The ray solution of the Helmholtz equation represents a high frequency approximation,
which is based on the solution of the
Eikonal equation for the phase of the propagating
wave,
and of the
transport equation,
which accounts for the wave's amplitude.
Ray tracing models are fast to compute and allow to incorporate in a easy way the
effects of variations of sound speed and boundary variations over range.
The WKBJ approximation can be further used to construct ray-like modes of propagation.
Major drawbacks of ray models are related to the break downs of the solution in the vicinity
of focal points and caustics and also due to being unable to handle diffraction.
Normal mode models expand the solution of the Helmholtz equation on an orthogonal
basis of eigenfunctions,
which are obtained by solving the so-called
normal mode equation.
In general each mode represents a pair of opposite oblique waves,
with a vertical component which is a standing wave and a horizontal component which
propagates away from the source.
Those modes can be grouped in two categories:
- Traped modes: modes, whose energy does not leave the boundaries of the waveguide.
- Evanescent modes: modes, with an amplitude which decreases exponentially along range.
The normal mode expansion of the pressure field is valid as long as sound speed does not
depend on range;
otherwise,
the solution of the Helmholtz equation requires solving the normal mode equation at
different ranges and matching of the expansions along range as the wave propagates.
That can be done by single mode-to-mode coupling
(a solution known as the adiabatic approximation)
or by calculation of the coupling coefficients,
which is a time consuming procedure.
An alternative to the mode expansion consists in the Fourier decomposition of the acoustic field
into an infinite set of horizontal waves,
which can be evaluated numerically using the fast Fourier transform.
Such numerical solution of the Helmholtz equation lies at the basis of all fast field models,
which can be considered as an extension of normal mode models,
since the normal mode expansion represents an asymptotic approximation
of the fast field integral far from the source.
Fast field models provide extremely accurate descriptions of the sound
field at low and high frequencies,
at short and large ranges from the source,
but at the cost of an intensive computational effort.
When the acoustic waveguide is changing slowly with range the Helmholtz equation can be
substituted by the so-called
parabolic equation,
which differs from the Helmholtz equation by only the first derivative in the
r direction.
Such substitution has the main numerical advantage that allows an efficient integration
of the solution by marching progressively along range.
However,
as a price to paid for the substitution it is important to keep in mind that
every parabolic equation model provides a solution,
which is valid only within a narrow cone against the horizontal.
BELLHOP:
part of the Acoustic Toolbox.
RSS Feed |
Contact | Accessibility | Products |
Disclaimer | CSS
and XHTML