c
TU Delft, The Netherlands, 2006
ON SOME APPROACH TO ACTIVE NOISE CONTROL
Sergei V. Utyuzhnikov∗, Victor S. Ryaben’kii† and Ali A. Turan∗
∗University of Manchester
School MACE, P.O. Box 88, Manchester, U.K., M60 1QD e-mail: s.utyuzhnikov@manchester.ac.uk, a.turan@manchester.ac.uk web page: http://www.mace.manchester.ac.uk/aboutus/staff/academic/
†Keldysh Institute for Applied Mathematics
Russian Academy of Sciences, Miusskaya Sq., Moscow 124047, Russia e-mail: ryab@keldysh.ru
Key words: Active Noise Shielding, Noise Control, Difference Potential Method, Green’s matrix
Abstract. The problem of active shielding and noise control is considered. In this problem one domain is shielded from the noise generated in another domain via an imple-mentation of additional sound sources situated outside of the first domain. The general solution of the problem in a finite–difference formulation can be obtained using the differ-ence potential method. This solution appears to be valid for arbitrary domains, medium and boundary conditions. It only requires the information on total sound (both ”friendly” and ”adverse”) at the perimeter of the domain to be shielded. Currently all analysis and applications of the difference potential solution have been limited by the Helmholtz equa-tion. In contrast to the previous publications, in the current paper the mechanism of the active shielding solution is analysed. For this purpose, the case of monochromatic wave propagation in a duct with an end termination is studied. The difference potential method is first applied to the system of acoustics equations. The correspondence between the finite–difference solution and continuous solution based on Green’s function is shown for the case of a uniform medium. Different possible representations of the active shielding terms are analysed.
1 INTRODUCTION
assumed to be in the protected area. In our paper we consider the general formulation of the AS problem when internal (”friendly”) sound is allowed to be in the shielded domain. First theoretical papers in the field of active noise shielding appeared about 30 years ago and belong to Jessel, Malyuzhinets and Fedoryuk [1], [2], [3]. Meanwhile first publications related to some possible realistic implementation arose much later (see, e.g., [4], [5]). These approaches have a number of principal limitations. Some of the noise suppression techniques provide sound control only in selected discrete [4], [5], [6] or directional [7] areas. Other techniques, in particular developed by Kincaid et al. [8], assume detailed information regarding the sources and nature of noise. A number of publications are devoted to optimization of the strengths of spatially distributed secondary sources to minimize a quadratic pressure cost function [9], [10]. The most comprehensive theoretical and practical reviews can be found in books [11], [12] and [13].
In the standard approaches to AS, it is necessary to know the characteristics of ”ad-verse” sources including their location. The approach based on the Difference Potential Method (DPM) [15], [16] allows us to obtain the general solution of the AS problem in the finite–difference formulation if the boundary value problem is linear and correct. This solution is applicable to quite arbitrary geometric configurations, medium properties and boundary conditions. The information on total sound (both ”friendly” and ”adverse”) is required only at the perimeter of the domain to be shielded. It is very important to em-phasize that knowledge of both the ”adverse” and ”friendly” components is not required. There are only two principal conditions for the obtained solution. The problem must be linear and have a unique solution. Though the ultimate AS solution is achieved in a finite-difference form, from a practical standpoint this may not be necessarily treated as a drawback because the implementation of the AS assumes some discrete distribution of the AS sources. The optimization of this solution in application to the Helmholtz equation has been analysed in [17], [18], [19], [20]. A comprehensive analysis of continuous and finite-difference surface potentials mostly appropriate for the Helmholtz equation is done by Tsynkov in [14]. The AS solution obtained in [14] is also valid for the linear analogue of the Hemholtz equation with variable coefficients.
2 STATEMENT OF THE AS PROBLEM
The AS problem can be formulated in the following form. Let us assume that noise propagation is described by some linear boundary value problem in domain D0:
Lw = S (1)
w ∈ UD0, (2)
where UD0 is a linear subspace of functions defined on D0such that the solution of problem
(1), (2) exists and unique. Equation (1) can correspond to either the Helmholtz equation or the acoustics equations.
Let us consider now some domain D such that D ⊂ D0. The sources on the right-hand
side can be situated both in D and outside of D:
S = Sf + Sa, (3)
supp Sf ⊂ D,
supp Sa ⊂ D0 D.
Here, Sf is the source of ”friendly” sound, while Sa is the source of ”adverse” noise.
We assume that the distribution of function w∂D at the boundary of D is known.
It is important to emphasize that only this information is assumed to be available. In particular, the distribution of the sources S at the right-hand side is unknown. The AS problem is reduced to seeking additional sources g in D0 D such that the solution of
problem
Lw = S + g, (4)
w ∈ UD0 (5)
coincides with the solution of problem (1), (2) on subdomain D with S ≡ Sf. It is worth
noting that an ”obvious” solution g = −Sa is not appropriate because the distribution of
Sa is unknown. Moreover, if the density Sa is known, the trivial solution g = −Sa is not
realistic for a practical realization.
3 THE GENERAL FINITE–DIFFERENCE SOLUTION
Following the formalism of the DPM [16], let us introduce some grid M0 in D 0. We
also introduce subsets of grid M0 as follows: M+ = M0 ∩ D, M− = M0\M+. Assume
that equation (1) is approximated on some stencil by equation Lhw(h)|m ≡ X n amnw(h)n = S (h) |m, m ∈ M 0 . (6)
w(h) ∈ UD(h)
0, (7)
where UD(h)0 is a linear discrete space that is a discrete counterpart of space UD0.
The extensions of the sets M0, M+, M− by the stencil we denote as N0, N+, N−,
respectively. It is clear that the sets N+ and N− intersect each other. We consider their
intersection as the grid boundary γ of the domain D: γ = N+T N−. The grid boundary
γ is split into two nonintersecting sub-boundaries: γ = γ+∪ γ−, where γ+ = γT M+ and
γ− = γT M−. Now we seek the finite-difference solution of the AS problem (4), (5).
The general solution of the AS problem in the discrete formulation is given by the following main theorem [15]:
gh = −Lhv(h)|M−, m ∈ M− (8)
gh = 0, m ∈ M+, (9)
where v(h) is an arbitrary function such that v(h) ∈ UD(h)
0, v
(h)
γ = wγ(h). (10)
The proof of this theorem can be found in [16]. It is important to emphasize that solution (8) is obtained for the general statement of the problem formulated in the previous section and not based on the knowledge of Green’s function, in particular.
It is clear that the function v(h) in (8) is not unique. A partial case of this function
corresponds to v(h)|M0\γ = 0. In this case the AS source term is only situated on a minimal
possible area. In [17], it is formulated as a proposition that such sources are to be minimal in L1.
4 GENERAL AS SOLUTION FOR 1D ACOUSTICS EQUATIONS
Let us implement the DPM based solution to the acoustic equations. It allows us to obtain some additional properties of the solution. In order to understand better the nature and characteristics of the solution, it is investigated in a 1D case with application to a duct flow. The primary sources are assumed to be in the interval of 0 < x ≤ L, while the secondary source is situated at x = 0 to shield the area x < 0. The locations of the primary sources are unknown. The duct is assumed to be closed at x = L. For the sake of simplicity we assume that the duct is closed at x = −L by an absolutely absorbing wall.
The 1D acoustic system for isentropic flow can be written as follows:
pt+ ρc2ux = fp, (11)
ut+ px/ρ = fu.
We assume that the acoustic sources are time-harmonic:
fp = ρc2fbpeiωt, (12)
fu = c bfueiωt.
Hence, the dependent variables can be rewritten in the following form:
p = ρcpeb iωt, (13)
u =ueb iωt. Then, the governing equations become as follows:
ikp +b bux = bfp, (14)
iku +b pbx = bfu.
This system can be easily written in the following characteristic form: b
L+Rb+ = bf+, (15)
b
L−Rb− = bf−,
where bL+ = ik + ∂x∂ , bL−= ik −∂x∂ , bR+=p +b u, bb R− =bp −u, bb f+ = bfp+ bfu, bf−= bfp− bfu.
The functions bR+ and bR− are the counterparts of the Riemann invariants of acoustics system (11) propagating along the characteristics dxdt = c and dxdt = −c, respectively.
In order to consider a finite-difference formulation of the problem, let us introduce a uniform grid with a constant step h = L/M . Assume that the grid M0 is represented by nodes i = −M, ..., M , then the set M+ corresponds to nodes i = −M, ..., −1.
We approximate these equations using the ”upwind” scheme: ik + 1 h∇ b R+m = bfm+, (16) ik − 1 h∆ b R−m = bfm−, m ∈ M0, (17) where ∇si ≡ si− si−1, ∆si ≡ si+1− si.
It is easy to see that in the case of equation (16) for bR+ the boundary γ ≡ γ+ is
a single-layer and corresponds to m = −1. In turn, in the case of equation (17) the boundary γ ≡ γ− also is a single-layer and coincides with the point m = 0.
System (16), (17) is completed by the following boundary conditions: b
Here, rl is the reflection coefficient of the wall on the right-hand side. The reflection
coefficient rl= 0 corresponds to a fully absorbing wall while rl = 1 represents the case of
an absolutely reflecting wall.
It is possible to prove that the solution of problem (16), (17), (18) is given by the formulas: b Rm+ = X p≤m h bfp+ (1 + ikh)m−p+1, (19) b R−m = rlRb + M (1 + ikh)M −m + X p≥m h bfp− (1 + ikh)1−m+p. (20)
5 DISCRETE SOLUTION OF AS PROBLEM FOR A DUCT
Let us now derive a discrete solution of the AS problem. Using the main theorem we are able to find the AS source acting only at the boundary γ. Consider a partial case of the function v(h) from the main theorem and assume that the fields bR+ and bR−in M−\ γ
equal to 0. Then, the appropriate system of equations including the AS source terms becomes: ik + 1 h∇ b R+m = bfm++bg+0δm,0, (21) ik − 1 h∆ b R−m = bfm−+gb−0δm,0, m ∈ M0. (22)
Here, δm,0 = 1 if m = 0, otherwise δm,0 = 0. The AS source terms are given by
b g+0 = b e R + −1 h , (23) b g−0 = −(ik + 1 h) bRe − 0
via the appropriate values of bR+and bR−to be taken from the measurements. In our case,
Let us now consider the two extreme cases of absolutely reflecting (rl = 1) and
absorb-ing (rl = 0) walls on the right-hand side respectively. We assume that in both cases the
AS solutions coincide each other. In the case of the absorbing wall, it can be achieved if we include the following additional source term on the right-hand side of equation (17) in order to compensate the absorption of the wave bR+:
b
fM− = (1 − rl)
b R+M
h . (26)
Then, in both cases we have the same bRe
+
−1 and bRe
−
0 in the measurements, hence we obtain
the same AS source terms. According to the main theorem, sources (23) must completely provide AS. Meanwhile, the AS filters the wave generated by bfM− (if rl= 0) since this wave
is related to the external source. On the other hand, formally the same AS remains the wave related to the reflection (if rl = 1) and having the characteristics coinciding with the
previous wave. Such a paradoxical situation, at first glance, can be explained as follows. The solution of problem (16), (17), (18) can be represented via matrix Green’s function. For this purpose, the following two auxiliary problems are to be solved:
ik + 1 h∇ b G(1)1,i|0 = δi,0, (27) ik − 1 h∆ b G(1)2,i|0 = 0, m ∈ M0, with the following boundary conditions:
b G(1)1,−M |0 = 0, Gb (1) 2,M |0 = rlGb (1) 1,M |0 (28) and ik + 1 h∇ b G(2)1,i|0 = 0, (29) ik − 1 h∆ b G(2)2,i|0 = δi,0, m ∈ M0,
subject to the same kind of boundary conditions: b G(2)1,−M |0 = 0, Gb (2) 2,M |0 = rlGb (2) 1,M |0. (30)
Here, m ∈ M0 and α h = 1 + ikh, θ(m) = ( 0 if m < 0 1 if m ≥ 0. Thus, the appropriate Green’s matrix-function is as follows:
b Gm|0 = θ(m)α −(m+1) h 0 rlα−2M +m−1h θ(−m)α m−1 h (33) Having introduced the following vectors
c Wi = ( bR+i , bR − i ) T, (34) b Fi = ( bfi+, bf − i ) T + δ i,0(bg + 0,bg − 0) T ≡ ( bF+, bF− )T, (35)
system (16), (17) can be written in the following form: c
LhWch = cFh, (36)
where cLh is the appropriate finite-difference operator of the system.
The solution of problem (16),(17),(18) can be represented via the following convolution operation:
c
W = bG ∗ bF . (37)
On the definition, this expression means as follows:
c Wm = M X −M b Gm−p|pFbph. (38)
If m < 0, for bR+ we have the following solution:
b
R+m =Xθ(m − p)h bfp+α−m+p−1h =X
p≤m
h bfp+αh−m+p−1. (39)
Thus, if m < 0, the additional term bg+0δ0 in (21) provides the field coinciding with the
solution without the effect of external noise (19). It is important to note that the result does not depend on the value of bg+0. Thus, it can be set as bg+0 = 0. This means that no AS is required for the function bR+. This conclusion also follows from the property of equation (16) and the bR+–Riemann invariant. Nevertheless, under some conditions the
If m < 0, the solution of problem (17), (18) is given by: b R−m = rlRb+Mαm−Mh + rlh b g0+α−2M −1+mh +X p≥m h bfp−αm−1−ph + hbg0−αm−1h (40)
From the exact solution (20) we have: b e R − 0 = rlRb+Mα−Mh + X p≥0 h bfp−α−1−ph . (41)
Hence, if m < 0, from (23) it follows that
b R−m = rlRbe + −1α −2M −1+m h + X m≤p<0 h bfp−αhm−1−p. (42)
Thus, solution (42) coincides with the solution of (16), (17), (18) without the ”adverse” noise, i.e. bf+
p = 0, bf −
p = 0 for p ≥ 0. It is possible to see that the first term in (42) is
responsible for the ”echo” of the ”friendly” sound. Thus, the function of the additional sources in the AS solution provided by the main theorem is not limited by noise elimination but can also include restoration of the echo of ”friendly” sound. It is important to note for further discussion that in (42) only the echo of the ”friendly” sound explicitly includes the reflection coefficient rl.
Thus, even if a wave arising from the right-hand side of the shielded domain is the same and the AS source is the same, the filtering procedure might be different due to the influence of the boundary condition. It is important to note that the AS solution requires neither the knowledge of the reflection coefficient r nor the knowledge of a noise source location. All the required information is included in the value of the total field at the boundary of the shielded domain to be obtained from an experiment. This information can include the total contribution of both internal and external sources. Taking into account the grid resolution requirement of kh << 1 all the measurements can be done at only one point.
From the analysis of solution (40), the following important conclusions can be formu-lated. If the sound echo is substantial and the AS is realized by a point source, then it is uniquely determined. If the reflection of ”friendly” sound is not important, the optimal AS corresponds to bg+0 = 0. In this case, if the source bg0+ exists, it vainly operates. From (25), it follows that if ”friendly” sources are absent, bg+0 automatically equals zero. In contrast tobg+0, the other source bg0− is uniquely determined.
6 CONTINUOUS CASE
b
G(x, 0) =
e−ikxθ(x) 0
rle−ikxe−i2k(L−x) eikxθ(−x)
(43) It is easy to prove that this continuous matrix corresponds to the ultimate matrix (33) if h → 0.
The vector of the AS point source term is represented by
b q0(x) = δ(x) b e R + 0, − bRe − 0 T , where δ(x) is the delta-function.
If x ≤ 0, the bR+– Riemann invariant is given by
b
R+(x) = e−ikx Z x
−L
eikξfb+(ξ)dξ. (44)
Hence, in the continuous case: b e R + (0) = Z 0 −L eikξfb+(ξ)dξ (45)
If x ≤ 0, bR−– Riemann invariant is as follows:
b R−(x) = rleik(x−L)Rb+L + eikx Z L x e−ikξfb−(ξ)dξ + rleik(x−2L)Rbe + 0 − e ikxb e R − 0 = (46) rleik(x−2L)Rbe + 0 + e ikx Z L x e−ikξfb−(ξ)dξ.
Let us introduce the transition matrix A from the Riemann invariants to the original variables such that
( bR+, bR−)T = A(p,b u)b T. Then,
A = 1 1
1 − 1
and the source term in the original variables is given by b
q0(x) = A−1bg0(x) = δ(x)(ub0, pb0)
T, (47)
Under the requirement of approximation kh << 1, the finite-difference AS source term is as follows:
b
q0(h) = A−1bg0(h) = 1
h(bu0, pb0)
If x < 0, the contribution to the solution in the original variables made by the AS is given by c Wc = eikx 2 ( bR − 0 − rle−i2kLRb+0)(−1, 1)T (49) This field provides AS if we have the delta–function source (47). In reality we are not able to realize such a source. Instead, we consider the finite–difference solution (48) and assume the source has a physical extent of ∆:
b
q(x) = 1
hθ(x + ∆/2) [1 − θ(x − ∆/2)] (ub0, pb0)
T (50)
This source term generates the following solution: c Wh = bG ∗q =b 2cWc kh sin k∆ 2 , (51)
where bG = A−1GA. In order to generate the same field cb Wc, the parameter h is approxi-mately to be equal to the thickness of the source ∆:
h = 2 ksin
k∆
2 ≈ ∆, (52)
and the thickness of the source must be small enough to satisfy the requirement k∆ << 1. In the original variables, the governing equations including the AS are as follows:
pt+ ρc2ux = ρc2qvol+ fp (53) ut+ px ρ = bvol ρ + fu, where qvol(x) = u0δ(x), bvol(x) = p0δ(x), u0 =bu0e
iωt, p
0 = ρcpb0e
iωt.
Here, qvol and bvol are volumetric monopole and dipole sources, accordingly. They
appropriately alter the acoustic balance of mass and momentum in the system [18]. If the reflection of ”friendly” sound is not substantial, the AS solution minimal in L1
and L2 is represented by the following sources:
qvol|opt = −(p0− ρcu0) δ(x) 2ρc , (54) bvol|opt = (p0− ρcu0) δ(x) 2 . (55)
In this case only the combination ρcqvol− bvol of the sources qvol and bvol is substantial.
Therefore, some simplified realizations are also available.
If both the ”friendly” sound and reflection on the left-hand side are absent, then the measurement of either the pressure or the velocity can be omitted. In the latter case, the source terms can be represented in the following form:
qvol|opt = −
2p0
7 CONCLUSION
The single–layer AS solution has been obtained for the case of 1D monochromatic wave propagation. The solution only requires the measurement results of the total acoustic field (both ”friendly” and ”adverse”) at the boundary of the shielded area. It does not require either the knowledge of the medium where the acoustic field is propagated or the boundary reflection coefficient values. The mechanism of the AS action has been revealed. The correspondence between the finite–difference solution and continuous solution is shown for the case of a uniform medium. In practice, the AS source can be implemented via an extended uniform source corresponding to a discrete solution containing the parameter h. It is shown that the optimal value of h corresponds to the thickness of the source which is to be much less than the wave length. The optimal solution has been found.
8 ACKNOWLEDGMENT
This research was supported by the Engineering and Physics Sciences Research Council (EPSRC) under grant GR/26832/01.
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