Fourier-Stieltjes transforms

In this paper the noise level produced by a macroscopically steady bubbly layer on a hydrofoil is predicted analytically, based on the stochastic properties of the fluctuating quantities in the layer. This analysis uses the technique of Fourier-Stieltjes transformation as used by O.M. Phillips. In this way, the sound spectrum outside the bubble layer can be correlated to the covariance of the fluctuating quantities, as gasfraction, velocity etc., assuming for instance that the bubble layer is stochastically stationairy and almost homogeneous. 

After Fourier-StieltJes transformation the general solution for the different Fourier components d7(K,y,(<>) of is... 

In this equation the gasfraction in point (x,y,t) is correlated to the gasfraction in a point (x, y, t ). A condition for application of the Fourier-StieltJes transform was, that the wave field must be homogeneous. This means that all probability-densities are invariant under the addition of a constant vector to all space points. Strictly speaking this condition is not fulfilled for a wavy wall. But when the length scale on which the mean quantities change, is small in respect with the variation of Z in the separation variable r, then the wave field can be assumed to be almost homogeneous. A second condition on the transformation yields the stationarity of the wave field, so the second moment function will be only dependent on the separation variable x. This condition is satisfied when the mean quantities are independent in time. In this way the frequency, wave-number spectrum can be written as. 

To carry out the spectral analysis, it is necessary to determine the spectral properties of the involved functions. These properties may be determined through the Fourier-Stieltjes transform (Priestley 1999). 

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