акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en

inhomogeneities. Equations (8.57) and (8.58) can be studied without specifying the form of the correlation coefficient N(r ) in the following two cases which correspond to limiting values of the wave parameter: (1) D 1 and (2) D 1. This study then reduces to an evaluation of the integral

0

in the two limiting cases mentioned.

This is the region of large values of wave parameter. Here the integrand in equation (8.59) is different from zero for value of r that do not exceed the correlation distance a in order of magnitude. In this region of relevant values of r , the argument ν of the integrand sine does not exceed the value ka2/ 4L , that is, it does not exceed the value 1/D

In the case of a large wave parameter, ν will be small. Therefore, siν − π2 , and equation (8.59) takes the form

The integral I can be neglected compared to N(ξ ,0,0), then equations (8.56) and (8.57) can be written as

Thus, in the case of the farfield zone or large wave parameter (D 1) the mean square amplitude and phase fluctuation are the same and increase in proportion to the distance.

8.3.3Correlation of Fluctuations

The mean square amplitude and phase fluctuations still do not give the complete characteristics of the statistical properties of the wavefield. The statistical properties of the fluctuation of the wavefield can be characterized more completely using a correlation function.

For L a, equation (8.66) can be further simplified. Introducing the coordinates 2x = ξ1 + ξ2 and ξ = ξ1 − ξ2 we can then integrate with respect to ξ between the limits –∞ and +∞. Equation (8.66) can be written as

0−∞

Since N(r ) is an even function of ξ , while 2(ξ , ρ) is an odd function of ξ , the second integral vanishes. In the first integral, we can integrate with respect to x. Thus, the problem is reduced

to calculating the integral

Transforming to polar coordinates (ρ, ) in the (η, ζ ) plane, we obtain

∞∞

00

Equation (8.71) gives the general solution of the problem of the correlation of amplitude and phase fluctuations at the receiver.

Here, we assume that the correlation coefficient N depends on the coordinate and time difference, that is, we assume that the process is stationary in time and homogeneous in space.

The time difference is

The correlation coefficient N is different from zero if r1 − r2 does not exceed the correlation distance a in order of magnitude. This means that the time difference t − t does not exceed the quantity ac in order of magnitude, that is,

The quantity ac defines the time it takes the wave to go a distance equal to the correlation distance (the scale of the inhomogeneities). If this time is small compared to the correlation time of the refractive index, then the time difference t − t can be set equal to zero and any explicit time dependence in the above equation vanishes. Consequently the quasi-static condition takes the form of the inequality

The quasi-static condition (equation (8.85)) for average quantities is much weaker than the quasi-static condition for unaveraged quantities, which can clearly be written as

In all actual cases, the quasi-static condition (equation (8.85)) for averaged quantities is evidently met with a margin. This justifies the quasi-static assumptions that we have made from the beginning.

8.3.5The Time Autocorrelation of the Amplitude Fluctuations

The change of the inhomogeneities in time produces a change in the frequency of the scattered waves and broadens the frequency bandwidth of the incident radiation. The nature of this broadening can be inferred from the form of the time autocorrelation function of the amplitude and phase fluctuations.

An appreciable weakening of the autocorrelation of the amplitude fluctuations can be expected after a time interval τ commensurate with the correlation time T of the refractive index. If the quasi-static condition is met, then the time interval τ is also large compared to the time ac ; that is,

c

158 Acoustical Imaging: Techniques and Applications for Engineers

Writing equation (8.86b) for the time t1 and t2, assuming that they are separated by the interval

τ :

Denoting the time autocorrelation fluctuation of the amplitude by F(τ ), we have, by definition

F (τ ) = B (L , 0, 0, t1 ) B(L , 0, 0, t2 )

Multiplying with equations (8.87) and (8.88) and averaging, we obtain

From (8.89) we have

Mintzer [11] started with the assumption that the correlation coefficient N(r ,t) separates into two factors, one of which depends only on the coordinates and the other only on the time – that is, the correlation coefficient can be represented in the form

where the left-hand side represents the time correlation function F(τ ) in the coordinate system fixed with respect to the observer, while the right-hand side represents the space correlation

function F1 ( ) l .

We then obtain

8.3.6Experimental Verification

The above consideration of moving inhomogeneities in acosutical imaging is the first application of relativity to acoustical imaging. In this case is the Galilean relativity. Experimental verification of the application of either the fluctuation theory or the statistical theory to acoustical imaging is as follows.

So far, to the author’s knowledge, there have been no experiments on the application of fluctuation theory or statistical theory to the acoustical imaging of solids. Experiments have been performed only for sound propagation in the atmosphere and in the ocean.

The final experimental studies of amplitude and phase fluctuations for sound propagation in the atmosphere are due to Krasilnikov and Ivanov-Shyts [12, 13]. They compared their experimental data with the theoretical results of Krasilnikov [14] using the ray approach. They discovered that the theoretical result concerning the phase fluctuation was in satisfactory agreement with experiment, while the theoretical formula for the amplitude fluctuations was not supported by the experimental data. Figure 8.1 shows a logarithmic plot of the dependence of the mean square phase fluctuations (over various time intervals) on the distance between the

√ ¯2

B as the distance changes. As can be seen from the figure, the amplitude fluctuations, like

¯2 = =

Figure 8.1 Dependence of S on L for the frequency ν 3 kcps (logarithmic scales): , t 0.04 s; 0, t = 0.08 s; , t = 0.2 s (Krasilnikov and Ivanov-Shyts [12])