Acoustic propagation for phased array analysis can be analyzed in the context of geometrical optics (Whitham 1974). Consider a stationary, compact source of sound in a wind tunnel, and suppose that the sound is suddenly turned on at t = O. Acoustic waves will propagate outward from the source region and eventually reach each point in the test chamber. For a point x, let a{x) denote the time at which the sound first reaches x. The surface t = a{x) is called the wavefront. It moves outward from the source as t increases. In the frequency domain, the wavefront is a surface of constant phase.
Uniform Flow
Suppose the wind tunnel flow is uniform with speed Vo in the direction of the +x axis. The acoustic pressure,p, obeys the convected wave equation
(2.1) The geometrical optics approach is to seek a high frequency asymptotic ex-pansion of the form
p - e-iw(t-a(x))
L
<I>n{x) (-iw)-nn=O (2.2)
Substituting Equation 2.2 into Equation 2.1 and isolating powers of w, the travel time function, a{x), is seen to obey the eikonal equation
1 [
oa]2
2 1 - Vo - = (Va) . (Va)
c
ox
(2.3)wind
x
Fig. 2.1. The wavefront t = a(x) in the case of uniform flow. The source is at location xs. In time t, the center of expansion of the wavefront, xe , has convected downstream a distance Mct while the wavefront has expanded to a radius of ct. The ray path connecting the source with the microphone at x makes an angle Br with the upstream direction. The line segment connecting Xe with x is parallel to the wavefront normal vector, and makes an angle Be with the upstream axis. This is called the "emission angle" because it resembles the emission angle in the case of a moving source in a stationary medium. In the present case, energy is actually emitted at the ray angle, Br
The leading amplitude term <1>0 is a solution to
[ 1 iJa] iJ<I>o [ i J 2a ]
2M --M- --+2(Va)·(V<I>o)
=
M2-2 -(Va).(Va) <1>0c iJxiJx iJx
(2.4) The Mach number of the wind tunnel flow is M = volc. Solutions to Equation 2.3 can be found graphically. In the case of a compact source at the origin, the wavefront is a spherical surface that expands outward at the speed of sound while the center of the sphere is simultaneously convected downstream at speed Vo. Figure 2.1 shows the wavefront at time t = a. The radius is of the sphere is ct, and the center is at Xe = (Mct, 0, 0).
To find the arrival time, a,for a given point x, use is made of the triangle Xs'Xe'
x in Figure 2.1. Side XXs is the receiver radius, r. Side XeX is the wavefront radius, ca(x). Finally, XXe is the convection distance, Mca(x). The angle of the ray con-necting the source to the receiver is denoted Or. The angle Oe' which characterizes the normal vector of the wavefront, is known, somewhat confusingly, as the
emis-sion angle. (The name emisemis-sion angle would be more appropriate for er , since the ray xsx is the path of energy propagation. However, use of term for ee is standard in the literature.) One application of the law of sines to the triangle gives
ca siner
(2.5) r sin ee
Another use of the law of sines and some manipulation produces a M coser + ~ 1-M2 sin2er
ric 1-M2 (2.6)
or 1 R-Mx
a=---::-c 1-M2 (2.7)
where
It can be verified Equation 2.7 is a solution to Equation 2.3.
The geometrical optics approximation for the amplitude of the wave is given by the solution to Equation 2.4:
<po(x) = 1 ( ) (2.9)
4rr ca x
The complete first term of the asymptotic expansion is
e-iw(t-a(x»
Po(x, t) = 4rrca x ( ) where a(x) is given by Equation 2.7.
(2.10)
Equation 2.10 is not an exact solution of Equation 2.1. Substituting Po into the convected wave equation and normalizing by (iJ2PoliJt2 ) to give a non-di-mensional error term, the result is
V2po-
:2 (;
+ Vo~r
Po 2M 12 = - - 2 --2 (M + cos e)
iJpo 1-M (aw) (2.11)
iJt2
This shows that Po solves the convected wave equation accurately for small M andlor large phase, aw.
An exact solution to Equation 2.1 is obtained by replacing ca by R in the de-nominator of Equation 2.10 to give the Green's function (Mosher 1991)
e-iw(t -a(x»
p(x, t) = -4-rr-R-- (2.12)
Amplitude Approximations for Nonuniform Flow
The exact solutions to Equations 2.1 and 2.4 are not generally available for cases with nonuniform flow, such as an open jet wind tunnel with the micro-phones placed outside the jet. It will be seen below that the travel time, a, can be computed numerically in these cases if the flow field is known. An approx-imation for the amplitude is required for the phased array method. One ap-proach is to use Equation 2.10. The relative error that would be introduced in the uniform case is
Po
PI-M2 x
---",I+M-x
R
I-M-R
(2.13)
Another approximation results from replacing ca by r in the denominator of Equation 2.10. This gives
e-iw(t -a(x))
Pr{x, t) = -4nr
The relative error for uniform flow in this case is
(2.14)
(2.15) For angles near broadside, such that xlR
<
MI2, Equation 2.10 is the better ap-proximation. For other angles, Equation 2.14 is preferred. Equation 2.15 also shows that wind tunnel flow reduces the apparent strength of a monopole source, relative to the case of no flow, for both upstream and downstream directions.Ray Tracing for Travel Time
In open jet wind tunnel testing for acoustics, the model is placed inside a large jet, and the microphones are typically positioned outside the jet (Figure 2.2).
Part of the acoustic path is in the flow and part is outside of it. In a nonuniform problem like this, it is not easy to find the entire wavefront, but it suffices to compute trajectories of individual points on the wavefront, i. e., rays, that con-nect source points with microphone locations.
Following Pierce, (1989) let such a ray be denoted xp(t). Its motion is the su-perposition of convection and propagation relative to the fluid, or
(2.16)
Fig. 2.2. A representation of an air-craft model in an open jet wind tun-nel with a planar phased array lo-cated outside the jet. The dimensions of the model, the jet, the array, and the space between the jet and the ar-ray are all comparable. The acoustic wavelength (not shown) is substan-tially smaller than these dimensions
where n (xp) is the unit normal vector to the wavefront. The wavefront normal vector is parallel to the slowness vector, s, which is defined by
s='I1a (2.17)
The length of s is given by the eikonal equation, Equation 2.3, which is still valid in nonuniform flow. Let Q be defined by
Q = 1 - Vo . s = 1 - Vo . '11 a The eikonal equation becomes
and so
n=-es
Q
(2.18)
(2.19)
(2.20) The ray path, xp (t), and the slowness vector, s (xp), can computed by solving the ray tracing equations
(2.21a) ds Q
-= - - 'I1e - s x ('11 x vo) - (s· '11) Vo
dt e (2.21b)
Equation 2.21 a is a restatement of Equation 2.16, and Equation 2.21 b can be derived from Equation 2.19 (Pierce, 1989).
An iterative method for determining the ray path between a source point and a microphone begins by setting xp(O) to the source point and choosing an initial guess for 5(t=0) as a vector with direction pointing toward the micro-phone from the source and length given by solving the quadratic Equation 2.19. Equation 2.21 is then integrated by a numerical algorithm such as the Runge-Kutta method of order four (Burden,et al.1978) to find the components of xp and 5 as functions of t. The integration continues until the distance be-tween the source point and xp(t) equals the distance from the source point to the microphone. Unless the flow is uniform, the stopping value of xp will prob-ably not coincide with the microphone location. Some measure of the error vector, Xp(tfinal) - Xmicrophone is taken as the objective function in an optimiza-tion procedure. Two angles describing the direcoptimiza-tion of the initial ray, s(t= 0), are then adjusted and the integration is repeated until the ray intersects
50==
- '
61 58 I 50 53
Fig. 2.3. Sample ray paths from a point on a model to microphones in a phased array lo-cated outside of an open jet. The Mach number is 0.2. A thin shear layer has been assumed for the ray tracing calculations
the microphone. The final value of t for this ray is the required travel time,
a(Xmicrophone)' This somewhat time consuming calculation is repeated for each microphone and each point in the beamforming grid.
Sample ray paths are shown in Figure 2.3. In this case, the shear layer was taken to be relatively thin, and the flow was assumed to be uniform inside and outside the jet, so the ray paths are broken lines with changes of direction at the shear layer. The use of a thick shear layer produces a curved ray path within the shear layer, but does not significantly change the propagation time.
It can be seen that the example in the figure is a very three-dimensional prob-lem. It is unlikely that a two-dimensional approximation to the geometry could produce travel times that are accurate enough for successful beamform-ing in the case that the model and the array have dimensions that are compa-rable to the radius of the jet, as in the figure.
As noted above, this method does not directly produce the wave amplitude.
The Blokhintzev invariant, which relates the amplitude to the flow velocity and the ray-tube area (Pierce, 1989) may be useful in estimating the amplitude. Be-cause the amplitude usually has a weak role in the beamforming process, it is adequate to apply an approximation as in Equation 2.10 or 2.12.
Array Source-Receiver Model Temporal Considerations
The phased array technique for wind tunnel acoustic measurements begins with simultaneous time series measurements of the pressure at each of N mi-crophones. The time series are partitioned into blocks oflength TB.An analy-sis frequency,!
=
wl2n, is selected, and a digital filtering technique is used to compute a complex, frequency-domain pressure amplitude for each block and each microphone. The analysis bandwidth, lITB is chosen to be no larger than a certain fraction, say 1110, of the analysis frequency. This ensures that phase measurements will be well defined. (When a series of frequencies is processed to create a spectrum, the analysis bandwidth can be kept constant, or it can be made proportional to the analysis frequency). The resulting un-steady narrowband signals are denoted un(t), n = 1, ... , N. In this context, tis an index for the analysis blocks. The unsteady narrowband signals vary on a time scale of TB•The signals, un(t), are modeled as Gaussian random functions of t. Suppose, for the moment, that only a single source is present in the wind tunnel and that it has location Xs. The source strength is also modeled as a complex Gaussian random function, S (xs, t). A quasi-steady viewpoint is adopted in which un(t), and S(xs,t) are related by a complex constant of proportionality, Cn(xs) for each microphone, n. Since the same Cn(xs) is applied for every t, random vari-ations in the source are exactly tracked by the narrowband microphone sig-nals. The propagation time is, of course, represented by the phase of Cn(xs).
Since the narrowband microphone signals are all proportional to the same source function in this case, they are also proportional to each other. In other words, they are perfectly correlated. This is only true for the unusual case of a single source.
At first glance, this quasi-steady assumption would seem to require that the source-receiver propagation times, characterized by a, must be much less than the analysis block length so that the narrowband signals can keep up with changes in S (xs, t) without significant delay. Such delays, however, do not in-validate the technique as long as they are approximately the same for all the microphones. The requirement for the validity of the quasi-steady model is that the differences in the propagation times from each part of the source re-gion to the various microphones must be a small fraction of TB• Stated another way, the analysis bandwidth must be small enough that the reciprocal of this bandwidth is much larger than the difference between the shortest and the longest propagation times from any given source point to each of the micro-phones. For an extended source region, this requirement must be met for each individual source point.
It is usually not required that the difference in propagation times be-tween different source points and a single microphone be much smaller than the block length. Since an uncorrelated source distribution will usually be assumed anyway, any additional decor relation caused by wide separa-tion of source points will not be harmful!. In the special case of an extended, coherent, source distribution, the analysis bandwidth is constrained so that the analysis block length is much larger than the difference in travel times between the farthest and nearest part of the source. This consideration can be important in measuring tone noise produced by large rotors.
It is sometimes possible to compensate for the array shape by introducing delays between the blocks of data for the various microphones. This process is called pre-steering. It is useful, for example, when a large planar array is ap-plied for a source region that is located at a large angle from the boresight direction. Pre-steering permits increased analysis bandwidth by correcting for the large difference in propagation times between microphones nearer the source and those located father from the source. Conversely, use of a wide bandwidth without pre-steering will cause the array to naturally reject inter-ference that arrives at a large oblique angle.
Distributed Sources
Suppose that there is a distribution of sources, and that each point in the source region, xs, is characterized by a distinct source function, S(xs, t). It is
I The fact that sources are permitted to be remote from each other and the array without restricting the analysis bandwidth is graphically illustrated in radio astronomy, where the travel times are measured in millions of years!
assumed that the sources associated with distinct points are uncorrelated2, i. e.,
(2.22) where q (xs) is the mean square source strength at xs, and the time average, < ), is defined by
1 T
<f(t» = -
J
f(t) dtT 0 (2.23)
The integration time, T, which is usually the data acquisition time, is as-sumed to be much longer than the block length, TB so that the time averages will approximate the mean values. If the integration time is insufficient, then an expression like Equation 2.22 has to be augmented with another term:
<s* (Xs,
t) S(xs, t» = q(xs) c5(xs -x~)
+-V q~xs)
TITB (2.24)
It will be assumed that T is large enough for convergence to mean values.
The propagation factors for array processing are usually assumed to have the form
(2.25) where a(xn, xs) is the acoustic travel time from Xs to microphone n, and D(xn, xs) is a magnitude factor. Examples of D(xn, xs) are 4nR(xn -xs), 4nr(xn-xs), and 4n ca(xn,xs), as discussed above.
The array processing technique is based on the assumption that the acoustic part of the narrowband time history for each microphone, un(t), can be described as a sum or integral of terms Cn(xs) S(xs, t). This essentially defines S(xs, t). Equation 2.25 describes a monopole radiation pattern. This does not mean that the processing technique applies only to monopole sources. Complicated sources will require several terms in the integral or sum.
2 By making different assumptions, it is possible to extend the method to various types of extended, correlated source distributions. One method is discussed near the end of this chapter. Of course, it is also possible to apply the method for uncorrelated sources to problems in which extended, correlated sources are present, and to use judgment inter-preting the results.
If the distance from the source region to the array is much greater than the dimensions of the array, then D (xn, xs) will not vary significantly over the ar-ray. In such cases, it is sufficient to replace D (xn, xs) by 1. If the flow is uniform and the source region is in the far field of the array, i. e., much farther than the square of the maximum array dimension divided by the wavelength, then it is accurate to approximate wo (xn, xs) in Equation 2.25 by k· Xn where k is a wavenumber with magnitude w/c and a direction pointing from the source to the array.
Each microphone measures the superposition of the sounds from the vari-ous parts of the source distribution. In addition, each microphone signal is as-sumed to contain a channel noise term,En(t). For microphones in the flow or mounted in the wall of the wind tunnel adjacent to the flow, this term is pri-marily caused by turbulence in the vicinity of the sensor. It may be consider-ably stronger than the signal received from the acoustic sources! It is assumed that the channel noise terms are Un correlated with each other and with the acoustic sources. The composite signal at each microphone is
(2.26)
Beamforming
--> -->
If un{t), Cn{xs), and En{t) are placed On the rows of N-vectors u(t), C{t) and
-->
E(t), then Equation 2.26 can be expressed in vector form
(2.27)
Microphone Weight Vectors
Beamforming makes use of a microphone weight vector,
W
(Xb), that is defined for each potential source point of interest. It is assumed that W{Xb) is normal-ized by(2.28) Another term for the microphone weight vector that occurs in the literature is
"steering vector:' In the simplest version of beamforming, the microphone outputs are scaled by the elements of the weight vector and summed, with the result that the array is mathematically "steered" to xb' Microphone weight vec-tors are closely related to array propagation vecvec-tors, as discussed below.
Beamforming Expressions
For each point in a grid, the beamforming expression
(2.29) is computed and plotted. It is intended that locations Xb corresponding to ac-tual sources should be assigned a value b (Xb) that represents the time-average square source strength, q, multiplied by the sum of the squares of the propa-gation factors
(2.30) In other words, b (Xb) is an estimate of the sum over the array microphones of the acoustic pressure-squared at each microphone caused by the source at a single point.
The second version of the beamforming expression in Equation 2.29 refers to the array Cross Spectral Matrix (CSM)
A = (u(t) ut(t) with elements
1 T
Ann'
= - f
un(t) U;' (t) dt, n, n'=
1, ... , N ToThe most basic choice of W{Xb) is evident from Equation 2.27:
(2.31)
(2.32)
(2.33) To maximize the contribution of S (Xb, t) t<!. this steered signal, W{Xb) should be chosen to give the largest value of wt{Xb) C{xb)that is consistent with the con-straint, Equation 2.28. This occurs when the weight vector is parallel to the propagation vector:
(2.34)
Performance Analysis
Suppose that there is a significant point source at xo, so that the source distri-bution can be written
(2.35)
where the term S'(xs, t) represents the sources other than the specially desig-nated one at Xo. When the array is steered to Xo using Equation 2.31, the result is
Taking the time average of the magnitude squared, using the assumed in-coherence of the various sources, and using Equation 2.32 gives the beam-forming result
b(xo) = IIC(xo) 112 qo +
J
Iwt(xo) w(xsW IIC(xs) 112 q'(xs)d3xs + F (2.37) where F is the averaged square channel noise termF
=
(Iwt(xo) E(tW) "" (IEt(tW) (2.38)The expression on the far right hand side of Equation 2.38 gives the value of F in the case that all of the microphones are similar in terms of channel noise and distance from Xo. The level of this noise is comparable to level of the chan-nel noise for each individual microphone. The level of the first term on the right hand side of Equation 2.37 is the intended measurement of the time-average square strength of the source at xo, as observed at the array. As noted in Equation 2.30, the level of this term is equal to the sum over the micro-phones of the levels created there by the source at Xo. The array gain has in-creased the signal-to-noise ratio by a factor of about N.
The middle term of Equation 2.37 represents sources other than the one at
The middle term of Equation 2.37 represents sources other than the one at