Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers

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13.3 Linear Dispersive Waves 541

Figure 13.2.2 The solution u (x, t) for diﬀerent times t =0,τ and 2τ ;the

characteristics are shown by the dotted lines; two of them from x = ξ

and

x = ξ

intersect at t>τ.

substituted in the equation, ∂/∂t, ∂/∂x, ∂/∂y,and∂/∂z produce factors

−iω, ik, il,andim respectively, and the solution exists provided ω and κ

are related by an equation

P (−iω, ik, il, im)=0. (13.3.3)

This equation is known as the dispersion relation. Evidently, we have a

direct correspondence between equation (13.3.1) and the dispersion relation

(13.3.3) through the correspondence

∂

∂t

↔−iω,



∂

∂x

∂

∂y

∂

∂z



↔ i (k, l,m) . (13.3.4)

Equation (13.3.1) and the corresponding dispersion relation (13.3.3) in-

dicate that the former can be derived from the latter and vice-versa by us-

ing (13.3.4). The dispersion relation characterizes the plane wave motion.

In many prob lems, the dispersion relation can be written in the explicit

form

ω = W (k, l,m) . (13.3.5)

The phase and the group velocities of the waves are deﬁned by

(κ

κ)=

κ, (13.3.6)

(κ

κ)=∇

ω, (13.3.7)

542 13 Nonlinear Partial Diﬀerential Equations with Applications

where

κ is the unit vector in the direction of wave vector κ

κ.

In the one-dimensional case, (13.3.5)–(13.3.7) become

ω = W (k) ,C

dω

. (13.3.8)

The one-dimensional waves given by (13.3.2) are called dispersive if the

group velocity C

≡ ω

′

(k) is not constant, that is, ω

′′

(k) = 0. Physically,

as time progresses, the diﬀerent waves disperse in the medium with the

result that a single hump breaks into wavetrains.

Example 13.3.1.

(i) Linearized one-dimensional wave equation

− c

=0,ω=+

ck. (13.3.9)

(ii) Linearized Korteweg and de Vries (KdV) equation for long water waves

+ αu

+ βu

xxx

=0,ω= αk − βk

. (13.3.10)

(iii) Klein–Gordon equation

− c

+ α

u =0,ω=+



+ α



. (13.3.11)

(iv) Schr¨odinger equation in quantum mechanics and de Broglie waves

i ψ

−



V −



∇



ψ =0,  ω =



+ V, (13.3.12)

where V is a constant potential energy, and h =2π is the Planck

constant.

The grou p velocity of de Broglie wave is (κ

κ/m), and through the corre-

spondence principle, ω is to be interpreted as the total energy,





/2m



as the kinetic energy, and κ

κ as the particle momentum. Hence, the group

velocity is the classical particle velocity.

(v) Equation for vibration of a beam

+ α

xxxx

=0,ω=+

αk

. (13.3.13)

(vi) The dispersion relation for water waves in an ocean of depth h

= gk tanh kh, (13.3.14)

where g is the acceleration due to gravity.

13.3 Linear Dispersive Waves 543

(vii) The Boussinesq equation

− α

∇

u − β

∇

=0,ω=+

ακ



1+β

. (13.3.15)

This equ ation arises in elasticity for longitudinal waves in bars, long

water waves, and plasma waves.

(viii) Electromagnetic waves in dielectrics



+ ω



− c



− ω

=0,



− ω



− c



− ω

=0, (13.3.16)

where ω

is the natural frequency of the oscillator, c

is the speed of

light, and ω

is the plasma frequency.

In view of the superpositi on principle, the general solution can be obtained

from (13.3.2) with the dispersion solution (13.3.3). For th e one-dimensional

case, the general solution has the Fourier integral representation

u (x , t)=



∞

−∞

F (k) e

i[kx−tW (k)]

dk, (13.3.17)

where F (k) is chosen to satisfy the initial or boundary data provided the

data are physically realistic enough to have Fourier transforms.

In many cases, as cited in Example 13.3.1, there are two modes ω =

W (k) so that the solution (13.3.17) has the form

u (x , t)=



∞

−∞

(k) e

i[kx−tW (k)]

dk +



∞

−∞

(k) e

i[kx−tW (k)]

dk, (13.3.18)

with the initial data at t =0

u (x , t)=φ (x) ,u

(x, t)=ψ (x) . (13.3.19)

The initial conditions give

φ (x)=



∞

−∞

(k)+F

(k)] e

ikx

dk,

ψ (x)=−i



∞

−∞

(k)+F

(k)] W (k) e

ikx

dk.

Applying the Fourier inverse transformations, we have

(k)+F

(k)=Φ (k)=

√

2π



∞

−∞

φ (x) e

−ikx

dx,

−iW (k)[F

(k) − F

(k)] = Ψ (k)=

√

2π



∞

−∞

ψ (x) e

−ikx

dx,

544 13 Nonlinear Partial Diﬀerential Equations with Applications

so that

(k)+F

(k)] =



Φ (k)+

iΨ(k)

W (k)



. (13.3.20)

The asymptotic behavior of u (x, t) for large t with ﬁxed (x/t)canbe

obtained by the Kelvin stationary phase approximation. For real φ (x),

ψ (x), Φ (−k)=Φ

∗

(k)andΨ (−k)=Ψ

∗

(k), where the asterisk denotes a

complex conjugate. It follows from (13.3.20) that, for W (k)even

(−k) ,F

(−k)]=[F

∗

(k) ,F

∗

(k)] , (13.3.21)

and for W (k) odd,

(−k) ,F

(−k)]=[F

∗

(k) ,F

∗

(k)] . (13.3.22)

In particular, when φ (x)=δ (x)andψ (x) ≡ 0, then F

(k)=F

(k)=

√

8π, and the solution (13.3.18) reduces to the form

u (x , t)=





∞

cos kx cos {tW (k)}dk. (13.3.23)

In order to obtain the asymptotic approximation by the Kelvin station-

ary phase method (see Section 12.7) for t →∞, we consider both cases

when W (k)iseven(W

′

(k) is odd) and when W (k)isodd(W

′

(k)iseven)

and make an extra reasonable assumption that W

′

(k) is monotonic and

positive for k>0. It turns out that the asymptotic solution for t →∞is

u (x , t) ∼ 2Re

(k)



2π

t |W

′′

(k)|



exp

(

θ (x, t) −

sgn W

′′

(k)

+ O





=Re

a (x, t) e

iθ(x,t)

, (13.3.24)

where k (x, t) is the positive root of the equation

′

(k)=

,ω= W (k) ,

> 0, (13.3.25ab)

θ (x, t)=xk(x, t) − tω(x, t) , (13.3.26)

and

a (x, t)=2F

(k)



2π

t |W

′′

(k)|



exp



−

iπ

sgn W

′′

(k)



. (13.3.27)

It is important to point out that solution (13.3.24) has a form similar

to that of the elementary plane wave solution, but k, ω,anda are no

13.4 Nonlinear Dispersive Waves and Whitham’s Equations 545

longer constants; they are functions of space variable x and time t.The

solution still represents an oscillatory wavetrain with the phase function

θ (x, t) describing the variations between local maxima and minima. Unlike

the elementary plane wavetrain, the present asymptotic result (13.3.24)

represents a nonuniform wavetrain in the sense that the amplitude, the

distance, and the time between successive maxima are not constant.

It also follows from (13.3.25a) that

= −

′

(k)

′′

(k) t

∼ O





, (13.3.28)

= −

′′

(k)

∼ O





. (13.3.29)

These results indicate the k (x, t) is a slowly varying function of x and t as

t →∞. Applying a similar argument to ω and a, we conclude that k (x, t),

ω (x, t), and a (x, t) are slowly varying functions of x and t as t →∞.

Finally, all these results seem to provide an imp ortant clue for natural

generalization of the concept of nonlinear and nonuniform wavetrains.

13.4 Nonlinear Dispersive Waves and Whitham’s

Equations

To describe a slowly varying nonlinear an d nonuniform oscillatory wavetrain

in a medium (see Whith am, 1974), we assume the existence of a solution

in the form (13.3.24) so that

u (x , t)=a (x, t) e

iθ(x,t)

+c.c., (13. 4.1)

where c.c. stands for the complex conjugate, a (x, t) is the complex ampli-

tude given by (13.3.27), an d the phase function θ (x, t)is

θ (x, t)=xk(x, t) − tω(x, t) , (13.4.2)

and k, ω,anda are slowly varying function of x and t.

Due to slow variations of k and ω, it is reasonable to assume that these

quantities still satisfy the dispersion relation

ω = W (k) . (13.4.3)

Diﬀerentiating (13.4.2) with respect to x and t respectively, we obtain

= k + {x − tW

′

(k)}k

, (13.4.4)

= −W (k)+{x − tW

′

(k)}k

. (13.4.5)

In the neighborhood of stationary points deﬁned by (13.3.25a), these

results become

546 13 Nonlinear Partial Diﬀerential Equations with Applications

= k (x, t) ,θ

= −ω (x, t) . (13.4.6)

These results can be used as a deﬁnition of local wavenumber and local

frequency of the slowly varying nonlinear wavetrain.

In view of (13.4.6), relation (13.4.3) gives a nonlinear partial diﬀerential

equation for the phase θ in the form

∂θ

∂t

+ W



∂θ

∂x



=0. (13.4.7)

The solution of this equation determines the geometry of the wave pattern.

However, it is convenient to eliminate θ from (13.4.6) to obtain

∂k

∂t

∂ω

∂t

=0. (13.4.8)

This is known as the Whitham equation for the conservation of waves, where

k represents the density of waves and ω is the ﬂux of waves.

Using the dispersion relation (13.4.3), we obtain

∂k

∂t

+ C

(k)

∂k

∂x

=0, (13.4.9)

where C

(k)=W

′

(k) is the group velocity. This represents the simplest

nonlinear wave (hyperbolic) equation for the propagation of k with the

group velocity C

(k).

Since equation (13.4.9) is similar to (13.2.1), we can use the analysis of

Section 13.2 to ﬁnd the general solution of (13.4.9) with the initial condition

k (x, 0) = f (x)att = 0. In this case, the solution has the form

k (x, t)=f (ξ) ,x= ξ + tF (ξ) , (13.4.10)

where F (ξ)=C

(f (ξ)). This further conﬁrms the propagation of k with

the velocity C

. Some physical interpretations of this kind of solution have

already been discussed in Section 13.2.

Equations (13.4.9) and (13.4.3) reveal that ω also satisﬁes the nonlinear

wave (hyperbolic) equation

∂ω

∂t

+ W

′

(k)

∂ω

∂x

=0. (13.4.11)

It follows from equations (13.4.9) and (13.4.11) that both k and ω remain

constant on the characteristic curves deﬁned by

= W

′

(k)=C

(k) , (13.4.12)

in the (x, t) plane. Since k and ω is constant on each curve, the characteristic

curves are straight lines with slope C

(k). The solution for k is given by

(13.4.10).

13.4 Nonlinear Dispersive Waves and Whitham’s Equations 547

Finally, it follows from the above analysis that any constant value of

the phase θ propagates according to θ (x, t) = constant, and hence,





=0, (13.4.13)

which gives, by (13.4.6),

= −

= C

. (13.4.14)

Thus, the phase of the waves propagates with the phase speed C

.Onthe

other hand, (13.4.9) ensures that the wavenumber k propagates with the

group velocity C

(k)=(dω/dk)=W

′

(k).

We next investigate how the wave energy propagates in the dispersive

medium. We consider the following integral involving the square of the wave

amplitude (energy) given by (13.3.24) between any two points x = x

and

x = x

(0 <x

)

Q (t)=



|a|

dx =



∗

dx, (13.4.15)

=8π



(k) F

∗

(k)

t |W

′′

(k)|

dx, (13.4.16)

which is, due to a change of variable x = tW

′

(k),

=8π



(k) F

∗

(k) dk, (13.4.17)

where k

= tW

′

), r =1, 2.

When k

is kept ﬁxed as t varies, Q (t) remains constant so that



|a|

dx,



∂

∂t

|a|

dx + |a|

′

) −|a|

′

) . (13.4.18)

In the limit x

− x

→ 0, this result reduces to the partial diﬀerential

equation

∂

∂t

|a|

∂

∂x

′

(k) |a|

=0. (13.4.19)

This represents the equation for the conservation of wave energy, where |a|

and |a|

′

(k) are the energy density and energy ﬂux respectively. It al so

follows that the energy propagates with the group velocity W

′

(k). It has

been shown that the wavenumber k also propagates with the group velocity.

Evidently, the group velocity plays a double role.

548 13 Nonlinear Partial Diﬀerential Equations with Applications

The above analysis reveals another imp ortant fact; equations (13.4.3),

(13.4.8), and (13.4.19) constitute a closed set of equations for the three

quantities k, ω,anda. Indeed, these are the fundamental equations for

nonlinear dispersive waves and are known as Whitham’s equations.

13.5 Nonlinear Instability

For inﬁnitesimal waves, the wave amplitude (ak ≪ 1) is very small, so that

nonlinear eﬀects can be neglected altogether. However, for ﬁnite ampli-

tude waves the terms i nvolving a

cannot be neglected, and the eﬀects of

nonlinearity become important. In the theory of water waves, S tokes ﬁrst

obtained the connection due to inherent nonlinearity between the wave-

proﬁle and the frequency of a steady periodic wave system. According to

the Stokes theory, the remarkable fact is the dependence of ω on a which

couples (13.4.8) to (13.4.19). This leads to a new nonlinear phenomenon.

For ﬁnite amplitude waves, the frequency ω has the Stokes expansion

ω = ω

(k)+a

(k)+...= ω



k, a



. (13.5.1)

This can be regarded as the nonlinear dispersion relation which depends on

both k and a

. In the linear case, the amplitude a → 0, (13.5.1) gives the

linear dispersion relation (13.4.3), that is, ω = ω

(k).

In order to discuss nonlinear instability, we substitute (13.5.1) into

(13.4.8) and retain (13.4.19) in the linear approximation to obtain the fol-

lowing coupled system:

∂k

∂t

∂

∂x

(k)+ω

(k) a

=0, (13.5.2)

∂a

∂t

∂

∂x

′

(k) a

=0, (13.5.3)

where W (k) ≡ ω

(k).

These equations can be further approximated to obtain

∂k

∂t

+ ω

′

∂k

∂x

+ ω

∂a

∂x

= O





, (13.5.4)

∂a

∂t

+ ω

′

∂a

∂x

+ ω

′′

∂k

∂x

=0. (13.5.5)

In matrix form, these equations read

⎛

⎝

′

′′

′

⎞

⎠

⎛

⎝

∂k

∂x

∂a

∂x

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

∂k

∂t

∂a

∂t

⎞

⎠

=0. (13.5.6)

Hence, the eigenvalues λ are th e roots of the determinant equation

13.6 The Traﬃc Flow Model 549

− λb

| =



′

− λω

′′

′

− λ



=0, (13.5.7)

where a

and b

are the coeﬃcient matrices of (13.5.6). This equation gives

the characteristic velocities

λ =

= ω

′

(ω

′′

)

a + O





. (13.5.8)

If ω

′′

> 0, the characteristics are real and the system is hyperbolic.

The double characteristic velocity splits into two separate real velocities.

This provides a new extension of the group velocity to nonlinear problems. If

the disturbance is initially ﬁnite in extent, it would eventually split into two

disturbances. In general, any initial disturbance or modulating source would

introduce d istur bances in both families of characteristics. In the hyperbolic

case, compressive modulation will progressively distort and steepen so that

the question of breaking will arise. These results are remarkably diﬀerent

from those found in linear theory, where there is only one characteristic

velocity and any hump may distort, due to the dependence of ω

′

(k)onk,

but would never split.

On the other hand, if ω

′′

< 0, the characteristics are complex and the

system is elliptic. This leads to ill-p osed problems. Any small perturbations

in k and a will be given by the solutions of the form exp [iα (x − λt)] where λ

is calculated from (13.5.8) for unperturbed values of k and a. In this elliptic

case, λ is complex, and the perturbation will grow as t →∞. Hence, the

original wavetrain will become un stable. In the linear theory, the elliptic

case does not arise at all.

Example 13.5.1. For Stokes waves in deep water, the dispersion relation is

ω =(gk)





, (13.5.9)

so that ω

(k)=(gk)

and ω

(k)=

√

In thi s case, ω

′′

(k)=−

√

−

so that ω

′′

= −

k<0. The con-

clusion is that Stokes waves in deep water are deﬁnitely unstable. This is

one of the most remarkable results in the theory of nonlinear water waves

discovered during the 1960’s.

13.6 The Traﬃc Flow Model

We consider the ﬂow of cars on a long highway under the assumptions that

cars do not enter or leave the highway at any one of its p oi nts. We take the

x-axis along the highway and assume that the traﬃc ﬂows in th e positive

550 13 Nonlinear Partial Diﬀerential Equations with Applications

direction. Suppose ρ (x, t) is the density representing the numb er of cars

per unit l ength at the point x of the highway at time t,andq (x, t)isthe

ﬂow of cars per unit time.

We assume a conservation law which states that the change in the total

amount of a physical quantity contained in any region of space must be

equal to the ﬂux of that quantity across the boundary of that r egion. In

this case, the time rate of change of the total number of cars in any segment

≤ x ≤ x

of the highway is given by



ρ (x, t) dx =



∂ρ

∂t

dx. (13.6.1)

This rate of change must be equal to the n et ﬂux across x

and x

given by

q (x

,t) − q (x

,t) (13.6.2)

which measures the ﬂow of cars entering the segment at x

minus the ﬂow

of cars leaving the segment at x

. Thus, we have the conservation equation



ρ (x, t) dx = q (x

,t) − q (x

,t) , (13.6.3)



∂ρ

∂t

dx = −



∂q

∂x

dx,





∂ρ

∂t

∂q

∂x



dx =0. (13.6.4)

Since the integrand in (13.6.4) is continuous, and (13.6.4) holds for every

segment [x

], it follows that the integrand must vanish so that we have

the partial diﬀerential equation

∂ρ

∂t

∂q

∂x

=0. (13.6.5)

We now introduce an additional assumption which is supported by both

theoretical and experimental ﬁndings. According to this assumption, the

ﬂow rate q depends on x and t only through the density, that is, q = Q (ρ)

for some function Q. This assumption seems to be reasonable in the sense

that the density of cars surrounding a given car indeed controls the speed of

that car. The functional relation between q and ρ depends on many factors,

including speed limits, weather conditions, and road characteristics. Several

speciﬁc relations are suggested by Haight (1963).

We consider here a particular relation q = ρv where v is the average

local velocity of cars. We assume that v is a function of ρ toaﬁrstapproxi-

mation. In view of this relation, (13.6.5) reduces to the nonlinear hyperbolic

equation