
2.8
Helmholtz resonator array as acoustic metamaterial
It has been shown, as stated, in Wang and Mak [25] that a periodic array of Helmholtz resonators can attenuate a signal. Fang et al. [16] show that this can be interpreted as if the acoustic medium through wich the sound wave propagates exhibits a complex modulus, as explained below. The propagation of the input pressure wave over a distance along the waveguide is given in equation 12. As the given waveguide, in Wang and Mak [25] and Wang and Mak[26], is divided into pieces of length D, any discrete distance may as well be taken to be x = nD. Due to this substitution, the wavenumber k must be replaced by the Bloch wavenumber q. This way, the sound in the waveguide behaves as if propagating with a complex wavenumber. A complex wavenumber corresponds to a locally complex wave velocity c ≡ ω
k. Therefor, a waveguide with an array of Helmholtz resonators
attached to it shows a complex wave velocity which cannot be traced back to the properties of the materials (that is, the propagating medium) alone, making it a metamaterial [4]. As the wave velocity in a medium is found by
c = s
E
ρ (40)
an effective complex wave velocity can either follow from a complex density ρ (which is non-physical) or a complex elasticity modulus E [5]. Fang et al. [16] interprets the complex wave
Table 1: Dimensions of Helmholtz Resonator-array used by Wang and Mak [26]. The resulting f,TL-graph is shown in figure 2. Note that the model represented a semi-infinite model: whereas a finite number (5) of Helmholtz resonators was used as a band-stop filter, the cylindrical waveguide itself stretches to infinity. This is done using the method described in section 3.3.
Dimensions Size Duct cross-section 13.2 cm2 Cavity radius 4.7 cm Cavity length 4 cm Neck length 4.55 cm Neck radius 1.7 cm Inter-resonator distance 40 cm
velocity as following from a complex elasticity modulus, as is often the case for acoustic metama-terials [4]. Besides, Fang et al. [16] shows that this complex modulus can have the wave velocity be higher than the normal 343 m s−1 for air or 1440 m s−1 for water.
3
Method
The previous section dealt with an array of Helmholtz resonators attached to a cylindrical waveg-uide. As the goal of the project is to use this array of Helmholtz resonators to equalize the signal from a horn, it must be shown that the array has the same effect on a horn as on a waveguide with constant cross-section. In order to demonstrate the effect of an array of Helmholtz resonators attached to a horn, an exponential horn has been modelled in COMSOL - first, the horn has been modelled as is; then, an array of Helmholtz resonators has been added to the horn. In order to show the direct influence of the Helmholtz resonators, a graph has been made, for both cases, of the transmission loss as a function of the frequency.
In order to affirm the working of COMSOL and the nonreflective boundary, the exact waveguide as described in Wang and Mak [26] has been modelled in COMSOL. This waveguide consists of a straight cylindrical pipe with an array of five Helmholtz resonators connected to it, with a period D. This configuration has already been shown in figure 2. The dimensions of the Helmholtz resonators used in this first simulation, as well as those of the main waveguide, are the same as in Wang and Mak [26] and given in table 1.
3.1
The exponential horn model
A model has been made of an exponential horn described in figure 3. Exponential horns pose two advantages as opposed to other horns: firstly, due to the high flaring at the mouth of the horn, fewer reflections of the outgoing sound wave as described in section 2.4 from the mouth into the horn form, even when ka 6= 1 [7]. Secondly, when reflections do form, most of the sound wave is reflected close to the borders of the mouth as opposed to close to the axis of the horn [7].
Figure 2: Array of Helmholtz resonators used by Wang and Mak[26].
Figure 3: Contour of the exponential horn used as a reference. The length of the horn is 11 cm, whereas the mouth radius of the horn throat is 0.5 cm and the radius of the horn is 11.5 cm. The contour of the horn follows the parametrised curve {x, r} = {s, e0.285045·s; 0 ≤ s ≤ 11}. In this figure, the element denoted by ”a” is the driver (in reality placed in the horn itself, at x = 0 - here placed outside for clarity). ”b” is the throat part of the horn delivering a large impedance to the driver, whereas ”c” is the mouth part of the horn, being very wide. Here, the flaring is large, which means that the radius of the horn increases rather quickly. This in turn decreases standing radial waves [7].
These reflections in turn may excite radial as opposed to axial standing waves [9] as described by Meykens et al. [11]. These radial standing waves would decrease the horn efficiency as acoustic power is lost to the radial movement of air; besides, radial standing waves are another cause for frequency-dependence of the transmission loss. This way, the flaring of the exponential horn again reduces transmission loss. Furthermore, Hall [7] states that wider wall angles decrease the forming of these higher propagation modes. The contour of the horn has been choosen so that for a frequency of 2000 Hz, the mouth radius is the same as the wavelength in water. This means that for frequencies larger than 2000 Hz, the effect of reflections from the mouth should decrease.
3.2
Enhanced horn model
In order to study the effects of the array of Helmholtz resonators as described in 2.6, the same horn as described in figure 3 has been modelled with an array of specifically designed Helmholtz resonators connected to the edges of the horn. As seen in figure 4, four Helmholtz Resonators have been attached to the horn, all of the same size and thus of the same resonant frequency ω0.
The resonators have been designed by parametrizing the dimensions in terms of the effective neck length Lef f. The parametrized dimensions are shown in table 2. The effective neck length was
then set at 1.21 cm, corresponding to a resonance frequency of 3370 Hz. The number of Helmholtz resonators thus used follows from the dimensions of the array: as Wang and Mak [26] show, the quality of a band-stop filter increases with the number of resonators used. However, in a straight pipe, as the band-stop filter is originally implemented, the number of resonators only depends on the length of the pipe and the length of each resonator-pipe element. In the case of a horn, the flaring of the horn may leave it impossible to add another resonator without the resonator’s cavity and the horn to overlap spatially. Because of this, a maximum of four resonators has been added to the horn.
3.3
Nonreflective boundary
In order to simulate the models, COMSOL reduces the models to a mesh of nonzero-length vertices. However, COMSOL is only able to provide an approximation to a solution of the Helmholtz equation for a finite space. On the contrary, the emission of sound from a horn is taken to be in an infinite space, so no part of the outgoing wave from the horn mouth is reflected back toward the horn. Normally, this is done using a Perfectly Matched Layer implemented already in COMSOL; due to license issues, it was not possible to use this feature in this research, so another approach was used. In order to simulate nonreflectivity, simply setting the pressure or its gradient to any finite number is not sufficient for this, as this imposes a boundary condition on the outgoing wave that is not there physically. In order to solve this, an additional region is marked in COMSOL where the wavespeed is set to be complex[10] (as opposed to 1440 m s−1 on the real axis): as the angular frequency of any generated power wave is by definition real and taking the well-known disperion relation
c = ω
Figure 4: 3D-figure of the enhanced horn. On the left, the first Helmholtz resonator is placed directly at the horn throat. On the right, the number of Helmholtz resonators is limited due to the flaring of the horn: adding another Helmholtz resonator would have the resonator cavity and horn geometry overlap each other spatially. This may be solved by increasing the neck of the resonator and adapting the resonator cavity as to maintain the same resonance frequency. However, it is unclear if the increased friction due to the neck may influence the band-stop filter’s response, which further research may work out.
Table 2: Dimensions of the Helmholtz Resonator used in the band-stop filter array as seen in figure 4. As shown, all dimensions were parametrized as a function of the effective neck length Lef f, which in turn
equals 1.21 cm.
Dimension Size Specification
Cavity volume L3ef f The cavity is cube, with cube vertices of a length of Lef f
Neck length Lneck≡ Lef f −π4Rneck Rneckis the neck radius of the resonator. Note
that Chen et al. [24], as well as Dosch and Hauck [8] cite different values, 1.7 and3π8 mul-tiplied by the neck radius respectively, the last of which is in the case of the neck flaring in the vicinity of the cavity. Here, the entire sur-face of the cavity and the neck are orthogonal, so the effective length from Wang and Mak [8] has been used.
Neck radius 101Lef f Neck radius is set small compared to the cavity
dimensions, as to ensure the validity of the end corrections by the conditions of Dosch and Hauck[8]
it follows that the wave number k is complex. Considering the equation for a plane wave, equation 6, having a complex valued wavenumber k in the form of
k = A + iB (42)
yields a decaying amplitude term in front of the propagating wave:
p(~r, t) = p0e−Bxe(iAx−iωt) (43)
The parameter A in equation 42 is the real part of the wave number and is so equal to the wavenumber as it would be in a nondecaying wave (k = ωc with c=1440 m s−1). The real parameter determining the strength of the decay is then B. This parameter has been choosen so that the over the length of the thickness of the aforementioned region the wave decays at least to 1 % of its amplituded when entering the region. Thus, a wave leaving the ”real simulation space” and entering the nonreflective region attenuates to 1 % of its strength before encountering the boundary of the entire model, which may as well be fully reflective. The reflected wave, being 1 % as large as the initial wave, is attenuated another time by the complex wave number before re-entering the ”real” simulation space. This way, any sound wave is reflected with an amplitude of only one 10.000th that of the initial wave. This way, the ”nonreflective” region approximates a
4
Results
From the simulations in the previous section, three f,TL-diagrams were generated. All three will be discussed in this section, as well as some corresponding graphs to explain more about the results.
4.1
Waveguide with Helmholtz resonator array
Firstly, a straight waveguide has been modelled with the same dimensions as the one simulated in Wang and Mak [26]. The transmission loss found in this simulation is shown in figure 5.
4.2
Bare horn
Secondly, a bare exponential horn as described in section 3.1 has been simulated. The transmission loss as a function of the signal frequency is shown in figure 6. Here, the periodicity of the peaks in transmission loss are visible. The first such peak occurs at 2000 Hz. Note that due to high mouth impedance for low frequencies, the transmission loss increases for decreasing frequencies when lower than 2000 Hz. Note that for frequencies higher than 6000 Hz, some of the mesh vertices in the simulation mesh are of similar size as the corresponding wavelength. This greatly decreases the accuracy of the simulation.
4.3
Enhanced horn
Finally, to the bare exponential horn described in section 3.1, an array of Helmholtz resonators has been added as described in section 3.2. The transmission loss as a function of frequency is shown in figure 7. Here, the same periodicity is clear as in figure 6. However, a peak is visible around 3370 Hz - the resonance frequency of the Helmholtz resonators in the resonator array. This implies that the transmission loss is increased at that particular frequency. This is also visible in figure 8, where both the transmission loss of the bare horn as well as the transmission loss of the enhanced horn have been plotted. Here, around 3370 Hz, a sharp peak in the difference in transmission loss (red) can be seen. Note that for frequecies higher than 6000 Hz, the graph shows more peaks - these may result from the decreased simulation accuracy due to the minimum vertex size being of similar magnitude as the wavelength.
5
Discussion
The f,TL-graph of the straight waveguide lined with five Helmholtz resonators as seen in figure 5 and figure 9 share the same peak in transmission loss, corresponding to the resonance frequency of the Helmholtz resonators. However, the shapes of the stop bands differ - in fact, they appear mirrored in the way that in figure 5, the main peak occurs for lower frequencies than the broader but less attenuative stop-band, whereas in figure 9 this is the other way around. It is unclear why this is. On the other hand, in figure 10, it can be seen that the smaller stop band occurs for lower frequencies, as was found in the simulations. Wang and Mak[26] explicitly state not to have an
Figure 5: Transmission loss as a function of frequency for the straight waveguide lined with five identical Helmholtz resonators as described in section 3.
Figure 6: Transmission loss as a function of frequency for the exponential horn described in section 3.1. Here, it is clearly visible that for very low frequencies, the transmission loss is very high - this is due to the reflections from the mouth, as for these frequencies, the mouth radius is smaller than the wavelength of the incoming sound wave. From 2000 Hz on, this effect decreases and the effect of standing waves is visible, with a high transmission loss for every frequency where the horn length is an integer number of wavelenths.
Figure 7: Transmission loss as a function of frequency for the exponential horn enhanced with an array of Helmholtz resonators.
Figure 8: Transmission loss for a bare exponential horn (blue), exponential horn with resonator array attached (green) and the difference (T Lbarehorn − T Lenhancedhorn) (red). It can be seen that up to
3370 Hz, the array poses little difference to the transmission loss of the model. However, at around 3370 Hz, the effect of the Helmholtz resonators is visible as a dip in the difference factor. However, this peak in transmission loss is placed at a somewhat higher frequency than the dip in TL, which resulted in the ”zigzag” beginning at 3000 Hz.
Figure 9: Transmission loss as found by Wang and Mak[26] for the straight cylindrical waveguide lined with five identical Helmholtz resonators as described in section 3 and section 2.6.
explanation for the occurence of these two bands, instead of only one attenuation peak. Therefor, as the resonance peak found by simulation is the same as found in the literature, it is taken that the simulation has been accurate.
As visible in figure 8, the transmission loss is greatly increased at the resonance frequencies of the Helmholtz resonators used. This implies that the Helmholtz resonators attached to a flaring waveguide exhibit the same attenuating effect as a band-stop filter as the same resonator array would have on a waveguide of continuous cross-section. However, as the resonance frequency is designed to be 3370 Hz, the resonance peak in figure 8 seems to deviate from the resonance frequency. Two causes for this are suggested: first of all, this effect may simply be because of the finite resolution of the mesh as well as the finite number of frequencies evaluated for. This may be found out by decreasing the maximum length of the mesh elements (increasing runtime) and interval between frequencies (also increasing runtime) but evaluating over a very limited frequency band (compensating for the increased runtime). Another explanation may be that the end-correction taken as in Dosch and Hauck [8] of π4 may deviate from the actual end-correction. This can be found by simulating a Helmholtz resonator alone and finding the resonance frequency. Comparing this to the calculated frequency should confirm or quantify the deviation of the used end-correction. In the conclusion that the array of Helmholtz resonator may be used one-to-one on a horn as it can be used on a waveguide of continuous cross-section, though, it should be noted that the simulations were only done using one exponential horn. No extensive research has been done to the effect of the pressure wave inside a horn diverging, safe that Hall [7] states that ususally within a highly flaring waveguide, no higher order modes than the plane wave exist. For the presented application of the resonator array, this may be neglectible as the resonators can attenuate the signal individually - that is, not relying on the periodicity of the array as described below in subsection 5.4. However, it is useful to investigate this more in order to use the periodicity of resonators for attenuating a signal.
5.1
Research suggestions
Multiple strategies for follow-up studies can be suggested. One of these strategies is to further exploit the periodicity of the output signal in ω-space; another to use, opposed to the ”vanilla” Helmholtz resonator, a subsequent form of resonator. Both will be discussed briefly hereafter. However, the application of an array of ”vanilla” Helmholtz resonators, as in this thesis, requires some more design research, as will be suggested afterwards.
5.2
Resonator variations
As possible extensions of the ”vanilla” Helmholtz resonator, I suggest two forms: the extended-neck resonator as described in Selamet and Lee [14] and the double Helmholtz resonator as described in Zu et al. [15]. The extended-neck resonator is a resonator in which the neck of the resonator is extended into the resonator cavity. This resonator has been described in Selamet and Lee [14]. The extension of the neck alone alters the resonance frequency of a single Helmholtz resonator only. Perforating this neck extension, as well as changing the neck extension angle (having either
diverging or converging neck extension), also influences the waveguide’s transmission loss due to the resonator. This may be useful as another tool to tune the resonator design.
The other resonator that may be used as a band-stop filter is the Double Helmholtz Resonator as described by Zu et al.[15]. In this configuration, a resonator cavity is, like a regular Helmholtz resonator, placed on a neck of small cross-section. Opposite to this neck, another neck-resonator combination is placed into the cavity. A cross-section of the combination if shown in figure 11 as taken from Zu et al. [15]. Zu et al. [15] show this configuration shows two resonance frequencies
f± = c0 2√2π s 1 + (ω2 ω1 )2(1 +V2 V1 ± r [1 + (ω2 ω1 )2(1 +V2 V1 )]2− 4(ω2 ω1 )2 (44)
with f± the higher f+and lower f− resonance frequencies, ω1and ω2 the resonant frequencies
as known from a vanilla Helmholtz resonator. V1and V2denote the cavity volume of the primary
and secondary resonator, respectively.
As the dual Helmholtz resonator used as a dual band-stop filter provides two resonance peaks of transmission loss to a waveguide, it may be used for attenuating two tranmission peaks at will. If these transmission peaks are adjacent to each other, an equalized band may be formed.
5.3
Further tuning of filter design
As stated, the design described in section 3 and 5 does not equalize a signal completely. This is partly due to the attenuation of the filter being to strong: to much of the acoustic power is ”leaked” away via the array of Helmholtz resonators. In any practical application, this is unwanted: due to the this attenuation maximum, a peak in transmission loss is seen, which actually opposes equalization. Therefor, a way should be found to regulate the amount of power lost to the moving of an air column in a single resonator. This may be done by designing the neck of the Helmholtz resonator in such way that the friction between the air column and the neck material decreases the maximum speed of the ”rod”, thus decreasing the volume velocity lost to moving this ”rod”.
5.4
Use of periodicity
As mentioned before, the presented model of an exponential horn lined with four identical Helmholtz resonators has been designed with only the power dissipated to the resonators in mind, whereas an array of Helmholtz resonators as shown in section 2.6 can also act as a band-stop filter due to the periodicity of the Helmholtz resonators. Wang and Mak [25] show that these stop-bands are near ωm=mπcD0 with m any positive integer. From this it follows that the stop-bands occur
periodically over the frequency range, with a stop-band occuring every c0
2D Hz. Whereas the
de-sign of the band-stop filter used in this research only exhibited one stop-band in the frequency range evaluated for (the first stop band due to the periodicity of the resonator array occurs at 20 000 Hz), exploiting this periodicity well may mean that the stop-bands occur at the frequencies where the horn shows a transmission peak, thus equalizing the transmission for a theoretically infinite frequency range.
Figure 10: Transmission loss as a function of frequency for an array of Helmholtz resonators lining a straight cylindrical waveguide for multiple numbers of identical Helmholtz resonators.
Figure 11: Cross-section of a dual Helmholtz resonator as described by Zu et al. [15]. In the bottom fo the figure, a straight waveguide is placed. Connected to the waveguide is a narrow neck (middle) which ends in the primary cavity, to which another neck-and-cavity (the secondary cavity, top) is connected.
6
Conclusion
From the simulations and theory given it follows that an array of Helmholtz resonators can reduce the transmission of a horn for a select range of frequencies. Although the design presented does not show an equalized transmission, more tools are available to design a horn that does shown an equalized transmission. These include the use of the periodic placement of the Helmholtz resonators as well as using friction within the Helmholtz resonators. This way, an acoustical horn can be designed with an equalized transmission for a large range of frequencies for underwater sound transmission.
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