One-way wave equation

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

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

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:

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

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]

${\displaystyle {\biggl (}{\partial ^{2} \over \partial t^{2}}-c^{2}{\partial ^{2} \over \partial x^{2}}{\biggr )}s={\biggl (}{\partial \over \partial t}-c{\partial \over \partial x}{\biggr )}{\biggl (}{\partial \over \partial t}+c{\partial \over \partial x}{\biggr )}s=0,}$

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

${\textstyle {\begin{aligned}&{{\frac {\partial s}{\partial t}}-c{\frac {\partial s}{\partial x}}=0}\\[6pt]&{{\frac {\partial s}{\partial t}}+c{\frac {\partial s}{\partial x}}=0}\end{aligned}}}$

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 ${\displaystyle p=p(x,t)}$ and particle velocity ${\displaystyle v=v(x,t)}$:

${\displaystyle {\frac {p}{v}}=\rho c,}$

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

The conversion of the impedance equation leads to:

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

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):[7]

${\displaystyle p:=E{\partial s \over \partial x}}$ [This is in full analogy to stress ${\displaystyle \sigma }$ in mechanics: ${\displaystyle \sigma =E\epsilon }$, with strain being defined as ${\displaystyle \epsilon ={\frac {\Delta L}{L}}}$]

${\displaystyle v={\partial s \over \partial t}}$

These relations inserted into the equation above (*) yield:

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

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

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

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

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

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

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

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

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

[8]

• wave equation

• standing wave