The proposed model studies the dependence of the average echo power I () (where is the time delay) resulting from the reflection of a quasi-monochromatic acoustic pulse from the multiscale random crack surface modeled as a fractal with Hausdorff-Besicovitch dimension (D+1) between 2 and 3. Final analysis gives the fractal dimension of the crack surface from the expression for the average echo power.

We represent the crack surface by an ensemble of Gaussian random fractal "height" functions h(R) giving its deviation from the plane R = (X, Y). Denoting ensemble averages by angle brackets, we specify the statistics of h by the mean height <h>, which we take to be zero, the mean square height H² and its mean square increment as

<(h (R0 + R) - h (R0 ) )2> = (R)

(1)

The fractality of h is embodied in the behavior of (R) as R 0.For a (D+1) dimensional surface, we have

(R) L(2D -2) R(4-2D)

(2)

Here L is the length characterizing the strength of the fractal roughness and equal to the distance (R) over which chords joining points of the surface have an RMS slope of one radian.

A source-receiver S, situated at height Z above the R plane defining a mean surface, emits a spherical pulse Ø(r. t) whose dependence on time t and distance r from S, is

where c is the wave speed. For F we take the quasi-monochromatic form

F(t) = exp (it)a(t)

(4)

where is the carrier frequency and a(t) to be the Gaussian envelope function, quasi-mono chromaticity will be ensured by taking >> .

Using the Kirchoff approximation in conjunction with the paraxial approximation, we arrive at the following equation

(5)

where the prime denotes differentiation of F with respect to its argument.

Considering time delays which exceed the roughness - broadened pulse duration /c, we arrive at the following expression for intensity I () as

(6)

By deforming the integration contour and expanding the exponential in the above equation, we obtain the following expression.

I() 1/(3-D)

(7)

From the above equation the fractal dimension of the crack surface can be evaluated. REFERNCES

1. B. B. Mandelbrot, "The Fractal Geometry of Nature", San Fracisco: Freeman, 1982