One-way wave equation

A one-way wave equation is a partial differential equation used in scientific fields such as geophysics, whose solutions include only waves that propagate in one direction. In the one-dimensional case, the one-way wave equation allows wave propagation to be calculated without the complication of having both an outgoing and incoming wave (e.g. destructive or constructive interference). Several approximation methods use the 1D one-way wave equation for 3D seismic calculations. 
One-dimensional case.
The standard 2nd-order wave equation in one dimension can be written as:
where formula_2 is the coordinate, formula_3 is time, formula_4 is the displacement, and formula_5 is the wave velocity.
Due to the ambiguity in the direction of the wave velocity, formula_6, the equation does not constrain the wave direction and so has solutions propagating in both the forward (formula_7) and backward (formula_8) directions. The general solution of the equation is the solutions in these two directions is: 
where formula_10 and formula_11 are equal and opposite displacements. 
When the one-way wave problem is formulated, the wave propagation direction can be arbitrarily selected by keeping one of the two terms in the general solution. 
Factoring the operator on the left side of the equation yields a pair of one-way wave equations, one with solutions that propagate forwards and the other with solutions that propagate backwards.
The forward- and backward-travelling waves are described respectively,
The one-way wave equations (in a homogeneous medium) can also be derived directly from the characteristic specific acoustic impedance. In a longitudinal plane wave, the specific impedance determines the local proportionality of pressure formula_14 and particle velocity formula_15: 
The conversion of the impedance equation leads to:
A longitudinal plane wave of angular frequency formula_19 has the displacement formula_4. The pressure formula_21 and the particle velocity formula_22 can be expressed in terms of the displacement formula_23 (formula_24: Elastic Modulus):
These relations inserted into the equation above (*) yield:
With the local wave velocity definition (speed of sound): 
directly follows the 1st-order partial differential equation of the one-way wave equation: 
The wave velocity formula_5 can be set within this wave equation as formula_34 or formula_35 according to the direction of wave propagation.
For wave propagation in the direction of formula_34 the unique solution is 
and for wave propagation in the formula_35 direction the respective solution is
There also exists a spherical one-way wave equation describing the wave propagation of a monopole sound source in spherial coordinates, i.e., in radial direction. By a modification of the radial nabla operator an inconsistency between spherical divergence and laplace operators is solved and the resulting solution does not show Bessel functions (in contrast to the known solution of the conventional two-way approach).
Three-dimensional Case.
The solution in the three-dimensional case is assumed to be similar way as for the one-dimensional case. There is a recent approach to use an hypothetical impulse flow equilibrium in a field point instead of an force equilibrium of a volume element to derive a 3D one-way wave equation. 