3. Underlying theories

Waves are described by partial differential equations.  These equations refer to the special field (electrodynamics, mechanics, chemistry, biology, etc.).  The most well known is the wave equation, but several other partial differential equations (heat equation, reaction-diffusion equation, etc.) have solutions of the traveling wave form.
 

Wave Equation

DY-(1/v²)(¶²Y/¶t²) = 0    (3.1)

D: Laplace operator
v: velocity

If v is a constant then the function Y= S(t-(x/v)), where S is an arbitrary one variable function, is a solution of (eq.4).  This solution describes a planar wave propagating in the positive x direction with the velocity v.  The arbitrary function S describes the wave form: for harmonic waves S is sinusoidal.

Now let v be a given function of space.  In this case (inhomogenious medium) exact analyitic solutions are not known in general.

We are looking for solutions of the traveling wave form:  Y(r, t) = A(r)S(t-g(r)).  In this case, after substitution, we get the following formulas:

ÑY=ÑA×S-S'×AÑg  (eq. 3.2)
DY=DAS-2S'ÑA×Ñg+S"A((Ñg)²-(1/v²))  (eq. 3.3)

Consequently,the wave equation takes the form:

0=AS"((Ñg)²-(1/v²))-S'(2ÑA×Ñg+ADg)+S×DA (eq. 3.4)
 

Waves With Arbitrary Form

(Ñg)²-(1/v²) = 0   (3.5)

2ÑA×Ñg+ADg = 0  (3.6)

DA = 0  (3.7)

then the traveling wave form (1) is a solution of the wave equation, S being arbitrary.  One might expect this case to be very exceptional since two fuctions must simultaneously satisfy three equations.  However, there are important solutions belonging to this case which follow.
 

Spherical Symmetry

Ñ. = (r/r)¶./¶r
D. = ¶²./¶v²+((n-1)/r)¶./¶r
 

Proposition

a) n=1, then one solution is: Y=S(t-r/v)  (planar wave)
b) n=3, then one solution is: Y=(A₀/r)×S(t-r/v)   (spherical wave)

Proof:  the solution of (3.5) with spherical symmetry is g=±(r/v). The sign + belongs to waves propagating toward the origin, while the  sign - belongs to waves propagating outward from the origin.  Choosing g=-r/v, equations (3.6) and (3.7) take the form:

2A' + ((n-1)/r)A = 0  (3.6a )

A"+ ((n-1)/r)A' = 0  (3.7a)

After differentiating (3.6a) and subatracting from  (3.7a) we get

A" = ((n-1)/r²)A

Using that in (3.7a)  we get

((n-1)/r)(A'+A/r)=0   (3.8)

And so equations (3.6a) and (3.8) can be solved when n = 1 giving A'=0 and A=const.

On the other hand, if we assume n¹1 then:

2A' + ((n-1)/r)A = 0
A' + A/r = 0

Combining these equations, we get
((n-3)/r)A=0,
from which n=3  follows. For this case, we can see that
A' + A/r = 0          ®   A=A₀/r
which completes the proof.

Remarks.

Those are the best known soltutions of the wave equations. For n=1 we  got the wave traveling in one direction with constant velocity v, undamped and undistorted. The wave form given by  the function S is arbitrary.

For n=3 we got a wave propagating from  the origin outward with  constant velocity v. The amplitude of this wave decreases (inversely proportional to r).The wave form is undistorted, it is given by the arbitrary function S.

There is no similar wave for other values of n, that is the reason  cylindrical waves, in general,  are not discussed in textbooks.

*******************

If we remove the restriction v=const., equations (8,9, and 10) enable us to construct solutions with spherical symmetry for any n.  From equation (10) we calculate A(r), then from equation (9) we calculate g(r) and lastly from equation (8) a velocity function v(r) is yeilded.  For such a v traveling wave solutions are given.
 

Geometrical Optics Approximation

ÑY»A(r)ikÑG×e^i(kG(r)-wt)    (3.9)

that is the amplitude is assumed to be slowly varying, it's derivative was neglected here.  This way we deduce the equation:

(ÑG)²-n²=0   (3.10)

where n=w/kv is the refractive index.  This is called the eikonal equation and is essentially identical with equation (3.5).

Sine-Gordon Equation

Reaction-diffusion equation

¶c_i/¶t = Su_irJ_r + SD_ikDc_k    (3.11)

c_i: the concentration of the i^th component
t: time
u_ir:  stoichiometric coefficient
J_r:  reaction rates
D_ik:  diffusion coeffeicients
D:  Laplace operator

If the reaction rates depend nonlinearly on the concentrations (a typical case), the equation (3.11) is a nonlinear partial differential equation.

For a one component system equation (3.11) reduces to the form which is known as the Fischer Equation (Murray, 1990) if
f(c) = c(1-c).  We are looking for traveling waves in 1-D, that is, solutions of the form

c(x,t) = S(t-x/c)    (3.12)

Where S is a one variable function.  Introducing a new variable V=S' the Fischer Equation reduces to the system
S'= V    (13a)
V' = (c²/D)(V-kS(1-S))    (13b)

The Jacobian matrix:  J:    0                                 1
                                          k(c²/D)(2S-1)             c²/D
 
 

The trace of J: TrJ = c²/D > 0
The determinant of J = k(c²/D)(1-2S)
The discriminant of J:  d = Ö((TrJ)²-4detJ)
                                       = Ö((c²/D)²(1-4(D/c²)k)(1-2S)

This dynamical system has two equilibrium points: (0,0) and (1,0).  The former is an unstable node(if c²>4kD)  or unstable focus (if c²<4kD), the latter is always a saddle.  It can be proved (Murray, 1989) that there exists a unit trajectory (lying entirely in the positve quadrant) from (0,0) to (1,0) for c ³ c_min = 2Ö(kD).  This solution of (3.13) belongs to a wave traveling with velocity c in the positive x direction.  The wave profile is a decreasing function of x which tends to 1 as x tends to negative ¥ and tends to zero as x tends to positive ¥. Although such traveling wave solution exists for any c ³ c_min, only the one belonging to c_min is stable.  The others are unstable.  Starting from an arbitrary initial wave front profile (from S = 1 to S = 0) the front approaches a specific form determined by the above mentioned unique trajectory belonging to the velocity c_min.  That is the asymptotically the front will propagate with c_min.  Mathematically precise theorems were deduced by Kolmogoroff, Petrovsky and Piscounoff (1937) (Murray, 1990).

A huge amount of published papers have dealt with further developments of the Reaction-Diffusion equation, including two or three dimentions, two or more components, and several other types of kinetics.

An outstanding part of the Reaction-Diffusion theories, deals with the curvature effect.  As a matter of fact the latest theories utilize an approximation procedure.  The front line in two dimentions is parameterized by its curvature.  From the Reaction-Diffusion equation it was deduced that the wave velocity must depend on the curvature of the front, the dependance is linear in first approximation:

v = v₀+Dg     (14)

v₀:  wave velocity of plane fronts
v:  wave velocity at the point where the curvature of the front is g
D:  diffusion coefficient

The relation (14) means that the wave velocity is greater than v₀ if the front is concave and smaller than v₀ if the front is convex.  Theoretical and experimental results show that curvature effect can be observed only for very large values of g, otherwise it may be neglected.  The curvature effect reflects a certain role of the amplitude:  when the wave front is expanding (convex front) it must decrease necissarily.  Some experiments show that the wave front may split into two parts in these cases.
 

Heat equation

¶T/¶t = DD

T: temperature
D: heat diffusivity

Reaction-Diffusion equation of a one component medium reduces to the heat equation (or in other words, diffusion equation) when there is no reaction.  It has been known for a long time that the heat equation admits a traveling wave sollution:

T(x,t) = A₀e^-axcosw(t-ax)       (15)

where a = w/(Ö(2D))

This is a damped traveling wave.  The harmonic oscillations at a specified place propagate in the positive x direction with a velocity 1/a while the amplitude decreases exponentially.  This solution describes the observed phenomena related to the diurnal an annual variations of temperature inside the earth near to the surface.  (Carslaw and Jaeger, 1959)
********************************
¶c/¶t = f(c) + DDc
c = u(x-ct)
-cu' = f(u) = Du"
U = u
V = u'
DV' = f(U)-cU
U'=V