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Mode conversion of Rayleigh and Lamb waves to compression waves at a metal-liquid interface
Mode conversion of Rayleigh and Lamb waves to compreksi6n waves at a metal-liquid interface M.O. DEIGHTON,
A.B. GILLESPIE,
R.B. PIKE and R.D. WATKINS
A theoretical treatment is given of a potentially useful ultrasonic mode-conversion process, from Rayleigh or Lamb waves (SAW’s) to angled compression waves in an adjacent liquid. Since the compression waves can form a well-defined beam, with typical angular width of only 1 o or 2”, it is possible to use this technique in underliquid viewing applications. Another use, in liquid level measurement, has advantages over conventional ultrasonic methods. Attractive features are the high power efficiency of the mode-conversion, in both directions, and its inherent separation of electromechanical transducer from the liquid, which may be vital in certain hazardous environments. With such applications in mind, the emphasis here is on ultrasonic processes in the liquid. After a descriptive account of pressure wave generation in the liquid, the main part of the paper is a general treatment of far-field beam profiles and includes transverse beam-width and discussion of pulsed operation. Finally we examine inverse mode conversion (compression wave to SAW), to identify the main physical processes occurring and arrive at a figure for power efficiency. This turns out to be 80% (max) against nearly 100% for transmission. Two papers by the same authors describing practical applications version are scheduled for inclusion in the next issue.
Introduction An ultrasonic mode-conversion process is described, which occurs at a metal surface carrying surface acoustic waves (SAW)* when immersed in liquid, resulting in angled compression waves in the liquid. Although the inverse process, at a solid/solid interface, has been applied in the wedge-transducer,’ for generating surface acoustic waves, the solid/liquid mode conversion does not seem wellknown. It is hoped therefore that this paper, with its companions,2Y3 will draw attention to important potential applications, in such fields as liquid-level measurement and remote under-liquid viewing. The advantages stem mainly from the high energy efficiency of the conversion, together with the relatively narrow beams that arise, whilst maintaining a useful isolation‘of the electromechanical transducer from the liquid. The last feature is specially important when dealing with hot and/or hazardous liquids. This paper is concerned with theoretical aspects of the mode-conversion process. A qualitative picture of the mechanism shows how surface waves are converted into oblique compression waves in an adjacent liquid. Being essentially a plane parallel wave system, the latter form a reasonably well-defined ultrasonic beam, leaving the
ULTRASONICS.
NOVEMBER
1981
/060249-l
A following section analyses the far-field angular beam profile in a liquid, produced by a rectangular radiator of defined dimensions. A general expression is derived for the beam profile and numerical examples show how to calculate expected beam width in the vertical plane. Calculation of lateral beam-width leads to the condition for an approximately circular beam cross-section. Pulsed operation is also considered. Finally, inverse mode-conversion is dealt with, from compression waves to surface waves, since this is utilized in most applications. The treatment is intuitive rather than analytical, but it suggests a remarkably high efficiency.
Pressure
The authors are at the Instrumentation and Applied Physics Division, AERE Harwell, Oxon, OX1 1 ORA, UK. Paper received 27 January 1981. Revised 12 June 1981.
0041-624X/81
radiating surface at a known angle and easily detected at some distance. Simple energy considerations show that exponential attenuation of surface waves occurs along the interface, nearly all their energy being transformed into pressure wave energy in the liquid, within typically several surface wavelengths.
Formation
l The terms ‘surface acoustic wave’, SAW or ‘surface wave’ used in this paper include Lamb waves as well as Rayleigh waves.
0 $02.00
of the mode con-
wave
of angled pressure waves in liquid generation
in a liquid
Fig. la illustrates the conversion of surface waves in a plane metal surface to pressure waves in an adjacent liquid. A, B and C are three points on the surface, which carries a SAW represented by the small undulation. This wave travels at velocity vb along the metal/liquid boundary and can be a Rayleigh wave or any suitable Lamb wave mode, provided its velocity exceeds that of sound in the liquid (CL). As any 0 1981
IPC Business Press 249
crest of the wave passes A it sets up a local disturbance in the liquid, which spreads out at velocity cL, forming a circular wavefront as shown, by the time the crest reaches C. Similarly, in passing B, it generates a similar wavelet of smaller radius, shown centred at B, by the time the interface crest reaches C. Clearly every point of the boundary generates a wavelet and, when the original crest reaches C, all these circles touch an inclined straight line PQ passing through C. In fact, the wavelets combine as a single plane wavefront PQC. The angle of inclination of this plane to the interface, 8, in Fig. la, is given by AP
c&e,=-_=
BQ -=
CL -,
AC BC v,, where cL is the velocity of sound in the liquid. For steel immersed in water (using Rayleigh waves), B,,, is typically about 30’.
(1)
= sin 8,
= cL/vb
(2)
as expected. It should be added that this simple treatment assumes twodimensional geometry. In practice, of course, any plate or strip carrying SAW has finite width, which will modify the spread in three dimensions of the radiated compression wave, a point touched on in the next section. Such finite width, incidentally, may also modify the primary Lamb wave itself, through edge effects. A more rigorous two-dimensional solution4 of the wave equations in liquid confirms the above result. Assuming a normal displacement wave at the boundary: v,
= +cos (kbx - at)
2.4= P/z,
(3)
(7)
the positive direction being that of wave propagation. The spatial distributions at any instant of pressure and particle velocities are sketched in Fig. lb, wherein the slanting lines represent peak pressures (solid lines positive, dashed lines negative) and the short arrows indicate particle velocities. Energy considerations. Attenuation of surface wave
It is readily shown that the mean power loss into the liquid from the surface (W rnm2) is: P = G;,
Likewise, the next following crest of the SAW generates an identical plane wavefront at HK in Fig. la. Thus a continuous wave-train along the surface sets up plane parallel compression waves in the adjacent liquid, which propagate away from the interface, at velocity CL and in a direction inclined at angle 19~ to the normal of the interface. The wavelengths of the compression wave (AL) and of the SAW at the boundary &) are related by: x&b
particle velocity is given by:
Z,~COS
em
(8)
which is proportional to the square of the peak (vertical) velocity 6,, at the point in question. This loss is at the expense of power in the Rayleigh or Lamb wave (that is, kinetic energy t elastic strain energy of the metal), which canAte quantified in W m-l width, and is also proportional to uYO. Thus SAW power, being subject to a proportionate loss per unit length at every point, must decay exponentially along the interface, resulting in a corresponding exponential attenuation of surface wave amplitude with distance. For Rayleigh waves the attenuation, in dB wavelength-1, depends only on the two materials and is independent of frequency or wavelength. Typical figures are 0.5 dB wavelength-’ for steel in water, and 1.5 dB, wavelength-’ for aluminium in water. For Lamb waves the situation is more complicated and attenuation in dB wavelength” , of a given mode in a given plate, varies with frequency over quite a wide range. Moreover with Lamb waves of given frequency, there are two possible values of attenuation (dB wavelength-‘), depending on whether the liquid is in contact with one or both surfaces of the plate: the latter figure is twice the former. The magnitude of attenuation rate, as we shall see, is an important parameter influencing angular width of the radiated beam.
The wave in the adjacent liquid has tangential and normal displacement components U, V, respectively, given by:
h/
u-
P
COS(kbx
+ fly - ot)
‘v = f&OS (kbx + @’ - tit) where p2
=
ki - kZ,
(5)
and kL , kb are respective wave-numbers of compression waves in liquid and of the SAW, at radian frequency o. Here co-&nate x is measured along the direction of SAW propagation andy normally to the interface. It can be shown that the plane waves in (4) are inclined at angle em to the interface. Particle displacement and velocity are always normal to the wavefronts and the wave system propagates at velocity CL through the liquid. The acoustic pressure p at points in the liquid is given by: p=-
Z&G
SlIl(kbX+fly - Cd)
where Z, is the acoustic impedance
250
b
.
(6)
cos em
of the liquid, and
a
lb
SAW
L
Metal
-
a - Formation of pressure wave in a liquid; b - instanFig. 1: taneous pressure and velocity distributions in liquid: continuous lines, peak positive pressure; broken lines, peak negative prassure; arrows, particle velocity
ULTRASONICS.
NOVEMBER
1981
A more precise wave analysis should take account of SAW attenuation along the interface and also viscosity of the liquid; the latter would oppose the horizontal slip at the surface implied by Fig. lb, and both effects may become important with high-angle waves (f3, large).
where A is a constant and I in the denominator represents the 1/r variation in a spherically spreading wavelet. For this purpose we neglect the difference between LP and r, but it must be included in the relative phases of signals arriving at P from different elements of the radiator, by including the time-delay LP/cL .
Angular distribution
Resultant pressure at P is given by the integral of (10) over the whole of OA, that is,
of ultrasound
in the far-field
Here we calculate the acoustic pressure field, at a considerable distance from a small immersed radiating surface of defined dimensions, a and b. In Fig. 2, OA represents a finite radiating rectangle of metal/liquid interface, of length a measured along the direction of SAW propagation, and width b normal to the plane of the diagram. Thus, initially, we consider a point P in the median plane at distance r from origin 0, in a direction making angle 13with the normal ON to the interface.
p(r,e,t)
=
[
From the previous section the ultrasound intensity should be maximum in a certain direction B,,, , equal to sin-’ (c~/v~). Now we consider the intensity decrease as 6 deviates either way from B,,, ; this determines the effective beam-width. The width is clearly crucial in any underliquid viewing application, and should be considered both in the plane of the diagram and in the perpendicular plane.
A* r
x
eax
cos[kbx-wjr,
+y)]d.x
(11)
In this expression, note that for convenience time (to) is measured from an instant later than that used in (9) and (lo), by the amount OP/cL. Instead of subtracting LP/cL from t, we therefore add OM/cL to to. The square bracket in the integral can be written compactly as (kOx - wto), where 0 sin e = kb (1 ke = k,, - -
-[she),
(12)
CL
Beam profile
in median
plane
approach is to divide the radiating surface into a large number of elementary strips of length b perpendicular to the plane of Fig. 2, and width du. One such strip located at L, a distance x from the left-hand edge of the radiator, is shown in the diagram. We then sum (or integrate) the individual contributions from all the strips to obtain the resultant pressure amplitude at P due to the whole radiator OA. This phasor sum must take account of amplitude and phase variation of the SAW along OA and also variation of the path length, such as LP, of acoustic signals in transit to P. The
To simplify matters, r is supposed sufficiently can assume :
large that we
Denoting the integral in (11) by 1(0, to), we have a
Ice, to) =
J0..
eecuc cos (kox - oto)
dx
[eew /ks sin (koa - ato)
= &e
(i) all poipts along the length b of an elementary strip are equidistant from P and (ii)
and.g = vb/cL . The physical meaning of ke is the net rate of change of phase of signals arriving at P, with position on OA of their point of origin. Clearly, if sin 0 = cL/vb, then ke = 0, so all such signals are in phase with one another; mutual reinforcement then gives maximum resultant intensity in the direction em.
--CXCOS (kea - oto)j
t ke sin wlo + a coswto]
Separating the sin wto and cos oto components of this expression, and squaring and adding their amplitudes
the lines OP, LP are nearly parallel, so, if we draw LM perpendicular to OP, then LP, MP are almost equal and path differences OP - LP = OM=xsine.
IP
These approximations restrict the analysis to points well beyond the Fresnel zone of the radiator. Assume the acoustic pressure distribution, adjacent to the radiator surface, is: p.
= I;Oe4r
po, immediately
cos (libX - ot)
(9)
where $, is the amplitude at 0, (Yis the attenuation coefficient of the SAW along OA, kb its wave-number and o the radian frequency. In effect, for this calculation, we have a linear radiator with an asymmetric exponential taper function, as sketched in Fig. 2, combined with a progressive linear phase shift along it. The contribution dp to acoustic pressure at P, from the element at L, has amplitude independent of 0 and is: Distance-
+, = -.Abdx r
ULTRASONICS.
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NOVEMBER
1981
Fig. 2 Geometry of radiator field point P (median plane)
for ultrasound
calculation
at a far-
251
.
attenuation, so we assume a + m , while (Yremains finite. Both exponentials in (15) are zero and it becomes
yields the squared pressure amplitude 6) at P, which is the quantity of prime importance. Details of the analysis are given elsewhere4 and one obtains:
1 - 2 cos (k*a). eTora + em20m .
CY’+k;
(13)
I: A-
__ i
Pmax
1
2 =
1
=
using (17).
(14)
(Y2 profile of the beam is:
1 - 2 cos (k@a). e-“’ + e-2aa
This is plotted for Rayleigh waves on stainless steel in water (n = 15, E = 2),as normalized i, in the solid curve of Fig. 3. As expected, the peak is at r3 = 30”, but the total beam width (between - 3 dB points) is only 0.7”.
(Y’ *------ct2 +k;
(1 _eaa)l
In general, for an unlimited radiator, the - 3 dB directions occur where X = + 1, whence, from (17)
(15) where ke = kb (1 - ,$ sin e), .$= vb/cL. A different form of this result, more convenient application, introduces three new quantities:(i)
sine=! I*-!t i 2nn
for general
= nhb,
S0sine”.sinemtc0sem.6e=~tc0sem.6e. (16)
Comparison with (21) then yields the?.wo - 3 dB directions:
6e=+
’ 277n&OSem ’
so the total 3 dB beam-width
@e)SdB
the ‘direction variable’, X, defined by: X = ke/cY = nhbkb (I- [Sill e)
=
(17)
Although this involves the attenuation, its prime purpose is as a directional variable, such that X = 0 at the peak of any beam profile (because sin 0 = l/g there), regardless of the value of angle em for particular materials.
(ii)
a = mnhb or &a = ?‘??
a
I\ \ Pmax i
(15) becomes
= 2 - 2 cos (kOa) ke2a2
(18) that is , j
(15) becomes:
2 = 1 - 2esmcos (mx) t eszm (1 - e-m)2
(22)
1 =a’
aLi? l_e-”
_ sin (G/2)
Pmax
/;
_ tanem ’ 77n
Zero attenuation, finite aperture limits. Here we assume (Y+ 0 while a is finite. Using the limit:
the aperture parameter m, such that total attenuation along aperture is emm; equivalent definitions are:-
With these substitutions,
’ 77n.$c0s em
(in radians) is:
approximately.
= 2m2(1 - .!jsin 0).
(iii)
(21)
+ 60, where 68 is small,
0 = 8,
and we see from the previous section that, for a given solid/liquid interface and assuming Rayleigh waves, n should be constant and independent of frequency, unlike (Y.For stainless steel in water, n = 15 typically, corresponding to attenuation 0.58 dB wavelength-’ . With Lamb waves, however, n varies with frequency. (ii)
!
With a narrow beam one can write
the ‘attenuation number’ n, defined as that length of radiator, in units hb, in which SAW amplitude decreases by a factor e. It is related to (Yby: l/o
(20)
= 1 t (27~1)~ (1 - .$sin e)2
(1 - eaa)l intensity
(ke/ol)2 1
In the direction 6,,, , ke = 0 and the second fraction above has its maximum value, giving:
The normalized
1 +
1 ‘CT’
I $12
Wa) I
as expected, where
(23b)
which simplifies to
In a hypothetical case, suppose .$ = 2 and a = 15 hb . Then @~/2= 15 rr (1 - 2 sin 0) and the plot of (23a) is now the dashed curve in Fig. 3. Some special cases: numerical
(9
252
examples
Aperture limited by attenuation only. Here the interface OA extends for a considerable distance, but the SAW amplitude decays effectively to zero at a moderately small distance from the edge at 0. The active portion of the radiator is then limited solely by
This differs considerably from the other, having a main peak some three times broader (at - 3 dB points), with a series of secondary peaks on either side. These are separated from one another by null points, the first pair at 0 = 27.8” and 0 = 32.2”, on each side of the main peak at 30”. More generally, the - 3 dB points of the profile in (23a) occur at
ULTRASONICS.
NOVEMBER
1981
t i/i
mox
-a=ca,n=l5 (0.5848
1.0 ---a=
wavelength“)
15&,,n=W (Zero attenuation) (0,58dBwavelength-‘1
25
26
27
28
29
31
30
32
33
34
0C’l Fig. 3 Specimen beam-profiles vb=3kmo-1 (cR),cL=15kms-1
for a stainlesssteel/water
interface:
+ 80’ = + 4n/9, so (23b) yields the directions as solutions of
$1/2 z
Writing 0 = 13m + 6 8 and using the same approximation as in Case (i), we obtain a total beam-width: (M),dB
=
8b 9a t cos 8,
Comparing with (22), one sees that, if a& here is equal to the attenuation number n of Case (i), then the beam width is always 2.8 times that of Case (i). (iii)
The general case: finite aperture and attenuation. This case more accurately represents the practical situation, but unfortunately does not give a simple explicit expression for beam width, unlike the two preceding cases. We must therefore resort to a graphical solution, or equivalent method. Equation (19) provides the best basis for this and Fig. 4 shows B/S max as a function of X, for various aperture parameters m. For narrow beams, X is nearly proportional to the (small) angular deviation of 0 from em.
The narrowest peak shown is that for m = 00, the curve From (19) it is seen that all the $lfi InU = (1 + x2)-‘. curves lie above this one, except that each touches the base curve at points where X is a positive or negative multiple of 2n/m. The secondary lobes appear (between the contacts) to a small extent as soon as finite apertures are used, and become more pronounced with severe truncation, as can be seen. Simultaneously the main lobe of the profile becomes much broader (in X-units). One of the curves (form = 1) has been transferred to Fig. 3, using (17) with n = 15, t = 2, to translate X-values into angles. This dot-dashed curve is for the same aperture (15?Q,) as the dashed curve, but with overall attenuation 8.7 dB instead of 0 dB. It is remarkable that almost no difference of 3 dB beam-width exists between these cases, the main effect of attenuation being to fill in the nulls which otherwise occur (and to reduce the absolute value of&,x). The main use of Fig. 4, however, is for comparing beam characteristics of different apertures a, having the same
ULTRASONICS.
NOVEMBER
1981
35
attenuation (dB wavelength-’ or n), since the nearly linear relation between X and 0 is then a fxed one. Normal practice alms to maximize power radiated in the direction err,, which requires that m should be as large as possible; hence the important curves in Fig. 4 are the inner ones. The exception occurs in some short-range applications, where restriction of the aperture may be needed to keep point P sufficiently beyond the Fresnel zone. An example, with stainless steel in liquid sodium, shows how to calculate beam-width, and highlights the limitations of the beam-forming process. Assuming Rayleigh waves (vb = 3 km s-r) and taking cL = 2.4 km s-l, we find 0, = 53 .l” . Although the characteristic acoustic impedance (2,) of liquid Na is somewhat greater than that of water, for these calculations we assume the same attenuation as before, n = 15. With t = 1.25 now, (21) or (22) yields (Ae), dn = 1.6”, for an unlimited aperture. Fig. 4 shows that limiting the aperture to, say, 30 wavelengths of SAW, that is, m = 2, increases beam width by a factor 1.55 to some 2.5”. Particularly noteworthy is the fact that, even with power attenuation eT4 = 0.018 and hence a radiation efficiency over 98%, we nevertheless have truncation broadening of the beam and small sidelobes are present, the fust pair at X = +4, that is, 0 = 50” and 56.5’. Note that an important condition for achieving the predicted beam-widths is that range r must substantially exceed the Fresnel distance. b2 rF = -.
(25)
2AL
Taking b = 15 hb as a reasonable radiator width for a nearly circular beam (see following), and XL = 0.8 hb , We find IF = 140 xb, SO an Operat@ range Of,500 it,,, at least, would be needed for a 2.5” beam width. Increasing aperture length to say 60 &, , in order to approach the 1.6” limit, would involve increasing b by 50% and thus the range to over 1000 Xb. In either case, the beam width (in cm) at the range in question exceeds the radiator width. Lateral
width
of beam
We consider the intensity change when P moves normally to the plane of Fig. 2 to a point P’, such that angle POP’ has a small value J/. The object is to achieve an approximately circular beam cross-section, by a suitable choice of aspect ratio a/b of the radiator. Assuming OP is along the axis of the beam, signals received at P from all strip-elements of the radiator are in phase and directly additive. The same is true of signals received at the offset point P’, to a high accuracy if J/ is small. Hence one can determine the lateral profile by condensing the radiator surface into a single uniform line-source at the edge 0. This profile has a well-known form:
r;= Pmax
I
sin (@‘/2) 4’12
I -
(26a)
where @’ = 2 AL
. bsin$.
(26b)
253
Xz2lLn(l-~slne) total attenuation along aperture = e-“, or 0 = mnhb
B/B mm t .^
that, provided 7 exceeds a modest minimum, ultrasound pulses detected at points in the beam all have flat-topped envelopes, whose amplitudes agree with the cw characteristics derived above. This object can be achieved even with spatial pulse lengths less than the aperture length. Rayleigh waves present the simplest case, being nondispersive. However one can allow for velocity dispersion in a Lamb wave pulse as in the example following, provided that no envelope distortion arises from dispersion* and the adjacent liquid attenuates the pulse exactly as it does a continuous wave.
I
I
I
I
I
I
-10
-8
-6
-4
-2
0
I
I
2
4
I
6
6
10
X Fig. 4 Generalized beam-profile X for various aperture truncation
as a function factors m
of direction
variable
It has the same characteristic multi-lobed shape as the dashed curye in Fig. 3, but of course centred at $J = 0. Since #‘/2 = + 4n/9 in the - 3 dB directions, total transverse beam width is: @+kde
(27)
= 2.
Assuming m = 2, as in the example above, the width in the median plane is 1.55 @%idB
=
7rnlcos
Equating (A$)3 dB a _=-
b
to
= Bm
3.1 Xb nagcos
(28)
Bm
(A@ 3 dB and using hL = hb/t, yields
1.11
(294
cos 8,
Fig. 5 illustrates a general method of approach with a specific case. In this space-time diagram, the diagonal band represents progress at group velocity +,(pr) of a pulse (of width T) along the aperture shown at the left of the figure. The close-spaced interior (phase) lines represent individual Wave-Crests, travelling at phase velocity +,(ph). A velocity ratio vb(pr)/vb@h) = 1.25 is used here, being typical of a zero-order antisymmetric Lamb wave, and this is therefore also the ratio of the slope of the two ‘edge’ lines in the figure to that of the phase lines. Aperture limits are shown at either 15 Xb or 30 &, . Notice in this case that, although a pulse of 12 cycles was introduced at the aperture (the same number would be detected at any aperture point), there are nevertheless 1.5 waves in transit in the complete pulse at any instant; this is a consequence of velocity dispersion. Observe also that new wave crests appear at intervals at the pulse leading edge and others are disappearing similarly from the trailing edge. Pulse characteristics of the radiated beam are related to the ‘view’ from point P in Fig. 2 at each moment. Because LP is less than OP (by x sin e), P sees events at 0 simultaneously with later events at L. Thus at any time (T/c~) + to, measured from the start of the pulse at 0, P has a simdtaneous view of all (x, t) combinations at the radiator such that
which gives the required aspect ratio of the radiator. This result can be written acos0, 7
_ -
(29b)
1.11
b
that is, the aspect ratio, as viewed from P, should be slightly over-square, a conclusion which is not surprising. With m-values other than 2, this result is not much altered, unless m is fairly large. This is expected, since the effective aperture length is then dominated by attenuation. Values of the numerical constant in (29) are tabulated in Table 1, for a few values of m. Pulsed operation
Since most systems operate with pulses, one must consider possible beam degradation when the SAW comprises only a few sine-wave cycles. In the near-field, the effect is broadly to limit the number of plane parallel waves produced in the liquid by each pulse and is not considered further, since interest centres on the far-field beam. We confine attention to a rectangular pulse envelope (width T), injected at the radiator, and conclude Table 1.
(30)
CL
These points constitute a moving ‘view-line’ in Fig. 5, through the varying point to on the time-axis and with constant slope cL /sin e . Instantaneous pressure at P is proportional to aperture surface pressures** integrated along this line, the integral being generally equivalent to that in (11). For axial points in the beam (0 = e,), the slope of the view-line is vb(&) and it is parallel with the phase-lines in the figure; four successive positions, at a small negative value of to and at to = 0, t 1 and tz, are shown by dashed lines, t1 and tz being chosen so their lines intersect the pulse ‘trailing edge”at the respective aperture limits a, and a*. The dotted lines at to = t2, with slightly altered slopes, represent the views from typical off-axis points (the - 6 dB directions, with a = 15 Xb, n = 15) at this instant. In pulsed operation, as distinct from cw, one evaluates the above integral at each instant to only over that portion of the view-line within the (shaded) pulse area in Fig. 5.
Aspect ratio of radiator for circular beam
m
1.5
2
2.5
a cosen;l/b
1.075
1.11
1.19
254
f=fo+XSln
3 -1.29
4
‘This requires that group velocity of frequencies.
1.50
l
be constant over a suitable range
* Each point (x, t) in Fig. 5 has a unique pressure value po,given by (9) inside the pulse band; outside this region pg = 0 everywhere.
ULTRASONICS.
NOVEMBER
1981
Clearly, if 19= B,,, and to is anywhere in the range 0 to tl for aperture al (or 0 to t2 for a = a2 ), then the integral limits are x = 0 and a, as in (1 l), thus yielding a result identical to that for continuous waves. This therefore defines the constant-amplitude part of the received pulse at P, which is constructed from only those waves that travel from end to end of the selected aperture. Such a flat-topped envelope occurs provided the pulse width exceeds the difference between group and phase transit times along the aperture, that is
r>
a ‘b@h)
a (31)
‘b(gr)
from the geometry of the diagram. Thus we find minimum usable pulse widths in our example of six cycles for a = 30 &,, or three cycles for a = 15 hb ; these figures are modified by only about 0.6 cycle in each case, if the receiver is moved to the - 6 dB off-axis points. Evidently the 12-cycle pulse illustrated would be satisfactory in both cases. Notice that the aperture 30 b is effective here, even though never more than half-filled by the actual pulse. A different situation arises when t, is a little outside the limits above; the view-line now intersects one or other of the pulse ‘edges’ shown in Fig. 5, with consequent truncation of the integral at one of its limits. These te-dependent limits are responsible for the rising or falling edges of the received pulse envelope. Short of a complete analysis, certain useful generalizations are possible when 13= 19, :-
(4
the duration of each transient hand side of inequality (3 l),
is equal to the right-
@I the
shape of each varies, with pulse-phase attenuation along the aperture, from nearly linear with low (Yto very curved with high (IL,
(cl
the direction of any curvature is opposite, when vb(.& exceeds vi,(@), to that when it is less.
If ti departs from B,,, , the transient durations change little, but the effect on their shapes is less easy to predict.
Non-dispersive (for example, Rayleigh) waves are a special case, in which phase-lines in Fig. 5 run parallel with the pulse ‘edges’. Axial points in the beam receive pulses identical in shape and width to those fed to the aperture, while off-axis points see short-duration transient edges to the envelope. The amplitude of the major flat portion, however, should still accord with the cw theory. To summarize, the effective beam widths under pulsed conditions are the same as those derived for continuous waves, provided:-
(9
the (rectangular) pulse envelope width fed to the radiating aperture somewhat exceeds the difference between SAW phase and group transit times along the aperture, and
(ii)
only the resulting flat top of the received pulse envelope is used in a plot of beam amplitude against direction 0.
Mode conversion from liquid compression waves to Rayleigh or Lamb waves Having produced an ultrasonic beam in a well-defined direction, one would look for reflections from a target. object. Important information might be the presence or absence of the object, or its direction from the transmitter. Using short pulses, the echo delay would also determine the object distance. In such cases, the receiver could be similar to the transmitter. Often, indeed, the same assembly can serve both functions, depending on the capacity of the generating transducer to detect SAW’s arriving from elsewhere. Here, our concern is with the other link of the reception process, namely conversion of ultrasonic waves in liquid to SAW on a metal plate or strip positioned to receive them. One would expect this to be most efficient when the almost plane waves arriving from a distant object are inclined at angle em to the interface, since intersections of consecutive fronts with the surface then match the appropriate SAW wavelength (see Fig. 1). Thus, pressure variations in the incident wave can build up the required surface-wave. Questions that arise are: How is SAW amplitude altered if the wave arrives from a different direction? Also, how does it depend on the length a of plate immersed in the insonified liquid? The reciprocity principle provides answers to these questions and also throws interesting light on the detailed mechanism of SAW build-up in the strip. Use of the reciprocity principle The reciprocity
A2
4 Fig. 5
Space-time
ULTRASONICS.
Time,
t-
diagram of pulse transit at aperture
NOVEMBER
1981
principle states that a complete system with two (linear) transducers Tr and T2, as sketched in Fig. 6, behaves as a linear passive four-terminal electrical network; it therefore has identical transfer impedances in both directions, for any given configuration. It follows that the e-dependence of output voltage from one pair of terminals, with a given current drive to the other pair and all other parameters fixed, is independent of which transducer is the transmitter. We deduce that the directional properties of the Tr /strip combination, when receiving plane waves from any direction 0, are the same as its transmitting characteristics, as illustrated in Figs 3 and 4. One consequence is that, if one uses a single transducer/ strip assembly to ‘view’ a small object at P, using ultra-
255
T2
3
Jc
/p
/
&;i’ A!r
Mox~mum omplltude-
1 ----
1
Liquid
/r
’ SAW amplitude
Fig. 6 Transmitter/receiver principle
system for application
of reciprocity
sound reflected or scattered back along the transmission path, then the overall dependence of received signal (that is, voltage at terminals 1 and 2) on direction of the object is proportional to the squares of ordinates of the curves in Figs 3 and 4. Thus the visibility angle is rather smaller than the beam-width calculated for transmission (or reception) alone. Next consider the effect of varying length a of the strip, while keeping 0 constant at 8, (for simplicity). With T1 as transmitter, the output of Tz is proportional to pmax and hence to 1 - e-w, from (14). By reciprocity, if Tz instead transmits quasi-plane waves towards OA at the optimum angle, then the output of T1 is likewise proportional to 1 - emm. Thus SAW amplitude, at the exit point from the liquid, saturates as a increases beyond three or four times n&, . Hence there is no point in making a receiving strip longer than this. We infer also that the build-up of amplitude along a fixed length of strip is a similar exponential function of position, starting from zero at the free end; this is shown in Fig. 6. In this case, a is about 2 nhb and the emerging SAW amplitude is a little less than the saturation value; thereafter the strip is in air and the wave reaches T1 without further change. Since reciprocity still applies in pulsed conditions, important pulse characteristics of a receiving assembly can be deduced from results in the previous section. For example, the response to a finite rectangular pulse of incident compression waves will be a flat-topped envelope at T1 output, provided the pulse width exceeds the limit set by (31). SAW build-up
and conversion
4
interface, the pressure of reflected wave (2) is in phase with that of incident wave (1) everywhere at the reflecting surface. So positive pressure planes of wave (1) intersect the surface at the same points as positive planes of wave (2), and similarly for negative planes. Thus net pressure applied to the metal surface is at first twice that in the incident wave, since the amplitudes of waves (1) and (2) are equal. This combined pattern travels along the interface at velocity vb , thereby inducing a surface wave. The crests of the SAW will be about a quarter-wavelength ahead of the pressure peaks, as shown, thus allowing positive pressure to act at falling points and negative pressure at rising points. In this way maximum energy is transferred to the metal surface. Once a finite SAW has built up, the surface-radiated wave (3), whose wavefronts are shown separately, also appears, as described earlier. This is simply superimposed on the other two waves, by virtue of the superposition theorem, since we have in effect combined the respective boundary conditions. An important feature of wave system (3) is that its positive pressure planes (long dashes) intersect the surface at the leading slopes of the SAW undulation, while the negative planes (dot-dashed) are associated with trailing slopes of the SAW, in accord with Fig. lb. (These lines are displaced slightly above those of wave (2) in Fig. 7, for clarity, though they are actually coincident.) The significant conclusion is that wave-systems (2) and (3) are oppositely phased, or have a relative displacement of half a wavelength. Thus they tend to cancel each other. Suppose the SAW at a point gives rise to a wave (3) amplitude just equal to the amplitude $ of wave (1) or wave (2). Mutual cancellation of waves (2) and (3) then occurs and only wave (1) remains. In this small region of the surface, the entire energy of the local incident wave is converted to surface-wave energy, with 100% efficiency, and added to SAW power already present. Elsewhere at the surface varying amounts of energy are returned to the liquid, either as mainly reflected waves (nearer the end A) or as mainly radiated waves (nearer the exit point 0). This is illustrated in Fig. 8. Thus overall, the power efficiency of a metal strip, for converting uniform liquid pressure waves to SAW, cannot be 100%. It is pertinent to ask what efficiency is achieved in reception. Before answering this, consider conditions at 0, where SAW amplitude has reached its saturation value, assuming a is long enough. It is seen that pressure amplitude in wave
efficiency
The central role played by attenuation coefficient (Yin the SAW build-up suggests that the process is a balance between the incident wave-train, the ordinary reflected wave (which is always present) and the re-radiated wave (which depends on SAW amplitude at the point in question). This balance varies from point to point along AO. Fig. 7 shows the three component waves in the immediate vicinity of a point on AO. The arrows indicate propagation directions of the wave-systems, all inclined at em to the normal of the interface. Positive pressure wavefronts for waves (1) and (2) are indicated by solid lines and negative pressure by broken lines, as before. Assuming a motionless
256
~ ------
Positive pressure fronts NegMlve II II
11 u
‘8 *’
lncldentond reflected waves @ ond @
_
-
Positive
-
-
Negative
4
DirectIon of surface velocity
t
Fig. 7
Radloted wove @
Detail of three wave-systems at a point of the strip
ULTRASONICS.
NOVEMBER
1981
surface only. If liquid is adjacent to both sides of the strip, one must allow for a wave radiated from the lower surface as well. The modifications to the theory are mostly straightforward, but note that (Y is doubled, while the maximum attainable SAW amplitude is halved. In the latter condition, near to the exit point, the whole of the incident power is removed by the wave radiated from the lower surface. (Waves (2) and (3) leaving upper surface cancel.) The expression for overall power efficiency is found to be similar to that in (32), but it lacks the factor 2 and uses the new value for 01.Hence maximum power efficiency is now only 40%. Thus there is a real advantage both in transmission and reception, at least for remote viewing applications, in confining liquid contact to one side of the strip only.
(3)is here 2;, so the resultant of waves (2) and (3) has amplitude ;: phased as (3). Power leaving the surface here is equal to incident power and no further increase of SAW amplitude occurs. Alternatively, when pressure in wave (3) is 2;, this exactly cancels the resultant pressure 2: at the metal surface, from waves (1) and (2) acting together. With zero net pressure at the interface, from all three waves, the surface acoustic wave proceeds along the surface with unchanging amplitude, as though the metal were in contact with air or vacuum. Assume the incident wave peak pressure is ; and mean power arriving at the surface is uniformly distributed and Pi per unit length. At a point distant s from the end A of the strip:amplitude
of wave (2) = I;, independent
amplitude
of wave (3) = 2$(1 - ewm),
since this is pfpportional approaches 2p.
Conclusions
of s,
to SAW amplitude
The basic theory underlying a useful ultrasonic modeconversion process has been given. This involves energy transfer from a surface acoustic wave (Rayleigh or Iamb wave) in a plane solid surface to an angled compression wave in a surrounding liquid. The main emphasis is on the beam-forming properties of the compression wave, including analysis of median and transverse beam-width and discussion of pulsed operation. We have also examined the converse process, that is, conversion from pressure waves to SAW, since this occurs in many applications.
and ultimately
Therefore net amplitude of wave leaving surface =p - 2;(1 - eds) =l(2eeQS - 1). Since power density in a wave is proportional to (pressure amplitude)’ , net power per unit length leaving the metal surface at the point distant s from A is: Pi(2e-“S
- 1)2 .
The variation of all these quantities in Fig. 8. Integrating
The importance of the subject is twofold; firstly because conversion power efficiency of both processes is very high (100% for the first, up to 80% for the converse), secondly for the practical reason that,the electro-mechanical transducer can be separated from the liquid by a length of metal strip or plate bearing the SAW. The technique is thus attractive in hazardous or high-temperature environments, as for example in instrumentation of the Fast Reactor.
along A0 is sketched
over the length Q of the immersed surface,
total incident
power = Pia,
a Pi(2eeas - 1)2 ds
total power returned to liquid = s
0
= pia _
2pi
(1 _ e-aa)2
a!
Therefore power leaving as SAW via strip = (2pJo) (1 - e-(Y0)2
SAW amplitude
(Note that this expression is proportional to squared SAW amplitude at 0; maximum available SAW power is represented by the shaded area at the bottom of Fig. 8.) So efficiency (7)) =
total incident
Y
---------
SAW power power
--------= 2(1 - e-m)2
(32) . I
m wherem
=(~a.
A plot of this (Fig. 9) reveals the surprising result that power efficiency of the pressure-wave-to-surface-wave conversion process can exceed 80%, if a is about 1.25 nhb, and r7 > 60% for 0.5 <
nQ
71”
1 ---------- :$ Point of 100% conversionefficiency/ +P^
Pressure amplitude wave @[negative sign indicates hase opposite to wave 62 1
Net pressure amplitude Waves @ + @
< 3
This does not prejudice earlier findings on the dependence of SAW amplitude on length of receiving strip; in particular, this amplitude is only about 0.7 of its maximum possible value, when the conversion (power) efficiency is optimum. It should be said that the arguments above tacitly assume a Rayleigh wave, or a Lamb wave with liquid against one
ULTRASONICS.
Pressure amplitude Wave (ij arwave @
0
NOVEMBER
1981
Power densities orrivmg departing Distance s from end A
_
Fig. 8 Distribution along A0 of pressure in three wave-systems and resulting power densities
257
IO
Cond,+,ons
described in the companion papers. It is anticipated that other applications could well arise in the future. Meanwhile, further theoretical work is needed on aspects not covered in the present paper, notably those mentioned at the end of the second section, since high-angle waves will often occur in liquid-sodium work.
Llqwd I” contact with one surface only (Angleof lncldence = em
0.6 F
References
04
1 2
Y
OO
2 o/blhtJ
Fig.9
Total power efficiency
I 3
I
I I
J
4
3
or m
of an immersed stripas a receiver
258
one a new type of liquid-level gauge, are
Rayleigh and Lamb Waves, Plenum Press, New York (1967) Gillespie, AS., Deighton, M.O., Pike, R.B., Watkins, RD.,
A New Ultrasonic Technique for the Measurement of Liquid Level. Ultrasonics (1982) in press Watkins, RD., Deighton, M.O., Gillespie, AS., Pike, R.B., A Proposed Method for Generating and Receiving Narrow Beams of Ultrasound in the Fast Reactor Liquid Sodium Environment,
4
Two applications,
Viktorov, I&
Ultrasonics (1982)
in press
Deighton, M.O., Gillespie, A.B., Pike, R.B., Watkins, R.D., Mode Conversion of Rayleigh and Lamb Waves to Comp ression Waves at a Metal-Liquid Interface, AERE-R.9963
ULTRASONICS.
NOVEMBER
1981
A.B. GILLESPIE,
R.B. PIKE and R.D. WATKINS
A theoretical treatment is given of a potentially useful ultrasonic mode-conversion process, from Rayleigh or Lamb waves (SAW’s) to angled compression waves in an adjacent liquid. Since the compression waves can form a well-defined beam, with typical angular width of only 1 o or 2”, it is possible to use this technique in underliquid viewing applications. Another use, in liquid level measurement, has advantages over conventional ultrasonic methods. Attractive features are the high power efficiency of the mode-conversion, in both directions, and its inherent separation of electromechanical transducer from the liquid, which may be vital in certain hazardous environments. With such applications in mind, the emphasis here is on ultrasonic processes in the liquid. After a descriptive account of pressure wave generation in the liquid, the main part of the paper is a general treatment of far-field beam profiles and includes transverse beam-width and discussion of pulsed operation. Finally we examine inverse mode conversion (compression wave to SAW), to identify the main physical processes occurring and arrive at a figure for power efficiency. This turns out to be 80% (max) against nearly 100% for transmission. Two papers by the same authors describing practical applications version are scheduled for inclusion in the next issue.
Introduction An ultrasonic mode-conversion process is described, which occurs at a metal surface carrying surface acoustic waves (SAW)* when immersed in liquid, resulting in angled compression waves in the liquid. Although the inverse process, at a solid/solid interface, has been applied in the wedge-transducer,’ for generating surface acoustic waves, the solid/liquid mode conversion does not seem wellknown. It is hoped therefore that this paper, with its companions,2Y3 will draw attention to important potential applications, in such fields as liquid-level measurement and remote under-liquid viewing. The advantages stem mainly from the high energy efficiency of the conversion, together with the relatively narrow beams that arise, whilst maintaining a useful isolation‘of the electromechanical transducer from the liquid. The last feature is specially important when dealing with hot and/or hazardous liquids. This paper is concerned with theoretical aspects of the mode-conversion process. A qualitative picture of the mechanism shows how surface waves are converted into oblique compression waves in an adjacent liquid. Being essentially a plane parallel wave system, the latter form a reasonably well-defined ultrasonic beam, leaving the
ULTRASONICS.
NOVEMBER
1981
/060249-l
A following section analyses the far-field angular beam profile in a liquid, produced by a rectangular radiator of defined dimensions. A general expression is derived for the beam profile and numerical examples show how to calculate expected beam width in the vertical plane. Calculation of lateral beam-width leads to the condition for an approximately circular beam cross-section. Pulsed operation is also considered. Finally, inverse mode-conversion is dealt with, from compression waves to surface waves, since this is utilized in most applications. The treatment is intuitive rather than analytical, but it suggests a remarkably high efficiency.
Pressure
The authors are at the Instrumentation and Applied Physics Division, AERE Harwell, Oxon, OX1 1 ORA, UK. Paper received 27 January 1981. Revised 12 June 1981.
0041-624X/81
radiating surface at a known angle and easily detected at some distance. Simple energy considerations show that exponential attenuation of surface waves occurs along the interface, nearly all their energy being transformed into pressure wave energy in the liquid, within typically several surface wavelengths.
Formation
l The terms ‘surface acoustic wave’, SAW or ‘surface wave’ used in this paper include Lamb waves as well as Rayleigh waves.
0 $02.00
of the mode con-
wave
of angled pressure waves in liquid generation
in a liquid
Fig. la illustrates the conversion of surface waves in a plane metal surface to pressure waves in an adjacent liquid. A, B and C are three points on the surface, which carries a SAW represented by the small undulation. This wave travels at velocity vb along the metal/liquid boundary and can be a Rayleigh wave or any suitable Lamb wave mode, provided its velocity exceeds that of sound in the liquid (CL). As any 0 1981
IPC Business Press 249
crest of the wave passes A it sets up a local disturbance in the liquid, which spreads out at velocity cL, forming a circular wavefront as shown, by the time the crest reaches C. Similarly, in passing B, it generates a similar wavelet of smaller radius, shown centred at B, by the time the interface crest reaches C. Clearly every point of the boundary generates a wavelet and, when the original crest reaches C, all these circles touch an inclined straight line PQ passing through C. In fact, the wavelets combine as a single plane wavefront PQC. The angle of inclination of this plane to the interface, 8, in Fig. la, is given by AP
c&e,=-_=
BQ -=
CL -,
AC BC v,, where cL is the velocity of sound in the liquid. For steel immersed in water (using Rayleigh waves), B,,, is typically about 30’.
(1)
= sin 8,
= cL/vb
(2)
as expected. It should be added that this simple treatment assumes twodimensional geometry. In practice, of course, any plate or strip carrying SAW has finite width, which will modify the spread in three dimensions of the radiated compression wave, a point touched on in the next section. Such finite width, incidentally, may also modify the primary Lamb wave itself, through edge effects. A more rigorous two-dimensional solution4 of the wave equations in liquid confirms the above result. Assuming a normal displacement wave at the boundary: v,
= +cos (kbx - at)
2.4= P/z,
(3)
(7)
the positive direction being that of wave propagation. The spatial distributions at any instant of pressure and particle velocities are sketched in Fig. lb, wherein the slanting lines represent peak pressures (solid lines positive, dashed lines negative) and the short arrows indicate particle velocities. Energy considerations. Attenuation of surface wave
It is readily shown that the mean power loss into the liquid from the surface (W rnm2) is: P = G;,
Likewise, the next following crest of the SAW generates an identical plane wavefront at HK in Fig. la. Thus a continuous wave-train along the surface sets up plane parallel compression waves in the adjacent liquid, which propagate away from the interface, at velocity CL and in a direction inclined at angle 19~ to the normal of the interface. The wavelengths of the compression wave (AL) and of the SAW at the boundary &) are related by: x&b
particle velocity is given by:
Z,~COS
em
(8)
which is proportional to the square of the peak (vertical) velocity 6,, at the point in question. This loss is at the expense of power in the Rayleigh or Lamb wave (that is, kinetic energy t elastic strain energy of the metal), which canAte quantified in W m-l width, and is also proportional to uYO. Thus SAW power, being subject to a proportionate loss per unit length at every point, must decay exponentially along the interface, resulting in a corresponding exponential attenuation of surface wave amplitude with distance. For Rayleigh waves the attenuation, in dB wavelength-1, depends only on the two materials and is independent of frequency or wavelength. Typical figures are 0.5 dB wavelength-’ for steel in water, and 1.5 dB, wavelength-’ for aluminium in water. For Lamb waves the situation is more complicated and attenuation in dB wavelength” , of a given mode in a given plate, varies with frequency over quite a wide range. Moreover with Lamb waves of given frequency, there are two possible values of attenuation (dB wavelength-‘), depending on whether the liquid is in contact with one or both surfaces of the plate: the latter figure is twice the former. The magnitude of attenuation rate, as we shall see, is an important parameter influencing angular width of the radiated beam.
The wave in the adjacent liquid has tangential and normal displacement components U, V, respectively, given by:
h/
u-
P
COS(kbx
+ fly - ot)
‘v = f&OS (kbx + @’ - tit) where p2
=
ki - kZ,
(5)
and kL , kb are respective wave-numbers of compression waves in liquid and of the SAW, at radian frequency o. Here co-&nate x is measured along the direction of SAW propagation andy normally to the interface. It can be shown that the plane waves in (4) are inclined at angle em to the interface. Particle displacement and velocity are always normal to the wavefronts and the wave system propagates at velocity CL through the liquid. The acoustic pressure p at points in the liquid is given by: p=-
Z&G
SlIl(kbX+fly - Cd)
where Z, is the acoustic impedance
250
b
.
(6)
cos em
of the liquid, and
a
lb
SAW
L
Metal
-
a - Formation of pressure wave in a liquid; b - instanFig. 1: taneous pressure and velocity distributions in liquid: continuous lines, peak positive pressure; broken lines, peak negative prassure; arrows, particle velocity
ULTRASONICS.
NOVEMBER
1981
A more precise wave analysis should take account of SAW attenuation along the interface and also viscosity of the liquid; the latter would oppose the horizontal slip at the surface implied by Fig. lb, and both effects may become important with high-angle waves (f3, large).
where A is a constant and I in the denominator represents the 1/r variation in a spherically spreading wavelet. For this purpose we neglect the difference between LP and r, but it must be included in the relative phases of signals arriving at P from different elements of the radiator, by including the time-delay LP/cL .
Angular distribution
Resultant pressure at P is given by the integral of (10) over the whole of OA, that is,
of ultrasound
in the far-field
Here we calculate the acoustic pressure field, at a considerable distance from a small immersed radiating surface of defined dimensions, a and b. In Fig. 2, OA represents a finite radiating rectangle of metal/liquid interface, of length a measured along the direction of SAW propagation, and width b normal to the plane of the diagram. Thus, initially, we consider a point P in the median plane at distance r from origin 0, in a direction making angle 13with the normal ON to the interface.
p(r,e,t)
=
[
From the previous section the ultrasound intensity should be maximum in a certain direction B,,, , equal to sin-’ (c~/v~). Now we consider the intensity decrease as 6 deviates either way from B,,, ; this determines the effective beam-width. The width is clearly crucial in any underliquid viewing application, and should be considered both in the plane of the diagram and in the perpendicular plane.
A* r
x
eax
cos[kbx-wjr,
+y)]d.x
(11)
In this expression, note that for convenience time (to) is measured from an instant later than that used in (9) and (lo), by the amount OP/cL. Instead of subtracting LP/cL from t, we therefore add OM/cL to to. The square bracket in the integral can be written compactly as (kOx - wto), where 0 sin e = kb (1 ke = k,, - -
-[she),
(12)
CL
Beam profile
in median
plane
approach is to divide the radiating surface into a large number of elementary strips of length b perpendicular to the plane of Fig. 2, and width du. One such strip located at L, a distance x from the left-hand edge of the radiator, is shown in the diagram. We then sum (or integrate) the individual contributions from all the strips to obtain the resultant pressure amplitude at P due to the whole radiator OA. This phasor sum must take account of amplitude and phase variation of the SAW along OA and also variation of the path length, such as LP, of acoustic signals in transit to P. The
To simplify matters, r is supposed sufficiently can assume :
large that we
Denoting the integral in (11) by 1(0, to), we have a
Ice, to) =
J0..
eecuc cos (kox - oto)
dx
[eew /ks sin (koa - ato)
= &e
(i) all poipts along the length b of an elementary strip are equidistant from P and (ii)
and.g = vb/cL . The physical meaning of ke is the net rate of change of phase of signals arriving at P, with position on OA of their point of origin. Clearly, if sin 0 = cL/vb, then ke = 0, so all such signals are in phase with one another; mutual reinforcement then gives maximum resultant intensity in the direction em.
--CXCOS (kea - oto)j
t ke sin wlo + a coswto]
Separating the sin wto and cos oto components of this expression, and squaring and adding their amplitudes
the lines OP, LP are nearly parallel, so, if we draw LM perpendicular to OP, then LP, MP are almost equal and path differences OP - LP = OM=xsine.
IP
These approximations restrict the analysis to points well beyond the Fresnel zone of the radiator. Assume the acoustic pressure distribution, adjacent to the radiator surface, is: p.
= I;Oe4r
po, immediately
cos (libX - ot)
(9)
where $, is the amplitude at 0, (Yis the attenuation coefficient of the SAW along OA, kb its wave-number and o the radian frequency. In effect, for this calculation, we have a linear radiator with an asymmetric exponential taper function, as sketched in Fig. 2, combined with a progressive linear phase shift along it. The contribution dp to acoustic pressure at P, from the element at L, has amplitude independent of 0 and is: Distance-
+, = -.Abdx r
ULTRASONICS.
(10)
NOVEMBER
1981
Fig. 2 Geometry of radiator field point P (median plane)
for ultrasound
calculation
at a far-
251
.
attenuation, so we assume a + m , while (Yremains finite. Both exponentials in (15) are zero and it becomes
yields the squared pressure amplitude 6) at P, which is the quantity of prime importance. Details of the analysis are given elsewhere4 and one obtains:
1 - 2 cos (k*a). eTora + em20m .
CY’+k;
(13)
I: A-
__ i
Pmax
1
2 =
1
=
using (17).
(14)
(Y2 profile of the beam is:
1 - 2 cos (k@a). e-“’ + e-2aa
This is plotted for Rayleigh waves on stainless steel in water (n = 15, E = 2),as normalized i, in the solid curve of Fig. 3. As expected, the peak is at r3 = 30”, but the total beam width (between - 3 dB points) is only 0.7”.
(Y’ *------ct2 +k;
(1 _eaa)l
In general, for an unlimited radiator, the - 3 dB directions occur where X = + 1, whence, from (17)
(15) where ke = kb (1 - ,$ sin e), .$= vb/cL. A different form of this result, more convenient application, introduces three new quantities:(i)
sine=! I*-!t i 2nn
for general
= nhb,
S0sine”.sinemtc0sem.6e=~tc0sem.6e. (16)
Comparison with (21) then yields the?.wo - 3 dB directions:
6e=+
’ 277n&OSem ’
so the total 3 dB beam-width
@e)SdB
the ‘direction variable’, X, defined by: X = ke/cY = nhbkb (I- [Sill e)
=
(17)
Although this involves the attenuation, its prime purpose is as a directional variable, such that X = 0 at the peak of any beam profile (because sin 0 = l/g there), regardless of the value of angle em for particular materials.
(ii)
a = mnhb or &a = ?‘??
a
I\ \ Pmax i
(15) becomes
= 2 - 2 cos (kOa) ke2a2
(18) that is , j
(15) becomes:
2 = 1 - 2esmcos (mx) t eszm (1 - e-m)2
(22)
1 =a’
aLi? l_e-”
_ sin (G/2)
Pmax
/;
_ tanem ’ 77n
Zero attenuation, finite aperture limits. Here we assume (Y+ 0 while a is finite. Using the limit:
the aperture parameter m, such that total attenuation along aperture is emm; equivalent definitions are:-
With these substitutions,
’ 77n.$c0s em
(in radians) is:
approximately.
= 2m2(1 - .!jsin 0).
(iii)
(21)
+ 60, where 68 is small,
0 = 8,
and we see from the previous section that, for a given solid/liquid interface and assuming Rayleigh waves, n should be constant and independent of frequency, unlike (Y.For stainless steel in water, n = 15 typically, corresponding to attenuation 0.58 dB wavelength-’ . With Lamb waves, however, n varies with frequency. (ii)
!
With a narrow beam one can write
the ‘attenuation number’ n, defined as that length of radiator, in units hb, in which SAW amplitude decreases by a factor e. It is related to (Yby: l/o
(20)
= 1 t (27~1)~ (1 - .$sin e)2
(1 - eaa)l intensity
(ke/ol)2 1
In the direction 6,,, , ke = 0 and the second fraction above has its maximum value, giving:
The normalized
1 +
1 ‘CT’
I $12
Wa) I
as expected, where
(23b)
which simplifies to
In a hypothetical case, suppose .$ = 2 and a = 15 hb . Then @~/2= 15 rr (1 - 2 sin 0) and the plot of (23a) is now the dashed curve in Fig. 3. Some special cases: numerical
(9
252
examples
Aperture limited by attenuation only. Here the interface OA extends for a considerable distance, but the SAW amplitude decays effectively to zero at a moderately small distance from the edge at 0. The active portion of the radiator is then limited solely by
This differs considerably from the other, having a main peak some three times broader (at - 3 dB points), with a series of secondary peaks on either side. These are separated from one another by null points, the first pair at 0 = 27.8” and 0 = 32.2”, on each side of the main peak at 30”. More generally, the - 3 dB points of the profile in (23a) occur at
ULTRASONICS.
NOVEMBER
1981
t i/i
mox
-a=ca,n=l5 (0.5848
1.0 ---a=
wavelength“)
15&,,n=W (Zero attenuation) (0,58dBwavelength-‘1
25
26
27
28
29
31
30
32
33
34
0C’l Fig. 3 Specimen beam-profiles vb=3kmo-1 (cR),cL=15kms-1
for a stainlesssteel/water
interface:
+ 80’ = + 4n/9, so (23b) yields the directions as solutions of
$1/2 z
Writing 0 = 13m + 6 8 and using the same approximation as in Case (i), we obtain a total beam-width: (M),dB
=
8b 9a t cos 8,
Comparing with (22), one sees that, if a& here is equal to the attenuation number n of Case (i), then the beam width is always 2.8 times that of Case (i). (iii)
The general case: finite aperture and attenuation. This case more accurately represents the practical situation, but unfortunately does not give a simple explicit expression for beam width, unlike the two preceding cases. We must therefore resort to a graphical solution, or equivalent method. Equation (19) provides the best basis for this and Fig. 4 shows B/S max as a function of X, for various aperture parameters m. For narrow beams, X is nearly proportional to the (small) angular deviation of 0 from em.
The narrowest peak shown is that for m = 00, the curve From (19) it is seen that all the $lfi InU = (1 + x2)-‘. curves lie above this one, except that each touches the base curve at points where X is a positive or negative multiple of 2n/m. The secondary lobes appear (between the contacts) to a small extent as soon as finite apertures are used, and become more pronounced with severe truncation, as can be seen. Simultaneously the main lobe of the profile becomes much broader (in X-units). One of the curves (form = 1) has been transferred to Fig. 3, using (17) with n = 15, t = 2, to translate X-values into angles. This dot-dashed curve is for the same aperture (15?Q,) as the dashed curve, but with overall attenuation 8.7 dB instead of 0 dB. It is remarkable that almost no difference of 3 dB beam-width exists between these cases, the main effect of attenuation being to fill in the nulls which otherwise occur (and to reduce the absolute value of&,x). The main use of Fig. 4, however, is for comparing beam characteristics of different apertures a, having the same
ULTRASONICS.
NOVEMBER
1981
35
attenuation (dB wavelength-’ or n), since the nearly linear relation between X and 0 is then a fxed one. Normal practice alms to maximize power radiated in the direction err,, which requires that m should be as large as possible; hence the important curves in Fig. 4 are the inner ones. The exception occurs in some short-range applications, where restriction of the aperture may be needed to keep point P sufficiently beyond the Fresnel zone. An example, with stainless steel in liquid sodium, shows how to calculate beam-width, and highlights the limitations of the beam-forming process. Assuming Rayleigh waves (vb = 3 km s-r) and taking cL = 2.4 km s-l, we find 0, = 53 .l” . Although the characteristic acoustic impedance (2,) of liquid Na is somewhat greater than that of water, for these calculations we assume the same attenuation as before, n = 15. With t = 1.25 now, (21) or (22) yields (Ae), dn = 1.6”, for an unlimited aperture. Fig. 4 shows that limiting the aperture to, say, 30 wavelengths of SAW, that is, m = 2, increases beam width by a factor 1.55 to some 2.5”. Particularly noteworthy is the fact that, even with power attenuation eT4 = 0.018 and hence a radiation efficiency over 98%, we nevertheless have truncation broadening of the beam and small sidelobes are present, the fust pair at X = +4, that is, 0 = 50” and 56.5’. Note that an important condition for achieving the predicted beam-widths is that range r must substantially exceed the Fresnel distance. b2 rF = -.
(25)
2AL
Taking b = 15 hb as a reasonable radiator width for a nearly circular beam (see following), and XL = 0.8 hb , We find IF = 140 xb, SO an Operat@ range Of,500 it,,, at least, would be needed for a 2.5” beam width. Increasing aperture length to say 60 &, , in order to approach the 1.6” limit, would involve increasing b by 50% and thus the range to over 1000 Xb. In either case, the beam width (in cm) at the range in question exceeds the radiator width. Lateral
width
of beam
We consider the intensity change when P moves normally to the plane of Fig. 2 to a point P’, such that angle POP’ has a small value J/. The object is to achieve an approximately circular beam cross-section, by a suitable choice of aspect ratio a/b of the radiator. Assuming OP is along the axis of the beam, signals received at P from all strip-elements of the radiator are in phase and directly additive. The same is true of signals received at the offset point P’, to a high accuracy if J/ is small. Hence one can determine the lateral profile by condensing the radiator surface into a single uniform line-source at the edge 0. This profile has a well-known form:
r;= Pmax
I
sin (@‘/2) 4’12
I -
(26a)
where @’ = 2 AL
. bsin$.
(26b)
253
Xz2lLn(l-~slne) total attenuation along aperture = e-“, or 0 = mnhb
B/B mm t .^
that, provided 7 exceeds a modest minimum, ultrasound pulses detected at points in the beam all have flat-topped envelopes, whose amplitudes agree with the cw characteristics derived above. This object can be achieved even with spatial pulse lengths less than the aperture length. Rayleigh waves present the simplest case, being nondispersive. However one can allow for velocity dispersion in a Lamb wave pulse as in the example following, provided that no envelope distortion arises from dispersion* and the adjacent liquid attenuates the pulse exactly as it does a continuous wave.
I
I
I
I
I
I
-10
-8
-6
-4
-2
0
I
I
2
4
I
6
6
10
X Fig. 4 Generalized beam-profile X for various aperture truncation
as a function factors m
of direction
variable
It has the same characteristic multi-lobed shape as the dashed curye in Fig. 3, but of course centred at $J = 0. Since #‘/2 = + 4n/9 in the - 3 dB directions, total transverse beam width is: @+kde
(27)
= 2.
Assuming m = 2, as in the example above, the width in the median plane is 1.55 @%idB
=
7rnlcos
Equating (A$)3 dB a _=-
b
to
= Bm
3.1 Xb nagcos
(28)
Bm
(A@ 3 dB and using hL = hb/t, yields
1.11
(294
cos 8,
Fig. 5 illustrates a general method of approach with a specific case. In this space-time diagram, the diagonal band represents progress at group velocity +,(pr) of a pulse (of width T) along the aperture shown at the left of the figure. The close-spaced interior (phase) lines represent individual Wave-Crests, travelling at phase velocity +,(ph). A velocity ratio vb(pr)/vb@h) = 1.25 is used here, being typical of a zero-order antisymmetric Lamb wave, and this is therefore also the ratio of the slope of the two ‘edge’ lines in the figure to that of the phase lines. Aperture limits are shown at either 15 Xb or 30 &, . Notice in this case that, although a pulse of 12 cycles was introduced at the aperture (the same number would be detected at any aperture point), there are nevertheless 1.5 waves in transit in the complete pulse at any instant; this is a consequence of velocity dispersion. Observe also that new wave crests appear at intervals at the pulse leading edge and others are disappearing similarly from the trailing edge. Pulse characteristics of the radiated beam are related to the ‘view’ from point P in Fig. 2 at each moment. Because LP is less than OP (by x sin e), P sees events at 0 simultaneously with later events at L. Thus at any time (T/c~) + to, measured from the start of the pulse at 0, P has a simdtaneous view of all (x, t) combinations at the radiator such that
which gives the required aspect ratio of the radiator. This result can be written acos0, 7
_ -
(29b)
1.11
b
that is, the aspect ratio, as viewed from P, should be slightly over-square, a conclusion which is not surprising. With m-values other than 2, this result is not much altered, unless m is fairly large. This is expected, since the effective aperture length is then dominated by attenuation. Values of the numerical constant in (29) are tabulated in Table 1, for a few values of m. Pulsed operation
Since most systems operate with pulses, one must consider possible beam degradation when the SAW comprises only a few sine-wave cycles. In the near-field, the effect is broadly to limit the number of plane parallel waves produced in the liquid by each pulse and is not considered further, since interest centres on the far-field beam. We confine attention to a rectangular pulse envelope (width T), injected at the radiator, and conclude Table 1.
(30)
CL
These points constitute a moving ‘view-line’ in Fig. 5, through the varying point to on the time-axis and with constant slope cL /sin e . Instantaneous pressure at P is proportional to aperture surface pressures** integrated along this line, the integral being generally equivalent to that in (11). For axial points in the beam (0 = e,), the slope of the view-line is vb(&) and it is parallel with the phase-lines in the figure; four successive positions, at a small negative value of to and at to = 0, t 1 and tz, are shown by dashed lines, t1 and tz being chosen so their lines intersect the pulse ‘trailing edge”at the respective aperture limits a, and a*. The dotted lines at to = t2, with slightly altered slopes, represent the views from typical off-axis points (the - 6 dB directions, with a = 15 Xb, n = 15) at this instant. In pulsed operation, as distinct from cw, one evaluates the above integral at each instant to only over that portion of the view-line within the (shaded) pulse area in Fig. 5.
Aspect ratio of radiator for circular beam
m
1.5
2
2.5
a cosen;l/b
1.075
1.11
1.19
254
f=fo+XSln
3 -1.29
4
‘This requires that group velocity of frequencies.
1.50
l
be constant over a suitable range
* Each point (x, t) in Fig. 5 has a unique pressure value po,given by (9) inside the pulse band; outside this region pg = 0 everywhere.
ULTRASONICS.
NOVEMBER
1981
Clearly, if 19= B,,, and to is anywhere in the range 0 to tl for aperture al (or 0 to t2 for a = a2 ), then the integral limits are x = 0 and a, as in (1 l), thus yielding a result identical to that for continuous waves. This therefore defines the constant-amplitude part of the received pulse at P, which is constructed from only those waves that travel from end to end of the selected aperture. Such a flat-topped envelope occurs provided the pulse width exceeds the difference between group and phase transit times along the aperture, that is
r>
a ‘b@h)
a (31)
‘b(gr)
from the geometry of the diagram. Thus we find minimum usable pulse widths in our example of six cycles for a = 30 &,, or three cycles for a = 15 hb ; these figures are modified by only about 0.6 cycle in each case, if the receiver is moved to the - 6 dB off-axis points. Evidently the 12-cycle pulse illustrated would be satisfactory in both cases. Notice that the aperture 30 b is effective here, even though never more than half-filled by the actual pulse. A different situation arises when t, is a little outside the limits above; the view-line now intersects one or other of the pulse ‘edges’ shown in Fig. 5, with consequent truncation of the integral at one of its limits. These te-dependent limits are responsible for the rising or falling edges of the received pulse envelope. Short of a complete analysis, certain useful generalizations are possible when 13= 19, :-
(4
the duration of each transient hand side of inequality (3 l),
is equal to the right-
@I the
shape of each varies, with pulse-phase attenuation along the aperture, from nearly linear with low (Yto very curved with high (IL,
(cl
the direction of any curvature is opposite, when vb(.& exceeds vi,(@), to that when it is less.
If ti departs from B,,, , the transient durations change little, but the effect on their shapes is less easy to predict.
Non-dispersive (for example, Rayleigh) waves are a special case, in which phase-lines in Fig. 5 run parallel with the pulse ‘edges’. Axial points in the beam receive pulses identical in shape and width to those fed to the aperture, while off-axis points see short-duration transient edges to the envelope. The amplitude of the major flat portion, however, should still accord with the cw theory. To summarize, the effective beam widths under pulsed conditions are the same as those derived for continuous waves, provided:-
(9
the (rectangular) pulse envelope width fed to the radiating aperture somewhat exceeds the difference between SAW phase and group transit times along the aperture, and
(ii)
only the resulting flat top of the received pulse envelope is used in a plot of beam amplitude against direction 0.
Mode conversion from liquid compression waves to Rayleigh or Lamb waves Having produced an ultrasonic beam in a well-defined direction, one would look for reflections from a target. object. Important information might be the presence or absence of the object, or its direction from the transmitter. Using short pulses, the echo delay would also determine the object distance. In such cases, the receiver could be similar to the transmitter. Often, indeed, the same assembly can serve both functions, depending on the capacity of the generating transducer to detect SAW’s arriving from elsewhere. Here, our concern is with the other link of the reception process, namely conversion of ultrasonic waves in liquid to SAW on a metal plate or strip positioned to receive them. One would expect this to be most efficient when the almost plane waves arriving from a distant object are inclined at angle em to the interface, since intersections of consecutive fronts with the surface then match the appropriate SAW wavelength (see Fig. 1). Thus, pressure variations in the incident wave can build up the required surface-wave. Questions that arise are: How is SAW amplitude altered if the wave arrives from a different direction? Also, how does it depend on the length a of plate immersed in the insonified liquid? The reciprocity principle provides answers to these questions and also throws interesting light on the detailed mechanism of SAW build-up in the strip. Use of the reciprocity principle The reciprocity
A2
4 Fig. 5
Space-time
ULTRASONICS.
Time,
t-
diagram of pulse transit at aperture
NOVEMBER
1981
principle states that a complete system with two (linear) transducers Tr and T2, as sketched in Fig. 6, behaves as a linear passive four-terminal electrical network; it therefore has identical transfer impedances in both directions, for any given configuration. It follows that the e-dependence of output voltage from one pair of terminals, with a given current drive to the other pair and all other parameters fixed, is independent of which transducer is the transmitter. We deduce that the directional properties of the Tr /strip combination, when receiving plane waves from any direction 0, are the same as its transmitting characteristics, as illustrated in Figs 3 and 4. One consequence is that, if one uses a single transducer/ strip assembly to ‘view’ a small object at P, using ultra-
255
T2
3
Jc
/p
/
&;i’ A!r
Mox~mum omplltude-
1 ----
1
Liquid
/r
’ SAW amplitude
Fig. 6 Transmitter/receiver principle
system for application
of reciprocity
sound reflected or scattered back along the transmission path, then the overall dependence of received signal (that is, voltage at terminals 1 and 2) on direction of the object is proportional to the squares of ordinates of the curves in Figs 3 and 4. Thus the visibility angle is rather smaller than the beam-width calculated for transmission (or reception) alone. Next consider the effect of varying length a of the strip, while keeping 0 constant at 8, (for simplicity). With T1 as transmitter, the output of Tz is proportional to pmax and hence to 1 - e-w, from (14). By reciprocity, if Tz instead transmits quasi-plane waves towards OA at the optimum angle, then the output of T1 is likewise proportional to 1 - emm. Thus SAW amplitude, at the exit point from the liquid, saturates as a increases beyond three or four times n&, . Hence there is no point in making a receiving strip longer than this. We infer also that the build-up of amplitude along a fixed length of strip is a similar exponential function of position, starting from zero at the free end; this is shown in Fig. 6. In this case, a is about 2 nhb and the emerging SAW amplitude is a little less than the saturation value; thereafter the strip is in air and the wave reaches T1 without further change. Since reciprocity still applies in pulsed conditions, important pulse characteristics of a receiving assembly can be deduced from results in the previous section. For example, the response to a finite rectangular pulse of incident compression waves will be a flat-topped envelope at T1 output, provided the pulse width exceeds the limit set by (31). SAW build-up
and conversion
4
interface, the pressure of reflected wave (2) is in phase with that of incident wave (1) everywhere at the reflecting surface. So positive pressure planes of wave (1) intersect the surface at the same points as positive planes of wave (2), and similarly for negative planes. Thus net pressure applied to the metal surface is at first twice that in the incident wave, since the amplitudes of waves (1) and (2) are equal. This combined pattern travels along the interface at velocity vb , thereby inducing a surface wave. The crests of the SAW will be about a quarter-wavelength ahead of the pressure peaks, as shown, thus allowing positive pressure to act at falling points and negative pressure at rising points. In this way maximum energy is transferred to the metal surface. Once a finite SAW has built up, the surface-radiated wave (3), whose wavefronts are shown separately, also appears, as described earlier. This is simply superimposed on the other two waves, by virtue of the superposition theorem, since we have in effect combined the respective boundary conditions. An important feature of wave system (3) is that its positive pressure planes (long dashes) intersect the surface at the leading slopes of the SAW undulation, while the negative planes (dot-dashed) are associated with trailing slopes of the SAW, in accord with Fig. lb. (These lines are displaced slightly above those of wave (2) in Fig. 7, for clarity, though they are actually coincident.) The significant conclusion is that wave-systems (2) and (3) are oppositely phased, or have a relative displacement of half a wavelength. Thus they tend to cancel each other. Suppose the SAW at a point gives rise to a wave (3) amplitude just equal to the amplitude $ of wave (1) or wave (2). Mutual cancellation of waves (2) and (3) then occurs and only wave (1) remains. In this small region of the surface, the entire energy of the local incident wave is converted to surface-wave energy, with 100% efficiency, and added to SAW power already present. Elsewhere at the surface varying amounts of energy are returned to the liquid, either as mainly reflected waves (nearer the end A) or as mainly radiated waves (nearer the exit point 0). This is illustrated in Fig. 8. Thus overall, the power efficiency of a metal strip, for converting uniform liquid pressure waves to SAW, cannot be 100%. It is pertinent to ask what efficiency is achieved in reception. Before answering this, consider conditions at 0, where SAW amplitude has reached its saturation value, assuming a is long enough. It is seen that pressure amplitude in wave
efficiency
The central role played by attenuation coefficient (Yin the SAW build-up suggests that the process is a balance between the incident wave-train, the ordinary reflected wave (which is always present) and the re-radiated wave (which depends on SAW amplitude at the point in question). This balance varies from point to point along AO. Fig. 7 shows the three component waves in the immediate vicinity of a point on AO. The arrows indicate propagation directions of the wave-systems, all inclined at em to the normal of the interface. Positive pressure wavefronts for waves (1) and (2) are indicated by solid lines and negative pressure by broken lines, as before. Assuming a motionless
256
~ ------
Positive pressure fronts NegMlve II II
11 u
‘8 *’
lncldentond reflected waves @ ond @
_
-
Positive
-
-
Negative
4
DirectIon of surface velocity
t
Fig. 7
Radloted wove @
Detail of three wave-systems at a point of the strip
ULTRASONICS.
NOVEMBER
1981
surface only. If liquid is adjacent to both sides of the strip, one must allow for a wave radiated from the lower surface as well. The modifications to the theory are mostly straightforward, but note that (Y is doubled, while the maximum attainable SAW amplitude is halved. In the latter condition, near to the exit point, the whole of the incident power is removed by the wave radiated from the lower surface. (Waves (2) and (3) leaving upper surface cancel.) The expression for overall power efficiency is found to be similar to that in (32), but it lacks the factor 2 and uses the new value for 01.Hence maximum power efficiency is now only 40%. Thus there is a real advantage both in transmission and reception, at least for remote viewing applications, in confining liquid contact to one side of the strip only.
(3)is here 2;, so the resultant of waves (2) and (3) has amplitude ;: phased as (3). Power leaving the surface here is equal to incident power and no further increase of SAW amplitude occurs. Alternatively, when pressure in wave (3) is 2;, this exactly cancels the resultant pressure 2: at the metal surface, from waves (1) and (2) acting together. With zero net pressure at the interface, from all three waves, the surface acoustic wave proceeds along the surface with unchanging amplitude, as though the metal were in contact with air or vacuum. Assume the incident wave peak pressure is ; and mean power arriving at the surface is uniformly distributed and Pi per unit length. At a point distant s from the end A of the strip:amplitude
of wave (2) = I;, independent
amplitude
of wave (3) = 2$(1 - ewm),
since this is pfpportional approaches 2p.
Conclusions
of s,
to SAW amplitude
The basic theory underlying a useful ultrasonic modeconversion process has been given. This involves energy transfer from a surface acoustic wave (Rayleigh or Iamb wave) in a plane solid surface to an angled compression wave in a surrounding liquid. The main emphasis is on the beam-forming properties of the compression wave, including analysis of median and transverse beam-width and discussion of pulsed operation. We have also examined the converse process, that is, conversion from pressure waves to SAW, since this occurs in many applications.
and ultimately
Therefore net amplitude of wave leaving surface =p - 2;(1 - eds) =l(2eeQS - 1). Since power density in a wave is proportional to (pressure amplitude)’ , net power per unit length leaving the metal surface at the point distant s from A is: Pi(2e-“S
- 1)2 .
The variation of all these quantities in Fig. 8. Integrating
The importance of the subject is twofold; firstly because conversion power efficiency of both processes is very high (100% for the first, up to 80% for the converse), secondly for the practical reason that,the electro-mechanical transducer can be separated from the liquid by a length of metal strip or plate bearing the SAW. The technique is thus attractive in hazardous or high-temperature environments, as for example in instrumentation of the Fast Reactor.
along A0 is sketched
over the length Q of the immersed surface,
total incident
power = Pia,
a Pi(2eeas - 1)2 ds
total power returned to liquid = s
0
= pia _
2pi
(1 _ e-aa)2
a!
Therefore power leaving as SAW via strip = (2pJo) (1 - e-(Y0)2
SAW amplitude
(Note that this expression is proportional to squared SAW amplitude at 0; maximum available SAW power is represented by the shaded area at the bottom of Fig. 8.) So efficiency (7)) =
total incident
Y
---------
SAW power power
--------= 2(1 - e-m)2
(32) . I
m wherem
=(~a.
A plot of this (Fig. 9) reveals the surprising result that power efficiency of the pressure-wave-to-surface-wave conversion process can exceed 80%, if a is about 1.25 nhb, and r7 > 60% for 0.5 <
nQ
71”
1 ---------- :$ Point of 100% conversionefficiency/ +P^
Pressure amplitude wave @[negative sign indicates hase opposite to wave 62 1
Net pressure amplitude Waves @ + @
< 3
This does not prejudice earlier findings on the dependence of SAW amplitude on length of receiving strip; in particular, this amplitude is only about 0.7 of its maximum possible value, when the conversion (power) efficiency is optimum. It should be said that the arguments above tacitly assume a Rayleigh wave, or a Lamb wave with liquid against one
ULTRASONICS.
Pressure amplitude Wave (ij arwave @
0
NOVEMBER
1981
Power densities orrivmg departing Distance s from end A
_
Fig. 8 Distribution along A0 of pressure in three wave-systems and resulting power densities
257
IO
Cond,+,ons
described in the companion papers. It is anticipated that other applications could well arise in the future. Meanwhile, further theoretical work is needed on aspects not covered in the present paper, notably those mentioned at the end of the second section, since high-angle waves will often occur in liquid-sodium work.
Llqwd I” contact with one surface only (Angleof lncldence = em
0.6 F
References
04
1 2
Y
OO
2 o/blhtJ
Fig.9
Total power efficiency
I 3
I
I I
J
4
3
or m
of an immersed stripas a receiver
258
one a new type of liquid-level gauge, are
Rayleigh and Lamb Waves, Plenum Press, New York (1967) Gillespie, AS., Deighton, M.O., Pike, R.B., Watkins, RD.,
A New Ultrasonic Technique for the Measurement of Liquid Level. Ultrasonics (1982) in press Watkins, RD., Deighton, M.O., Gillespie, AS., Pike, R.B., A Proposed Method for Generating and Receiving Narrow Beams of Ultrasound in the Fast Reactor Liquid Sodium Environment,
4
Two applications,
Viktorov, I&
Ultrasonics (1982)
in press
Deighton, M.O., Gillespie, A.B., Pike, R.B., Watkins, R.D., Mode Conversion of Rayleigh and Lamb Waves to Comp ression Waves at a Metal-Liquid Interface, AERE-R.9963
ULTRASONICS.
NOVEMBER
1981