Pulse Propagation Analysis and the Helmholtz Equation 


            While time-domain analysis and modeling have been the approach for studying low-frequency seismic wave propagation in geophysics, underwater acousticians have traditionally favored a spectral analysis technique because only the mean signal energy seemed to have a predictable behavior at sonar frequencies of interest.  However, now we can observe a trend in sonar development towards lower frequency, which should lead to both higher signal stability and better predictability because low frequency signals are less affected by bottom attenuation, by rough interface scattering and by scattering from small scale inclusions.  Thus, powerful time-series analysis is becoming a valuable tool for studying complex propagation situations in the ocean.  Within the framework of linear acoustics there are fundamentally two approaches to this modeling problem.  The first one is to solve the pulse propagation problem via the frequency domain by Fourier synthesis of CW (continuous wave) results:

                                                                                                                                                                                    (1)

where   is the source spectrum and  is the spatial transfer function.  This approach is attractive since it requires little programming effort.  Once one has developed a model for the solution of the CW problem, one can generate the transfer function at a number of discrete frequencies within the frequency band of interest.  The evaluation of the integral in Eq. (1) can be then done by an FFT at each spatial position r for which the pulse response is desired.  Alternatively, one can solve the problem directly in the time domain, which, however, requires the development of a new set of propagation codes and does not allow the incorporation of developments achieved  earlier in underwater acoustics.  In my dissertation I follow the Fourier synthesis approach.  The strength of this method lies in its simplicity and versatility, so we can use the new CW model presented in Chapter 3 to produce a pulse response and can also compare the results with any existing CW computational codes, described later in this chapter.

            Since the coefficients of the two differential operators in the wave equation are independent of time, the dimension of the equation can be reduced to three by use of the frequency-time Fourier transform pair:

                              (2)

,                                    (3)

leading to the frequency-domain wave equation, or Helmholtz equation,

,                        (4)

where  is the medium wavenumber at angular frequency  and  is the source spectral function.  It should be pointed out that although the Helmholtz equation (4) is simpler to solve than the full wave equation, this simplification is achieved at the cost of having to evaluate the inverse Fourier transform, Eq. (2), if we have a process which differs from sinusoidal in its time dependence.  However, many ocean acoustic applications in the past were of narrow-band nature.  Thus, the Helmholtz equation has formed the theoretical basis for the most important numerical methods, including  the Ray, Wavenumber Integration (WI), PE, and NM approaches.

            Since  acoustic sources are local in nature, most of the time we deal with the homogeneous Helmholtz equation:

.                                  (5)

In spite of the relative simplicity of Eq. (5), there is no universal solution technique available.  The actual solution technique that can be applied depends on the following factors:

- Dimensionality of the problem.

- Sound speed variation, .

- Boundary conditions.

- Source-receiver geometry.

- Frequency and bandwidth.

For some problems the environment is so complex that only direct methods of the discretization of the differential operator in (5), such as the Finite-Difference Method (FDM) and Finite-Element Method (FEM), are applicable.  In general an optimal approach is a hybrid of analytical and numerical methods, and all the computational methods described in the following are of this category.  Although all these methods have the Helmholtz equation as the starting point, they differ in the degree to which  analytical and numerical components are utilized in the solution scheme.  Since  analytical methods are restricted to a few canonical problems with simple environmental and source parameters, the computational methods with a large analytical component are also restricted to problems where the actual environment is well represented by a simple idealized model.  In the following a review of the main computational methods for the solution of  the Helmholtz equation  and their mathematical bases  are  presented.