A one-way wave equation is a first-order
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
describing a
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
field resulting from superposition of two waves in opposite directions.
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 .
One-dimensional case
The scalar
second-order (two-way) wave equation describing a
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
field can be written as:
where
is the coordinate,
is time,
is the displacement, and
is the wave velocity.
Due to the ambiguity in the direction of the wave velocity,
, the equation does not contain information about the wave direction and therefore has solutions propagating in both the forward (
) and backward (
) directions. The general solution of the equation is the summation of the solutions in these two directions is:
where
and
are the displacement amplitudes of the waves running in
and
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.
The forward- and backward-travelling waves are described respectively,
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
and particle velocity
:
with
= density.
The conversion of the impedance equation leads to:
A longitudinal plane wave of angular frequency
has the displacement
.
The pressure
and the particle velocity
can be expressed in terms of the displacement
(
:
Elastic Modulus
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
):
for the 1D case this is in full analogy to
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
:
, with
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
being defined as
These relations inserted into the equation above () yield:
With the local wave velocity definition (
speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
):
directly(!) follows the 1st-order partial differential equation of the one-way wave equation:
The wave velocity
can be set within this wave equation as
or
according to the direction of wave propagation.
For wave propagation in the direction of
the unique solution is
and for wave propagation in the
direction the respective solution is
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 Nabla may refer to any of the following:
* the nabla symbol ∇
** the vector differential operator, also called del, denoted by the nabla
* Nabla, tradename of a type of rail fastening system (of roughly triangular shape)
* ''Nabla'' (moth), a ge ...
operator an inconsistency between spherical divergence and Laplace operators is solved and the resulting solution does not show
Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
(in contrast to the known solution of the conventional two-way approach).
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.
[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: a) derivation from impedance theorem
and b) derivation from a tensorial impulse flow equilibrium in a field point.
Inhomogeneous media
For inhomogeneous media with location-dependent elasticity module
, density
and wave velocity
an analytical solution of the one-way wave equation can be derived by introduction of a new field variable.
Further mechanical and electromagnetic waves
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.
See also
*
* {{annotated link, Standing wave
References
Geophysics
Wave mechanics
Acoustics
Sound
Continuum mechanics