The space-dependent form of the wave equation for a wave that is harmonic in time:
,
where
,
=angular frequency, and V=velocity.
Derivation of the Helmholtz equation
Given the homogeneous form of the scalar wave equation
.
Here
,
is time,
is the wavespeed, and
is the wave field.
If we replace
by its Fourier transform representation
,
noting that the second derivative of
with respect to time
the following Fourier integral form
results.
Because the only way for this Fourier integral representation to vanish is if its integrand vanishes,
the Helmholtz equation appears
.
External links
find literature about Helmholtz equation
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