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Acoustic saturation and the conversion efficiency of the parametric array
Journal of Sound and Vibration (1982) 82(4), 473-487
ACOUSTIC
SATURATION
EFFICIENCY
AND THE CONVERSION
OF THE PARAMETRIC
ARRAY
R. W. LARDNER Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada (Received 21 July 1980, and in revised form 13 October 1981)
The Mendousse solution of Burgers’ equation is generalized to the case when the boundary condition involves a mixture of two sinusoidal inputs, and the form of the old-age wave is extracted. At low and moderate amplitudes of the primary inputs, the frequency of this asymptotic wave is the difference frequency and its amplitude exhibits the phenomenon of acoustic saturation. For fixed input levels it is shown that there is an optimum combination of material parameters which maximize the amplitude of the difference frequency output. At high input amplitudes, acoustic saturation breaks down and the old-age wave changes character completely. In general its frequency is lower than the difference frequency. Only when the down-shift ratio is equal to an integer plus $ is the dominant asymptotic frequency equal to the difference frequency. In this case, at very high input amplitudes, the amplitude of the old-age wave becomes very sensitively dependent on the phase difference between the two primary inputs. When the two primaries are exactly in phase, amplification of the difference frequency wave by several orders of magnitude above the saturation level occurs.
1. INTRODUCTION Since the original papers of Lighthill [l, 21 and Westervelt [3,4] the parametric acoustic array has been the subject of a great many investigations. A review has recently been
given by Bjorno [5]. One aspect of current interest concerns the possibility of improvement in the conversion efficiency of the parametric array perhaps by changing the medium close to the transducer [5, pp. 54-551. The complete equations describing the parametric acoustic array are complex, involving the effects of non-linear compressibility, thermo-viscous damping and diffraction, and past investigations have been based on some simplified model. One such model that has been used by Fenlon [6] and Cary [7, 81 is a one-dimensional model based on Burgers’ equation. This model has the advantage of fully and correctly including both the non-linear effects and thermo-viscous damping, although diffraction effects are ignored. Fenlon [6] has developed a heuristic approach to allow diffraction effects to be included. Fenlon’s solution of Burgers’ equation involved treating the non-linearities by a direct perturbation method, and thus is valid only at distances from the source which are much less than the shock formation distance, or alternatively at signal levels which are much less than those necessary for shocks to form. (This same remark is also true of the original Westervelt theory [4] as well as subsequent extensions [9, lo] to include spherical spreading of the pimary beam.) In view of the importance of the problem, it seems worthwhile extending Fenlon’s analysis to include the non-linear terms without approximation, thus allowing the solution of the Burgers’ equation model to be examined above the shock formation threshold. Since diffraction is neglected, or at best included heuristically, it cannot be claimed that the results obtained from the Burgers’ equation model 473 0022-460X/82/120473+
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R. W. LARDNER
will be precisely applicable to a physical parametric array. Nevertheless, it is hoped that they may be at least qualitatively indicative of the behaviour of a physical array, particularly since they show some very striking and potentially useful features. The solution of Burgers’ equation generated by a sinusoidal input on the boundary was obtained by Mendousse [l l] and has been analyzed in great detail by Blackstock [12]. The approach of the present paper is to generalize the Mendousse solution to the case when the boundary condition involves a mixture of two sinusoidal inputs and then to extract the asymptotic form of that solution (i.e., the old-age solution). At low and moderate levels of input signals the asymptotic wave always has frequency equal to the difference of the two primary frequencies. At low amplitudes of the two sinusoidal inputs, this asymptotic signal corresponds to that which can be obtained by a direct perturbation solution of Burgers’ equation [6]. It can be related to the Westervelt solution by including a suitable factor to allow for spherical spreading. Its amplitude is proportional to the product of the amplitudes of the two primary inputs. At higher levels of the input amplitudes, a transitional state of saturation of the difference frequency signal occurs, that is, its amplitude approaches a constant maximum which is independent of the input amplitudes. Although this “saturation” eventually breaks down, there is a range of input levels for which the difference frequency amplitude remains approximately constant. In fact this “saturation amplitude” is the same as the saturation amplitude which would be generated by a transducer driven by a single sinusoidal input at the difference frequency itself. It turns out that in this range of input levels, the amplitude of the difference frequency wave, u& depends on the medium through a single combination of material parameters which is denoted here by uoo.Furthermore UAapproaches zero both as uoo+ 0 and uoo+ co, and has a maximum of some intermediate value of urn. Provided the ratio ul/uz of the two primary amplitudes lies between 4 and 2, the value of uco at which the maximum u,, occurs is given approximately by the formula uoo= 2.1 u1u2/(u1+u2). Thus, for given input levels, there is an optimum combination of material parameters which will permit the maximum generation of difference frequency signal. With a transducer operating at a single input frequency, acoustical saturation is permanent no matter how high the amplitude of the input signal is made (until the Burger’s equation model ceases to apply). On the other hand for a combination of two inputs at different frequencies, saturation is only a transitional stage, and at higher input levels, higher order inter-modular products enter the asymptotic solution and change its character completely. In most cases, the dominant frequency of the old-age wave is no longer the difference frequency but is some lower frequency. For example, when the down-shift ratio is equal to an integer, the dominant frequency becomes equal to half the difference frequency. Only when the down-shift ratio is equal to II +& where II is an integer, does the old-age wave have a frequency equal to the difference frequency. In this latter case another interesting phenomenon which occurs is that the amplitude UAbecomes dependent on the phase difference between the two primary inputs. At input levels which are only moderately higher than is necessary to cause “saturation”, this dependence is not very strong, and it is possible to enhance UAby only a few percent (depending on n) above the saturation level. However at high input levels, the results indicate that considerable amplification of the difference frequency amplitude will occur. The conditions necessary to achieve this are that the two input signals on the face of the transducer should be exactly in phase, the down-shift ratio should be exactly n +i for some integer n and u1 and u2 should be sufficiently high that 2u1u2/(u1 + u2) > n2u,. (Note that “saturation” of the difference frequency signal can be said to occur when ul,
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
475
u2 = uoo. Thus to achieve the indicated increase in conversion efficiency, it is necessary to have input amplitudes greater by a factor of about n * than those which cause saturation.)
2. BASIC
THEORETICAL
RESULTS
It is well-known that the propagation of uni-directional (i.e., simple) plane acoustic waves in a thermo-viscous fluid is governed to a good approximation by Burgers’s equation. Derivations and discussions of this equation can be found in the review article by Bjarno [13] or in the book by Whitham [14, chapters 3 and 41. It can be written, in Blackstock’s notation [ 121, in the form
avjau- vav/ay=r-'a2v/ay2.
(1)
Here V(u, y) = u/u0 is the re-scaled particle velocity, u being the physical velocity and u0 a characteristic velocity amplitude. The other variables are defined by y = o(t-x/c),
u = .$wx/c,
((2)
where o is a characteristic radian frequency, c is the linear sound speed, E = Q/C is the acoustic Mach number and /3 is the non-linearity parameter of the medium (often denoted by (1 +B/2A)). Finally, r is the Gol’dberg constant, defined by r =
113)
EpW/CYC,
where [Yis the linear absorption coefficient corresponding to the radian frequency w. The problem of interest is to find the solution of equation (1) in the region x > 0 when the boundary x = 0 is excited by a combination of two primary frequencies o1 and w2:
where u1 and u2 are the primary amplitudes and 4 is the phase difference between the primary inputs. If one defines w = i(w, + w2) as the average of the primary frequencies and introduces the inverse down-shift ratio S as S = (ol - w2)/w, then one can write the boundary condition in the form V(0, y) = (ul/uo) sin (1 +$S)y +(u~/uo) sin [(l -$)y
+4], y >O.
The general solution of equation (1) can be obtained by the Cole-Hopf [I% 161,
Vh
Y) =(2/m
In 4%
5(a, Y) =
$
transformation
y)/ay,
which transforms equation (1) into a linear conduction this latter can be written down immediately [12] as
(4)
(5)
equation for & The solution of
[_m
LXO, y + mq) edq2 dq,
(6)
a3
where m = J4a/T.
From equation (5) one has that
[’I_:V(O, y’) dy’)
5(0, Y) = exp 2r
and substituting from equation (4) one finds that
m, Y I=
exp{C-~~c0s(l+~S)y-r2c0s~(1-~~)~+~I~,~~0
1
LY
17
(7)
476
R. W. LARDNER
where ri = E4l/2U0(1 +$s,,
r, = ru~/2u~(l
c=r,+r2cos+
-$s,,
(8)
Substituting from equation (7) into equation (6) and taking y/m >>1 in order to eliminate transients, one obtains the solution in the form 5(g, y)=$eCIm
exp I-r,
y + mq)]
cos [(l +b)(
PC.2
-r2cos[(1-$S)(y+ms)+~]-~2}d~. At this stage it is convenient to introduce the representation e
--rcos~= ,Z,e,(-1)7,(r)
where e. = 1 and E, = 2 for n 3 1 and the I,(T) enables one to write
ib,y)=eC f
cos d,
are modified Bessel functions. This
(9)
EkEI(-i)k+‘Ik(rl)II(r2)Mkl,
k,/=O
where m M*i=$
s
_-a3
cos[k(l+~S)(y+mq)]cos[1(1-~S)(y+mq)+l~]e~q2dq
= t exp [-(m2/4)n+(k, +$ exp [-(m2/4)n_(k,
1)2] cos [n+(k, l)y + @] 1)2] cos [n_(k, 1)~ - @I,
with n,(k, I) = k(l +$S)*II(l -is). The factor ec can be dropped from equation (9) since it cancels when V is calculated via equation (5). With m* = 4a/I’, the solution (9) then takes the final form C(U, y) = $ ,E, EkEI(-I)k+‘Ik(r,)I,(r2){e-“‘““‘cos
(n+y + @)+e -n’rr”Ycos(n. y -@)}. (IO)
This is the equivalent of the solution obtained by Mendousse [ 1 l] and discussed at length by Blackstock [12] for the case when the boundary is excited at a single frequency. So far the solution is exact. Thus equation (10) can be used in order to estimate the eventual form of the wave at large distances from the boundary: that is, for large values of u/r. (Note from equations (2) and (3) that u/r = CU.) All the terms in equation (10) except the k = I= 0 term decay exponentially with u, so one should pick out the terms which decay the slowest. When 6 is small compared with 1, which is the case of main interest, n+(k, I) is of order 1 for all k, 1, so the terms which involve it+ decay relatively fast. On the other hand n_(k, I) can be of order 8, and such terms dominate the solution for u sufficiently large. In particular the term k = I = 1 gives n_(k, 1) = 6 and provides the difference frequency contribution to the solution. Retaining only the (k, I) = (0,O) and (1,l) terms, one has r(~, Y I-
Iouxdr2~
+ wrlvl~r2~
e-“‘2’r
cos (8~ - 4).
(11)
Other terms could be retained in equation (10). For example the (0, 1) and (1,O) terms would allow one to examine the decay of the two primary amplitudes while the (2,2)
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
477
term would give the amplitude of the first harmonic of the difference frequency, and so on. Since the main concern here is with the generation of the difference frequency signal, only the term relating to this frequency has been kept. It should also be observed that there are certain n_ terms in equation (10) which decay more slowly than the difference frequency term retained in expression (11). These arise when k(1 + $) = 1(1 -is). However when 6 <<1, such terms occur only for large values of k and 1 (of order 8-l). Since, for fixed arguments, the Bessel functions Ik(ri) and II tend to zero exponentially as k and I become large, such terms have small amplitudes and can be ignored (provided S is small). These terms do however become important when the amplitudes u1 and u2 are sufficiently large, and their investigation will be the subject of sections 4-6. From expressions (5) and (11) one gets V(fl, Y) - -(46/T)H(Ti)H(r2)
e -C’2’rsin (Sy - c$),
where H(z) = I1(z)/IO(z). Returning to physical variables, one then finally obtains u - -6u,H(2ul/u,)H(2~Z/~as)
ePszux sin [So(t-x/c)
-41,
u m = 4cYc2/po.
(12) (13)
3. DISCUSSION OF THE DIFFERENCE FREQUENCY AMPLITUDE In the asymptotic solution (12) one can denote by uA the amplitude of the difference frequency wave: uA = Su,H(2ul/u,)H(2u2/u,).
(14)
This amplitude uA can now be examined in the cases when u1 and u2 are small or large compared with uoo. When z -0, one has I&) - 1, Ii(z) -z/2 and so H(z) - z/2. Therefore, when ul, u2cc
Urn, UA
--&dlU2/Um
=
8@&/4ac2.
(15)
This result exactly duplicates that obtained by Fenlon [6, equation (lob)] if one bears in mind that in that author’s notation (Ye= 2a and ffTr >>1. Fenlon’s direct perturbation approach in which the non-linear term in equation (1) is treated as a small perturbation is of course applicable only to the case when u1 and u2 are small. When z + ob, one has I,(z) -e*/JZm and so H(z) - 1. Therefore, when ul, u2 >>uoo, UA-&.&x, = 4&c2/&k
(16)
It is clear that in this limit acoustic saturation is achieved: the limiting amplitude becomes independent of the primary amplitudes ui and ~2. It is interesting to note that the saturated amplitude (16) of the difference frequency signal is exactly the same as would be obtained had the boundary been excited by a monofrequent excitation of frequency 6~ [12]. Thus the output limitation of a saturated parametric array is the same as that of a single frequency transducer operating at the difference frequency. It has been pointed out by Fenlon [6] that it is possible to recover Westervelt’s original expression [4] for the axial difference frequency signal from expression (15) by including an additional factor of (&&/2mr), where So is the area of the source aperture and r is the radial distance from the source (r = x on the axis). This factor may be rationalized on the basis that it is equal to (rA/r) where rA= &&,/27rc is the collimation distance for the difference frequency; such a factor would account for spreading losses of a difference
478
R. W.LARDNER
frequency signal spreading from the range rA. If one speculates that a similar factor is appropriate in the general case, one can conclude from expression (14) that the axial difference frequency amplitude accounting for spherical spreading losses in three dimensions would be given by uA = (2s2aS,/~~r)H(pwul/2Sac2)H(PWU2/2~~c2).
(17)
In particular, the saturation amplitude is (2S2&0/7r@). Returning to more solid ground, one can examine uA as a function of uco for fixed ul, u2 and S. One observes from expression (14) that uA depends on material parameters only through uo3,and thus the dependence of uAon uoois vital to the question of improving the conversion efficiency of the parametric array, for example by surrounding the transducer by a second medium [5]. For large values of urn, uA is given by expression (15) while for small values of uA it is given by expression (16). From this behaviour, and the functional information that can be deduced from expression (14), its graph must therefore have the form shown in Figure 1. The most important feature of this graph is the existence of a maximum, which implies that, for given input strengths u1 and u2, there is an optimum medium within which to locate the transducer in order to generate the largest difference frequency signal.
Figure 1. Form of ud
In order to calculate the position of the maximum, one can set z = 2ui/u, and write expression (14) in the form (&/2ui8)
= z-‘H(z)H(kz).
and F = u2/u1 (18)
Computed results for the value of z at which the right-hand side of this equation is maximum are given for several values of CL in Table 1. (Note that, because of the symmetry of expression (14) one can take u2s ui-i.e., p 3 l-without any loss of generality.) TABLE
1
Values of z for which expression (18) is maximum, for several values of p CL
z
PZI(N + 1)
1 1.25 1.5 2 3 4
1.90 1.69 1.56 1.34 1.10 0.96
0.95 0.94 0.92 0.89 O-83 0.77
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
479
The last column of Table 1 shows the quantity p.z/(p + 1) which, as can be seen, remains approximately constant for p up to about 2. Motivation to consider this quantity comes from an examination of the asymptotic approximation to the right-hand side of expression (18). For large z, I&)-(e’/&){1+(1/8r)+(9/128.r2)+. II(z) - (e'/&){l
- (3/8z) - (15/128z2) -. . .},
from which it follows that H(z) - 1 -(l/22) u4/2u,S -(l/t)-[(p
. .},
- (1/8z2) - . . . . Hence, from expression (18), + 1)/2~~2~]-(~
- 1)‘/8p2z3.
(19)
The error in this approximation is not greater than about 5% provided z and ~LZ exceed 1. When p is close to 1, the last term in expression (19) is small. Using the first two terms to estimate the position of the maximum gives z = (p + 1)/p. One can see from Table 1 that this formula is not too far out for CL< 2. A more accurate approximation on the basis of the numerical results would be 2 = 0*95(p +1)/p,
(20)
which is accurate to within about 6% for 2 c CL~2. For values of p outside this range this formula becomes inaccurate principally because the value of z at the maximum is decreasing out of the range of validity of the asymptotic approximations. From expression (20) the optimum condition for difference signal generation is that U oa=2~lu1u2/(u1+u2), $< uJur62. One can substitute from expression (13) for urn and also use the fact that [12] CY= bw2/2pc3, where b is the material dissipation constant?, to write this condition in the form bw/&c = 1~05u~u~/(u~ + u2),
(21)
spreading losses being ignored. To the extent that expression (17) can be used as a guide to the effect of such losses, one can conclude that the difference frequency output will be maximized by making the extra factor (So&/c) as large as possible while at the same time satisfying condition (21). 4. THE EFFECT OF HIGH AMPLITUDES AND MODERATE DOWN-SHIFT RATIOS In section 2, the general solution (10) for f(a, y) was approximated by retaining only the k = I= 0 term and the II_ term corresponding to k = 1 = 1. However, as was pointed out, there are other terms in the sum for which the factors n_(k, 1) which occur in the exponents can be of magnitude 6 or less. When the down-shift ratio 6-l is large, the factors Ik(J’i)Il(f2) which also occur in these terms are very small, and the terms can legitimately be ignored. However when S-’ is only moderately large, or when the input levels are high, some at least of these terms must be included. Their inclusion has a considerable effect on the form of the old-age solution. n+(k, 1) is always of order 1 or more, as also is n_(k, 1) in the cases when k = 0, 1 # 0 and I = 0, k # 0. Dropping these terms and cancelling an overall factor which has no + b = (4/3)7 + [ + k(y - 1)/c, where v and b are the shear and bulk viscosities, y is the ratio of specific heats and c, is the specific heat at constant pressure.
k is the thermal
conductivity,
480
R. W. LARDNER
effect on V, one can therefore write expression (10) in the form 4”(a, y)- I+2 ,F,
(-l)k+‘ffk(rl)&(r2)
exp
021cos [n-k
[-(dOdk,
I)Y
-@I, (22)
Hk
(r)
=
n_(k,/)=k(l+$Sj-/(l-$6).
Ik (~)/I,(~),
In order for n- to be of order 6, it is necessary that I> k. With I= k + d, where d is some positive integer, n_(k, 1) = S(k +$d -da-‘).
(23)
Since n-(1,1) = S, one need only consider those terms in expression
(22) for which It follows
n_(k, E) lies in the interval [-S, S] (as far as the old-age solution is concerned). from expression (23) that -S < n_(k, I) s S if and only if
(6-l-&)d-laks(K’-+)d+l.
(24)
Hence the values of k which are of significance are of order (6-l - $)d and the values of I are of order (S-’ +$)d. When 6-l is very large, the functions Hk(rr) and Z$(rJ which occur as coefficients in expression (22) have large orders, and thus are negligible provided their arguments are not large. Using standard asymptotic formulae [17] gives, for v>>l,
(25) and for x D 1, lo(x) - (l/J2*x)
e”[l+ 0(1/x)],
(26)
and therefore for v >>1 and x >>1, H.(x)-(&)“‘exp[J,‘+,’
-x-vln
vf (
J v2+x x
2)1-
In the further particular case when x >>v >>1, this reduces to EL(x) -exp {-v*/~x}.
(27)
It is clear therefore that, for large V, K(x) remains very small until x is of the order Y’. Using this result shows that for k, I = da-‘, the kl- term in expression (22) is negligible provided that ri c $(dS-I)‘,
i = 1,2.
(28)
Setting d = 1 to get the strongest bound, one concludes that the approximation used in section 2 is valid provided that ui < (u,/10)SP2, i = 1, 2. The difference frequency solution (11) or (12) is effectively saturated when ri =2. When the down-shift ratio is 10, the conditions (28) are not violated until ri = 20. Thus it requires input signal strengths greater by a factor of 10 than those necessary to produce acoustic saturation in order that the other terms in expression (22) should become significant besides the k = 1= 1 term. However when the down-shift ratio S-’ is equal to 5. this factor is reduced to 2.5.
PARAMETRIC
ARRAY
CONVERSION
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EFFICIENCY
As the ri are increased or as S -’ is decreased, the conditions (28) are first violated for d = 1. Thus the first of the additional terms to become significant are those for which d = 1. At higher values of ri or at a lower value of 6-l , the d = 2 terms become significant. Thus as the ri are increased one must progressively include higher and higher values of d. In the range #S-‘G r, s :S-‘, only the d = 1 terms need be included, and their effect can next be examined. Let 6-i lie in the interval n -$< S-’ < n +& where n is some integer. Then the values of k which satisfy condition (24) for d = 1 are k = II - 1 and k = n, corresponding respectively to 1= it and 1= n + 1. The corresponding values of n_(k, 1) are II = s(n-$-6-l) and n_=6(n+$-6-l). The first of these has the smallest value of III. i, and hence has the slowest decaying exponential factor, when II -$< 6-l < n. The second has the smallest In-1 for IE
(rl)fJ”+l(r2)
cm
l-2H,(rl)H,+1(T2)
BY
-(n
cos ($Sy + n4)
+ lMl],
exp [-~p~62/~]cos
[p+Sy -(n + l)~$],
wherep,=n*$-6-l. The remaining case is when S -’ = n + $ for some integer it. Then condition (24) for d = 1 has three solutions: k = n - 1, n, n + 1 (1= n, n + 1, n +2). The corresponding values of n_(k, 1) are -6, 0, +6 respectively. In this case it is necessary to include also the k = 1= 1 term in the asymptotic solution, and one gets r(~,YY)-[l-2H,(~,)~“+1(~2)cos(n+1)91
+2 exp [-crS2/T]{H1(T1)H1(r2)
cos
(Sy -4)
-H”-,(Tl)H,(r2)COS(~y+n~)-H,+,(I’l)H,,+2(~2)COs[~Y-(~+2)~11.
From these results, the following expressions are obtained via the transformation for the leading asymptotic behaviour of V(a, y): II -+
V((T, y)-p_s(4/r)H,_I(Tl)H,(r2)
6-l = n : V(u, y) - 6(2/r)
ePus2’4rWp~(Tl)H,(~2)
+H,(rl)Kt1(r2)
sin BY
n
xe ~“““r{H1(~l)Hl(T2)
(29)
sin ($6~ +M) -(n
(30)
+ 1MlI,
e~~“PZ’2!rsin [p+Sy -(n + l)+l,
V(V,,) --6(4/r)[1-2f&(Tl)H,+l(T*)
-H,+l(~lW,+2(f2)
e~~rrP’s2’rsin(p-sy -m$),
(5)
(31)
(33s (n + u41r’
sin (Sy -4)
sin [SY -(n
+
-Hn--1(T1)Hn(T2)
3411.
sin (6y f nc$)
(32)
The most striking result to emerge from these formulae is that the dominant frequency in the old-age wave is not always the difference frequency SW, but may be anywhere
482
R. W. LARDNER
between So and zero depending on the down-shift ratio S-‘. Only when S-’ equals an integer plus $ is the dominant frequency equal to SW. (When p* are small, the results (29) and (31) lose accuracy except at very large values of u, and further terms should be retained in the solution.) A second property of these asymptotic waves is that in the cases S-’ = n or rz +$, the amplitude of the wave depends on the phase difference 4 between the two primary inputs. In these two cases, if one expresses V in the form
V(0;y)- rS(4/T)P e-rrr2S*‘r sin (ray - $1) (with r = $ when S as follows:
-1
=n andr=lwhenS-’
(33)
= rz +$) then the amplitude factor P is given
Sp'=n:P=[A2,_1+A:+2A,_1A,cos(2n+l)#‘; S-‘=n+;:P=[l-2A,cos(n+
l)d]-‘[l
+A;_l+A:+l
-2(A,_1+A,+l)cos(nil)+ +~A,-~A,+~cos~(~+~)~I"~; (34) here A,,=H,(~I)H,+~(~~), and also H1(T,) = 1, which is appropriate for the region concerned (ri - rr*). In the case (34), P is rather sensitively dependent on 4, Since A,,varies from 0 to 1 as the ri vary, it even appears that P can be made infinitely large by a suitable choice of the ri and 4. This is an illusion however, since, before the & become large enough to make A, equal to 1, the terms in expression (22) corresponding to d = 2 become significant, and the approximate solutions (29)-(32) become invalid. (From expression (28), the d = 2 terms become significant when r, --$(K1)* = 4n2/5. From expression (27), at this value of ri one has H,(r,) ~e-“~ and so A, =e-5'4<$. Equations (33) and (34) cannot be used when A, exceeds this value.) Nevertheless the dependence of P on 4 is of sufficient importance to warrant a further investigation, which is presented in the next section.
5. THE DEPENDENCE OF AMPLITUDE ON PHASE DIFFERENCE In this section the case S-’ = II +i is considered and the dependence of the amplitude factor P,given by expression (34), on the phase difference 4 is examined. In order to obtain sufficiently accurate expressions for A, and A,*I,the approximation (27) for H”(x) seems unlikely to be exact enough. A better approximation can be obtained by retaining the next order terms in expressions (25) and (26) [17], to obtain? K(x)
- exp [-(v2/2x)
+ (v4/24x3) - (v2/4x2)],
x >>Y >>1.
(35)
Using this gives, for m = 0, *l,
_exp (n+m)'+(n+m)' (n+m)' (n+m+1)2+(n+m+l)4 A "frill ____ i 2r1 24r:-724r: 2r2 t The approximation an error of (l/600)%,
-
(35) appears to be astonishingly accurate. For example, it reproduces HS(lO) with an error of l/10% and even H5(5) with an error of 0.6%.
(n+m+l)' 4r: I. HIO(lOO) with
PARAMETRIC
ARRAY
CONVERSION
483
EFFICIENCY
For the region of interest, in which the ri are 0(n2), one can write ri = yin’. Then retaining terms in the exponent only up to order it P2, one can write A “+m _ q e--Xm--Bm2,
(34)
4=exp(-~[I+f(~-~)]-~[l+fi+f(l+~-~)]},
(37) @,L _L+1 2n2 ( y1 J.
x=X+;(l+%
(38, 39)
Substituting expression (36) into expression (34) and retaining only terms of order n -’ gives P= l+K2R(lj),
(40)
R(~)-n2[1-2~cos~]-2{2q2~2-q~~~~(~2-28)-4~~2cos2~), where (I,= (n + l)& Consistent with this order of approximation l3= A/2n2, A
=
one can then write
q = eehj2,
x = Aln,
(4.1)
1*
&o(Ul+ u2)
(L+L >qp-y [ Yl
Y2
2Ll,u2
(42)
This finally leads to the expression R ($) - hq[2hq + (1 - A) cos sl,- 29 co? $I/( 1 - 29 cos +Q2.
(43)
As remarked at the end of the last section, the d = 2 terms should be included when the ri become as large as 4n2/5: that is, when the yi become as large as 0.8. The above results therefore hold only when A is greater than about 2.5. Graphs of R(G) for different values of A are shown in Figure 2. As A becomes large, corresponding to relatively low input levels (but, of course, still above the conventional saturation levels), R(4) becomes very small. In this limit, the theory of section 2 holds.
Figure
2. Graphs
of R(I)) for A 3 2 based on the approximation
of retaining
d = 1 terms only.
484
R. W. LARDNER
As A decreases, Z?(4) becomes significantly different from zero, indicating a departure from this lowest order theory. For A b 3.46, the maximum amplitude occurs when II/ = 180”: that is, when the phase difference between the two primary inputs is 180”/(n + 1). For smaller values of A the value of (I, at which R is maximum decreases towards zero; when A = 3 the maximum occurs at I/J= 120” and when A = 2 at 11% 60”. Within the range A 2 2, the value of R (0) is negative, indicating that when the two input signals are exactly in phase, the asymptotic difference frequency signal is somewhat lower than might be expected on the basis of the lower order theory of section 3. On the other hand a small amount of magnification can be achieved at these input levels by choosing an appropriate phase difference between the two primaries. However, because of the factor n -’ in expression (40), the departures from the theory of section 2 are rather small. For example, when n = 10 (down-shift ratio = 10.5) the total variation of P between its maximum and minimum values is about 3% when A = 6 and just over 4% when A = 2. The graph for the case A = 2 shown in Figure 2 is not to be taken too seriously, since it is outside the range of validity of expression (32). It has been included since it does give a qualitative indication that, at smaller values of A, R(4) may develop a maximum value of sizeable magnitude at a value of 4 which is at or close to zero. If this is the case, then there would be significant enhancement of the difference frequency signal above the saturation level at these higher input strengths. In order to investigate this point, it is necessary to return to equation (22) and include terms for which d > 1. This is the subject of the next section.
6. THE INCLUSION
OF HIGHER
ORDER
TERMS
According to the criterion (28), the d = 2 terms become significant for yI 3 4/5 and the d = 3 terms are significant for y, 2 9/5. Thus for 2.5 2 A 2 1.1 it is sufficiently accurate to include only the d = 2 term in the solution (22). From expression (23) with d = 2 and K’=n+$,onefindsthatn=-6fork=2d-1, =Ofork=2d,and =+6fork=2d+l; otherwise In.-\ > 6. Including these terms and proceeding as in section 4 one readily obtains that, as u + CO, V is given asymptotically by expression (33) with r = 1, where the amplitude factor P is now given by P= [l-2A, +[(A,_,
cos 4+2Bz, -A,+,)
cos 2$]
‘{[l-CA,
sin I,?-(B2n__1 -Bz,,+l)
-,+A,+,)
cos t,b+(&n~~+Bz,,+,)
cos 2&
sin 2+]*}““,
where A,, = H,(TI)H,,+I(I’~) and B, = H,,(~I)H,,+~(I’~). One can approximate A,, and B, by using expression expression (36) as before for A, and also
(35) and one then
recovers
B 2n+m - q2 e -2xm mBm’, where x and 0 are given by expressions
q*=exp
(44)
(38) and (39) and
(-&[4+$(;-&)]
(45)
+[4+;+-$(4+;-4&
Substituting these formulae into the above expression for P, expanding and 8 and retaining only terms up to order l/n2, one obtains P = (1 + af($)-‘[(2e +4f(1k)~(q where
-x2)4
sin I+!-2~
cos * - (20 -4x2)4* sin 2$)*+
f(G) = 1 - 2q cos !/I+ 2q2 cos 2JI. To this order
in powers
of x
cos 241
O(l/n3)}“2, one can use the approximations
PARAMETRIC
ARRAY
CONVERSION
485
EFFICIENC\
expressions (45) and (37) shows also that q2=em2* +O(K’):= one can write P in the form (40) to within an error of order y1m3,
(41). Comparing
q4 + O(n -‘). Therefore
where -A)q
R(4) = Af($)-2{f($)[(1
cos CL--(1 -4A)q4
cos 2&]+2A(q sin *-2q4
sin 249)‘},
with f(9) = 1 - 2q cos r/l+ 2q4 cos 24. Graphs of R(4) for several values of A between 1 and 3 are shown in Figure 3. The graph for A = 3 is very similar to that shown in Figure 2. For A = 2 the graph is qualitatively similar to that in Figure 2, but is quantitatively rather different, especially near (I,= 0. Thus the approximation of retaining only the d = 1 terms has begun to break down at this value of A. The graph for A = 1 in Figure 3 is on the border of the region of validity of the present approximation, and again can be expected to show some numerical departures from the correct result. The most striking feature in Figure 3 is the development of a large maximum of R (4) at 4 = 0 as A decreases below 1.5. This indicates that rather substantial amplification of the difference frequency signal will occur at these levels of signal input provided that the two primary signals are exactly in phase.
6
Figure
3. Graphs
of R(G) for 1 s A s 3 based
on the approximation
of retaining
d = 1 and rl = 2 terms only.
In order to examine the extent to which such amplification is possible one must include terms in l((~, y) corresponding to d > 2 so that A can be reduced below 1. For a general d, one sees from expression (23) that k = nd - 1, nd and nd + 1 lead respectively to n- = -8, 0 and +6. Including terms from d = 1 up to d = D one obtains, from expressions (22) and (S), therefore, that
1 -1
1+
X sin (6y -g)+ 1
+ffnd+l(rl)H(n+l)d+,
i d=l
(-1)dH,d(rl)H(n+lid(r2)
CoS
i
(-l)d[H”d--~(r,)Pr,,+l)d-l
WA
d=l
(r,)
sin
(&‘--@J
-&)I].
d$
sin (6y -6 +d$) (46)
486
R. W. LARDNER
The asymptotic approximation (35) may be used for all of the terms occurring here provided that ri 2 2nD: that is, provided D c $yin t. With this limit on D, the ratio of the second term to the first term in the exponent of expression (35) is v2/12x2= D2/12yfn2< l/48. Thus the second term may be ignored (as also may be the third) without causing too great an error. This provides a significant simplification, and one can obtain approximation formulae analogous to expressions (36) and (44): namely, H nd+m(~dH(n+wi+m W2)-qd
expC-xdm-0m2),
where x and 19are given by expressions (38) and (39), and qd = exp {-$[rT’ + r;l(l+
n-‘)‘]d*} = qd2[1 + 0(1/n)].
Proceeding as before to expand the terms in V in powers of B and x one obtains V in the form (33) with r = 1 where P has the form (40) up to O(K3) and R (4) = fMC2{f($)
-f($)’ -f($)f”($)
f(4) = 1 + 2 ;
d=l
+ V($)},
(47)
(-l)dqd’ cos di,b
primes denoting derivatives with respect to 4. Expression (48) can be recognized as the expansion of the theta function 0&,, q). Provided q is not too close to 1, one can therefore allow D + COand set f(4) = 0&, q). (Note that if one allowed D -, COin expression (46), one would not be able to use the asymptotic approximation (35) for the large values of d which would occur in the sum. It is essential to maintain the bound D G iyin in the argument leading from expression (46) to expression (47).) The numerical results obtained from expressions (47) and (48) support our earlier =0 tentative conclusion that R(4) develops a maximum of appreciable magnitude at I++ as A is decreased. For A < 1, the graph of R($,) is qualitatively similar to that shown in Figure 4 for A = 1. R(180) becomes very small as A is decreased, but the maximum at r,Q= 0 becomes very large. In Figure 4 the graph of log R (0) is shown as a function of A for 0.3 s A s 1.3. As can be seen, R(0) increases by several orders of magnitude
45 k Y 2-
0_ 0
Figure
0.5
4. Graph
x
I-O
of Log R(0) as a function
15
of A
t From expression (28), the terms up to d = D need to be included only when r, *iD’n*. assumed condition will hold provided nD z 10, which will generally be satisfied for all D 2 1.
Therefore
the
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
487
as A decreases
below 1. Although the approximations made in deriving the result contained in expressions (40) and (47) will not remain valid for such large values of R as are shown in Figure 4 for A G 0.5, nevertheless there is clearly very substantial amplification of the difference frequency signal when A is decreased below 1. The condition A < 1 is equivalent to 2u1uz/(u1 + u2) > (6-l -$)‘u~. When the two primaries have equal amplitudes this becomes u1 = u2 > (6-l --~)“u~.
ACKNOWLEDGMENT
The work in this paper owed its inception to conversations with Mr Adrian Jongens and Professor Ron New of the Central Acoustical Laboratory at the University of Cape Town, and it is with pleasure that I should like to thank both of them for their continued interest and assistance.
REFERENCES 1. M. J. LIGHTHILL 1952 Proceedings of the Royal Society A211,564-587. On sound generated aerodynamically, I. General theory. 2. M. J. LIGHTHILL 1954 Proceedings of the Royal Society A222, l-32. On sound generated aerodynamically, II. Turbulence as a source of sound. 3. P. J. WESTERVELT 1957 Journal of the Acoustical Society of America 29,934-935. Scattering of sound by sound. 4. P. J. WESTERVELT 1963 Journal of the Acoustical Society of America 35,535-537. Parametric acoustic array. 5. L. BJ~RNIZ)1977 Aspects of SignalProcessing, Part I (editor G. Tacconi). Dordrecht: D. Reidel Publishing Co., pp. 33-59. Parametric acoustic arrays. 6. F. H. FENLON 1974 Journal of the Acoustical Society of America 55, 35-46. On the performance of a dual frequency parametric source via matched asymptotic solution of Burgers’ equation. 7. B. B. CARY 1973 Journal of Sound and Vibration 30,455-464. An exact shock wave solution to Burgers’ equation for parametric excitation of the boundary. 8. B. B. CARY 1975 Journal of Sound and Vibration 42, 235-241. Asymptotic Fourier analysis of a “sawtooth like” wave for dual frequency source excitation. 9. H. 0. BERKTAY 1965 Journal of Sound and Vibration 2, 435-461. Possible exploitation of nonlinear acoustics in underwater transmitting applications. 10. T. G. MUIR 1971 Ph.D. Thesis, University of Texas at Austin. An analysis of the parametric acoustic array for spherical wave fields. 11. J. S. MENDOUSSE 1953 Journal of the Acoustical Society of America 25, 51-54. Nonlinear dissipative distortion of progressive sound waves at moderate amplitudes. 12. D. T. BLACKSTOCK 1964 Journal of the Acoustical Society of America 36, 534-542. Thermoviscous attenuation of plane, periodic, finite amplitude sound waves. 13. L. BJBRN~ 1976 in Acoustics and Vibrations Progress, Volume II (editors R. W. B. Stephens and H. G. Leventhall). London: Chapman & Hall, pp. 101-203. Nonlinear acoustics. 14. G. B. WHITHAM 1974 Linear and Nonlinear Waves. New York: John Wiley-Interscience. 15. E. HOPF 1950 Communications in Pure and Applied Mathematics 3, 201-230. The partial differential equation uI + uu, = pu,,. 16. J. D. COLE 1951 Quarterly of Applied Mathematics 9, 225-236. On a quasilinear parabolic equation occurring in aerodynamics. 17. M. ABRAMOWITZ and I.A. STEGUN 1965 Handbook of Mathematical Functions. New York: Dover Publications.
ACOUSTIC
SATURATION
EFFICIENCY
AND THE CONVERSION
OF THE PARAMETRIC
ARRAY
R. W. LARDNER Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada (Received 21 July 1980, and in revised form 13 October 1981)
The Mendousse solution of Burgers’ equation is generalized to the case when the boundary condition involves a mixture of two sinusoidal inputs, and the form of the old-age wave is extracted. At low and moderate amplitudes of the primary inputs, the frequency of this asymptotic wave is the difference frequency and its amplitude exhibits the phenomenon of acoustic saturation. For fixed input levels it is shown that there is an optimum combination of material parameters which maximize the amplitude of the difference frequency output. At high input amplitudes, acoustic saturation breaks down and the old-age wave changes character completely. In general its frequency is lower than the difference frequency. Only when the down-shift ratio is equal to an integer plus $ is the dominant asymptotic frequency equal to the difference frequency. In this case, at very high input amplitudes, the amplitude of the old-age wave becomes very sensitively dependent on the phase difference between the two primary inputs. When the two primaries are exactly in phase, amplification of the difference frequency wave by several orders of magnitude above the saturation level occurs.
1. INTRODUCTION Since the original papers of Lighthill [l, 21 and Westervelt [3,4] the parametric acoustic array has been the subject of a great many investigations. A review has recently been
given by Bjorno [5]. One aspect of current interest concerns the possibility of improvement in the conversion efficiency of the parametric array perhaps by changing the medium close to the transducer [5, pp. 54-551. The complete equations describing the parametric acoustic array are complex, involving the effects of non-linear compressibility, thermo-viscous damping and diffraction, and past investigations have been based on some simplified model. One such model that has been used by Fenlon [6] and Cary [7, 81 is a one-dimensional model based on Burgers’ equation. This model has the advantage of fully and correctly including both the non-linear effects and thermo-viscous damping, although diffraction effects are ignored. Fenlon [6] has developed a heuristic approach to allow diffraction effects to be included. Fenlon’s solution of Burgers’ equation involved treating the non-linearities by a direct perturbation method, and thus is valid only at distances from the source which are much less than the shock formation distance, or alternatively at signal levels which are much less than those necessary for shocks to form. (This same remark is also true of the original Westervelt theory [4] as well as subsequent extensions [9, lo] to include spherical spreading of the pimary beam.) In view of the importance of the problem, it seems worthwhile extending Fenlon’s analysis to include the non-linear terms without approximation, thus allowing the solution of the Burgers’ equation model to be examined above the shock formation threshold. Since diffraction is neglected, or at best included heuristically, it cannot be claimed that the results obtained from the Burgers’ equation model 473 0022-460X/82/120473+
15 $03.00/O
@ 1982 Academic
Press Inc. (London)
Limited
474
R. W. LARDNER
will be precisely applicable to a physical parametric array. Nevertheless, it is hoped that they may be at least qualitatively indicative of the behaviour of a physical array, particularly since they show some very striking and potentially useful features. The solution of Burgers’ equation generated by a sinusoidal input on the boundary was obtained by Mendousse [l l] and has been analyzed in great detail by Blackstock [12]. The approach of the present paper is to generalize the Mendousse solution to the case when the boundary condition involves a mixture of two sinusoidal inputs and then to extract the asymptotic form of that solution (i.e., the old-age solution). At low and moderate levels of input signals the asymptotic wave always has frequency equal to the difference of the two primary frequencies. At low amplitudes of the two sinusoidal inputs, this asymptotic signal corresponds to that which can be obtained by a direct perturbation solution of Burgers’ equation [6]. It can be related to the Westervelt solution by including a suitable factor to allow for spherical spreading. Its amplitude is proportional to the product of the amplitudes of the two primary inputs. At higher levels of the input amplitudes, a transitional state of saturation of the difference frequency signal occurs, that is, its amplitude approaches a constant maximum which is independent of the input amplitudes. Although this “saturation” eventually breaks down, there is a range of input levels for which the difference frequency amplitude remains approximately constant. In fact this “saturation amplitude” is the same as the saturation amplitude which would be generated by a transducer driven by a single sinusoidal input at the difference frequency itself. It turns out that in this range of input levels, the amplitude of the difference frequency wave, u& depends on the medium through a single combination of material parameters which is denoted here by uoo.Furthermore UAapproaches zero both as uoo+ 0 and uoo+ co, and has a maximum of some intermediate value of urn. Provided the ratio ul/uz of the two primary amplitudes lies between 4 and 2, the value of uco at which the maximum u,, occurs is given approximately by the formula uoo= 2.1 u1u2/(u1+u2). Thus, for given input levels, there is an optimum combination of material parameters which will permit the maximum generation of difference frequency signal. With a transducer operating at a single input frequency, acoustical saturation is permanent no matter how high the amplitude of the input signal is made (until the Burger’s equation model ceases to apply). On the other hand for a combination of two inputs at different frequencies, saturation is only a transitional stage, and at higher input levels, higher order inter-modular products enter the asymptotic solution and change its character completely. In most cases, the dominant frequency of the old-age wave is no longer the difference frequency but is some lower frequency. For example, when the down-shift ratio is equal to an integer, the dominant frequency becomes equal to half the difference frequency. Only when the down-shift ratio is equal to II +& where II is an integer, does the old-age wave have a frequency equal to the difference frequency. In this latter case another interesting phenomenon which occurs is that the amplitude UAbecomes dependent on the phase difference between the two primary inputs. At input levels which are only moderately higher than is necessary to cause “saturation”, this dependence is not very strong, and it is possible to enhance UAby only a few percent (depending on n) above the saturation level. However at high input levels, the results indicate that considerable amplification of the difference frequency amplitude will occur. The conditions necessary to achieve this are that the two input signals on the face of the transducer should be exactly in phase, the down-shift ratio should be exactly n +i for some integer n and u1 and u2 should be sufficiently high that 2u1u2/(u1 + u2) > n2u,. (Note that “saturation” of the difference frequency signal can be said to occur when ul,
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
475
u2 = uoo. Thus to achieve the indicated increase in conversion efficiency, it is necessary to have input amplitudes greater by a factor of about n * than those which cause saturation.)
2. BASIC
THEORETICAL
RESULTS
It is well-known that the propagation of uni-directional (i.e., simple) plane acoustic waves in a thermo-viscous fluid is governed to a good approximation by Burgers’s equation. Derivations and discussions of this equation can be found in the review article by Bjarno [13] or in the book by Whitham [14, chapters 3 and 41. It can be written, in Blackstock’s notation [ 121, in the form
avjau- vav/ay=r-'a2v/ay2.
(1)
Here V(u, y) = u/u0 is the re-scaled particle velocity, u being the physical velocity and u0 a characteristic velocity amplitude. The other variables are defined by y = o(t-x/c),
u = .$wx/c,
((2)
where o is a characteristic radian frequency, c is the linear sound speed, E = Q/C is the acoustic Mach number and /3 is the non-linearity parameter of the medium (often denoted by (1 +B/2A)). Finally, r is the Gol’dberg constant, defined by r =
113)
EpW/CYC,
where [Yis the linear absorption coefficient corresponding to the radian frequency w. The problem of interest is to find the solution of equation (1) in the region x > 0 when the boundary x = 0 is excited by a combination of two primary frequencies o1 and w2:
where u1 and u2 are the primary amplitudes and 4 is the phase difference between the primary inputs. If one defines w = i(w, + w2) as the average of the primary frequencies and introduces the inverse down-shift ratio S as S = (ol - w2)/w, then one can write the boundary condition in the form V(0, y) = (ul/uo) sin (1 +$S)y +(u~/uo) sin [(l -$)y
+4], y >O.
The general solution of equation (1) can be obtained by the Cole-Hopf [I% 161,
Vh
Y) =(2/m
In 4%
5(a, Y) =
$
transformation
y)/ay,
which transforms equation (1) into a linear conduction this latter can be written down immediately [12] as
(4)
(5)
equation for & The solution of
[_m
LXO, y + mq) edq2 dq,
(6)
a3
where m = J4a/T.
From equation (5) one has that
[’I_:V(O, y’) dy’)
5(0, Y) = exp 2r
and substituting from equation (4) one finds that
m, Y I=
exp{C-~~c0s(l+~S)y-r2c0s~(1-~~)~+~I~,~~0
1
LY
17
(7)
476
R. W. LARDNER
where ri = E4l/2U0(1 +$s,,
r, = ru~/2u~(l
c=r,+r2cos+
-$s,,
(8)
Substituting from equation (7) into equation (6) and taking y/m >>1 in order to eliminate transients, one obtains the solution in the form 5(g, y)=$eCIm
exp I-r,
y + mq)]
cos [(l +b)(
PC.2
-r2cos[(1-$S)(y+ms)+~]-~2}d~. At this stage it is convenient to introduce the representation e
--rcos~= ,Z,e,(-1)7,(r)
where e. = 1 and E, = 2 for n 3 1 and the I,(T) enables one to write
ib,y)=eC f
cos d,
are modified Bessel functions. This
(9)
EkEI(-i)k+‘Ik(rl)II(r2)Mkl,
k,/=O
where m M*i=$
s
_-a3
cos[k(l+~S)(y+mq)]cos[1(1-~S)(y+mq)+l~]e~q2dq
= t exp [-(m2/4)n+(k, +$ exp [-(m2/4)n_(k,
1)2] cos [n+(k, l)y + @] 1)2] cos [n_(k, 1)~ - @I,
with n,(k, I) = k(l +$S)*II(l -is). The factor ec can be dropped from equation (9) since it cancels when V is calculated via equation (5). With m* = 4a/I’, the solution (9) then takes the final form C(U, y) = $ ,E, EkEI(-I)k+‘Ik(r,)I,(r2){e-“‘““‘cos
(n+y + @)+e -n’rr”Ycos(n. y -@)}. (IO)
This is the equivalent of the solution obtained by Mendousse [ 1 l] and discussed at length by Blackstock [12] for the case when the boundary is excited at a single frequency. So far the solution is exact. Thus equation (10) can be used in order to estimate the eventual form of the wave at large distances from the boundary: that is, for large values of u/r. (Note from equations (2) and (3) that u/r = CU.) All the terms in equation (10) except the k = I= 0 term decay exponentially with u, so one should pick out the terms which decay the slowest. When 6 is small compared with 1, which is the case of main interest, n+(k, I) is of order 1 for all k, 1, so the terms which involve it+ decay relatively fast. On the other hand n_(k, I) can be of order 8, and such terms dominate the solution for u sufficiently large. In particular the term k = I = 1 gives n_(k, 1) = 6 and provides the difference frequency contribution to the solution. Retaining only the (k, I) = (0,O) and (1,l) terms, one has r(~, Y I-
Iouxdr2~
+ wrlvl~r2~
e-“‘2’r
cos (8~ - 4).
(11)
Other terms could be retained in equation (10). For example the (0, 1) and (1,O) terms would allow one to examine the decay of the two primary amplitudes while the (2,2)
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
477
term would give the amplitude of the first harmonic of the difference frequency, and so on. Since the main concern here is with the generation of the difference frequency signal, only the term relating to this frequency has been kept. It should also be observed that there are certain n_ terms in equation (10) which decay more slowly than the difference frequency term retained in expression (11). These arise when k(1 + $) = 1(1 -is). However when 6 <<1, such terms occur only for large values of k and 1 (of order 8-l). Since, for fixed arguments, the Bessel functions Ik(ri) and II tend to zero exponentially as k and I become large, such terms have small amplitudes and can be ignored (provided S is small). These terms do however become important when the amplitudes u1 and u2 are sufficiently large, and their investigation will be the subject of sections 4-6. From expressions (5) and (11) one gets V(fl, Y) - -(46/T)H(Ti)H(r2)
e -C’2’rsin (Sy - c$),
where H(z) = I1(z)/IO(z). Returning to physical variables, one then finally obtains u - -6u,H(2ul/u,)H(2~Z/~as)
ePszux sin [So(t-x/c)
-41,
u m = 4cYc2/po.
(12) (13)
3. DISCUSSION OF THE DIFFERENCE FREQUENCY AMPLITUDE In the asymptotic solution (12) one can denote by uA the amplitude of the difference frequency wave: uA = Su,H(2ul/u,)H(2u2/u,).
(14)
This amplitude uA can now be examined in the cases when u1 and u2 are small or large compared with uoo. When z -0, one has I&) - 1, Ii(z) -z/2 and so H(z) - z/2. Therefore, when ul, u2cc
Urn, UA
--&dlU2/Um
=
8@&/4ac2.
(15)
This result exactly duplicates that obtained by Fenlon [6, equation (lob)] if one bears in mind that in that author’s notation (Ye= 2a and ffTr >>1. Fenlon’s direct perturbation approach in which the non-linear term in equation (1) is treated as a small perturbation is of course applicable only to the case when u1 and u2 are small. When z + ob, one has I,(z) -e*/JZm and so H(z) - 1. Therefore, when ul, u2 >>uoo, UA-&.&x, = 4&c2/&k
(16)
It is clear that in this limit acoustic saturation is achieved: the limiting amplitude becomes independent of the primary amplitudes ui and ~2. It is interesting to note that the saturated amplitude (16) of the difference frequency signal is exactly the same as would be obtained had the boundary been excited by a monofrequent excitation of frequency 6~ [12]. Thus the output limitation of a saturated parametric array is the same as that of a single frequency transducer operating at the difference frequency. It has been pointed out by Fenlon [6] that it is possible to recover Westervelt’s original expression [4] for the axial difference frequency signal from expression (15) by including an additional factor of (&&/2mr), where So is the area of the source aperture and r is the radial distance from the source (r = x on the axis). This factor may be rationalized on the basis that it is equal to (rA/r) where rA= &&,/27rc is the collimation distance for the difference frequency; such a factor would account for spreading losses of a difference
478
R. W.LARDNER
frequency signal spreading from the range rA. If one speculates that a similar factor is appropriate in the general case, one can conclude from expression (14) that the axial difference frequency amplitude accounting for spherical spreading losses in three dimensions would be given by uA = (2s2aS,/~~r)H(pwul/2Sac2)H(PWU2/2~~c2).
(17)
In particular, the saturation amplitude is (2S2&0/7r@). Returning to more solid ground, one can examine uA as a function of uco for fixed ul, u2 and S. One observes from expression (14) that uA depends on material parameters only through uo3,and thus the dependence of uAon uoois vital to the question of improving the conversion efficiency of the parametric array, for example by surrounding the transducer by a second medium [5]. For large values of urn, uA is given by expression (15) while for small values of uA it is given by expression (16). From this behaviour, and the functional information that can be deduced from expression (14), its graph must therefore have the form shown in Figure 1. The most important feature of this graph is the existence of a maximum, which implies that, for given input strengths u1 and u2, there is an optimum medium within which to locate the transducer in order to generate the largest difference frequency signal.
Figure 1. Form of ud
In order to calculate the position of the maximum, one can set z = 2ui/u, and write expression (14) in the form (&/2ui8)
= z-‘H(z)H(kz).
and F = u2/u1 (18)
Computed results for the value of z at which the right-hand side of this equation is maximum are given for several values of CL in Table 1. (Note that, because of the symmetry of expression (14) one can take u2s ui-i.e., p 3 l-without any loss of generality.) TABLE
1
Values of z for which expression (18) is maximum, for several values of p CL
z
PZI(N + 1)
1 1.25 1.5 2 3 4
1.90 1.69 1.56 1.34 1.10 0.96
0.95 0.94 0.92 0.89 O-83 0.77
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
479
The last column of Table 1 shows the quantity p.z/(p + 1) which, as can be seen, remains approximately constant for p up to about 2. Motivation to consider this quantity comes from an examination of the asymptotic approximation to the right-hand side of expression (18). For large z, I&)-(e’/&){1+(1/8r)+(9/128.r2)+. II(z) - (e'/&){l
- (3/8z) - (15/128z2) -. . .},
from which it follows that H(z) - 1 -(l/22) u4/2u,S -(l/t)-[(p
. .},
- (1/8z2) - . . . . Hence, from expression (18), + 1)/2~~2~]-(~
- 1)‘/8p2z3.
(19)
The error in this approximation is not greater than about 5% provided z and ~LZ exceed 1. When p is close to 1, the last term in expression (19) is small. Using the first two terms to estimate the position of the maximum gives z = (p + 1)/p. One can see from Table 1 that this formula is not too far out for CL< 2. A more accurate approximation on the basis of the numerical results would be 2 = 0*95(p +1)/p,
(20)
which is accurate to within about 6% for 2 c CL~2. For values of p outside this range this formula becomes inaccurate principally because the value of z at the maximum is decreasing out of the range of validity of the asymptotic approximations. From expression (20) the optimum condition for difference signal generation is that U oa=2~lu1u2/(u1+u2), $< uJur62. One can substitute from expression (13) for urn and also use the fact that [12] CY= bw2/2pc3, where b is the material dissipation constant?, to write this condition in the form bw/&c = 1~05u~u~/(u~ + u2),
(21)
spreading losses being ignored. To the extent that expression (17) can be used as a guide to the effect of such losses, one can conclude that the difference frequency output will be maximized by making the extra factor (So&/c) as large as possible while at the same time satisfying condition (21). 4. THE EFFECT OF HIGH AMPLITUDES AND MODERATE DOWN-SHIFT RATIOS In section 2, the general solution (10) for f(a, y) was approximated by retaining only the k = I= 0 term and the II_ term corresponding to k = 1 = 1. However, as was pointed out, there are other terms in the sum for which the factors n_(k, 1) which occur in the exponents can be of magnitude 6 or less. When the down-shift ratio 6-l is large, the factors Ik(J’i)Il(f2) which also occur in these terms are very small, and the terms can legitimately be ignored. However when S-’ is only moderately large, or when the input levels are high, some at least of these terms must be included. Their inclusion has a considerable effect on the form of the old-age solution. n+(k, 1) is always of order 1 or more, as also is n_(k, 1) in the cases when k = 0, 1 # 0 and I = 0, k # 0. Dropping these terms and cancelling an overall factor which has no + b = (4/3)7 + [ + k(y - 1)/c, where v and b are the shear and bulk viscosities, y is the ratio of specific heats and c, is the specific heat at constant pressure.
k is the thermal
conductivity,
480
R. W. LARDNER
effect on V, one can therefore write expression (10) in the form 4”(a, y)- I+2 ,F,
(-l)k+‘ffk(rl)&(r2)
exp
021cos [n-k
[-(dOdk,
I)Y
-@I, (22)
Hk
(r)
=
n_(k,/)=k(l+$Sj-/(l-$6).
Ik (~)/I,(~),
In order for n- to be of order 6, it is necessary that I> k. With I= k + d, where d is some positive integer, n_(k, 1) = S(k +$d -da-‘).
(23)
Since n-(1,1) = S, one need only consider those terms in expression
(22) for which It follows
n_(k, E) lies in the interval [-S, S] (as far as the old-age solution is concerned). from expression (23) that -S < n_(k, I) s S if and only if
(6-l-&)d-laks(K’-+)d+l.
(24)
Hence the values of k which are of significance are of order (6-l - $)d and the values of I are of order (S-’ +$)d. When 6-l is very large, the functions Hk(rr) and Z$(rJ which occur as coefficients in expression (22) have large orders, and thus are negligible provided their arguments are not large. Using standard asymptotic formulae [17] gives, for v>>l,
(25) and for x D 1, lo(x) - (l/J2*x)
e”[l+ 0(1/x)],
(26)
and therefore for v >>1 and x >>1, H.(x)-(&)“‘exp[J,‘+,’
-x-vln
vf (
J v2+x x
2)1-
In the further particular case when x >>v >>1, this reduces to EL(x) -exp {-v*/~x}.
(27)
It is clear therefore that, for large V, K(x) remains very small until x is of the order Y’. Using this result shows that for k, I = da-‘, the kl- term in expression (22) is negligible provided that ri c $(dS-I)‘,
i = 1,2.
(28)
Setting d = 1 to get the strongest bound, one concludes that the approximation used in section 2 is valid provided that ui < (u,/10)SP2, i = 1, 2. The difference frequency solution (11) or (12) is effectively saturated when ri =2. When the down-shift ratio is 10, the conditions (28) are not violated until ri = 20. Thus it requires input signal strengths greater by a factor of 10 than those necessary to produce acoustic saturation in order that the other terms in expression (22) should become significant besides the k = 1= 1 term. However when the down-shift ratio S-’ is equal to 5. this factor is reduced to 2.5.
PARAMETRIC
ARRAY
CONVERSION
481
EFFICIENCY
As the ri are increased or as S -’ is decreased, the conditions (28) are first violated for d = 1. Thus the first of the additional terms to become significant are those for which d = 1. At higher values of ri or at a lower value of 6-l , the d = 2 terms become significant. Thus as the ri are increased one must progressively include higher and higher values of d. In the range #S-‘G r, s :S-‘, only the d = 1 terms need be included, and their effect can next be examined. Let 6-i lie in the interval n -$< S-’ < n +& where n is some integer. Then the values of k which satisfy condition (24) for d = 1 are k = II - 1 and k = n, corresponding respectively to 1= it and 1= n + 1. The corresponding values of n_(k, 1) are II = s(n-$-6-l) and n_=6(n+$-6-l). The first of these has the smallest value of III. i, and hence has the slowest decaying exponential factor, when II -$< 6-l < n. The second has the smallest In-1 for IE
(rl)fJ”+l(r2)
cm
l-2H,(rl)H,+1(T2)
BY
-(n
cos ($Sy + n4)
+ lMl],
exp [-~p~62/~]cos
[p+Sy -(n + l)~$],
wherep,=n*$-6-l. The remaining case is when S -’ = n + $ for some integer it. Then condition (24) for d = 1 has three solutions: k = n - 1, n, n + 1 (1= n, n + 1, n +2). The corresponding values of n_(k, 1) are -6, 0, +6 respectively. In this case it is necessary to include also the k = 1= 1 term in the asymptotic solution, and one gets r(~,YY)-[l-2H,(~,)~“+1(~2)cos(n+1)91
+2 exp [-crS2/T]{H1(T1)H1(r2)
cos
(Sy -4)
-H”-,(Tl)H,(r2)COS(~y+n~)-H,+,(I’l)H,,+2(~2)COs[~Y-(~+2)~11.
From these results, the following expressions are obtained via the transformation for the leading asymptotic behaviour of V(a, y): II -+
V((T, y)-p_s(4/r)H,_I(Tl)H,(r2)
6-l = n : V(u, y) - 6(2/r)
ePus2’4rWp~(Tl)H,(~2)
+H,(rl)Kt1(r2)
sin BY
n
xe ~“““r{H1(~l)Hl(T2)
(29)
sin ($6~ +M) -(n
(30)
+ 1MlI,
e~~“PZ’2!rsin [p+Sy -(n + l)+l,
V(V,,) --6(4/r)[1-2f&(Tl)H,+l(T*)
-H,+l(~lW,+2(f2)
e~~rrP’s2’rsin(p-sy -m$),
(5)
(31)
(33s (n + u41r’
sin (Sy -4)
sin [SY -(n
+
-Hn--1(T1)Hn(T2)
3411.
sin (6y f nc$)
(32)
The most striking result to emerge from these formulae is that the dominant frequency in the old-age wave is not always the difference frequency SW, but may be anywhere
482
R. W. LARDNER
between So and zero depending on the down-shift ratio S-‘. Only when S-’ equals an integer plus $ is the dominant frequency equal to SW. (When p* are small, the results (29) and (31) lose accuracy except at very large values of u, and further terms should be retained in the solution.) A second property of these asymptotic waves is that in the cases S-’ = n or rz +$, the amplitude of the wave depends on the phase difference 4 between the two primary inputs. In these two cases, if one expresses V in the form
V(0;y)- rS(4/T)P e-rrr2S*‘r sin (ray - $1) (with r = $ when S as follows:
-1
=n andr=lwhenS-’
(33)
= rz +$) then the amplitude factor P is given
Sp'=n:P=[A2,_1+A:+2A,_1A,cos(2n+l)#‘; S-‘=n+;:P=[l-2A,cos(n+
l)d]-‘[l
+A;_l+A:+l
-2(A,_1+A,+l)cos(nil)+ +~A,-~A,+~cos~(~+~)~I"~; (34) here A,,=H,(~I)H,+~(~~), and also H1(T,) = 1, which is appropriate for the region concerned (ri - rr*). In the case (34), P is rather sensitively dependent on 4, Since A,,varies from 0 to 1 as the ri vary, it even appears that P can be made infinitely large by a suitable choice of the ri and 4. This is an illusion however, since, before the & become large enough to make A, equal to 1, the terms in expression (22) corresponding to d = 2 become significant, and the approximate solutions (29)-(32) become invalid. (From expression (28), the d = 2 terms become significant when r, --$(K1)* = 4n2/5. From expression (27), at this value of ri one has H,(r,) ~e-“~ and so A, =e-5'4<$. Equations (33) and (34) cannot be used when A, exceeds this value.) Nevertheless the dependence of P on 4 is of sufficient importance to warrant a further investigation, which is presented in the next section.
5. THE DEPENDENCE OF AMPLITUDE ON PHASE DIFFERENCE In this section the case S-’ = II +i is considered and the dependence of the amplitude factor P,given by expression (34), on the phase difference 4 is examined. In order to obtain sufficiently accurate expressions for A, and A,*I,the approximation (27) for H”(x) seems unlikely to be exact enough. A better approximation can be obtained by retaining the next order terms in expressions (25) and (26) [17], to obtain? K(x)
- exp [-(v2/2x)
+ (v4/24x3) - (v2/4x2)],
x >>Y >>1.
(35)
Using this gives, for m = 0, *l,
_exp (n+m)'+(n+m)' (n+m)' (n+m+1)2+(n+m+l)4 A "frill ____ i 2r1 24r:-724r: 2r2 t The approximation an error of (l/600)%,
-
(35) appears to be astonishingly accurate. For example, it reproduces HS(lO) with an error of l/10% and even H5(5) with an error of 0.6%.
(n+m+l)' 4r: I. HIO(lOO) with
PARAMETRIC
ARRAY
CONVERSION
483
EFFICIENCY
For the region of interest, in which the ri are 0(n2), one can write ri = yin’. Then retaining terms in the exponent only up to order it P2, one can write A “+m _ q e--Xm--Bm2,
(34)
4=exp(-~[I+f(~-~)]-~[l+fi+f(l+~-~)]},
(37) @,L _L+1 2n2 ( y1 J.
x=X+;(l+%
(38, 39)
Substituting expression (36) into expression (34) and retaining only terms of order n -’ gives P= l+K2R(lj),
(40)
R(~)-n2[1-2~cos~]-2{2q2~2-q~~~~(~2-28)-4~~2cos2~), where (I,= (n + l)& Consistent with this order of approximation l3= A/2n2, A
=
one can then write
q = eehj2,
x = Aln,
(4.1)
1*
&o(Ul+ u2)
(L+L >qp-y [ Yl
Y2
2Ll,u2
(42)
This finally leads to the expression R ($) - hq[2hq + (1 - A) cos sl,- 29 co? $I/( 1 - 29 cos +Q2.
(43)
As remarked at the end of the last section, the d = 2 terms should be included when the ri become as large as 4n2/5: that is, when the yi become as large as 0.8. The above results therefore hold only when A is greater than about 2.5. Graphs of R(G) for different values of A are shown in Figure 2. As A becomes large, corresponding to relatively low input levels (but, of course, still above the conventional saturation levels), R(4) becomes very small. In this limit, the theory of section 2 holds.
Figure
2. Graphs
of R(I)) for A 3 2 based on the approximation
of retaining
d = 1 terms only.
484
R. W. LARDNER
As A decreases, Z?(4) becomes significantly different from zero, indicating a departure from this lowest order theory. For A b 3.46, the maximum amplitude occurs when II/ = 180”: that is, when the phase difference between the two primary inputs is 180”/(n + 1). For smaller values of A the value of (I, at which R is maximum decreases towards zero; when A = 3 the maximum occurs at I/J= 120” and when A = 2 at 11% 60”. Within the range A 2 2, the value of R (0) is negative, indicating that when the two input signals are exactly in phase, the asymptotic difference frequency signal is somewhat lower than might be expected on the basis of the lower order theory of section 3. On the other hand a small amount of magnification can be achieved at these input levels by choosing an appropriate phase difference between the two primaries. However, because of the factor n -’ in expression (40), the departures from the theory of section 2 are rather small. For example, when n = 10 (down-shift ratio = 10.5) the total variation of P between its maximum and minimum values is about 3% when A = 6 and just over 4% when A = 2. The graph for the case A = 2 shown in Figure 2 is not to be taken too seriously, since it is outside the range of validity of expression (32). It has been included since it does give a qualitative indication that, at smaller values of A, R(4) may develop a maximum value of sizeable magnitude at a value of 4 which is at or close to zero. If this is the case, then there would be significant enhancement of the difference frequency signal above the saturation level at these higher input strengths. In order to investigate this point, it is necessary to return to equation (22) and include terms for which d > 1. This is the subject of the next section.
6. THE INCLUSION
OF HIGHER
ORDER
TERMS
According to the criterion (28), the d = 2 terms become significant for yI 3 4/5 and the d = 3 terms are significant for y, 2 9/5. Thus for 2.5 2 A 2 1.1 it is sufficiently accurate to include only the d = 2 term in the solution (22). From expression (23) with d = 2 and K’=n+$,onefindsthatn=-6fork=2d-1, =Ofork=2d,and =+6fork=2d+l; otherwise In.-\ > 6. Including these terms and proceeding as in section 4 one readily obtains that, as u + CO, V is given asymptotically by expression (33) with r = 1, where the amplitude factor P is now given by P= [l-2A, +[(A,_,
cos 4+2Bz, -A,+,)
cos 2$]
‘{[l-CA,
sin I,?-(B2n__1 -Bz,,+l)
-,+A,+,)
cos t,b+(&n~~+Bz,,+,)
cos 2&
sin 2+]*}““,
where A,, = H,(TI)H,,+I(I’~) and B, = H,,(~I)H,,+~(I’~). One can approximate A,, and B, by using expression expression (36) as before for A, and also
(35) and one then
recovers
B 2n+m - q2 e -2xm mBm’, where x and 0 are given by expressions
q*=exp
(44)
(38) and (39) and
(-&[4+$(;-&)]
(45)
+[4+;+-$(4+;-4&
Substituting these formulae into the above expression for P, expanding and 8 and retaining only terms up to order l/n2, one obtains P = (1 + af($)-‘[(2e +4f(1k)~(q where
-x2)4
sin I+!-2~
cos * - (20 -4x2)4* sin 2$)*+
f(G) = 1 - 2q cos !/I+ 2q2 cos 2JI. To this order
in powers
of x
cos 241
O(l/n3)}“2, one can use the approximations
PARAMETRIC
ARRAY
CONVERSION
485
EFFICIENC\
expressions (45) and (37) shows also that q2=em2* +O(K’):= one can write P in the form (40) to within an error of order y1m3,
(41). Comparing
q4 + O(n -‘). Therefore
where -A)q
R(4) = Af($)-2{f($)[(1
cos CL--(1 -4A)q4
cos 2&]+2A(q sin *-2q4
sin 249)‘},
with f(9) = 1 - 2q cos r/l+ 2q4 cos 24. Graphs of R(4) for several values of A between 1 and 3 are shown in Figure 3. The graph for A = 3 is very similar to that shown in Figure 2. For A = 2 the graph is qualitatively similar to that in Figure 2, but is quantitatively rather different, especially near (I,= 0. Thus the approximation of retaining only the d = 1 terms has begun to break down at this value of A. The graph for A = 1 in Figure 3 is on the border of the region of validity of the present approximation, and again can be expected to show some numerical departures from the correct result. The most striking feature in Figure 3 is the development of a large maximum of R (4) at 4 = 0 as A decreases below 1.5. This indicates that rather substantial amplification of the difference frequency signal will occur at these levels of signal input provided that the two primary signals are exactly in phase.
6
Figure
3. Graphs
of R(G) for 1 s A s 3 based
on the approximation
of retaining
d = 1 and rl = 2 terms only.
In order to examine the extent to which such amplification is possible one must include terms in l((~, y) corresponding to d > 2 so that A can be reduced below 1. For a general d, one sees from expression (23) that k = nd - 1, nd and nd + 1 lead respectively to n- = -8, 0 and +6. Including terms from d = 1 up to d = D one obtains, from expressions (22) and (S), therefore, that
1 -1
1+
X sin (6y -g)+ 1
+ffnd+l(rl)H(n+l)d+,
i d=l
(-1)dH,d(rl)H(n+lid(r2)
CoS
i
(-l)d[H”d--~(r,)Pr,,+l)d-l
WA
d=l
(r,)
sin
(&‘--@J
-&)I].
d$
sin (6y -6 +d$) (46)
486
R. W. LARDNER
The asymptotic approximation (35) may be used for all of the terms occurring here provided that ri 2 2nD: that is, provided D c $yin t. With this limit on D, the ratio of the second term to the first term in the exponent of expression (35) is v2/12x2= D2/12yfn2< l/48. Thus the second term may be ignored (as also may be the third) without causing too great an error. This provides a significant simplification, and one can obtain approximation formulae analogous to expressions (36) and (44): namely, H nd+m(~dH(n+wi+m W2)-qd
expC-xdm-0m2),
where x and 19are given by expressions (38) and (39), and qd = exp {-$[rT’ + r;l(l+
n-‘)‘]d*} = qd2[1 + 0(1/n)].
Proceeding as before to expand the terms in V in powers of B and x one obtains V in the form (33) with r = 1 where P has the form (40) up to O(K3) and R (4) = fMC2{f($)
-f($)’ -f($)f”($)
f(4) = 1 + 2 ;
d=l
+ V($)},
(47)
(-l)dqd’ cos di,b
primes denoting derivatives with respect to 4. Expression (48) can be recognized as the expansion of the theta function 0&,, q). Provided q is not too close to 1, one can therefore allow D + COand set f(4) = 0&, q). (Note that if one allowed D -, COin expression (46), one would not be able to use the asymptotic approximation (35) for the large values of d which would occur in the sum. It is essential to maintain the bound D G iyin in the argument leading from expression (46) to expression (47).) The numerical results obtained from expressions (47) and (48) support our earlier =0 tentative conclusion that R(4) develops a maximum of appreciable magnitude at I++ as A is decreased. For A < 1, the graph of R($,) is qualitatively similar to that shown in Figure 4 for A = 1. R(180) becomes very small as A is decreased, but the maximum at r,Q= 0 becomes very large. In Figure 4 the graph of log R (0) is shown as a function of A for 0.3 s A s 1.3. As can be seen, R(0) increases by several orders of magnitude
45 k Y 2-
0_ 0
Figure
0.5
4. Graph
x
I-O
of Log R(0) as a function
15
of A
t From expression (28), the terms up to d = D need to be included only when r, *iD’n*. assumed condition will hold provided nD z 10, which will generally be satisfied for all D 2 1.
Therefore
the
PARAMETRIC
ARRAY
CONVERSION
EFFICIENCY
487
as A decreases
below 1. Although the approximations made in deriving the result contained in expressions (40) and (47) will not remain valid for such large values of R as are shown in Figure 4 for A G 0.5, nevertheless there is clearly very substantial amplification of the difference frequency signal when A is decreased below 1. The condition A < 1 is equivalent to 2u1uz/(u1 + u2) > (6-l -$)‘u~. When the two primaries have equal amplitudes this becomes u1 = u2 > (6-l --~)“u~.
ACKNOWLEDGMENT
The work in this paper owed its inception to conversations with Mr Adrian Jongens and Professor Ron New of the Central Acoustical Laboratory at the University of Cape Town, and it is with pleasure that I should like to thank both of them for their continued interest and assistance.
REFERENCES 1. M. J. LIGHTHILL 1952 Proceedings of the Royal Society A211,564-587. On sound generated aerodynamically, I. General theory. 2. M. J. LIGHTHILL 1954 Proceedings of the Royal Society A222, l-32. On sound generated aerodynamically, II. Turbulence as a source of sound. 3. P. J. WESTERVELT 1957 Journal of the Acoustical Society of America 29,934-935. Scattering of sound by sound. 4. P. J. WESTERVELT 1963 Journal of the Acoustical Society of America 35,535-537. Parametric acoustic array. 5. L. BJ~RNIZ)1977 Aspects of SignalProcessing, Part I (editor G. Tacconi). Dordrecht: D. Reidel Publishing Co., pp. 33-59. Parametric acoustic arrays. 6. F. H. FENLON 1974 Journal of the Acoustical Society of America 55, 35-46. On the performance of a dual frequency parametric source via matched asymptotic solution of Burgers’ equation. 7. B. B. CARY 1973 Journal of Sound and Vibration 30,455-464. An exact shock wave solution to Burgers’ equation for parametric excitation of the boundary. 8. B. B. CARY 1975 Journal of Sound and Vibration 42, 235-241. Asymptotic Fourier analysis of a “sawtooth like” wave for dual frequency source excitation. 9. H. 0. BERKTAY 1965 Journal of Sound and Vibration 2, 435-461. Possible exploitation of nonlinear acoustics in underwater transmitting applications. 10. T. G. MUIR 1971 Ph.D. Thesis, University of Texas at Austin. An analysis of the parametric acoustic array for spherical wave fields. 11. J. S. MENDOUSSE 1953 Journal of the Acoustical Society of America 25, 51-54. Nonlinear dissipative distortion of progressive sound waves at moderate amplitudes. 12. D. T. BLACKSTOCK 1964 Journal of the Acoustical Society of America 36, 534-542. Thermoviscous attenuation of plane, periodic, finite amplitude sound waves. 13. L. BJBRN~ 1976 in Acoustics and Vibrations Progress, Volume II (editors R. W. B. Stephens and H. G. Leventhall). London: Chapman & Hall, pp. 101-203. Nonlinear acoustics. 14. G. B. WHITHAM 1974 Linear and Nonlinear Waves. New York: John Wiley-Interscience. 15. E. HOPF 1950 Communications in Pure and Applied Mathematics 3, 201-230. The partial differential equation uI + uu, = pu,,. 16. J. D. COLE 1951 Quarterly of Applied Mathematics 9, 225-236. On a quasilinear parabolic equation occurring in aerodynamics. 17. M. ABRAMOWITZ and I.A. STEGUN 1965 Handbook of Mathematical Functions. New York: Dover Publications.