One-way wave equation

The scalar second-order (two-way) wave equation describing a standing wavefield can be written as:

$$\backslash frac\{\backslash partial^2\; s\}\{\backslash partial\; t^2\}\; -\; c^2\; \backslash frac\{\backslash partial^2\; s\}\{\backslash partial\; x^2\}\; =\; 0,$$

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

Due to the ambiguity in the direction of the wave velocity, $c^2=(+c)^2=(-c)^2$, the equation does not contain information about the wave direction and therefore has solutions propagating in both the forward ($+x$) and backward ($-x$) directions. The general solution of the equation is the summation of the solutions in these two directions:

$$s(x,t)=s\_\{+\}(t\; -x/c)\; +\; s\_\{-\}\; (t\; +x/c)$$

where $s\_\{+\}$ and $s\_\{-\}$ are the displacement amplitudes of the waves running in $+c$ and $-c$ direction.

When a one-way wave problem is formulated, the wave propagation direction has to be (manually) 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.{{citation| last1=Baysal | first1=Edip |title=A two‐way nonreflecting wave equation|date=February 1984| volume=49|issue=2|periodical=Geophysics|at=pp. 132–141| doi=10.1190/1.1441644|issn=0016-8033|last2=Kosloff|first2=Dan D.|last3=Sherwood| first3=J. W. C.| bibcode=1984Geop...49..132B}}{{citation| last=Angus|first=D. A.| title=The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena|date=2013-08-17|volume=35| issue=2|periodical=Surveys in Geophysics| at=pp. 359–393| doi=10.1007/s10712-013-9250-2|bibcode=2014SGeo...35..359A | s2cid=121469325|issn=0169-3298| url=http://eprints.whiterose.ac.uk/77377/8/SIG_2013_with_coversheet.pdf}}{{Cite journal |last1=Bschorr|first1=Oskar| last2=Raida| first2=Hans-Joachim|date=December 2021 |title=Factorized One-Way Wave Equations |journal=Acoustics |language=en|volume=3 |issue=4 |pages=717–722 |doi=10.3390/acoustics3040045 |doi-access=free}}

$$\backslash left(\{\backslash partial^2\backslash over\backslash partial\; t^2\}-c^2\{\backslash partial^2\backslash over\backslash partial\; x^2\}\backslash right)s=$$

\left({\partial\over\partial t}-c{\partial\over\partial x}\right)

\left({\partial\over\partial t}+c{\partial\over\partial x}\right)s=0,

The backward- and forward-travelling waves are described respectively (for $c\; >\; 0$),

\begin{align}

& {\frac{\partial s}{\partial t} - c \frac{\partial s}{\partial x} = 0} \\[6pt]

& {\frac{\partial s}{\partial t} + c \frac{\partial s}{\partial x} = 0}

\end{align}

The one-way wave equations can also be physically derived directly from specific acoustic impedance.

In a longitudinal plane wave, the specific impedance determines the local proportionality of pressure $p=\; p(x,t)$ and particle velocity $v=\; v(x,t)$:{{Cite web| title=Sound - Impedance |url=https://www.britannica.com/science/sound-physics|access-date=2021-05-20 |website=Encyclopedia Britannica| language=en}}

$$\backslash frac\{p\}\{v\}=\backslash rho\; c\; ,$$

with $\backslash rho$ = density.

The conversion of the impedance equation leads to:

{{NumBlk||$$v\; -\; \backslash frac\{1\}\{\backslash rho\; c\}\; p\; =\; 0$$|{{EquationRef|⁎}}}}

A longitudinal plane wave of angular frequency $\backslash omega$ has the displacement $s\; =\; s(x,t)$.

The pressure $p$ and the particle velocity $v$ can be expressed in terms of the displacement $s$ ($E$: Elastic Modulus){{Cite web| title=elastic modulus |url=https://www.britannica.com/science/elastic-modulus|access-date=2021-12-15 |website=Encyclopedia Britannica| language=en}}{{Better source needed|reason=Recently published primary source|date=May 2021}}:

$$p:=E\; \{\backslash partial\; s\backslash over\backslash partial\; x\}$$ for the 1D case this is in full analogy to stress $\backslash sigma$ in mechanics: $\backslash sigma\; =\; E\; \backslash varepsilon$, with strain being defined as $\backslash varepsilon\; =\; \backslash frac\{\backslash Delta\; L\}\{L\}$ {{Cite web|title=Young's modulus {{!}} Description, Example, & Facts |url=https://www.britannica.com/science/Youngs-modulus|access-date=2021-05-20| website=Encyclopedia Britannica| language=en}}

$$v\; =\; \{\backslash partial\; s\; \backslash over\backslash partial\; t\}$$

These relations inserted into the equation above ({{EquationNote|⁎}}) yield:

$$\{\backslash partial\; s\; \backslash over\backslash partial\; t\}\; -\; \{E\backslash over\; \backslash rho\; c\}\; \{\backslash partial\; s\backslash over\backslash partial\; x\}\; =\; 0$$

With the local wave velocity definition (speed of sound):

$$c=\backslash sqrt\{E(x)\; \backslash over\; \backslash rho(x)\}\; \backslash Leftrightarrow\; c\; =\; \{E\backslash over\; \backslash rho\; c\}$$

directly(!) follows the 1st-order partial differential equation of the one-way wave equation:

$$\{\backslash frac\{\backslash partial\; s\}\{\backslash partial\; t\}-c\; \backslash frac\{\backslash partial\; s\}\{\backslash partial\; x\}\; =\; 0\}$$

The wave velocity $c$ can be set within this wave equation as $+c$ or $-c$ according to the direction of wave propagation.

For wave propagation in the direction of $+c$ the unique solution is

$$s(x,t)=s\_\{+\}(t\; -x/c)$$

and for wave propagation in the $-c$ direction the respective solution is{{Cite web|url=https://mathworld.wolfram.com/WaveEquation1-Dimensional.html|title=Wave Equation--1-Dimensional}}

$$s(x,t)=s\_\{-\}(t+x/c)$$

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).{{Cite journal |last1=Bschorr|first1=Oskar| last2=Raida| first2=Hans-Joachim|date=March 2021 |title=Spherical One-Way Wave Equation |journal=Acoustics |language=en|volume=3 |issue=2 |pages=309–315 |doi=10.3390/acoustics3020021 |doi-access=free}}

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.The mathematics of PDEs and the wave equation https://mathtube.org/sites/default/files/lecture-notes/Lamoureux_Michael.pdf In fact, the 3D one-way wave equation can be derived from first principles:

• derivation from impedance theorem and
• derivation from a tensorial impulse flow equilibrium in a field point.

It is also possible to derive the vectorial two-way wave operator from synthesis of two one-way wave operators (using a combined field variable). This approach shows that the two-way wave equation or two-way wave operator can be used for the specific condition $\backslash nabla\; \backslash mathbf\; c=0$, i.e. for homogeneous and anisotropic medium, whereas the one-way wave equation resp. one-way wave operator is also valid in inhomogeneous media.{{Cite journal |last1=Raida |first1=Hans-Joachim|date=March 2022|title=One-Way Wave Operator |journal=Acoustics|language=en| volume=4|issue=4| pages=885–893| doi=10.3390/acoustics4040053|doi-access=free}}

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

The method of PDE factorization can also be transferred to other 2nd or 4th order wave equations, e.g. transversal, and string, Moens/Korteweg, bending, and electromagnetic wave equations and electromagnetic waves.

• {{annotated link|Wave equation}}
• {{annotated link|Standing wave}}
• {{annotated link|Continuity equation}}

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