bicharacteristics of the wave equation - kinematical...

Bicharacteristics of the Wave Equation -Kinematical Consideration, and Special theory

of Relativity

PHOOLAN PRASAD

DEPARTMENT OF MATHEMATICSINDIAN INSTITUTE OF SCIENCE, BANGALORE

25 September, 2012

bicharacteristics

P. Prasad

Simplest wave equation

ut + c ux = u , a = real constant (1)

The characteristic curves are

φ(x, t) l x− at = constant (2)

along which we have a compatibilitycondition

du

dtl

(∂

∂t+ a

∂

∂x

)u = u (3)

Therefore, general solution is

u = etf(x− at), f ∈ C1(R) (4)

bicharacteristics P. Prasad Department of Mathematics 2 / 23

bicharacteristics

P. Prasad

Simplest wave equation

Note that the function φ satisfies thecharacteristic PDE

φt + aφx = 0 (5)

Thus we see that for the PDE (1),1 we have a characteristic equation for the

characteristic function φ, which satisfiescharacteristic PDE and

2 along the characteristic curves thereexists a compatibility condition, whichgives the solution.


bicharacteristics

P. Prasad

One-dimensional wave equation

We start with the wave equation in two independentvariable:

utt − a2 uxx = 0. (6)

We note(∂2

∂t2− a2 ∂

2

∂x2

)u =

(∂

∂t+ c

∂

∂x

)(∂

∂t− a ∂

∂x

)u (7)

Define characteristic varaibles r = ut + ux ands = ut + ux, then(

∂

∂t− c ∂

∂x

)r = 0 and

(∂

∂t+ c

∂

∂x

)s = 0


bicharacteristics

P. Prasad

One-dimensional wave equation ... continued

The characteristic PDE for φ giving bothcharacteristics satisfies

Q(ϕx, ϕt) l ϕ2t − a2 |ϕt|2 = 0 (8)

(8) is a nonlinear first order PDE but in this case it isequivalent to two first order PDEs

φt + aφx = 0 and φt − aφx = 0

which immediately give the two families ofcharacteristic curves:

x− at = α, x+ at = β; α, β = constants. (9)


bicharacteristics

P. Prasad

One-dimensional wave equation ... continued

The general solution of 1-D wave equation is

u = f(x− at) + g(x+ at) ; f, g ∈ C2(R) (10)

⇒ two waves of translations, one moves with velocitya and another with velocity −a.Thus we see that for the 1-D wave equation,

1 we have a characteristic equation for thecharacteristic function φ, which satisfies thenonlinear characteristic PDE,

2 which gives two families of characteristic curvesand

3 along first (second) family of characteristic curvesthere exists a compatibility condition:characteristic variable r (s) is constant, whichgives the solution.


bicharacteristics

P. Prasad

Introduction to Bicharacteristics

I indicate only the concepts by considering thewave equation in Rm+1:

utt − a2 ∆mu = 0. (11)

∆m =m∑i=1

∂2

∂x2i

, a = constant > 0

Characteristic PDE is a first order PDE forthe characteristic surfaces Ω : ϕ(x, t) =constant

Q(∇ϕ, ϕt) l ϕ2t − a2 |∇ϕ|2 = 0 (12)


bicharacteristics

P. Prasad

Introduction to Bicharacteristics ... continued

Equation (12) is equivalent to two first orderPDEs:

φt + a |∇ϕ| = 0 (13)

andφt − a |∇ϕ| = 0, (14)

where |∇ϕ| =√

(ϕ2x1

+ ϕ2x2

+ ...+ ϕ2xm

).The characteristic equations are nownonlinear but we have a theory in whichsolution is obtained with the help ofCharpit’s equations.Bicharateristics are characteristics curvesin space-time of a characteristic equation.


bicharacteristics

P. Prasad

Bicharacteristic equations from a General EikonalEquation

The Charpit’s equations of the PDE (inwhich the dependent variable φ does notappear)

Q(∇ϕ, ϕt) = 0 (15)

aredt

dσ= Qϕt,

dxαdσ

= Qϕxα (16)

dϕtdσ

= −Qt,dϕxαdσ

= −Qxα. (17)

These are bicharacteristic equations from ageneral Eikonal Equation.


bicharacteristics

P. Prasad

Characteristic Surface and Wavefront

A characteristic surface isΩ : ϕ(x, t) = α, α = fixed.A Wavefront Ωt is the projection on(x)-space of a section of Ω by t=constantplane.

Unit normal of Ωt : n =∇ϕ|∇ϕ|

(18)

Velocity of Ωt : c = − ϕt|∇ϕ|

(19)

A ray is a projection on (x)-space of abicharacteristic curve in space-time.

bicharacteristics P. Prasad Department of Mathematics10 /23

bicharacteristics

P. Prasad

A Characteristic Conoid and Bicharacteristics of amodel wave equation

−1

0

1

2

3 −1

0

1

2

3

−1

−0.8

−0.6

−0.4

−0.2

0

yx

t

(xP, y

P, t

n+1)

Figure: Sections of the characteristic conoid are shown in red


bicharacteristics

P. Prasad

Wavefronts and Rays of a model wave equation

−1 0 1 2 3−1

−0.5

0

0.5

1

1.5

2

2.5

3

x

y

Figure: Sections of the characteristic conoid are shown in red


bicharacteristics

P. Prasad

Wave Equation: Bicharacteristic Equations

When

Q(∇ϕ, ϕt) = φt ± a |∇ϕ| (20)

Q is independent of t, xα, u and thebicharateristic equations reduce to

dt

dσ= 1,

dxαdσ

= ±a φxα|∇ϕ|

,dϕtdσ

= 0,dϕxαdσ

= 0

(21)which reduce to

dxαdt

= ±a φxα|∇ϕ|

,dϕtdt

= 0,dϕxαdt

= 0 (22)


bicharacteristics

P. Prasad

Wave Equation: Bicharacteristic Curves and Rays

For the wave equation the wave frontvelocity is c = ±a0 and most importantsolutions of (13-14) are

ϕ ≡ (t− t0)±1

a|x− x0| (23)

The bicharacteristic equations in terms of xand n become

dxαdt

= ±nαa,dnαdt

= 0 (24)

Bicharacteristic curves of the wave equationare straight lines in space-time given by:

x = x0 ± na(t− t0) (25)


bicharacteristics

P. Prasad

Wave Equation: Bicharacteristic Curves and Rays... continued

At a given time t (23) represents a circularwavefront with centre at (x0, t0) and radius= a(t− t0).Equation (25) gives a ray which moves in astraight line with velocity a from the point(x0, t0) in an arbitrarily chosen direction n.


bicharacteristics

P. Prasad

Wave Equation: Geometry of CharacteristicConoid and Bicharacteristis

Figure:


bicharacteristics

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Space-like plane and Time-like Direction

For m > 1, important role in theory is played bySpace-like surface R : for which value at anypoint P on R does not influence the solution atany other point on R.Example: The plane t = 0 is space-like.

Figure:bicharacteristics P. Prasad Department of Mathematics17 /23

bicharacteristics

P. Prasad

Space-like plane and Time-like Direction

For m > 1, important role in theory is played bya time like direction

Time-like direction : at a point P0(x0, t0)points into the interior of the characteristicconoid at P0. Example : the direction(0, 0, . . . , 0, 1)For m = 1, i.e., wave equation in one spacedimension, there is no distinction betweenspace-like planes and time-like diections.The characteristic conoid is known as nullcone in physics.Generators of the null cone give particlepaths moving with the velocity of light.


bicharacteristics

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Wave Equation: Space-like plane

Figure:


bicharacteristics

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Special Theory of Relativity

The two axioms of thespecial theory of relativity can be statedtogether as:

the propagation of light in an inertial frameis governed by the wave equation with a asthe same constant for all inertial frames

Every non-singular real lineartransformation, from one inertial frame toanother, of the variables t, x1, x2, ..., xm,under which the wave equation remainsinvariant are stated on the next slide.


bicharacteristics

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Special Theory of Relativity ... continued

1 A Lorentz transformation (this one is STR),

2 a translation of origin and

3 a similarity transformation (see G.Petrovsky, Lectures on partial differentialequations, 1954).

One important consequence of STR: The Lorentztransformation maps t-axis into a time-like lineand x1, x2, ..., xm axes into space-like lines.


bicharacteristics

P. Prasad

Kinematics of a Wavefront and its Dynamics

We have discussed only the kinematics of alinear wavefront.

Discussion of dynamics of a wavefront wouldrequire use of the transport of equation forthe wave amplitude along a ray.

For a nonlinear wavefront (including a shockfront), the ray equations and the transportequation get coupled. This leads to difficultand challenging research investigations,which are really beautiful.


bicharacteristics

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Example 3 — continued

Thank you


bicharacteristics of the wave equation - kinematical...

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