Bilateral Laplace transform

The explicit solution of Eq. (27), which uses a Fourier transform or a bilateral Laplace transform, is described in detail in Ref. 38. Its eigenvalues and eigenvectors are determined by the nonlinear eigenvalue equation... 

The Laplace transform can be alternatively defined as the bilateral Laplace transform or two-sided Laplace transform by extending the limits of integration to the entire real axis. In this case, the formulas become... 

One can consider a very general interface displacement given in terms of bilateral Laplace transform as,... 

All fluid dynamical systems are continuous system with infinite degrees of freedom and the governing equations depend continuously upon both space and time. While for any system, the time-dependent signal cannot move back in time, the space dependent signal can propagate in all directions with respect to the location of the source. This therefore requires that we develop a theory based on bilateral Laplace transform - a topic described in great details in Papoulis (1962) and Van der Pol Bremmer (1959). 

The description of the local solution and other details of selecting the Bromwich contours are given in Sengupta et al. (1994) and Sengupta Rao (2006). The near-held response created due to wall excitation is shown in these references as due to the essential singularity of the bilateral Laplace transform of the disturbance stream function. While the experiments of Schubauer Skramstad (1947) verihed the instability theory, the instability theory is incapable of explaining all the aspects of the experiments or... 

Here an explanation is provided on the structure of the near-field solution with the help of some fundamental theorems. These theorems provide the basis for interpreting both the near- and far-field solutions. These are due to Abel and Tauber and their utility was highlighted by Van der Pol Bremmer (1959) in connection with the properties of bilateral Laplace transform. In exploring relationships between the original in the physical plane and the image or transform in the spectral plane these two important... 

Following the above, let us explore the relation between the original and the image of bilateral Laplace transforms in relation to the evolution of small disturbances in boundary layers. In particular, one would be interested in the behaviour of the original in the neighbourhood of the exciter (x = 0) as determined by the image 4> y, a) for a —> oo according to the Tauber s theorem. 

In Fig. 2.32, the bilateral Laplace transform, (f>onormal derivative, have been shown that correspond to the imposed velocity boundary conditions of Fig.2.31. It is clearly evident that oo is much larger than cf) and this information is used here to compare different types of free stream excitations in the next subsection. 

Figm-e 2.32 The bilateral Laplace transform of imposed velocity disturbance components shown in Fig. 2.30... 

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