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Some fundamental transmission properties of impedance transitions
WAVE MOTION 6 (1984) 389--406 NORTH-HOLLAND
389
S O M E F U N D A M E N T A L T R A N S M I S S I O N P R O P E R T I E S OF IMPEDANCE TRANSITIONS L.-E. ANDERSSON Department of Mathematics, University of Linkiiping, S-581 83 Linkrping, Sweden B. LUNDBERG Department of Mechanical Engineering, University of Lulea, S-951 87 Luled, Sweden Received 11 July 1983
Reflection and transmission of waves by impedance transitions from a constant input to a constant output characteristic impedance are considered. Several fundamental properties are explored, primarily for impedance transitions with piece-wise constant characteristic impedance in an arbitrary number N of intervals of equal length. For example, the following properties are shown: (i) The relative momentum transmission depends only on the ratio of output to input characteristic impedance. (ii) For a given impedance transition there are at most, and generally exactly, 2 N different transitions, including the original one, with identical transmission properties. (iii) For monotoneous impedance transitions the efficiency of energy transmission is minimized by one with an abrupt change in characteristic impedance. (iv) There exists an optimal impedance transition, with a certain antisymmetry, which maximizes the efficiency of energy transmission for a given incident wave of finite duration and energy. Several of the results can be extended to more general classes of impedance transitions. Simple illustrative examples are given.
1. Introduction
Reflection and transmission of waves by a nonuniform transition between two uniform media is a problem of great interest in areas such as acoustics, optics, electromagnetic theory, geophysis, and mechanics of percussive drilling. Generally the local reflection and transmission of onedimensional waves in lossless media can be expressed in terms of the local variation of a suitably defined characteristic impedance function, which completely represents a transition in this respect. In this paper we deal with reflection and transmission of waves by an impedance transition from a constant input characteristic impedance ZIN to a constant output characteristic impedance ZOUT- The fundamental equations are formulated for onedimensional extensional waves in a linearly-elastic bar with variable cross-sectional area, Young's modulus, and density. However, interpretation for other physical systems can readily be made in terms of the wave travel time ~: and the characteristic impedance function Z(~:). Thus, any system is represented by Z(~) with Z(~)= ZIN for ~:~ ( - ~ , 0] and Z(~:)= Zou-r for ~:~ [d, oo). The impedance transition is represented by Z(~) for ~:s [0, d], which means that Z(0)= ZtN and Z(d)= ZOUT are included. Primarily we consider the class Dm of impedance transitions Z such that Z(~:) is piece-wise constant in N subintervals of (0, d) of equal length d~ N. From a matrix relation between the Fourier transforms of the incident, reflected and transmitted waves several general properties of the impedance transition and its transfer matrix are established. 0165-2125/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
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It is shown that the relative momentum transmission depends only on the characteristic impedance ratio ZOuT/ZIN. Thus, unlike the relative energy transmission (the efficiency of energy transmission) it depends neither on the incident wave shape nor on the profile of the impedance transition. This observation was made by Mao and Rader [1] from numerical results. The concept of transmission-equivalent impedance transitions is introduced. By this we mean transitions which, for the same ZIN and the same Zou-r, have identical transmission properties. It is shown that certain simple impedance transformations generate impedance transitions Z TM which are transmission-equivalent to a given transition Z. Also it is established that for a given impedance transition Z e DN there are at most, and generally exactly, 2 s different transmission-equivalent transitions in DN, including the original one. Thus, whereas an impedance transition is completely determined by its reflection properties [2], it is not uniquely defined by its transmission properties. The proof of this result also indicates a procedure for determining all impedance transitions in DN which are transmission-equivalent to a given transition Z e DN. Energy transmission through an impedance transition is studied and two fundamental results are established. First, the efficiency of energy transmission ~ is minimized for monotoneous transitions from a given Z,N to a given ZouT (that is, Z(~) is either nondecreasing or nonincreasing for ~e [0, d]). The transition which minimizes T/ is one with an abrupt change in Z(~) from ZIN to ZouT, as may perhaps be expected intuitively. This result is independent of the incident wave. Secondly, it is shown that for a given incident wave, and for given values of Z~N and ZouT, there exists an optimal impedance transition Z ° ~ e DN which maximizes the efficiency of energy transmission r/. This optimal impedance transition is antisymmetric in the sense that Z ° ~ ( ~ ) z ° P ' r ( d - ~ ) - - Z , NZouT for ~ e [0, d]. Several of the results obtained can be generalized to more extensive classes of impedance transitions. Impedance transitions in DN with N = 1 and 2, and also an exponential transition, are employed to illustrate the fundamental results. Investigations of impedance transitions have been carried out for a long time. Thus, at the end of last century Rayleigh [3] considered the propagation of extensional waves in an elastic wire with an abrupt change in characteristic impedance. Early work concerning waves in elastic bars with gradually changing characteristic impedance was carried out by Donnell [4]. More recently there has been much interest in minimizing reflected energy and maximizing transmitted energy. For example, Fischer [5], Lundberg et al. [6], and Gupta [7] have been concerned with the problem of maximizing energy transmission through joints between drill rods. Also, Pedersen et al. [8] have studied the matching of a piezoelectric transducer to a test object by means of a backing layer such that the reflected energy is minimized.
2. Theory For the sake of definiteness we consider propagation of onedimensional extensional waves in a straight linearly-elastic bar with cross-sectional area A, Young's modulus E, and density p. These quantities, as well as the elastic wave speed c = ( E / p ) 1/2 and the characteristic impedance Z - - A E / c , are functions of the coordinate x along the bar. Introducing the wave travel time ~:= ~o d x ' / c ( x ' ) we can represent the system by the characteristic impedance function Z(~:) which assumes the constant value ZIN for ~:e (--~, 0] and the constant value ZouT for ¢ e [d, oo). In the transition interval [0, d] the characteristic impedance Z(~:) varies from Z~N to ZouT. For impedance transitions Z e DN the interval (0, d) is divided into N subintervals (0, h), (h, 2 h ) , . . . , and (d - h, d) of equal length h = d~ N in which the characteristic impedance Z(~:) assumes the constant values Z~, Z2, . . . , and ZN, respectively. These subintervals will also be referred to as segments.
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The wave propagation is governed by the linear equation 0 where f(~, t) is the tensile force and v(~:, t) is the particle velocity in the ~:-direction at the cross-section and the time t. Interpretations of c, Z, f, and v for other physical systems are given in reference [6]. Introducing the Fourier transform to)
t)] e-i<°t dt r.'<<, L~(f, )]:s2 r,<<.,') Lv(s~,
(2.2)
we can express equation (2.1) as °
We next formulate a theory in discrete form for impedance transitions Z ~ Dr+. By a limiting process we then extend the theory to differential form for impedance transitions with continuous impedance variation. Finally, in this section, we illustrate the discrete and differential forms of the theory with some examples. A. Discrete f o r m
When dealing with impedance transitions Z ~ Dsv we particularly consider the points ~:k=(k-1/2)h,
k=0,1,...,N+l.
(2.4)
They are the point ~:o= - h / 2 at the input end of the transition with characteristic impedance Zo = ZXN, the midpoints ~:1, ~2, • •., and ~:swof the subintervals of the transition with characteristic impedances Z1, Z2 . . . . . and ZN, respectively, and the point ~N+, = d + h i 2 at the output end of the transition with characteristic impedance Z N + 1 : Z o u T. For the interval ( ~ k - - h / 2 , ~k + h / 2 ) equation (2.3) yields
[,<<,<,,)1 [ ' l~(f, w)J :
--IIZk
, ] llZk
L fk(aj)
e i°>(~-
(2.5)
where f~-(w) and fk(tO) represent forces associated with harmonic waves travelling in the directions of increasing and decreasing ~:, respectively, at the midpoint ~:k of the interval. Due to the required continuity of force f(¢, to) and particle velocity 13(~:,to) at ~ = ~k + h / 2 = ~k÷, -- h / 2 equation (2.5) gives ~2k= C(Olfk)~k+l,
(2.6)
where we have introduced the vector ~-k = ( Z k ) -'/2
(2.7)
_
and the matrix pz cosh(ak) C(ak) = I_ sinh(ak)
sinh(ak) ] (l/z) cosh(ak)J'
(2.8)
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with ak = (1/2) ln(Zk+l/Zk).
z = e i'°h,
(2.9)
From equations (2.8) and (2.9) it follows that the matrix C has the properties det(C) = 1,
Ct=C,
(2.10)
(~-- l(o~k) = C ( - Otk) ,
where C denotes the complex conjugate of C. By repeated use of equation (2.6) we obtain DZ0=Hk~k+,,
Hk=C(ct0)'''C(ak)
(2.11)
and, for k = N, ~-0=HN[-N+I,
HN = C ( a o ) ' ' '
C(aN).
(2.12)
HN is a transfer matrix which relates the vector @:oat the input end of the transition to the vector ~-N+~ at the output end of the transition. These vectors can be expressed as
where Fr =fl/(Z~n) '/z, Fr =fR/(Z,N) '/2, and F r =•/(ZouT) '/z represent the incident, reflected, and transmitted waves, respectively, at the impedance transition. By induction it follows from equations (2.8) and (2.12) that the transfer matrix H N is of the general form
F AN(Z)
LBN(I/z)
BN(Z) ] AN(1/Z)J'
(2.14/
where, for z = e i'h, AN(1/z) = AN(Z) and BN(I/z) = BN(Z) are the complex conjugates of AN(Z) and BN(Z), respectively. From equations (2.10), (2.12) and (2.14) it follows that the matrix H N(Z) also has the properties det[HN(Z)] = AN(Z)AN(1/ Z)-- BN(Z)BN(1/ Z) = 1,
O-O-N'(z)= [ AN(I/z) --BN(Z)]. L-BN(I/z) AN(Z)J By induction it also follows from equations (2.8), (2.12) and (2.14) that in the low-frequency limit z the matrix elements AN and BN are given by AN(I) = cosh(a),
BN(I) = sinh(a),
(2.15) =
e i''h .--) l
(2.16)
where N
a = Y. ak = (1/2) ln(Zovv/Z,N) k=O
(2.17)
represents the overall change in characteristic impedance of the transition. Thus, in the low-frequency limit the transfer matrix HN is independent of the profile of the impedance transition, as should be expected. From equations (2.9) and (2.12) to (2.14) we obtain the relations
fir = RFt,
R(w)
•~r = T/~I,
T(to) = l/AN(ei°'h),
= BN(e-i~°h)/ a N ( e i°~h)
(2.18)
and (2.19)
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between the reflected, transmitted, and incident waves. Thus, the reflection coeffÉcient R(to) depends on both matrix elements AN and BN, whereas the transmission coefficient T(to) depends on the matrix element AN only. According to equations (2.15), (2.18), and (2.19) the reflection and transmission coefficients are related through [R(to)[ 2 +l T(o,)l 2 = 1.
(2.20)
The momentum carried by the incident, reflected, and transmitted waves, respectively, can be expressed
p, = -(Z,N)l/2 f ~ooF,( t) dt = -(Z,N)'/2F,(O), PR (ZIN) 1/2 S~ooFR(t) dt = (Z,N)'/2FR(0),
(2.21)
=
PT = --(ZouT) 1/2 fToo Fr(t) dt = --(ZouT)
1/2
A
FT(O)-
Using equations (2.16) to (2.19) and (2.21) we obtain
pR/p. = - ( Z o u . - Z,N)/(ZouT + Z,N), (2.22)
PT/ P, = 2ZouT/ ( ZouT +Z,N), and also the relation PR +Pr = P~ which expresses conservation of momentum. Thus, we have shown that the relative reflection and transmission of momentum depend only on the characteristic impedance ratio
Z o ~ l Z,~. Equations (2.22) also show that -1 <~PR/Pl ~ 1 and 0 ~< PT/Pl ~ 2. In the particular case ZouT = Zm equations (2.21) and (2.22) give PR = 0 and S_~ooFR(t) dt = 0. Thus, in this case the reflected wave carries no momentum and it contains tension (FR > 0) and compression (FR <0) to equal extents. The energies carried by the incident, reflected and transmitted waves are, by Parseval's identity,
W~= WR =
IF,(t)12dt=(l/27r)
IF~(~o)l doJ,
I2 [FR(t)I 2 dt = (l/2~r) f ° IFR(tO)l2 dto,
(2.23)
--oo
w~--
ff~IFW)I=d,-- (1/2~) I7ooIF'(<'>)I=do,.
Using equations (2.18) to (2.20) and (2.23) we obtain
W~l w , =
IR(o,)rl,~,(o,)l ~ do,
I,~,(~,)l ~ do,, --o0
(2.24)
W~l Wl =
I T(,<,)l~l~/(
/S;
Ig(<,>)l ~ d , o ,
cO
and also the relation WR + WT = W~ which expresses conservation of energy. Thus, the relative reflection and transmission of energy depend on the characteristic impedance function Z(~:) as well as on the incident wave F~(t). Therefore, for the efficiency of energy transmission Wr/W~, which we shall return to in Section
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4, we introduce the notation
nEz, F,] =
IT(~0)12L~,(~o)l 2 d o
(2.25)
I~,(~o)12 do.
From equation (2.20) we also obtain 0 ~ Ie(. )l 1 and 0 Ir(o)l <- 1, and therefore from equations (2.24) 0 <~ WR <~ WI and 0<~ Wr<~ W~. Before turning to the differential form of the theory we give some theorems concerning properties of the elements AN(Z) and BN(Z) of the transfer matrix HN(Z). Several of these theorems will be used in Sections 3 and 4. The following proposition concerns the general form of the matrix elements AN(Z) and BN(Z).
Proposition 2.1. AN(Z )
and BN(Z) are o f the form
N A N ( Z ) = ~ akzN+l 2k,
N B N ( Z ) = ~ bkzN-2k,
k=O
k~O
where ak and bk are real. For N = 0 ao = cosh(ao),
bo = sinh(ao).
For N = 1 bo = cosh(ao) sinh(al),
ao = cosh(ao) cosh(aO, al = sinh(ao) sinh(aO,
b, = sinh(ao) cosh(al).
For N > 1 N ao = 1-[ cosh(ak),
N-I
bo = l-I cosh(ak) sinh(aN), k=0
k=0 N--|
aN = sinh(ao) I] cosh(ak) sinh(aN),
N bN = s i n h ( a o ) H cosh(ak). k=l
k=l
Also, for N >1 1 bo/ ao = aN / bN = tanh(aN ),
aN / bo = bN / ao = tanh(ao).
The proof, which can be carried out by induction, is omitted. Note that this proposition implies that ao > 0. The following theorem concerns the zeroes of the function AN(Z). Physically it means that no transmitted wave can appear at the output end until a disturbance has propagated through the transition. Theorem 2.2. The function AN(Z ) has all its zeroes in the unit disc [z[ < 1. A p r o o f of this theorem can be carried out by induction, using Rouch6's theorem on the number of zeroes of analytic functions. The complete p r o o f is given in reference [9]. The following l e m m a states that the function AN(Z) is uniquely determined by the function BN(Z). Lemma 2.3. Let BN(Z) be any function of the form N BN(Z) "= Z bk ZN-2k, k=O
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with real coefficients bk. Then there exists exactly one function AN(Z) of the form N AN(Z)= Z ak zN-I-zk, k=0
such that A N ( Z ) A N ( I / z ) = 1 + BN(Z)BN(1/Z) for all z # O, AN(Z) has all its zeroes in the unit disc Iz] < 1, and ao > O. Moreover, all the coefficients ak are real. Proof. Let BN(Z) = ~ bk zN-2k, k-m where 0<~ m ~< n ~< N and b,, ~ 0, b, # 0. Then Q(z)= zZ("-")[l + BN(Z)BN(I/z)] is a polynomial such that Q ( 0 ) # 0 and Q ( z i ) = o c : > Q ( l / z i ) = o . Define the function AN(Z) by AN(Z)=aoZPIIIz, I
with real coefficients b k. Then there exist unique real numbers {OLk}N-0 such that
C(~o)C(~,).-.
[ AN(Z)
c(o~N)= H,,, = /
L BN(1/z)
BN(2) ] / AN(I/z)J'
where A s ( z ) is a function of the form given in Proposition 2.1. We indicate briefly the main ideas of the proof. A complete p r o o f can be found in reference [9]. First the function AN(Z) =~k=0 N akzN+l-2k is uniquely determined with the aid of Lemma 2.3. Using the relations bo/ao = aN/bN = t a n h ( a N ) and aN/bo = bN/ao=tanh(ao) given in Proposition 2.1 we can construct the matrix C-~(t~o) or C-I(o~N). Multiplying HN(Z) from the left or right with C-l(ao) or C - I ( a N ) , respectively, we obtain a similar matrix HN-I. Repeating the process gives the desired representation.
B. Differential form In order to deal with impedance transitions with continuous variation in characteristic impedance we consider them as limits of impedance transitions Z c DN. Thus, by letting N ~ oo, k ~ oo, and h ~ 0 such that Nh = d and kh = ~: are constant we obtain ~-~k(ei~°h)--> H(~, to), HN(eia'h) --> H(d, to), mN(ei~°h)--> A(d, to), BN(ei'°h)--> B(d, to), etc. In this way, equations (2.12), for example, are replaced by ~-(0, to) = H(d, to)U':(d, to)
(2.26)
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and H(o, to)= 0,
(2.27)
a(~) = ( I / 2 ) Z ' ( ( ) / Z ( ~ ) .
(2.28)
oH(C, to)/a~ = H(~:, to)R(¢, to), where D is the 2 × 2 unit matrix and
R(~, to)=
ito a(~)
a(~)], -ito 3
Here the function a(¢), which can also be expressed a ( ¢ ) = (1/2)(d/d~:)ln[Z(¢)], corresponds to the numbers ak = (1/2)ln(Zk+l/Zk) defined in equation (2.9). In particular the quantities a(¢) and {Ctk}kN=t are nonnegative (nonpositive) for impedance transitions with nondecreasing (nonincreasing) characteristic impedance Z(~:) for ~:E [0, d]. Such impedance transitions are called monotoneous and will be considered further in Section 4. Results closely related to equations (2.26) to (2.28) have been given also by, for example, Miller [10] and Pedersen et al. [8]. Sometimes it may be convenient to determine the transfer matrix H via the matrix
re -'~
~(~:, to) = H(~, to)/L
0
0 1
(2.29)
ei,O~J.
This matrix, in turn, is determined from the differential equation
all(C, to)la~ = H(~, to)R(~, to),
H(0, to) = n,
(2.30)
where R(s¢' to)= [a(s¢)0e 2i0,~ a(~)O2i'°~ ] .
(2.31)
If there are discontinuities in the characteristic impedance function Z(~:) special care has to be taken as a(~:) contains the derivative Z'(~). In such cases, however, it may be convenient to determine the matrix P(d, to) introduced in [6] as an intermediate step. This matrix is determined from the differential equation
aP(¢,~)/a~:=ito[1/z(~) 0 Z~)]p(¢,m), P(O,w)=O,
(2.32)
which does not contain Z'(~). The transfer matrix H(d, to) is related to P(d, to) through r zo'J~
-Z'J~,T 1
J,'(,,.
r
7,/2
7,/~ 7
(2.33)
C. E x a m p l e s
In order to illustrate the theory presented in this section we give the elements of the transfer matrix for impedance transitions with N = l and 2, and also for an exponential transition. (a) One-segment transition, N = l: A J z ) = ao z2 +
a I -----cosh(ao) cosh(al)Z 2 +sinh(ao) sinh((~0, (2.34)
B l ( z ) = boz + b l z -1 = cosh(ao) sinh(at)z + sinh(ao) cosh(a0z -l.
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(b) Two-segment transition, N = 2: A2(z) = aoz3 + a l z + a2z-1
=
cosh(ao) cosh(cr 1) cosh(a2)z 3
+ sinh(a i) sinh(ao + a2)z + sinh(ao) cosh(a i) sinh(a2) z-l, (2.35) B2( z ) = boz 2 + bl + b2z -2 = cosh(ao) cosh(al) sinh(a2)z 2
+sinh(al) cosh(ao + a2) +sinh(ao) cosh(ai) cosh(a2)z -2. (c) Exponential transition, Z(~) = ZIN(ZouT/Zm)~/d: A(d, to) = cos(~b) +i~od sin(4b)/~b,
(2.36) B(d, aJ)= a sin(
Here a is given by equation (2.17) and 4b = [(oM) 2 - a2] 1/2.
(2.37)
We shall develop these examples further in Sections 3 and 4.
3. Transmission-equivalence Although two impedance transitions with the same Z~N and the same Z o u z but with different Z(~:) for ~c(O, d) always have different reflection properties they may sometimes have identical transmission properties. This circumstance justifies the following definition.
Definition 3.1. Two impedance transitions are called transmission-equivalent if they have the same Z m and the same ZOUT and if both give the same transmitted wave FT for the same arbitrary incident wave F~. This definition and equations (2.19) imply that two impedance transitions from Zm to Zouz are transmission-equivalent if and only if they have the same AN(e i'h) or, more generally, the same A(d, to) for all aJ. Determining all impedance transitions which are transmission-equivalent to a given transition generally requires some numerical effort. However, we shall present three simple impedance transformations which give impedance transitions that are transmission-equivalent to the given one. Also, some examples will indicate how to determine all impedance transitions which are transmission-equivalent to a given impedance transition Z s DN. A. Impedance transformations
We first define two simple impedance transformations.
Definition 3.2. The transformations zINV(sr) = Z 1 N Z o u T / Z ( ~ ) and zREV(sr) = Z ( d - ~), ~ ~ (0, d), are called inversion and reversion, respectively, of the impedance transition Z. The following theorem states how these impedance transformations can be used to generate impedance transitions which are transmission-equivalent to a given transition.
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Theorem 3.1. The impedance transitions Z and (zINV) REV always transmission-equivalent. Also the impedance transitions Z, Z INv and Z REV transmission-equivalent provided that Z~N = ZOUT. are
are
Proof. We prove the theorem for an impedance transition Z E DN with the transfer matrix HN given by equations (2.12) and (2.14). Combined inversion and reversion means that ao, a~, . . . , aN are replaced by aN, a N - i , a O. Therefore the transfer matrix becomes • .
.
,
(HINNV)REV~---C(aN) • • • C(o/o) = c t ( a N ) • • • Ct(o/o) = [C(ao) • • • C(aN)]t = HtN, where use has been made of the property C t ( a ) = C ( a ) . Hence (A~V) REv= AN. Reversion means that ao, a ~ , . . . , aN are replaced by --aN, --aN ,, • • •, --aO. The transfer matrix becomes H ~ ,~ v = C ( - - a , , , ) .
AN
HN= fin
• • C(--ao) = C-'(a,,).
BN ::::~NI_~
AN
• • C-~(ao) = [C(ao)-
• • C(a,,,)]-'
= H~'.
AN
-BN
AN "
Here we have used equations (2.10) and (2.15). Consequently A r E v = AN. Inversion may be considered as combined inversion and reversion followed by reversion. Hence, •A ~ v = [(A~V)REV] REv = [AN] REv = A N and the p r o o f is complete. The validity of this theorem can be extended to impedance transitions with continuous variation in characteristic impedance through a limiting process. A property of impedance transitions of special importance in Section 4 is defined as follows. Definition 3.3. The impedance transition Z is called antisymmetric (in a multiplicative sense) if ~ ) = Z~NZouT for ~:E (0, d).
Z(~)Z(d-
From Definitions 3.2 and 3.3 it follows that an antisymmetric transition has the property (z~NV)REV(~)= Z(~). Thus, for such an impedance transition, combined inversion and reversion generates a transition which is identical with the original one. The following theorem provides properties of impedance transitions Z c DN which are equivalent to antisymmetry. Theorem 3.2. For an impedance transition Z E DN the following properties are equivalent: (a) Z is antisymmetric. (b) ak = a N - k f o r k = 0 . . . . . N. (c) bk = bN k for k = 0 , . . . , N. (d) B N ( e iooh) is real f o r all to. (e) BN(Z) = B N ( 1 / Z ) f o r z ~ O. (f) HN(Z) is symmetric.
Proof. It is more or less obvious that (a)c:>(b) and (c)<::>(d)<=>(e)c:>(f). Also, in the first part of the p r o o f of T h e o r e m 3.1 we have ( H ~ v ) gEv= H% from which it follows that ( a ) ~ ( f ) . Therefore it only remains to prove that, for instance, ( c ) ~ ( b ) . We do this by induction. It follows from Proposition 2.1 that the statement is true for N = 0 and N = 1. We now suppose that the statement is true when the number of segments of the transition is N - 2 and prove that then it is true also when the number of segments is N.
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Consider for N > 1 the transfer matrix HN given by equation (2.14). It is symmetric, H~v -----HN,and the matrix element BN has the form given in Proposition 2.1 with bk = bN-k for k = 0 . . . . , N. From bo = b~ and the relations b o / a o = t a n h ( a N ) and b N / a o = t a n h ( a o ) , in the same proposition, there follows ao = aN. Then the matrix C ( a 0 " • • C(aN_j) = C-'(ao)HNC-~(~N) = C - ' ( a o ) H N C - ' ( a o ) is symmetric as HN and C -~ are symmetric. However, this matrix is the transfer matrix of an impedance transition which has N - 2 segments and is represented by a , , . . . , aN_~. Then, by the induction hypothesis, ak = aN-g for k = 1 , . . . , N - 1. This completes the proof. B. A theorem on transmission-equivalence The following theorem concerns the number of different impedance transitions in DN which are transmission-equivalent to a given transition in DN. Theorem 3.3. I f Z e DN is a given impedance transition, then there are at most, and in general exactly, 2 ~ different impedance transitions in Dlv, including Z, which are transmission-equivalent to Z. A complete p r o o f of the theorem can be found in reference [9]. We indicate the ideas of the p r o o f with the aid of the following examples. (i) Consider, for N = 2, the functions B~(z) = - c z - 2 ( z 2 - 1/2)(z 2 - 3), B22(z) = ( c / 2 ) z - 2 ( z 2 - 2)(z 2 - 3), B 3 ( z ) - - ( 3 c / 2 ) z - 2 ( z 2 - 2)(z 2 - 1/3), B4(z) --- 3cz-Z(z 2 - 1/2)(z 2 - 1/3). It is easy to verify that the expressions l + B ~ ( z ) B ~ ( l / z ) are identical for i - 1 , 2, 3 and 4, and that B~(I) = c = sinh(a). The function A2(z) determined by A2(z)A2(I / z) = 1 + B~(z)B~(1 / z) and L e m m a 2.3 is therefore the same for i = 1, 2, 3 and 4. Hence, the impedance transitions Z ~ determined by Theorem 2.4 are 2 2 = 4 different transmission-equivalent transitions. It is almost obvious that it is not possible to obtain other transmission-equivalent transitions. (ii) Consider, for N = 2, the functions B~(z) = ( c / 5 ) z - 2 ( z 4 +4), B~(z) = ( 4 c / 5 ) z - 2 ( z 4 + 1/4). We obtain two different transmission-equivalent transitions Z 1 and Z 2 in this case. The p r o o f of the previous theorem gives the following corollary.
Corollary 3.4.
Suppose that Z m ~ ZouT. Then a necessary and su~cient condition that Z e DN has no other transmission-equivalent impedance transition in DN is that the function BN( Z) has all its zeroes on the unit circle Izl = 1. C. Examples Some concepts and results of this section can be illustrated by considering the same examples as in Section 2C. For this purpose, let Z be a given impedance transition and let Z TE be a transition which is
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
400
transmission-equivalent to Z. Also, if Z c DN and Z a'E c DN, let these transitions be represented by {%k}~:0 and r~t%k TE~N J'k=0, respectively. The latter numbers are determined from the former by requiring that both AN(z) and a should be the same. (a) One-segment transition, N = 1: With the aid of the first of equations (2.34) we require
cosh(% TE) cosh(%TE) = cosh(ao) cosh(a 0, sinh(%orE) sinh(t~T E) = sinh(ao) sinh(%l),
(3.1)
%vE+%TE=%0+%. This system has exactly two solutions, namely, % ~ = %0,
%~E = %,
%T~ : %1,
%TE - %.
(3.2) Thus, in agreement with Theorem 3.3 there are 21= 2 impedance transitions zXE~ D~ which are transmission-equivalent to Z ~ Dr. These transitions are Z TE = Z and Z TE = (z1NV) aEV, respectively. They are different unless Z is antisymmetric, that is, unless %o= % (equivalently, Z 2 = Z~NZouT). (b) Two-segment transition, N - - 2: With the aid of the first of equations (2.35) we require c o s h ( %TE) cosh(%rE) c o s h ( % E) = cosh(%o) cosh(%) cosh(%), sinh(a~E) sinh(%TE + %E) = sinh(%1) sinh(% + %),
(3.3) sinh(% -E) cosh(%TE) sinh(% -E) = sinh(cto) cosh(%) sinh(%2), % ~ + % E + % E = % + % 1 +%2.
Two solutions of this system are found by inspection. They are TE Or'0 ~ %0,
TE %0 ~- %2,
% T E ~--. %1,
%1rE ~ %1,
TE %2 ~- %2,
%
(3.4)
TE ~ %,
and correspond to Z TE= Z and Z TE= (z~NV) REv, respectively. These transitions are different unless Z is antisymmetric, that is, unless % = % (equivalently, ZIZ2 = ZmZouT). Other solutions are less directly obtained. In the particular case Zm = ZOUT, that is, % + % + % = 0,
(3.5)
however, two additional solutions can be readily obtained. They are ~o~ = -%0,
%~E = _ % ,
TE /2/1 ~--- - - % 1 ,
%E
%E
%TE = - - % "
= __%,
~ --%1,
(3.6)
and correspond to Z TE = Z mv and Z TE = Z REv, respectively. The solutions (3.4) and (3.6) are all different unless % = a2 (equivalently, ZIZ2 = ZINZouT) or % = - - % (equivalently, ZI = Z2). Thus, in this special case, we have demonstrated that in agreement with Theorem 3.3 there are 2 2= 4 impedance transitions
L-E. Andersson, B. Lundberg/Properties of impedance transitions
401
zTEe D2 which are transmission-equivalent to Z s D2, and generally these transitions are all different. Also, these transitions are the ones which are predicted by Theorem 3.1. (c) Exponential transition, Z(~) = ZIN(ZouT/Zm)~/d : According to Definition 3.3 this impedance transition is antisymmetric. We notice from the second of equations (2.36) that B is real. Compare with Theorem 3.2, which, however, has been proved only for impedance transitions Z ~ DN. Further we note that B(d, to) has all its zeroes for real oJ, that is, [z[ = 1. In accordance with Corollary 3.4 this might indicate that there does not exist any different transmissionequivalent transition.
4. Transmission of energy
We now consider the efficiency of energy transmission 7/[Z, FI] defined by equation (2.25). Throughout the section Z(se) represents an impedance transition with given input impedance ZtN, output impedance ZOUT, and length d, whereas Fz(t) represents an incident wave of finite duration and energy. First we restrict ourselves to monotoneous impedance transitions and show that ,/ is always minimized by one which makes an abrupt jump from ZIN to ZOUT. Then we show that, for a given incident wave, there exists an optimal impedance transition in DN which maximizes 7. This optimal transition is antisymmetric. A. Minimization o f energy transmission through monotoneous impedance transitions
We start with the following lemma concerning the coefficients ak for a monotoneous impedance transition, that is, one for which Z(~:) is either nondecreasing or nonincreasing. Lemma 4.1. I f Z ~ DN is a monotoneous impedance transition, then the coefficients ak are nonnegative, that is, ak >~O for k = O , . . . , N.
Proof. The coefficients a o , . . . , aN are determined from the numbers ao . . . . , aN with the aid of equations (2.8), (2.12), (2.14), and Proposition 2.1. If we first suppose that Z is nondecreasing, then O~k ~" 0, k -- 0 . . . . , N, and for the coefficients in the matrices C(t~k) we have cosh(t~k) > 0 and sinh(ak)/> 0, k = 0 . . . . . N. Then, obviously, a k >10, k - - 0 , . . . , N. If instead Z is nonincreasing we consider the transition Z ~Nv which is nondecreasing. As AN = A ~ v by the proof of Theorem 3.1, we obtain ak>~O, k = 0 , . . . , N, also in this case. This completes the proof. With the aid of this lemma we can now prove the following theorem which provides a lower bound for energy transmission through monotoneous impedance transitions. Theorem 4.2. Let Z A ~ DN be an impedance transition which jumps abruptly from ZIN to ZOUT, and let Z be a monotoneous transition in D~vfrom ZIN tO Zou T which does not have thisproperty. Then ,/[Z A, FI] < r/[Z, Fl] for any incident wave FI o f finite duration and energy.
Proof. From equations (2.8), (2.12), and (2.14) we obtain for the transition Z A AA(e i'°h) = cosh(cr) e i~'~a+h), where t~ is given by equation (2.17). Therefore, by equations (2.19) and (2.25) r/[Z A, FI] = 1/cosh2(a).
L.-E. Andersson, B. Lundberg / Propertiesof impedance transitions
402
For the transition Z we obtain from Proposition 2.1 N
AN(eioJh): ~ akei~oh(N+l k=O
2k)
and, by the triangle inequality and Lemma 4.1, N
IAN(ei h)l
N
Y lakl-k=0
ak, k~0
with strict inequality except for countably many values of w with no finite point of accumulation. But according to Proposition 2.1 and the first of equations (2.16) N
v~ ak = AN(l) = cosh(a). k=0
Therefore, by equations (2.19), (2.25), and the results above
n[Z, F,]
=
11
IAN(ei'°h)l-=l#,(O.')l~ do.'
/I
lP'i(o.,)l ~ do., I> I/cosh~(a) =
nCZ A, F,].
co
Moreover, if F(t) is zero outside a finite interval [a, b] and if the energy ~ ] F ( t ) l 2 at < ~ , then the Fourier transform Fi(to) = ~ Fi(t) e -i~°t dt is an entire function of to. Therefore Fl(to ) can have at most finitely many zeroes on every bounded domain in the complex to-plane. Hence, we have strict inequality, rl[Z, F~] > rl[Z A, FI]. This completes the proof. Corollary 4.3. ]AN(ei'°h)[ ~
~[Z A, F,]<~ ~[Z, F,]. The proof can be carried out by approximating Z with transitions in DN and letting N ~ ~ .
B. Maximization of energy transmission The following theorem states that any impedance transition in DN, which is not antisymmetric, can be replaced by one which is antisymmetric and has higher efficiency of energy transmission for all incident waves. Theorem 4.5. Let Z c DN be a given impedance transition which is not antisymmetric. Then there exists an antisymmetric transition zAS ~ DN, from the same ZIN to the same ZouT, such that r/[Z As, FI]> r/[Z, Fl]
for any incident wave F~ of finite duration and energy. Proof. Take BgS(z)= (1/2)[BN(z)+ BN(I/z)] so that BAS(ei'°h)= Re BN(ei'°h). According to Theorems 2.4 and 3.2 B As corresponds to an antisymmetric impedance transition Z AS . Also BgS(l) = BN(I) and therefore Z As and Z have the same ZOuT/ZIN according to equations (2.16) and (2.17). We have IBAS(ei'°h)l<~ [BN(ei'°h)l and, according to the first of equations (2.15), IAAS(ei'°h)l<~IAN(ei~h)l. Also, unless BAS(ei"'h)---BN(ei°'h), that is Z As = Z, we have strict inequality everywhere except possibly for countably many values
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
403
of to with no finite point of accumulation. From equations (2.19) and (2.25) we also have r / rzAS L , FI] =
IAAS(e'~")l-=lF,(to)l= do)
/I
IFI(~)I dto oo
and
IA~(e'~)I-~IP,(~)I=
~7[Z, F,] =
dw
=
dto
co
and consequently r/[Z As, F|] i> ~/[Z, F~]. Moreover, if F~(t) is zero outside a finite interval [a, b] and if the energy S~ IF(t)l z dt < ~, then by the same arguments as in the proof of Theorem 4.2 we have strict inequality, B[Z As, F | ] > ~7[Z, F~]. This completes the proof. The following theorem provides a necessary condition for an impedance transition which maximizes the energy transmission for a given incident wave. Theorem 4.6. Let FI be a given incident wave of finite duration and energy. Then there exists an optimal impedance transition z°r'r e DN with the property r/[Z °r'r, FI]~ > r/[Z, FI] for all Z c DN. Furthermore Z °r'r is antisymmetric. Proof. Consider the function g(bo, b~,..., bN) given by g(bo, b ~ , . . . , bN)= f ~ I/~(~o)12(1 +[BN(ei'°h)12)-~ do),
where N
BN(e i'°h) = ~ bk e i'°h(N-2k). k=0
g is a continuous function of (bo, b l , . . . , bN)e R/v+l. For the rest of the proof we need the following lemma. Lemma. The function g has a maximum when (bo, b l , . . . , bN)eR N÷j. The proof of this lemma is omitted because of its length, but is given in reference [9]. Let (bo, b j , . . . , bN) be a point in R ~÷~ where g takes its maximum and define B%r'r(z) by N
B°PT(z) = ~ bkZ N-2k. k=0
Let z ° v r e DN be the corresponding impedance transition given by Theorem 2.4. Then r/[Z °vr, F | ] ~> r/[Z, F~] for all Z e D~. The antisymmetry of Z °vr follows from Theorem 4.5. This finishes the proof. Next we state a conjecture and a theorem regarding the optimal impedance transition Z °yr. Conjecture 4.7. Z °r'r is uniquely defined. Theorem 4.8. B°r'r(z) has all its zeroes on the unit circle Iz[ = 1.
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
404
Proof. Suppose, on the contrary, that B ° ~ ( z ) has a zero zi with Iz, I ~ 1. Then, by Corollary 3.4, there exists an impedance transition z T E c DN, with corresponding function B~vE(z), which is transmissionequivalent to Z °Pr. From the examples following Theorem 3.3 it is also clear that Z TE can be chosen so that it is not antisymmetric. Then, however, by Theorem 4.5, there exists an antisymmetric transition Z As such that
r/[Z As, F,]> n[Z TE, FI] = r/[Z °p'r, FI]~> rl[Z, FI] for all Z ~ DN. This is a contradiction. Consequently B°er(z) has all its zeroes on the unit circle ]z I = 1. Antisymmetry of the optimal impedance transition is, by Theorem 3.2, equivalent to B°PT(z) = B°Pa~(1/z) for z # 0. Thus, antisymmetry means that B°r'r(z~) = 0¢~ B°Vr(1/z~) = 0 which is a weaker statement than that of Theorem 4.8. The next theorem is a more general version of Theorem 4.6 where the class DN has been replaced by the class D of impedance transitions Z, from ZIN to ZOUT, such that ln[Z(~:)] is of bounded variation (that is, can be expressed as the difference between two functions which are nondecreasing and bounded). We have no p r o o f for the existence of an optimal transition Z °r'r in this case. However, it seems reasonable to conjecture it. We omit the p r o o f of the theorem.
Theorem 4.9. Let F~ be a given incident wave of finite duration and energy and suppose that z ° P r ~ D has the property r/[Z °r'r, FI]/> ~7[Z, F~] for all Z c D. Then Z °rr is antisymmetric. C. Examples In order to illustrate some results of this section we consider again the impedance transitions with N = 1 and 2. For these transitions in DN equations (2.19) and (2.25) give the efficiency r/=
[AN(e
itoh
-2
^
2
)l If,(a,)l da,
/I
I~,(a,)l~dto.
(4.1)
oo
(a) One-segment transition, N = 1: The first of equations (2.34) gives [A,(ei'°h)12 = cos2(,oh) cosh2(o0 + sin2(toh) cosh2(a0 - at).
(4.2)
According to equations (4.1) and (4.2) the efficiency r/ is maximized for any given incident wave FI by ao = a~ which implies the result Z 2 = Z~NZouT. By Theorem 3.2 this means that Z °ev is antisymmetric in agreement with Theorem 4.6. Also, Z °P'r is uniquely defined in accordance with Conjecture 4.7. Further, the second of equations (2.34) gives
B°~r(z) = c o s h ( a / 2 ) sinh(a/2)(z + 1/z).
(4.3)
Thus, the zeroes of B°Pr(z), z~ = i and z2 = - i , are on the unit circle as predicted by Theorem 4.8. (b) Two-segment transition, N = 2: The first of equations (2.35) gives
[A2(eio~h)[ ~ = {cos~(o~h) cosh(a) - sin2(wh) cosh[2(ct2 + ao) - c~]}2 + sin2(2wh) cosh2(a2 + ao - a ) cosh2(a2 - a0).
(4.4)
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
405
As a = ao + a~ + a2 is given this expression can be considered as a function of a 2 - a o and a~ + ao. From equations (4.1) and (4.4) it is evident that for any incident wave F~ a necessary condition for optimality is a2 - ao = 0, which implies ao = a2 o r Z I Z 2 = Z i N Z o u T. This means that Z ° ~ i s antisymmetric in agreement with Theorem 4.6. In this case a2 + ao, and therefore Z °rr, depends on the incident wave F~.
5. Concluding remarks Most results of the present paper have been established for transitions in D~, that is, transitions with piece-wise constant characteristic impedance functions. The main reason for this is increased simplicity. However, such transitions also have technological importance. The Theorem 3.3 on transmission-equivalence may have practical use in design. Consider, for example, a transition from one cross-sectional area to another on a drill steel and assume that this transition has favourable transmission properties. Then there are generally a number of different transitions which share these properties. However, some of these transitions may be superior in other respects such as low weight, high strength, and simple geometry. Thus, there may be a certain freedom for the designer to make improvements without affecting the transmission properties. The existence of an optimal impedance transition which is antisymmetric, as expressed by Theorem 4.6, has importance in optimization. First, it justifies optimization, and secondly it may reduce the computational efforts considerably, especially for large N. For transitions with one segment, N = 1, we have demonstrated that the efficiency of energy transmission is maximized for any given incident wave when the characteristic impedances are in geometric progression, that is, Z 1 / Z I N = Z O u T / Z 1. This result is in agreement with the required antisymmetry of the optimum transition, and, in fact, follows from this requirement alone. For an N-segment transition the requirement of antisymmetry for an optimum transition, ZjZN = Z2Z~t_I . . . . . ZtNZouT, does not imply that the characteristic impedances are in geometric progression. In fact, the optimum impedance transition depends on the incident wave and, under certain circumstances, may not even be monotoneous. This is illustrated by the following simple example: Consider an impedance transition with ZOuT/Z~N = 10 and N = 2, and a rectangular incident wave with duration 5h. Then the efficiency of energy transmission ~ is maximized by Z~/Z~N-~4.2 and Z 2 / Z m ~ 2 . 4 which gives r/max= 0.522. ThUS we have Z m < Z l > Z2< ZouT. Under certain circumstances an impedance transition with ZIN, Z~ . . . . . ZN, ZouT in geometric progression may be nearly optimal. This condition for optimality is obtained if contributions to the transmitted wave Fr from waves which have been reflected are neglected. For this case Fischer [5] derived the result r/max = {4/[(1 + r l / " ) ( l +r-I/n)]} n with n = N + I and r=ZouT/ZIN. We have considered maximization of r/ without functional constraints for Z ~ , . . . , ZN. For equal input and output impedances this leads to the trivial result Z~N = Z~ . . . . . ZN = ZOUT and '/']max 1. For this case, however, Gupta [7] considered the functional constraint ~k=l N Zk = M N / d , which means that the transition has a given mass M. Because of this constraint the optimum transitions obtained are not antisymmetric. :
References [1] M. Mao and D. Rader, "Longitudinal stress pulse propagation in nonuniform elastic and viscoelastic bars", Int. J. Solids Structures 6, 519-538 (1970).
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L.-E. Andersson, B. Lundberg / Properties of impedance transitions
[2] M.M. Sondhi and B. Gopinath, "Determination of vocal-tract shape from impulse response at the lips", J. Acoust. Soc. Am. 49, 1867-1873 (1971). [3] Lord Rayleigh, The Theory of Sound, 2nd edition, Vol. 1,250-251, Macmillan (1894). Reprinted by Dover, New York (1945). [4] L.H. Donnell, "Longitudinal wave transmission and impact", Trans. Am. Soc. Mech. Engrs 52, 153 (1930). [5] H.C. Fischer, "On longitudinal impact II. Elastic impact of bars with cylindrical sections of different diameters and of bars with rounded ends", Appl. sci. Res. A.8, 278-308 (1959). [6] B. Lundberg, R. Gupta and L.-E. Andersson, "Optimum transmission of elastic waves through joints", Wave Motion 1, 193-200 (1979). [7] R. Gupta, "Optimum design of wave transmitting joints", Wave Motion 4, 75-83 (1982). [8] P.C. Petersen, O. Tretiak and Ping He, "Impedance-matching properties of an inhomogeneous matching layer with continuously changing acoustic impedance", J. Acoust. Soc. Am. 72, 327-336 (1982). [9] L.-E. Andersson and B. Lundberg, Some Fundamental Properties oflmpedance Transitions, Technical Report 1984:11 T, University of Lule~i (1984). [10] N.B. Miller, "Reflections from gradual transition sound absorbers", J. Acoust. Soc. Am. 30, 967-973 (1958).
389
S O M E F U N D A M E N T A L T R A N S M I S S I O N P R O P E R T I E S OF IMPEDANCE TRANSITIONS L.-E. ANDERSSON Department of Mathematics, University of Linkiiping, S-581 83 Linkrping, Sweden B. LUNDBERG Department of Mechanical Engineering, University of Lulea, S-951 87 Luled, Sweden Received 11 July 1983
Reflection and transmission of waves by impedance transitions from a constant input to a constant output characteristic impedance are considered. Several fundamental properties are explored, primarily for impedance transitions with piece-wise constant characteristic impedance in an arbitrary number N of intervals of equal length. For example, the following properties are shown: (i) The relative momentum transmission depends only on the ratio of output to input characteristic impedance. (ii) For a given impedance transition there are at most, and generally exactly, 2 N different transitions, including the original one, with identical transmission properties. (iii) For monotoneous impedance transitions the efficiency of energy transmission is minimized by one with an abrupt change in characteristic impedance. (iv) There exists an optimal impedance transition, with a certain antisymmetry, which maximizes the efficiency of energy transmission for a given incident wave of finite duration and energy. Several of the results can be extended to more general classes of impedance transitions. Simple illustrative examples are given.
1. Introduction
Reflection and transmission of waves by a nonuniform transition between two uniform media is a problem of great interest in areas such as acoustics, optics, electromagnetic theory, geophysis, and mechanics of percussive drilling. Generally the local reflection and transmission of onedimensional waves in lossless media can be expressed in terms of the local variation of a suitably defined characteristic impedance function, which completely represents a transition in this respect. In this paper we deal with reflection and transmission of waves by an impedance transition from a constant input characteristic impedance ZIN to a constant output characteristic impedance ZOUT- The fundamental equations are formulated for onedimensional extensional waves in a linearly-elastic bar with variable cross-sectional area, Young's modulus, and density. However, interpretation for other physical systems can readily be made in terms of the wave travel time ~: and the characteristic impedance function Z(~:). Thus, any system is represented by Z(~) with Z(~)= ZIN for ~:~ ( - ~ , 0] and Z(~:)= Zou-r for ~:~ [d, oo). The impedance transition is represented by Z(~) for ~:s [0, d], which means that Z(0)= ZtN and Z(d)= ZOUT are included. Primarily we consider the class Dm of impedance transitions Z such that Z(~:) is piece-wise constant in N subintervals of (0, d) of equal length d~ N. From a matrix relation between the Fourier transforms of the incident, reflected and transmitted waves several general properties of the impedance transition and its transfer matrix are established. 0165-2125/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
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L.-E. Andersson, B. Lundberg / Properties of impedance transitions
It is shown that the relative momentum transmission depends only on the characteristic impedance ratio ZOuT/ZIN. Thus, unlike the relative energy transmission (the efficiency of energy transmission) it depends neither on the incident wave shape nor on the profile of the impedance transition. This observation was made by Mao and Rader [1] from numerical results. The concept of transmission-equivalent impedance transitions is introduced. By this we mean transitions which, for the same ZIN and the same Zou-r, have identical transmission properties. It is shown that certain simple impedance transformations generate impedance transitions Z TM which are transmission-equivalent to a given transition Z. Also it is established that for a given impedance transition Z e DN there are at most, and generally exactly, 2 s different transmission-equivalent transitions in DN, including the original one. Thus, whereas an impedance transition is completely determined by its reflection properties [2], it is not uniquely defined by its transmission properties. The proof of this result also indicates a procedure for determining all impedance transitions in DN which are transmission-equivalent to a given transition Z e DN. Energy transmission through an impedance transition is studied and two fundamental results are established. First, the efficiency of energy transmission ~ is minimized for monotoneous transitions from a given Z,N to a given ZouT (that is, Z(~) is either nondecreasing or nonincreasing for ~e [0, d]). The transition which minimizes T/ is one with an abrupt change in Z(~) from ZIN to ZouT, as may perhaps be expected intuitively. This result is independent of the incident wave. Secondly, it is shown that for a given incident wave, and for given values of Z~N and ZouT, there exists an optimal impedance transition Z ° ~ e DN which maximizes the efficiency of energy transmission r/. This optimal impedance transition is antisymmetric in the sense that Z ° ~ ( ~ ) z ° P ' r ( d - ~ ) - - Z , NZouT for ~ e [0, d]. Several of the results obtained can be generalized to more extensive classes of impedance transitions. Impedance transitions in DN with N = 1 and 2, and also an exponential transition, are employed to illustrate the fundamental results. Investigations of impedance transitions have been carried out for a long time. Thus, at the end of last century Rayleigh [3] considered the propagation of extensional waves in an elastic wire with an abrupt change in characteristic impedance. Early work concerning waves in elastic bars with gradually changing characteristic impedance was carried out by Donnell [4]. More recently there has been much interest in minimizing reflected energy and maximizing transmitted energy. For example, Fischer [5], Lundberg et al. [6], and Gupta [7] have been concerned with the problem of maximizing energy transmission through joints between drill rods. Also, Pedersen et al. [8] have studied the matching of a piezoelectric transducer to a test object by means of a backing layer such that the reflected energy is minimized.
2. Theory For the sake of definiteness we consider propagation of onedimensional extensional waves in a straight linearly-elastic bar with cross-sectional area A, Young's modulus E, and density p. These quantities, as well as the elastic wave speed c = ( E / p ) 1/2 and the characteristic impedance Z - - A E / c , are functions of the coordinate x along the bar. Introducing the wave travel time ~:= ~o d x ' / c ( x ' ) we can represent the system by the characteristic impedance function Z(~:) which assumes the constant value ZIN for ~:e (--~, 0] and the constant value ZouT for ¢ e [d, oo). In the transition interval [0, d] the characteristic impedance Z(~:) varies from Z~N to ZouT. For impedance transitions Z e DN the interval (0, d) is divided into N subintervals (0, h), (h, 2 h ) , . . . , and (d - h, d) of equal length h = d~ N in which the characteristic impedance Z(~:) assumes the constant values Z~, Z2, . . . , and ZN, respectively. These subintervals will also be referred to as segments.
L.-E. Andersson, B. Lundberg/Properties of impedance transitions
391
The wave propagation is governed by the linear equation 0 where f(~, t) is the tensile force and v(~:, t) is the particle velocity in the ~:-direction at the cross-section and the time t. Interpretations of c, Z, f, and v for other physical systems are given in reference [6]. Introducing the Fourier transform to)
t)] e-i<°t dt r.'<<, L~(f, )]:s2 r,<<.,') Lv(s~,
(2.2)
we can express equation (2.1) as °
We next formulate a theory in discrete form for impedance transitions Z ~ Dr+. By a limiting process we then extend the theory to differential form for impedance transitions with continuous impedance variation. Finally, in this section, we illustrate the discrete and differential forms of the theory with some examples. A. Discrete f o r m
When dealing with impedance transitions Z ~ Dsv we particularly consider the points ~:k=(k-1/2)h,
k=0,1,...,N+l.
(2.4)
They are the point ~:o= - h / 2 at the input end of the transition with characteristic impedance Zo = ZXN, the midpoints ~:1, ~2, • •., and ~:swof the subintervals of the transition with characteristic impedances Z1, Z2 . . . . . and ZN, respectively, and the point ~N+, = d + h i 2 at the output end of the transition with characteristic impedance Z N + 1 : Z o u T. For the interval ( ~ k - - h / 2 , ~k + h / 2 ) equation (2.3) yields
[,<<,<,,)1 [ ' l~(f, w)J :
--IIZk
, ] llZk
L fk(aj)
e i°>(~-
(2.5)
where f~-(w) and fk(tO) represent forces associated with harmonic waves travelling in the directions of increasing and decreasing ~:, respectively, at the midpoint ~:k of the interval. Due to the required continuity of force f(¢, to) and particle velocity 13(~:,to) at ~ = ~k + h / 2 = ~k÷, -- h / 2 equation (2.5) gives ~2k= C(Olfk)~k+l,
(2.6)
where we have introduced the vector ~-k = ( Z k ) -'/2
(2.7)
_
and the matrix pz cosh(ak) C(ak) = I_ sinh(ak)
sinh(ak) ] (l/z) cosh(ak)J'
(2.8)
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
392
with ak = (1/2) ln(Zk+l/Zk).
z = e i'°h,
(2.9)
From equations (2.8) and (2.9) it follows that the matrix C has the properties det(C) = 1,
Ct=C,
(2.10)
(~-- l(o~k) = C ( - Otk) ,
where C denotes the complex conjugate of C. By repeated use of equation (2.6) we obtain DZ0=Hk~k+,,
Hk=C(ct0)'''C(ak)
(2.11)
and, for k = N, ~-0=HN[-N+I,
HN = C ( a o ) ' ' '
C(aN).
(2.12)
HN is a transfer matrix which relates the vector @:oat the input end of the transition to the vector ~-N+~ at the output end of the transition. These vectors can be expressed as
where Fr =fl/(Z~n) '/z, Fr =fR/(Z,N) '/2, and F r =•/(ZouT) '/z represent the incident, reflected, and transmitted waves, respectively, at the impedance transition. By induction it follows from equations (2.8) and (2.12) that the transfer matrix H N is of the general form
F AN(Z)
LBN(I/z)
BN(Z) ] AN(1/Z)J'
(2.14/
where, for z = e i'h, AN(1/z) = AN(Z) and BN(I/z) = BN(Z) are the complex conjugates of AN(Z) and BN(Z), respectively. From equations (2.10), (2.12) and (2.14) it follows that the matrix H N(Z) also has the properties det[HN(Z)] = AN(Z)AN(1/ Z)-- BN(Z)BN(1/ Z) = 1,
O-O-N'(z)= [ AN(I/z) --BN(Z)]. L-BN(I/z) AN(Z)J By induction it also follows from equations (2.8), (2.12) and (2.14) that in the low-frequency limit z the matrix elements AN and BN are given by AN(I) = cosh(a),
BN(I) = sinh(a),
(2.15) =
e i''h .--) l
(2.16)
where N
a = Y. ak = (1/2) ln(Zovv/Z,N) k=O
(2.17)
represents the overall change in characteristic impedance of the transition. Thus, in the low-frequency limit the transfer matrix HN is independent of the profile of the impedance transition, as should be expected. From equations (2.9) and (2.12) to (2.14) we obtain the relations
fir = RFt,
R(w)
•~r = T/~I,
T(to) = l/AN(ei°'h),
= BN(e-i~°h)/ a N ( e i°~h)
(2.18)
and (2.19)
L-E. Andersson, B. Lundberg/ Propertiesof impedance transitions
393
between the reflected, transmitted, and incident waves. Thus, the reflection coeffÉcient R(to) depends on both matrix elements AN and BN, whereas the transmission coefficient T(to) depends on the matrix element AN only. According to equations (2.15), (2.18), and (2.19) the reflection and transmission coefficients are related through [R(to)[ 2 +l T(o,)l 2 = 1.
(2.20)
The momentum carried by the incident, reflected, and transmitted waves, respectively, can be expressed
p, = -(Z,N)l/2 f ~ooF,( t) dt = -(Z,N)'/2F,(O), PR (ZIN) 1/2 S~ooFR(t) dt = (Z,N)'/2FR(0),
(2.21)
=
PT = --(ZouT) 1/2 fToo Fr(t) dt = --(ZouT)
1/2
A
FT(O)-
Using equations (2.16) to (2.19) and (2.21) we obtain
pR/p. = - ( Z o u . - Z,N)/(ZouT + Z,N), (2.22)
PT/ P, = 2ZouT/ ( ZouT +Z,N), and also the relation PR +Pr = P~ which expresses conservation of momentum. Thus, we have shown that the relative reflection and transmission of momentum depend only on the characteristic impedance ratio
Z o ~ l Z,~. Equations (2.22) also show that -1 <~PR/Pl ~ 1 and 0 ~< PT/Pl ~ 2. In the particular case ZouT = Zm equations (2.21) and (2.22) give PR = 0 and S_~ooFR(t) dt = 0. Thus, in this case the reflected wave carries no momentum and it contains tension (FR > 0) and compression (FR <0) to equal extents. The energies carried by the incident, reflected and transmitted waves are, by Parseval's identity,
W~= WR =
IF,(t)12dt=(l/27r)
IF~(~o)l doJ,
I2 [FR(t)I 2 dt = (l/2~r) f ° IFR(tO)l2 dto,
(2.23)
--oo
w~--
ff~IFW)I=d,-- (1/2~) I7ooIF'(<'>)I=do,.
Using equations (2.18) to (2.20) and (2.23) we obtain
W~l w , =
IR(o,)rl,~,(o,)l ~ do,
I,~,(~,)l ~ do,, --o0
(2.24)
W~l Wl =
I T(,<,)l~l~/(
/S;
Ig(<,>)l ~ d , o ,
cO
and also the relation WR + WT = W~ which expresses conservation of energy. Thus, the relative reflection and transmission of energy depend on the characteristic impedance function Z(~:) as well as on the incident wave F~(t). Therefore, for the efficiency of energy transmission Wr/W~, which we shall return to in Section
L.-E. Andersson, B. Lundberg/Properties of impedance transitions
394
4, we introduce the notation
nEz, F,] =
IT(~0)12L~,(~o)l 2 d o
(2.25)
I~,(~o)12 do.
From equation (2.20) we also obtain 0 ~ Ie(. )l 1 and 0 Ir(o)l <- 1, and therefore from equations (2.24) 0 <~ WR <~ WI and 0<~ Wr<~ W~. Before turning to the differential form of the theory we give some theorems concerning properties of the elements AN(Z) and BN(Z) of the transfer matrix HN(Z). Several of these theorems will be used in Sections 3 and 4. The following proposition concerns the general form of the matrix elements AN(Z) and BN(Z).
Proposition 2.1. AN(Z )
and BN(Z) are o f the form
N A N ( Z ) = ~ akzN+l 2k,
N B N ( Z ) = ~ bkzN-2k,
k=O
k~O
where ak and bk are real. For N = 0 ao = cosh(ao),
bo = sinh(ao).
For N = 1 bo = cosh(ao) sinh(al),
ao = cosh(ao) cosh(aO, al = sinh(ao) sinh(aO,
b, = sinh(ao) cosh(al).
For N > 1 N ao = 1-[ cosh(ak),
N-I
bo = l-I cosh(ak) sinh(aN), k=0
k=0 N--|
aN = sinh(ao) I] cosh(ak) sinh(aN),
N bN = s i n h ( a o ) H cosh(ak). k=l
k=l
Also, for N >1 1 bo/ ao = aN / bN = tanh(aN ),
aN / bo = bN / ao = tanh(ao).
The proof, which can be carried out by induction, is omitted. Note that this proposition implies that ao > 0. The following theorem concerns the zeroes of the function AN(Z). Physically it means that no transmitted wave can appear at the output end until a disturbance has propagated through the transition. Theorem 2.2. The function AN(Z ) has all its zeroes in the unit disc [z[ < 1. A p r o o f of this theorem can be carried out by induction, using Rouch6's theorem on the number of zeroes of analytic functions. The complete p r o o f is given in reference [9]. The following l e m m a states that the function AN(Z) is uniquely determined by the function BN(Z). Lemma 2.3. Let BN(Z) be any function of the form N BN(Z) "= Z bk ZN-2k, k=O
L.-E. Andersson, B. Lundberg/Properties of impedance transitions
395
with real coefficients bk. Then there exists exactly one function AN(Z) of the form N AN(Z)= Z ak zN-I-zk, k=0
such that A N ( Z ) A N ( I / z ) = 1 + BN(Z)BN(1/Z) for all z # O, AN(Z) has all its zeroes in the unit disc Iz] < 1, and ao > O. Moreover, all the coefficients ak are real. Proof. Let BN(Z) = ~ bk zN-2k, k-m where 0<~ m ~< n ~< N and b,, ~ 0, b, # 0. Then Q(z)= zZ("-")[l + BN(Z)BN(I/z)] is a polynomial such that Q ( 0 ) # 0 and Q ( z i ) = o c : > Q ( l / z i ) = o . Define the function AN(Z) by AN(Z)=aoZPIIIz, I
with real coefficients b k. Then there exist unique real numbers {OLk}N-0 such that
C(~o)C(~,).-.
[ AN(Z)
c(o~N)= H,,, = /
L BN(1/z)
BN(2) ] / AN(I/z)J'
where A s ( z ) is a function of the form given in Proposition 2.1. We indicate briefly the main ideas of the proof. A complete p r o o f can be found in reference [9]. First the function AN(Z) =~k=0 N akzN+l-2k is uniquely determined with the aid of Lemma 2.3. Using the relations bo/ao = aN/bN = t a n h ( a N ) and aN/bo = bN/ao=tanh(ao) given in Proposition 2.1 we can construct the matrix C-~(t~o) or C-I(o~N). Multiplying HN(Z) from the left or right with C-l(ao) or C - I ( a N ) , respectively, we obtain a similar matrix HN-I. Repeating the process gives the desired representation.
B. Differential form In order to deal with impedance transitions with continuous variation in characteristic impedance we consider them as limits of impedance transitions Z c DN. Thus, by letting N ~ oo, k ~ oo, and h ~ 0 such that Nh = d and kh = ~: are constant we obtain ~-~k(ei~°h)--> H(~, to), HN(eia'h) --> H(d, to), mN(ei~°h)--> A(d, to), BN(ei'°h)--> B(d, to), etc. In this way, equations (2.12), for example, are replaced by ~-(0, to) = H(d, to)U':(d, to)
(2.26)
396
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
and H(o, to)= 0,
(2.27)
a(~) = ( I / 2 ) Z ' ( ( ) / Z ( ~ ) .
(2.28)
oH(C, to)/a~ = H(~:, to)R(¢, to), where D is the 2 × 2 unit matrix and
R(~, to)=
ito a(~)
a(~)], -ito 3
Here the function a(¢), which can also be expressed a ( ¢ ) = (1/2)(d/d~:)ln[Z(¢)], corresponds to the numbers ak = (1/2)ln(Zk+l/Zk) defined in equation (2.9). In particular the quantities a(¢) and {Ctk}kN=t are nonnegative (nonpositive) for impedance transitions with nondecreasing (nonincreasing) characteristic impedance Z(~:) for ~:E [0, d]. Such impedance transitions are called monotoneous and will be considered further in Section 4. Results closely related to equations (2.26) to (2.28) have been given also by, for example, Miller [10] and Pedersen et al. [8]. Sometimes it may be convenient to determine the transfer matrix H via the matrix
re -'~
~(~:, to) = H(~, to)/L
0
0 1
(2.29)
ei,O~J.
This matrix, in turn, is determined from the differential equation
all(C, to)la~ = H(~, to)R(~, to),
H(0, to) = n,
(2.30)
where R(s¢' to)= [a(s¢)0e 2i0,~ a(~)O2i'°~ ] .
(2.31)
If there are discontinuities in the characteristic impedance function Z(~:) special care has to be taken as a(~:) contains the derivative Z'(~). In such cases, however, it may be convenient to determine the matrix P(d, to) introduced in [6] as an intermediate step. This matrix is determined from the differential equation
aP(¢,~)/a~:=ito[1/z(~) 0 Z~)]p(¢,m), P(O,w)=O,
(2.32)
which does not contain Z'(~). The transfer matrix H(d, to) is related to P(d, to) through r zo'J~
-Z'J~,T 1
J,'(,,.
r
7,/2
7,/~ 7
(2.33)
C. E x a m p l e s
In order to illustrate the theory presented in this section we give the elements of the transfer matrix for impedance transitions with N = l and 2, and also for an exponential transition. (a) One-segment transition, N = l: A J z ) = ao z2 +
a I -----cosh(ao) cosh(al)Z 2 +sinh(ao) sinh((~0, (2.34)
B l ( z ) = boz + b l z -1 = cosh(ao) sinh(at)z + sinh(ao) cosh(a0z -l.
L-E. Andersson, B. Lundberg / Properties of impedance transitions
397
(b) Two-segment transition, N = 2: A2(z) = aoz3 + a l z + a2z-1
=
cosh(ao) cosh(cr 1) cosh(a2)z 3
+ sinh(a i) sinh(ao + a2)z + sinh(ao) cosh(a i) sinh(a2) z-l, (2.35) B2( z ) = boz 2 + bl + b2z -2 = cosh(ao) cosh(al) sinh(a2)z 2
+sinh(al) cosh(ao + a2) +sinh(ao) cosh(ai) cosh(a2)z -2. (c) Exponential transition, Z(~) = ZIN(ZouT/Zm)~/d: A(d, to) = cos(~b) +i~od sin(4b)/~b,
(2.36) B(d, aJ)= a sin(
Here a is given by equation (2.17) and 4b = [(oM) 2 - a2] 1/2.
(2.37)
We shall develop these examples further in Sections 3 and 4.
3. Transmission-equivalence Although two impedance transitions with the same Z~N and the same Z o u z but with different Z(~:) for ~c(O, d) always have different reflection properties they may sometimes have identical transmission properties. This circumstance justifies the following definition.
Definition 3.1. Two impedance transitions are called transmission-equivalent if they have the same Z m and the same ZOUT and if both give the same transmitted wave FT for the same arbitrary incident wave F~. This definition and equations (2.19) imply that two impedance transitions from Zm to Zouz are transmission-equivalent if and only if they have the same AN(e i'h) or, more generally, the same A(d, to) for all aJ. Determining all impedance transitions which are transmission-equivalent to a given transition generally requires some numerical effort. However, we shall present three simple impedance transformations which give impedance transitions that are transmission-equivalent to the given one. Also, some examples will indicate how to determine all impedance transitions which are transmission-equivalent to a given impedance transition Z s DN. A. Impedance transformations
We first define two simple impedance transformations.
Definition 3.2. The transformations zINV(sr) = Z 1 N Z o u T / Z ( ~ ) and zREV(sr) = Z ( d - ~), ~ ~ (0, d), are called inversion and reversion, respectively, of the impedance transition Z. The following theorem states how these impedance transformations can be used to generate impedance transitions which are transmission-equivalent to a given transition.
L.-E. Andersson, B. Lundberg / Propertiesof impedance transitions
398
Theorem 3.1. The impedance transitions Z and (zINV) REV always transmission-equivalent. Also the impedance transitions Z, Z INv and Z REV transmission-equivalent provided that Z~N = ZOUT. are
are
Proof. We prove the theorem for an impedance transition Z E DN with the transfer matrix HN given by equations (2.12) and (2.14). Combined inversion and reversion means that ao, a~, . . . , aN are replaced by aN, a N - i , a O. Therefore the transfer matrix becomes • .
.
,
(HINNV)REV~---C(aN) • • • C(o/o) = c t ( a N ) • • • Ct(o/o) = [C(ao) • • • C(aN)]t = HtN, where use has been made of the property C t ( a ) = C ( a ) . Hence (A~V) REv= AN. Reversion means that ao, a ~ , . . . , aN are replaced by --aN, --aN ,, • • •, --aO. The transfer matrix becomes H ~ ,~ v = C ( - - a , , , ) .
AN
HN= fin
• • C(--ao) = C-'(a,,).
BN ::::~NI_~
AN
• • C-~(ao) = [C(ao)-
• • C(a,,,)]-'
= H~'.
AN
-BN
AN "
Here we have used equations (2.10) and (2.15). Consequently A r E v = AN. Inversion may be considered as combined inversion and reversion followed by reversion. Hence, •A ~ v = [(A~V)REV] REv = [AN] REv = A N and the p r o o f is complete. The validity of this theorem can be extended to impedance transitions with continuous variation in characteristic impedance through a limiting process. A property of impedance transitions of special importance in Section 4 is defined as follows. Definition 3.3. The impedance transition Z is called antisymmetric (in a multiplicative sense) if ~ ) = Z~NZouT for ~:E (0, d).
Z(~)Z(d-
From Definitions 3.2 and 3.3 it follows that an antisymmetric transition has the property (z~NV)REV(~)= Z(~). Thus, for such an impedance transition, combined inversion and reversion generates a transition which is identical with the original one. The following theorem provides properties of impedance transitions Z c DN which are equivalent to antisymmetry. Theorem 3.2. For an impedance transition Z E DN the following properties are equivalent: (a) Z is antisymmetric. (b) ak = a N - k f o r k = 0 . . . . . N. (c) bk = bN k for k = 0 , . . . , N. (d) B N ( e iooh) is real f o r all to. (e) BN(Z) = B N ( 1 / Z ) f o r z ~ O. (f) HN(Z) is symmetric.
Proof. It is more or less obvious that (a)c:>(b) and (c)<::>(d)<=>(e)c:>(f). Also, in the first part of the p r o o f of T h e o r e m 3.1 we have ( H ~ v ) gEv= H% from which it follows that ( a ) ~ ( f ) . Therefore it only remains to prove that, for instance, ( c ) ~ ( b ) . We do this by induction. It follows from Proposition 2.1 that the statement is true for N = 0 and N = 1. We now suppose that the statement is true when the number of segments of the transition is N - 2 and prove that then it is true also when the number of segments is N.
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
399
Consider for N > 1 the transfer matrix HN given by equation (2.14). It is symmetric, H~v -----HN,and the matrix element BN has the form given in Proposition 2.1 with bk = bN-k for k = 0 . . . . , N. From bo = b~ and the relations b o / a o = t a n h ( a N ) and b N / a o = t a n h ( a o ) , in the same proposition, there follows ao = aN. Then the matrix C ( a 0 " • • C(aN_j) = C-'(ao)HNC-~(~N) = C - ' ( a o ) H N C - ' ( a o ) is symmetric as HN and C -~ are symmetric. However, this matrix is the transfer matrix of an impedance transition which has N - 2 segments and is represented by a , , . . . , aN_~. Then, by the induction hypothesis, ak = aN-g for k = 1 , . . . , N - 1. This completes the proof. B. A theorem on transmission-equivalence The following theorem concerns the number of different impedance transitions in DN which are transmission-equivalent to a given transition in DN. Theorem 3.3. I f Z e DN is a given impedance transition, then there are at most, and in general exactly, 2 ~ different impedance transitions in Dlv, including Z, which are transmission-equivalent to Z. A complete p r o o f of the theorem can be found in reference [9]. We indicate the ideas of the p r o o f with the aid of the following examples. (i) Consider, for N = 2, the functions B~(z) = - c z - 2 ( z 2 - 1/2)(z 2 - 3), B22(z) = ( c / 2 ) z - 2 ( z 2 - 2)(z 2 - 3), B 3 ( z ) - - ( 3 c / 2 ) z - 2 ( z 2 - 2)(z 2 - 1/3), B4(z) --- 3cz-Z(z 2 - 1/2)(z 2 - 1/3). It is easy to verify that the expressions l + B ~ ( z ) B ~ ( l / z ) are identical for i - 1 , 2, 3 and 4, and that B~(I) = c = sinh(a). The function A2(z) determined by A2(z)A2(I / z) = 1 + B~(z)B~(1 / z) and L e m m a 2.3 is therefore the same for i = 1, 2, 3 and 4. Hence, the impedance transitions Z ~ determined by Theorem 2.4 are 2 2 = 4 different transmission-equivalent transitions. It is almost obvious that it is not possible to obtain other transmission-equivalent transitions. (ii) Consider, for N = 2, the functions B~(z) = ( c / 5 ) z - 2 ( z 4 +4), B~(z) = ( 4 c / 5 ) z - 2 ( z 4 + 1/4). We obtain two different transmission-equivalent transitions Z 1 and Z 2 in this case. The p r o o f of the previous theorem gives the following corollary.
Corollary 3.4.
Suppose that Z m ~ ZouT. Then a necessary and su~cient condition that Z e DN has no other transmission-equivalent impedance transition in DN is that the function BN( Z) has all its zeroes on the unit circle Izl = 1. C. Examples Some concepts and results of this section can be illustrated by considering the same examples as in Section 2C. For this purpose, let Z be a given impedance transition and let Z TE be a transition which is
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
400
transmission-equivalent to Z. Also, if Z c DN and Z a'E c DN, let these transitions be represented by {%k}~:0 and r~t%k TE~N J'k=0, respectively. The latter numbers are determined from the former by requiring that both AN(z) and a should be the same. (a) One-segment transition, N = 1: With the aid of the first of equations (2.34) we require
cosh(% TE) cosh(%TE) = cosh(ao) cosh(a 0, sinh(%orE) sinh(t~T E) = sinh(ao) sinh(%l),
(3.1)
%vE+%TE=%0+%. This system has exactly two solutions, namely, % ~ = %0,
%~E = %,
%T~ : %1,
%TE - %.
(3.2) Thus, in agreement with Theorem 3.3 there are 21= 2 impedance transitions zXE~ D~ which are transmission-equivalent to Z ~ Dr. These transitions are Z TE = Z and Z TE = (z1NV) aEV, respectively. They are different unless Z is antisymmetric, that is, unless %o= % (equivalently, Z 2 = Z~NZouT). (b) Two-segment transition, N - - 2: With the aid of the first of equations (2.35) we require c o s h ( %TE) cosh(%rE) c o s h ( % E) = cosh(%o) cosh(%) cosh(%), sinh(a~E) sinh(%TE + %E) = sinh(%1) sinh(% + %),
(3.3) sinh(% -E) cosh(%TE) sinh(% -E) = sinh(cto) cosh(%) sinh(%2), % ~ + % E + % E = % + % 1 +%2.
Two solutions of this system are found by inspection. They are TE Or'0 ~ %0,
TE %0 ~- %2,
% T E ~--. %1,
%1rE ~ %1,
TE %2 ~- %2,
%
(3.4)
TE ~ %,
and correspond to Z TE= Z and Z TE= (z~NV) REv, respectively. These transitions are different unless Z is antisymmetric, that is, unless % = % (equivalently, ZIZ2 = ZmZouT). Other solutions are less directly obtained. In the particular case Zm = ZOUT, that is, % + % + % = 0,
(3.5)
however, two additional solutions can be readily obtained. They are ~o~ = -%0,
%~E = _ % ,
TE /2/1 ~--- - - % 1 ,
%E
%E
%TE = - - % "
= __%,
~ --%1,
(3.6)
and correspond to Z TE = Z mv and Z TE = Z REv, respectively. The solutions (3.4) and (3.6) are all different unless % = a2 (equivalently, ZIZ2 = ZINZouT) or % = - - % (equivalently, ZI = Z2). Thus, in this special case, we have demonstrated that in agreement with Theorem 3.3 there are 2 2= 4 impedance transitions
L-E. Andersson, B. Lundberg/Properties of impedance transitions
401
zTEe D2 which are transmission-equivalent to Z s D2, and generally these transitions are all different. Also, these transitions are the ones which are predicted by Theorem 3.1. (c) Exponential transition, Z(~) = ZIN(ZouT/Zm)~/d : According to Definition 3.3 this impedance transition is antisymmetric. We notice from the second of equations (2.36) that B is real. Compare with Theorem 3.2, which, however, has been proved only for impedance transitions Z ~ DN. Further we note that B(d, to) has all its zeroes for real oJ, that is, [z[ = 1. In accordance with Corollary 3.4 this might indicate that there does not exist any different transmissionequivalent transition.
4. Transmission of energy
We now consider the efficiency of energy transmission 7/[Z, FI] defined by equation (2.25). Throughout the section Z(se) represents an impedance transition with given input impedance ZtN, output impedance ZOUT, and length d, whereas Fz(t) represents an incident wave of finite duration and energy. First we restrict ourselves to monotoneous impedance transitions and show that ,/ is always minimized by one which makes an abrupt jump from ZIN to ZOUT. Then we show that, for a given incident wave, there exists an optimal impedance transition in DN which maximizes 7. This optimal transition is antisymmetric. A. Minimization o f energy transmission through monotoneous impedance transitions
We start with the following lemma concerning the coefficients ak for a monotoneous impedance transition, that is, one for which Z(~:) is either nondecreasing or nonincreasing. Lemma 4.1. I f Z ~ DN is a monotoneous impedance transition, then the coefficients ak are nonnegative, that is, ak >~O for k = O , . . . , N.
Proof. The coefficients a o , . . . , aN are determined from the numbers ao . . . . , aN with the aid of equations (2.8), (2.12), (2.14), and Proposition 2.1. If we first suppose that Z is nondecreasing, then O~k ~" 0, k -- 0 . . . . , N, and for the coefficients in the matrices C(t~k) we have cosh(t~k) > 0 and sinh(ak)/> 0, k = 0 . . . . . N. Then, obviously, a k >10, k - - 0 , . . . , N. If instead Z is nonincreasing we consider the transition Z ~Nv which is nondecreasing. As AN = A ~ v by the proof of Theorem 3.1, we obtain ak>~O, k = 0 , . . . , N, also in this case. This completes the proof. With the aid of this lemma we can now prove the following theorem which provides a lower bound for energy transmission through monotoneous impedance transitions. Theorem 4.2. Let Z A ~ DN be an impedance transition which jumps abruptly from ZIN to ZOUT, and let Z be a monotoneous transition in D~vfrom ZIN tO Zou T which does not have thisproperty. Then ,/[Z A, FI] < r/[Z, Fl] for any incident wave FI o f finite duration and energy.
Proof. From equations (2.8), (2.12), and (2.14) we obtain for the transition Z A AA(e i'°h) = cosh(cr) e i~'~a+h), where t~ is given by equation (2.17). Therefore, by equations (2.19) and (2.25) r/[Z A, FI] = 1/cosh2(a).
L.-E. Andersson, B. Lundberg / Propertiesof impedance transitions
402
For the transition Z we obtain from Proposition 2.1 N
AN(eioJh): ~ akei~oh(N+l k=O
2k)
and, by the triangle inequality and Lemma 4.1, N
IAN(ei h)l
N
Y lakl-k=0
ak, k~0
with strict inequality except for countably many values of w with no finite point of accumulation. But according to Proposition 2.1 and the first of equations (2.16) N
v~ ak = AN(l) = cosh(a). k=0
Therefore, by equations (2.19), (2.25), and the results above
n[Z, F,]
=
11
IAN(ei'°h)l-=l#,(O.')l~ do.'
/I
lP'i(o.,)l ~ do., I> I/cosh~(a) =
nCZ A, F,].
co
Moreover, if F(t) is zero outside a finite interval [a, b] and if the energy ~ ] F ( t ) l 2 at < ~ , then the Fourier transform Fi(to) = ~ Fi(t) e -i~°t dt is an entire function of to. Therefore Fl(to ) can have at most finitely many zeroes on every bounded domain in the complex to-plane. Hence, we have strict inequality, rl[Z, F~] > rl[Z A, FI]. This completes the proof. Corollary 4.3. ]AN(ei'°h)[ ~
~[Z A, F,]<~ ~[Z, F,]. The proof can be carried out by approximating Z with transitions in DN and letting N ~ ~ .
B. Maximization of energy transmission The following theorem states that any impedance transition in DN, which is not antisymmetric, can be replaced by one which is antisymmetric and has higher efficiency of energy transmission for all incident waves. Theorem 4.5. Let Z c DN be a given impedance transition which is not antisymmetric. Then there exists an antisymmetric transition zAS ~ DN, from the same ZIN to the same ZouT, such that r/[Z As, FI]> r/[Z, Fl]
for any incident wave F~ of finite duration and energy. Proof. Take BgS(z)= (1/2)[BN(z)+ BN(I/z)] so that BAS(ei'°h)= Re BN(ei'°h). According to Theorems 2.4 and 3.2 B As corresponds to an antisymmetric impedance transition Z AS . Also BgS(l) = BN(I) and therefore Z As and Z have the same ZOuT/ZIN according to equations (2.16) and (2.17). We have IBAS(ei'°h)l<~ [BN(ei'°h)l and, according to the first of equations (2.15), IAAS(ei'°h)l<~IAN(ei~h)l. Also, unless BAS(ei"'h)---BN(ei°'h), that is Z As = Z, we have strict inequality everywhere except possibly for countably many values
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
403
of to with no finite point of accumulation. From equations (2.19) and (2.25) we also have r / rzAS L , FI] =
IAAS(e'~")l-=lF,(to)l= do)
/I
IFI(~)I dto oo
and
IA~(e'~)I-~IP,(~)I=
~7[Z, F,] =
dw
=
dto
co
and consequently r/[Z As, F|] i> ~/[Z, F~]. Moreover, if F~(t) is zero outside a finite interval [a, b] and if the energy S~ IF(t)l z dt < ~, then by the same arguments as in the proof of Theorem 4.2 we have strict inequality, B[Z As, F | ] > ~7[Z, F~]. This completes the proof. The following theorem provides a necessary condition for an impedance transition which maximizes the energy transmission for a given incident wave. Theorem 4.6. Let FI be a given incident wave of finite duration and energy. Then there exists an optimal impedance transition z°r'r e DN with the property r/[Z °r'r, FI]~ > r/[Z, FI] for all Z c DN. Furthermore Z °r'r is antisymmetric. Proof. Consider the function g(bo, b~,..., bN) given by g(bo, b ~ , . . . , bN)= f ~ I/~(~o)12(1 +[BN(ei'°h)12)-~ do),
where N
BN(e i'°h) = ~ bk e i'°h(N-2k). k=0
g is a continuous function of (bo, b l , . . . , bN)e R/v+l. For the rest of the proof we need the following lemma. Lemma. The function g has a maximum when (bo, b l , . . . , bN)eR N÷j. The proof of this lemma is omitted because of its length, but is given in reference [9]. Let (bo, b j , . . . , bN) be a point in R ~÷~ where g takes its maximum and define B%r'r(z) by N
B°PT(z) = ~ bkZ N-2k. k=0
Let z ° v r e DN be the corresponding impedance transition given by Theorem 2.4. Then r/[Z °vr, F | ] ~> r/[Z, F~] for all Z e D~. The antisymmetry of Z °vr follows from Theorem 4.5. This finishes the proof. Next we state a conjecture and a theorem regarding the optimal impedance transition Z °yr. Conjecture 4.7. Z °r'r is uniquely defined. Theorem 4.8. B°r'r(z) has all its zeroes on the unit circle Iz[ = 1.
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
404
Proof. Suppose, on the contrary, that B ° ~ ( z ) has a zero zi with Iz, I ~ 1. Then, by Corollary 3.4, there exists an impedance transition z T E c DN, with corresponding function B~vE(z), which is transmissionequivalent to Z °Pr. From the examples following Theorem 3.3 it is also clear that Z TE can be chosen so that it is not antisymmetric. Then, however, by Theorem 4.5, there exists an antisymmetric transition Z As such that
r/[Z As, F,]> n[Z TE, FI] = r/[Z °p'r, FI]~> rl[Z, FI] for all Z ~ DN. This is a contradiction. Consequently B°er(z) has all its zeroes on the unit circle ]z I = 1. Antisymmetry of the optimal impedance transition is, by Theorem 3.2, equivalent to B°PT(z) = B°Pa~(1/z) for z # 0. Thus, antisymmetry means that B°r'r(z~) = 0¢~ B°Vr(1/z~) = 0 which is a weaker statement than that of Theorem 4.8. The next theorem is a more general version of Theorem 4.6 where the class DN has been replaced by the class D of impedance transitions Z, from ZIN to ZOUT, such that ln[Z(~:)] is of bounded variation (that is, can be expressed as the difference between two functions which are nondecreasing and bounded). We have no p r o o f for the existence of an optimal transition Z °r'r in this case. However, it seems reasonable to conjecture it. We omit the p r o o f of the theorem.
Theorem 4.9. Let F~ be a given incident wave of finite duration and energy and suppose that z ° P r ~ D has the property r/[Z °r'r, FI]/> ~7[Z, F~] for all Z c D. Then Z °rr is antisymmetric. C. Examples In order to illustrate some results of this section we consider again the impedance transitions with N = 1 and 2. For these transitions in DN equations (2.19) and (2.25) give the efficiency r/=
[AN(e
itoh
-2
^
2
)l If,(a,)l da,
/I
I~,(a,)l~dto.
(4.1)
oo
(a) One-segment transition, N = 1: The first of equations (2.34) gives [A,(ei'°h)12 = cos2(,oh) cosh2(o0 + sin2(toh) cosh2(a0 - at).
(4.2)
According to equations (4.1) and (4.2) the efficiency r/ is maximized for any given incident wave FI by ao = a~ which implies the result Z 2 = Z~NZouT. By Theorem 3.2 this means that Z °ev is antisymmetric in agreement with Theorem 4.6. Also, Z °P'r is uniquely defined in accordance with Conjecture 4.7. Further, the second of equations (2.34) gives
B°~r(z) = c o s h ( a / 2 ) sinh(a/2)(z + 1/z).
(4.3)
Thus, the zeroes of B°Pr(z), z~ = i and z2 = - i , are on the unit circle as predicted by Theorem 4.8. (b) Two-segment transition, N = 2: The first of equations (2.35) gives
[A2(eio~h)[ ~ = {cos~(o~h) cosh(a) - sin2(wh) cosh[2(ct2 + ao) - c~]}2 + sin2(2wh) cosh2(a2 + ao - a ) cosh2(a2 - a0).
(4.4)
L.-E. Andersson, B. Lundberg / Properties of impedance transitions
405
As a = ao + a~ + a2 is given this expression can be considered as a function of a 2 - a o and a~ + ao. From equations (4.1) and (4.4) it is evident that for any incident wave F~ a necessary condition for optimality is a2 - ao = 0, which implies ao = a2 o r Z I Z 2 = Z i N Z o u T. This means that Z ° ~ i s antisymmetric in agreement with Theorem 4.6. In this case a2 + ao, and therefore Z °rr, depends on the incident wave F~.
5. Concluding remarks Most results of the present paper have been established for transitions in D~, that is, transitions with piece-wise constant characteristic impedance functions. The main reason for this is increased simplicity. However, such transitions also have technological importance. The Theorem 3.3 on transmission-equivalence may have practical use in design. Consider, for example, a transition from one cross-sectional area to another on a drill steel and assume that this transition has favourable transmission properties. Then there are generally a number of different transitions which share these properties. However, some of these transitions may be superior in other respects such as low weight, high strength, and simple geometry. Thus, there may be a certain freedom for the designer to make improvements without affecting the transmission properties. The existence of an optimal impedance transition which is antisymmetric, as expressed by Theorem 4.6, has importance in optimization. First, it justifies optimization, and secondly it may reduce the computational efforts considerably, especially for large N. For transitions with one segment, N = 1, we have demonstrated that the efficiency of energy transmission is maximized for any given incident wave when the characteristic impedances are in geometric progression, that is, Z 1 / Z I N = Z O u T / Z 1. This result is in agreement with the required antisymmetry of the optimum transition, and, in fact, follows from this requirement alone. For an N-segment transition the requirement of antisymmetry for an optimum transition, ZjZN = Z2Z~t_I . . . . . ZtNZouT, does not imply that the characteristic impedances are in geometric progression. In fact, the optimum impedance transition depends on the incident wave and, under certain circumstances, may not even be monotoneous. This is illustrated by the following simple example: Consider an impedance transition with ZOuT/Z~N = 10 and N = 2, and a rectangular incident wave with duration 5h. Then the efficiency of energy transmission ~ is maximized by Z~/Z~N-~4.2 and Z 2 / Z m ~ 2 . 4 which gives r/max= 0.522. ThUS we have Z m < Z l > Z2< ZouT. Under certain circumstances an impedance transition with ZIN, Z~ . . . . . ZN, ZouT in geometric progression may be nearly optimal. This condition for optimality is obtained if contributions to the transmitted wave Fr from waves which have been reflected are neglected. For this case Fischer [5] derived the result r/max = {4/[(1 + r l / " ) ( l +r-I/n)]} n with n = N + I and r=ZouT/ZIN. We have considered maximization of r/ without functional constraints for Z ~ , . . . , ZN. For equal input and output impedances this leads to the trivial result Z~N = Z~ . . . . . ZN = ZOUT and '/']max 1. For this case, however, Gupta [7] considered the functional constraint ~k=l N Zk = M N / d , which means that the transition has a given mass M. Because of this constraint the optimum transitions obtained are not antisymmetric. :
References [1] M. Mao and D. Rader, "Longitudinal stress pulse propagation in nonuniform elastic and viscoelastic bars", Int. J. Solids Structures 6, 519-538 (1970).
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L.-E. Andersson, B. Lundberg / Properties of impedance transitions
[2] M.M. Sondhi and B. Gopinath, "Determination of vocal-tract shape from impulse response at the lips", J. Acoust. Soc. Am. 49, 1867-1873 (1971). [3] Lord Rayleigh, The Theory of Sound, 2nd edition, Vol. 1,250-251, Macmillan (1894). Reprinted by Dover, New York (1945). [4] L.H. Donnell, "Longitudinal wave transmission and impact", Trans. Am. Soc. Mech. Engrs 52, 153 (1930). [5] H.C. Fischer, "On longitudinal impact II. Elastic impact of bars with cylindrical sections of different diameters and of bars with rounded ends", Appl. sci. Res. A.8, 278-308 (1959). [6] B. Lundberg, R. Gupta and L.-E. Andersson, "Optimum transmission of elastic waves through joints", Wave Motion 1, 193-200 (1979). [7] R. Gupta, "Optimum design of wave transmitting joints", Wave Motion 4, 75-83 (1982). [8] P.C. Petersen, O. Tretiak and Ping He, "Impedance-matching properties of an inhomogeneous matching layer with continuously changing acoustic impedance", J. Acoust. Soc. Am. 72, 327-336 (1982). [9] L.-E. Andersson and B. Lundberg, Some Fundamental Properties oflmpedance Transitions, Technical Report 1984:11 T, University of Lule~i (1984). [10] N.B. Miller, "Reflections from gradual transition sound absorbers", J. Acoust. Soc. Am. 30, 967-973 (1958).