One-way wave equation

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A one-way wave equation is a first order partial differential equation used in scientific fields such as geophysics, whose solution is - in contrast to the well-known 2nd order two-way wave equations with a solution consisting of two waves travelling in opposite directions - a single propagating wave travelling in a pre-defined direction (the direction in 1D is defined by the sign of the wave velocity).^[1][2] In the one-dimensional case, the one-way wave equation allows wave propagation to be calculated without the mathematical complication of solving a 2nd order differential equation. Due to the fact that in the last decades no 3D one-way wave equation could be found numerous approximation methods based on the 1D one-way wave equation are used for 3D seismic and other geophysical calculations, see also the section § Three-dimensional Case.^[3][4][1]

One-dimensional case[]

The standard 2nd-order wave equation in one dimension can be written as:

where ${\displaystyle x}$ is the coordinate, ${\displaystyle t}$ is time, ${\displaystyle s=s(x,t)}$ is the displacement, and ${\displaystyle c}$ is the wave velocity.

Due to the ambiguity in the direction of the wave velocity, ${\displaystyle c^{2}=(+c)^{2}=(-c)^{2}}$, the equation does not constrain the wave direction and so has solutions propagating in both the forward (${\displaystyle +x}$) and backward (${\displaystyle -x}$) directions. The general solution of the equation is the solutions in these two directions is:

where ${\displaystyle s_{+}}$ and ${\displaystyle s_{-}}$ 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.^[5][6]

The forward- and backward-travelling waves are described respectively,

The one-way wave equations 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 ${\displaystyle p=p(x,t)}$ and particle velocity ${\displaystyle v=v(x,t)}$:^[7]

with ${\displaystyle \rho }$ = density.

The conversion of the impedance equation leads to:^[8]

$${\displaystyle v-{\frac {1}{\rho c}}p=0}$$

(⁎)

A longitudinal plane wave of angular frequency ${\displaystyle \omega }$ has the displacement ${\displaystyle s=s(x,t)}$. The pressure ${\displaystyle p}$ and the particle velocity ${\displaystyle v}$ can be expressed in terms of the displacement ${\displaystyle s}$ (${\displaystyle E}$: Elastic Modulus)^{[9][better source needed]}:

for the 1D case this is in full analogy to stress ${\displaystyle \sigma }$ in mechanics: ${\displaystyle \sigma =E\varepsilon }$, with strain being defined as ${\displaystyle \varepsilon ={\frac {\Delta L}{L}}}$ ^[10]

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 ${\displaystyle c}$ can be set within this wave equation as ${\displaystyle +c}$ or ${\displaystyle -c}$ according to the direction of wave propagation.

For wave propagation in the direction of ${\displaystyle +c}$ the unique solution is

and for wave propagation in the ${\displaystyle -c}$ direction the respective solution is^[11]

There also exists a spherical one-way wave equation describing the wave propagation of a monopole sound source in spherical 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).^[12]

Three-dimensional Case[]

The One-Way equation and solution in the three-dimensional case was assumed to be similar way as for the one-dimensional case by a mathematical decomposition (factorization) of a 2nd order differential equation.^[13] In fact, the 3D One-way wave equation can be derived from first principles: a) derivation from impedance theorem ^[8] and b) derivation from a tensorial impulse flow equilibrium in a field point.^[12]

Inhomogeneous Media[]

For inhomogeneous media with location-dependent elasticity module ${\displaystyle E(x)}$, density ${\displaystyle \rho (x)}$ and wave velocity ${\displaystyle c(x)}$ an analytical solution of the one-way wave equation can be derived by introduction of a new field variable.^[14]

Further Mechanical and Electromagnetic Waves[]

The method of PDE factorization can also be tranferred to other standard 2nd or 4th order wave equations, e.g. transversal, and string, Moens/ Korteweg, bending, and electromagnetic wave equations and electromagnetic waves. The related one-way equations and their solutions are given in.^[14]

See also[]

• Transport equation
• Wave equation – Second-order linear differential equation important in physics
• Standing wave

References[]