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DAYS on DIFFRACTION' 2008

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Wave mo des in p erio dic systems of thin tub es
Andrey V. Shanin, Maxim S. Dorofeev
Department of Physics, Moscow State University, Russia; e-mail: andrey shanin@ok.ru
The problem of wave propagation in a p erio dic system of thin tub es is studied. The waves are describ ed by the disp ersion diagrams, i.e. by the dep endencies of the propagation constants on the temp oral frequency. The main question under consideration is how many mo des can propagate in a particular system. The pap er presents a metho d of finding the disp ersion diagrams, and an inequality for the numb er of mo des, linking the numb er of mo des with the top ological prop erties of the system.

node

H

closednode

1

Introduction

cell w1 p1 w2 p2 w3 p3 input terminals

opennode

Consider an infinite periodic set of tubes filled with compressible lossless linear gas (or liquid). The tubes are assumed to be one-dimensional, i.e. only propagation of the piston mode is taken into account. The tubes are connected with each other by small nodes. We assume that for each node the sum of incoming flows is equal to zero. There can be dead ends (i.e. closed nodes), or open nodes. The flow from the close node is equal to zero, and the pressure at an open node is equal to zero. An example of the system is shown in Fig. 1. Here "periodic" means single-periodic, i.e. the system is infinite along only one dimension (from left to right in the figures). A stationary (time harmonic) problem is studied. The time dependence has form e-it and is omitted everywhere. The smallest period of the system will be called a cel l of the system. The cuts separating a cell will be called the terminals of the cell. The terminals are divided into input and output terminals (we assume that the positive direction of mode propagation along the cell is from the input terminals to the output terminals). The choice of the cell shape is not unique. We assume that each terminal is connected with only one tube within the cell, i.e. the cut points are chosen somewhere in middles of tubes before splitting the system into the cells, rather than cutting across the nodes connecting three or more tubes. Note that for many cells even the number of input/output terminals is not uniquely defined. For example, in Fig. 1 there is a possibility to split the system into cells having two or three input terminals. However, the number of output terminals

p' w' 1 1 p' w' 2 2 p' w' 3 3 output terminals

Figure 1: Periodic system of tubes (top) and a single cell (bottom) should always be equal to the number of input terminals. As a practical application we have in mind water supply or ventilation systems in tall buildings. A cell in such a system is naturally the set of ducts belonging to a single floor. Let the input terminals be numbered, and let the output terminals be numbered by the same numbers in a matching order. We associate with the input terminals the pressures pm , m = 1, . . . , M and the gas flows wm directed into the cell. Accordingly, we associate with the output terminals the pressures p and the flows wm directed from m the cell (see Fig. 1). A wave mode is a solution of the governing equations in the tubes and boundary conditions in the nodes obeying the relations p where = ( ) = exp{i ( )} (2)
m

= p m ,

wm = wm ,

(1)


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is common for all terminals. The value ( ) will be called a propagation constant. As usually, there can be propagating modes corresponding to real and evanescent modes corresponding to having a non-zero complex part. A propagating mode can propagate either in the positive or in the negative direction. The direction can be established by studying the group velocity v
gr

pressures at the ends the flows from corres Construct a matrix C pressures (i.e. a DtN w1 w2

of the tube by p1 and p2 , and ponding ends by w1 and w2 . connecting the flows with the mapping for a tube): =C p p
1 2

.

(5)

= H (d /d )

-1

,

(3)

In the general case this matrix can be obtained by solving the Webster's equation d2 p 1 dS dp + + k 2 p = 0, 2 dl S dl dl w(l) = S (l) dp , (6) i 0 dx

where H is the length of the cell. As we shall see below, for each mode having propagation constant there exists a mode having the propagation constant equal to - , i.e. a mode propagating or decaying similarly, but in the opposite direction. For complicated cells for a fixed time frequency there can exist different types of modes. The main task of the current work is to estimate the number of modes and to compute the dispersion diagrams, i.e. the dependencies j ( ) for different modes labelled by the index j . If there exists a matrix T connecting the pressures and flows at the output with the pressures and flows at the input, i.e.
(p , . . . , p , w1 , . . . , wM )t = 1 M

where k = /c0 , c0 and 0 are the speed of sound and the density of the medium. The flow in the second relation is taken with respect to the positive l direction. There are two linearly independent solutions of the Webster's equation, so one can find the integration constants from the boundary values (p1 , p2 ), and then reconstruct the boundary values (w1 , w2 ). Here we write down the form of matrix C for the case of a constant cross-section: C=- iS 0 c0 sin(k L) cos(k L) -1 -1 cos(k L) . (7)

T · (p1 , . . . , pM , w1 , . . . , wM )

t

(4)

Also it is easy to find such a matrix for the crosssection depending on l linearly or quadratically.

This relation is true for arbitrary shape of the tube, i.e. for an arbitrary function S (l). The proof follows immediately from the Green's theorem. This fact is important for what follows. Let us construct an equation for finding the parameter . Consider a waveguide mode in a cell described above. Denote the pressures at the nodes of the cell by p1 , . . . , pN . The first M values, i.e. p1 , . . . , pM are related to the input terminals, and 2 Mathematical formalism the values pM +1 , . . . , pN are the pressures at the inConsider a single tube of the length L (see Fig. 2). ternal nodes of the cell. The pressures at the output Introduce a coordinate l along the tube. The cross- terminals are not taken as independent unknowns section S of the tube may depend on l. Denote the since they are equal to p1 , . . . , pM . The pressures

then the values j would be simply the eigenvalues of T ( ). From this consideration it is clear that the number of modes cannot be more than 2M , and the number of modes running in the positive direction cannot be bigger than the number of input terminals. The problem is that in many cases it is impossible to construct such a matrix T . For example, for the cell shown in Fig. 1 such a matrix does not exist. A natural explanation is that instead of a cell with 3 terminals one could select a splitting for which each cell has only two terminals (for such a splitting a matrix T does exist). So one can suppose that the number of modes, propagating in the positive direction is equal to the minimum number of terminals available for some splitting. However this is not true, and below we discuss it in details. We develop another technique applicable to any cell and any splitting.

p1 w1 L p
2

w2
Figure 2: A single tube Note that the matrix is symmetrical, i.e.
1 2 C2 = C1 .

(8)


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at the open nodes are also not taken into account, since they are known to be zero. Thus, for the cell shown in Fig. 1 N = 7. We assume that no tube connects directly an input and an output terminal. If this happens, one can place an additional node somewhere on the tube. To determine and p1 , . . . , pN consider the flows going to each node (including the input terminals). The sum of these flows should be zero, and this constitutes N linear equations, which can be written in the matrix form as follows: p1 W1 . . (9) . G . = 0, . . W
N

This dependence reflects the fact that the tube connects nodes belonging to different cells in the chain. Thus, for the case of tubes having constant crosssections, the matrix G is defined explicitly. This matrix depends on and . If for some there exits a waveguide mode, then due to (9), det[G(, )] = 0. (14)

Being considered as a function of , the determinant (14) is a polynomial of and -1 . By construction and by taking into account (8), G(, ) = Y1 ( ) + Y2 ( ) +
-1

Y2t ( ),

(15)

p

N

where Y1 is a symmetrical matrix. This means that G() = Gt (-1 ) , and

where Wn is the sum of all flows into the node n. det[G(, )] = g ( + -1 ), (16) The matrix G can be obtained from (7) as follows: where g (z ) is a polynomial. Each root of this polyG= G . (10) nomial, if not equal to ±2, corresponds to a pair of different 1,2 , and thus to a pair of propagation The index runs over all tubes. The matrices G constants 1,2 , such that 2 = -1 . If all roots of are constructed by the following rules. Let C be g are simple and not equal to ±2, and the degree the matrix (5) found for the tube with index . Let of the polynomial in equal to , then there exist this tube connect the nodes n1 and n2 which are different wave modes propagating in each direction. either internal nodes of the cell or input terminals. In any case, the numbed of modes propagating in the positive (or negative) direction is not bigger Then than . The main aim of the rest of the paper is to (G )m1 = m1 ,n1 m2 ,n1 (C )1 + m1 ,n1 m2 ,n2 (C )1 + estimate . m2 2 1
m 1 ,n
2



m 2 ,n

1

2 (C )1 +

m 1 ,n

2



m 2 ,n

2

(C )2 . 2

(11)

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Some examples

Here and below is the Kronecker's delta. This formula means that the only non-zero elements of G stand on the crossings of rows and the columns with indices n1 , n2 and these elements are equal to corresponding elements of the matrix C . If the tube connects the internal node n with an open node, and the open node corresponds to the second node in Fig. 2, then (G )m1 = m2
m 1 ,n m 2 ,n

Here we study two simple examples, namely a cell with an open node, and a cell with a close node (see Fig. 3). For these cells we construct a matrix G by the algorithm described above and find the dispersion diagram.
input terminal L1 S
1

output terminal L2 S2 L3 S
3

input terminal L1 S
1

output terminal L2 S2 L3 S
3

(C )1 . 1

(12)

If the tube connects the output terminal n1 with the internal node n2 then the structure of the wave mode should be taken into account. This structure results in the non-diagonal terms' dependence on : (G )m1 = m2
m 1 ,n
2

open node

closed node

m 1 ,n

1



m 2 ,n

1

1 (C )1 +

m 1 ,n

1



m 2 ,n

2

(C )1 2 + (13)

Figure 3: Two simple examples of cells Consider a cell shown in Fig. 3 (left). According to the procedure outlined above, the G-matrix has



m 2 ,n

1

(C )2 + 1

m 1 ,n

2



m 2 ,n

2

(C )2 2


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form of i0 c0 G = S S
2 1

cot(k L1 ) - csc(k L1 )

- csc(k L1 ) cot(k L1 )

+ + (17)

cot(k L2 ) --1 csc(k L2 ) - csc(k L2 ) cot(k L2 ) S
3

0 0

0 cot(k L3 )

.

One can see that the polynomial g defined by (16) is linear for this case. This means that generally there are two modes, one in each direction. We have computed the propagation constants ( ) with the following set of dimensionless parameters: 0 = c0 = 1, S1 = S2 = S3 = 1, L1 = L2 = 1, L3 = 10. The real and imaginary parts of are shown in Fig. 4. Note that the real part takes values from - to .

The positive Re[ ] corresponds to the mode travelling in the positive direction, and the negative one corresponds to the mode travelling in the negative direction (note that positive values of correspond to positive d /d ). In the opacity regions there are also two modes, decaying in the positive and negative direction. The decay parameter tends to infinity at the resonance frequencies of the tube with an open end. Consider similarly the cell shown in Fig. 3, right. Again, performing the computations according to the method described above, cot(k L1 ) - csc(k L1 ) 0 i0 c0 G = S1 - csc(k L1 ) cot(k L1 ) 0 + 0 0 0 cot(k L2 ) --1 csc(k L2 ) 0 cot(k L2 ) 0 + S2 - csc(k L2 ) 0 0 0 0 0 0 S3 0 cot(k L3 ) - csc(k L3 ) (18) 0 - csc(k L3 ) cot(k L3 ) The propagation constants computed similarly to the previous case are shown in Fig. 5. The computations were conducted for the same parameters. The properties of the diagram are mostly the same as discussed above. A ma jor difference is that the first (i.e. low-frequency) zone is transparent. This means that in a closed system there is a possibility of propagation of low-frequency lowdispersive modes, whose properties are very close to usual longitudinal waves, but the velocity is lower than the velocity of sound in the medium. 4 Topological estimation of the number of modes

Let us estimate the degree of the polynomial g , i.e. find the maximum number of modes travelling or decaying in the positive direction. In the examples considered in the previous section this question was quite trivial: since there was only one input terminal per a cell, the maximum number of modes travelling or decaying in the positive direction was one. As we can see from the dispersion diagrams, Figure 4: Dispersion diagram for a cell with for almost all temporal frequencies this estimation an open node was exact. However, for bigger amount of input / The axis is split into the segments of trans- output terminals the problem becomes non-trivial. Let us formulate the main result of the section parency (where Im[ ] = 0) and opacity. The diagram starts with an opacity region. In the transpar- and the paper. Define a closed loop in a cell as folent regions there are two opposite values of Re[ ]. lows. Take a single cell of the system. Attach the


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one above another without having common points.
a +1
1

b

+1
1'

-1
2 2'

order=+1

order=0 +1

c
1

1'

2

2'

3

3'

+1 order=+2

Figure 6: Examples of closed loops on cells The main result is as follows: Theorem 1 The degree of g is not bigger than the maximum possible order of closed loop for a given cel l. Before proving this statement let us demonstrate that it is useful and non-trivial. Consider the cell shown in Fig. 7. It is not easy to guess the number of modes in such a cell, and the intuitive estimation is 3, since any cross-section of the system crosses at least three tubes. However, the maximum order of loop is 2, and the analysis based on the G matrix confirms this estimation of the number of modes.
1 5 2 3 4 7 6 2' 3' 4' 1'

Figure 5: Dispersion diagram for a cell with a closed node output terminals to corresponding input terminals. The result will be called a closed cel l. A closed loop is a closed oriented path along the closed cell, passing each node no more than one time. This path should go from node to node along the tubes. The path is not necessarily connected, i.e. it can consist of several closed oriented sub-loops. We allow as a possibility a path consisting of a single node, and a path consisting of two nodes connected by a tube (however both of these possibilities are not important for what follows). The main feature of the closed loop is that a double pass through a node is not allowed. Introduce the order of the closed loop as follows. At each point where the loop crosses a terminal, assign +1 if the loop goes from the output to the input terminal, and -1 if the loop goes in the opposite direction. The order of the loop is the sum of all assigned values. In Fig. 6 we represent several cells and loops in them. Note that in (c) only the points marked by circles are nodes. Simple intersections in the figure correspond to tubes passing

Figure 7: An example of a cell Let us prove the theorem. First, make a preparatory step. If an inner node is connected both with some input terminal and some output terminal, insert an additional node between the inner node and the input terminal. Obviously, putting a node on an existing tube changes only mathematical expressions, but not the physics and not the answer. For


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example, for the cell in Fig. 7 it is necessary to insert nodes between 1 and 5, 2 and 6, 3 and 7, 4 and 7. As the result, dimension of matrix G becomes bigger, but each non-zero non-diagonal element corresponds exactly to one tube. Consider the standard representation of det[G]: det[G] =
d 1 (d)Gd1 G12 G13 . . . G1N , d d d

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Conclusion

On the base of Dirihlet-to-Neumann approach the number of modes propagation through oneperiodical 1D pipe system is linked to the degree of the polynomial g . A topological estimation of the degree of the polynomial g is obtained.

(19)

Acknowledgements where N is the number of nodes (inner nodes plus input terminals), d is a permutation of the numbers The work is supported by the RFBR grant No 071 . . . N , = ±1 depending on the parity of the per- 02-00803-a and by a "Nauchnye shkoly" grant. mutation. Obviously, if is the maximum power of contained in det[G], then there exists such a permutation d that the product
1 1 Gd1 G12 Gd3 . . . G1N d d

(20)

is non-zero and contains . Note that each factor in (20) can be either a constant with respect to , or be proportional to ±1 . The statement of the theorem follows from the fact: Each non-zero product having form of (20) corresponds to a closed loop on the cel l. The power is equal to the order of the loop. To prove the last fact note that Gm = 0 means dm that there exists a tube connecting the nodes with the numbers m and dm . Split the permutation d into a set of non-intersecting cycles. Consider a cycle m 1 m 2 m 3 . . . mh m 1 . (21)

There should exist tubes connecting m1 with m2 , m2 with m3 , etc up to mh with m1 . So, we can define a loop composed of sub-loops connecting the nodes in the order (21). All conditions of the definition of a closed loop will be fulfilled. Each sub-loop is defined on a closed cell, since the input and output terminals are labelled by the same indices. If a tube connects two inner node or an inner node with the input terminal, then corresponding Gmn+1 is a constant with respect to . If the tube mn goes from an inner node m to an output terminal n, then corresponding Gm is proportional to . Note n that the same pass adds +1 to the order of the loop. If a tube goes from an output terminal n to an inner node m, then corresponding factor Gm is n proportional to -1 , and of course this pass adds -1 to the order of the loop. Thus, the proof of the theorem is completed. Although the theorem states an inequality, a less humble (and more adequate) formulation would be "almost always equal to" instead of "no bigger than".