Editing Wave equation (section)

==Vectorial wave equation in three space dimensions==
{{Primary sources|1=section|date=September 2024}}

The vectorial wave equation (from which the scalar wave equation can be directly derived) can be obtained by applying a force equilibrium to an infinitesimal volume element. If the medium has a modulus of elasticity <math>E</math> that is homogeneous (i.e. independent of <math>\mathbf{x}</math>) within the volume element, then its stress tensor is given by <math>\mathbf{T} = E \nabla \mathbf{u}</math>, for a vectorial elastic deflection <math>\mathbf{u}(\mathbf{x}, t)</math>. The local equilibrium of:
# the tension force <math>\operatorname{div} \mathbf{T} = \nabla\cdot(E \nabla \mathbf{u}) = E \Delta\mathbf{u}</math> due to deflection <math>\mathbf{u}</math>, and
# the inertial force <math>\rho \partial^2\mathbf{u}/\partial t^2</math> caused by the local acceleration <math>\partial^2\mathbf{u} / \partial t^2</math> 
can be written as <math>\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} - E \Delta \mathbf{u} = \mathbf{0}.</math>

By merging density <math>\rho</math> and elasticity module <math>E,</math> the sound velocity <math>c = \sqrt{E/\rho}</math> results (material law). After insertion, follows the well-known governing wave equation for a homogeneous medium:<ref name="br3">{{Cite journal |last1=Bschorr |first1=Oskar |last2=Raida |first2=Hans-Joachim |date=April 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}} [[File:CC-BY icon.svg|50px]] Text was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/  Creative Commons Attribution 4.0 International License].</ref>
<math display="block">\frac{\partial^2 \mathbf{u}}{\partial t^2} - c^2 \Delta \mathbf{u} = \boldsymbol{0}.</math>
(Note: Instead of vectorial <math>\mathbf{u}(\mathbf{x}, t),</math> only scalar <math>u(x, t)</math> can be used, i.e. waves are travelling only along the <math>x</math> axis, and the scalar wave equation follows as <math>\frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0</math>.)

The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions. From the quadratic velocity term <math>c^2 = (+c)^2 = (-c)^2</math> can be seen that there are two waves travelling in opposite directions <math>+c</math> and <math>-c</math> are possible, hence results the designation “two-way wave equation”.
It can be shown for plane longitudinal wave propagation that the synthesis of two [[one-way wave equation]]s leads to a general two-way wave equation. For <math>\nabla\mathbf{c} = \mathbf{0},</math> special two-wave equation with the d'Alembert operator results:<ref>{{Cite journal |last=Raida |first=Hans-Joachim |date=October 2022 |title=One-Way Wave Operator |journal=Acoustics |language=en |volume=4 |issue=4 |pages=885–893 |doi=10.3390/acoustics4040053 |doi-access=free}}</ref> 
<math display="block">\left(\frac{\partial}{\partial t} - \mathbf{c} \cdot \nabla\right)\left(\frac{\partial}{\partial t} + \mathbf{c} \cdot \nabla \right) \mathbf{u} =
 \left(\frac{\partial^2}{\partial t^2} + (\mathbf{c} \cdot \nabla) \mathbf{c} \cdot \nabla\right) \mathbf{u} =
 \left(\frac{\partial^2}{\partial t^2} + (\mathbf{c} \cdot \nabla)^2\right) \mathbf{u} = \mathbf{0}.</math>
For <math>\nabla \mathbf{c} = \mathbf{0},</math> this simplifies to
<math display="block">\left(\frac{\partial^2}{\partial t^2} + c^2\Delta\right) \mathbf{u} = \mathbf{0}.</math>
Therefore, the vectorial 1st-order [[one-way wave equation]] with waves travelling in a pre-defined propagation direction <math>\mathbf{c}</math> results<ref name="br2">{{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=714–722 |doi=10.3390/acoustics3040045 |doi-access=free}}</ref> as
<math display="block">\frac{\partial \mathbf{u}}{\partial t} - \mathbf{c} \cdot \nabla \mathbf{u} = \mathbf{0}.</math>