On the other hand, for the frequencies at which the group velocity ω0′ reaches an extremum,ω′′0 = 0. In this case, the cubic term in the dispersion relation (3.7.2) plays an important role. Consequently, equation (3.8.4) re- duces to a form similar to (3.8.7) withω′′0 = 0 in the exponential factor.
Once again, a simple calculation from the resulting integral (3.8.4) reveals thatA(x, t) satisfies thelinearized Korteweg–de Vries (KdV) equation
∂A
∂t +ω0′ ∂A
∂x +1
6ω′′′0 ∂3A
∂x3 = 0. (3.8.11)
By transferring to the new variablesξ=x−ω0′tandτ=twhich correspond to a reference system moving with the group velocity ω′0, we obtain the linearized KdV equation
∂A
∂τ +1
6ω′′′0 ∂3A
∂ξ3 = 0. (3.8.12)
This describes waves in a dispersive medium with a weak high frequency dispersion.
One of the remarkable nonlinear model equations is the Korteweg–de Vries (KdV) equation in the form
ut+αu ux+βuxxx = 0, −∞< x <∞, t >0. (3.8.13) This equation arises in many physical problems including water waves, ion acoustic waves in a plasma, and longitudinal dispersive waves in elastic rods.
The exact solution of this equation is called thesolitonwhich is remarkably stable. We shall discuss the soliton solution in Chapter 13.
Another remarkable nonlinear model equation describing solitary waves is known as thenonlinear Schr¨odinger (NLS) equationwritten in the stan- dard form
i ut+1
2ω′′0uxx+γ|u|2u= 0, −∞< x <∞, t >0. (3.8.14) This equation admits a solution called thesolitary wavesand describes the evolution of the water waves; it arises in many other physical systems that include nonlinear optics, hydromagnetic and plasma waves, propagation of heat pulse in a solid, and nonlinear instability problems. The solution of this equation will be discussed in Chapter 13.
3.9 Exercises
1. Show that the equation of motion of a long string is utt =c2uxx−g,
whereg is the gravitational acceleration.
2. Derive thedamped wave equation of a string utt+a ut=c2uxx,
where the damping force is proportional to the velocity and a is a constant. Considering a restoring force proportional to the displacement of a string, show that the resulting equation is
utt+aut+bu=c2uxx,
whereb is a constant. This equation is called thetelegraph equation.
3. Consider the transverse vibration of a uniform beam. Adopting Euler’s beam theory, the momentM at a point can be written as
M =−EI uxx,
where EI is called the flexural rigidity, E is the elastic modulus, and I is the moment of inertia of the cross section of the beam. Show that the transverse motion of the beam may be described by
utt+c2uxxxx= 0,
where c2 =EI/ρA, ρ is the density, andA is the cross-sectional area of the beam.
4. Derive the deflection equation of a thin elastic plate
∇4u=q/D,
where qis the uniform load per unit area,D is the flexural rigidity of the plate, and
∇4u=uxxxx+ 2uxxyy+uyyyy. 5. Derive the one-dimensional heat equation
ut=κuxx, where κ is a constant.
Assuming that heat is also lost by radioactive exponential decay of the material in the bar, show that the above equation becomes
ut=κuxx+he−αx, wherehandαare constants.
6. Starting from Maxwell’s equations in electrodynamics, show that in a conducting medium electric intensity E, magnetic intensity H, and current densityJsatisfy
∇2X=µεXtt+µσXt,
whereXrepresentsE,H, andJ,µis the magnetic inductive capacity, εis the electric inductive capacity, andσis the electrical conductivity.
3.9 Exercises 85 7. Derive thecontinuity equation
ρt+ div (ρu) = 0, andEuler’s equation of motion
ρ[ut+ (u·grad)u] + gradp= 0, in fluid dynamics.
8. In the derivation of the Laplace equation (3.6.11), the potential at Q which is outside the body is ascertained. Now determine the potential at Qwhen it is inside the body, and show that it satisfies thePoisson equation
∇2u=−4πρ, whereρis the density of the body.
9. Setting U = eiktu in the wave equation Utt =∇2U and setting U = e−k2tu in the heat equation Ut = ∇2U, show that u(x, y, z) satisfies theHelmholtz equation
∇2u+k2u= 0.
10. The Maxwell equations in vacuum are
∇ ×E=−∂B
∂t, ∇ ×B=µε∂E
∂t,
∇ ·E= 0, ∇ ·B= 0,
where µ and ε are universal constants. Show that the magnetic field B = (0, By(x, t),0) and the electric field E = (0,0, Ez(x, t)) satisfy the wave equation
∂2u
∂t2 =c2∂2u
∂x2,
whereu=By or Ez andc= (µε)−12 is the speed of light.
11. The equations of gas dynamics are linearized for small perturbations about a constant stateu= 0,ρ=ρ0, andp0=p(ρ0) withc20=p′(ρ0).
In terms of velocity potential φdefined by u=∇φ, the perturbation equations are
ρt+ρ0divu= 0,
p−p0=−ρ0φt=c20(ρ−ρ0), ρ−ρ0=−ρ0
c20 φt.
Show thatf andusatisfy the three dimensional wave equations ftt =c20∇2f, and utt=c20∇2u,
wheref =p,ρ, orφand
∇2≡ ∂2
∂x2 + ∂2
∂y2+ ∂2
∂z2.
12. Consider a slender body moving in a gas with arbitrary constant ve- locity U, and suppose (x1, x2, x3) represents the frame of reference in which the motion of the gas is small and described by the equations of problem 11. The body moves in the negativex1 direction, and (x, y, z) denotes the coordinates fixed with respect to the body so that the co- ordinate transformation is (x, y, z) = (x1+U t, x2, x3). Show that the wave equationφtt=c20∇2φreduces to the form
M2−1 Φxx=Φyy+Φzz,
whereM ≡U/c0is the Mach number andΦis the potential in the new frame of reference (x, y, z).
13. Consider the motion of a gas in a taper tube of cross section A(x).
Show that the equation of continuity and the equation of motion are ρ=ρ0
1− ∂ξ
∂x− ξ A
∂A
∂x
=ρ0
1− 1
A
∂
∂x(Aξ)
, and
ρ0
∂2ξ
∂t2 =−∂p
∂x,
where x is the distance along the length of the tube, ξ(x) is the dis- placement function,p=p(ρ) is the pressure-density relation,ρ0 is the average density, andρis the local density of the gas.
Hence derive the equation of motion ξtt=c2 ∂
∂x 1
A
∂
∂x(Aξ)
, c2=∂p
∂ρ.
Find the equation of motion whenAis constant. IfA(x) =a0exp (2αx) wherea0 andαare constants, show that the above equation takes the form
ξtt =c2(ξxx+ 2αξx).
14. Consider the current I(x, t) and the potential V(x, t) at a point x and time t of a uniform electric transmission line with resistance R,
3.9 Exercises 87 inductanceL, capacityC, and leakage conductanceGper unit length.
(a) Show that bothI andV satisfy the system of equations LIt+RI=−Vx,
CVt+GV =−Ix. Derive the telegraph equation
uxx−c−2utt−a ut−bu= 0, for u=I or V, wherec2= (LC)−1,a=RC+LGandb=RG.
(b) Show that the telegraph equation can be written in the form utt−c2uxx+ (p+q)ut+pq u= 0,
wherep= GC andq= RL. (c) Apply the transformation
u=vexp
−1
2(p+q)t
to transform the above equation into the form vtt−c2vxx= 1
4(p−q)2v.
(d) When p=q, show that there exists an undisturbed wave solution in the form
u(x, t) =e−ptf(x+ct),
which propagates in either direction, wheref is an arbitrary twice dif- ferentiable function of its argument.
Ifu(x, t) =Aexp [i(kx−ωt)] is a solution of the telegraph equation utt−c2uxx−αut−βu= 0, α=p+q, β =pq,
show that the dispersion relation holds ω2+iαω−
c2k2+β2 = 0.
Solve the dispersion relation to show that u(x, t) = exp
−1 2p t
exp
i
kx− t
2
4c2k2+ (4q−p2)
. Whenp2= 4q, show that the solution represents attenuated nondisper- sive waves.
(e) Find the equations forI andV in the following cases:
(i) Lossless transmission line (R=G= 0), (ii) Ideal submarine cable (L=G= 0),
(iii) Heaviside’s distortionless line (R/L=G/C= constant =k).
15. The Fermi–Pasta–Ulam model is used to describe waves in an anhar- monic lattice of length l consisting of a row of n identical masses m, each connected to the next by nonlinear springs of constant κ. The masses are at a distanceh=l/napart, and the springs when extended or compressed by an amountdexert a forceF =κ
d+α d2 whereα measures the strength of nonlinearity. The equation of motion of the ith mass is
m¨yi =κ"
(yi+1−yi)−(yi−yi−1) +α(
(yi+1−yi)2−(yi−yi−1)2)#
, where i = 1,2,3...n, yi is the displacement of the ith mass from its equilibrium position, andκ,αare constants withy0=yn = 0.
Assume a continuum approximation of this discrete system so that the Taylor expansions
yi+1−yi =hyx+h2
2!yxx+h3
3!yxxx+h4
4!yxxxx+o h5 , yi−yi−1 =hyx−h2
2!yxx+h3
3!yxxx−h4
4!yxxxx+o h5 , can be used to derive the nonlinear differential equation
ytt=c2[1 + 2αhyx]yxx+o h4 , ytt=c2[1 + 2αhyx]yxx+c2h2
12 yxxxx+o h5 , where
c2= κh2 m .
Using a change of variables ξ = x−ct, τ =cαht, show that u= yξ
satisfies the Korteweg–de Vries (KdV) equation uτ+uuξ+βuξξξ =o
ε2 , ε=αh, β= h 24α.
16. The one-dimensional isentropic fluid flow is obtained from Euler’s equa- tions (3.1.14) in the form
ut+u ux=−1
ρpx, ρt+ (ρu)x= 0, p=p(ρ).
3.9 Exercises 89 (a) Show thatuandρsatisfy the one-dimensional wave equation
u ρ
tt
−c2 u
ρ
xx
= 0, wherec2=dpdρ is the velocity of sound.
(b) For a compressible adiabatic gas, the equation of state isp=Aργ, whereA andγare constants; show that
c2= γp ρ .
17. (a) Obtain the two-dimensional unsteady fluid flow equations from (3.1.14).
(b) Find the two-dimensional steady fluid flow equations from (3.1.14).
Hence or otherwise, show that
c2−u2 ux−u v(uy+vx) +
c2−v2 vy = 0, where
c2=p′(ρ).
(c) Show that, for an irrotational fluid flow (u=∇φ), the above equa- tion reduces to the quasi-linear partial differential equations
c2−φ2x φxx−2φxφyφxy+
c2−φ2y φyy = 0.
(d) Show that the slope of the characteristic C satisfies the quadratic equation
c2−u2 dy
dx 2
+ 2uv dy
dx
+
c2−v2 = 0.
Hence or otherwise derive c2−v2
dv du
2
−2uv dv
du
+
c2−u2 = 0.
18. For an inviscid incompressible fluid flow under the body force, F =
−∇Φ, the Euler equations are
∂u
∂t +u· ∇u=−∇Φ−1
ρ∇p, divu= 0.
(a) Show that the vorticityωωω=∇ ×usatisfies the vorticity equation Dωωω
Dt = ∂ωωω
∂t +u· ∇ωωω=ωωω· ∇u.
(b) Give the interpretation of this vorticity equation.
(c) In two dimensions, show that DωDtωω = 0 (conservation of vorticity).
19. The evolution of the probability distribution functionu(x, t) in nonequi- librium statistical mechanics is described by the Fokker–Planck equa- tion (See Reif (1965))
∂u
∂t = ∂
∂x ∂u
∂t +x
u.
(a) Use the change of variables
ξ=x et and v=u e−t
to show that the Fokker–Planck equation assumes the form with u(x, t) =etv(ξ, τ)
vt=e2tvξξ.
(b) Make a suitable change of variable t to τ(t), and transform the above equation into the standard diffusion equation
vt=vξξ.
20. The electric field E(x) and the electromagnetic field H(x) in free space (a vacuum) satisfy the Maxwell equationsEt=ccurlH,Ht=
−ccurlH, divE= 0 = divH, wherecis the constant speed of light in a vacuum. Show that bothEandHthethree-dimensional wave equations
Ett=c2∇2E and Htt=c2∇2H,
wherex= (x, y, z) and∇2is the three-dimensional Laplacian.
21. Consider longitudinal vibrations of a free elastic rod with a variable cross section A(x) with x measured along the axis of the rod from the origin. Assuming that the material of the rod satisfies Hooke’s law, show that the displacement function u(x, t) satisfies the generalized wave equation
utt=c2uxx+ c2 A(x)
dA dx
ux,
wherec2= (λ/ρ),λis a constant that describes the elastic nature of the material, andρis the line density of the rod. When A(x) is constant, the above equation reduces to one-dimensional wave equation.
4
Classification of Second-Order Linear Equations
“When we have a good understanding of the problem, we are able to clear it of all auxiliary notions and to reduce it to simplest element.”
Ren´e Descartes
“The first process ... in the effectual study of sciences must be one of sim- plification and reduction of the results of previous investigations to a form in which the mind can grasp them.”
James Clerk Maxwell