Exercises

The Cauchy Problem and Wave Equations

5.12 Exercises

whereH₀⁽¹⁾(z) is the Hankel function given by H₀⁽¹⁾(z) = 2

πi _∞

0

exp (izcoshξ)dξ. (5.11.17) In view of the asymptotic expansion ofH₀⁽¹⁾(z) in the form

H₀⁽¹⁾(z)∼ 2

πz ¹₂

exp

"

i

z−π 4

#

, z→ ∞, (5.11.18) the asymptotic solution foru(R, t) in the limit_ωR

c → ∞is

u(R, t)∼ − iq₀

4

2c πωR

¹₂ exp

−i

ωt−ωR c −π

4

.

This represents the cylindrical wave propagating with constant velocityc.

The amplitude of the wave decays likeR⁻¹2 as R→ ∞.

Example 5.11.2.For a supersonic flow (M >1) past a solid body of revo- lution, the perturbation potentialΦsatisfies the cylindrical wave equation

Φ_RR+ 1

RΦ_R=N²Φ_xx, N²=M²−1,

where R is the distance from the path of the moving body and x is the distance from the nose of the body.

It follows from problem 12 in 3.9 Exercises thatΦsatisfies the equation Φyy+Φzz =N²Φxx.

This represents a two-dimensional wave equation withx↔tandN²↔

1

c². For a body of revolution with (y, z)↔(R, θ),_∂θ^∂ ≡0, the above equation reduces to the cylindrical wave equation

Φ_RR+ 1

RΦ_R= 1 c²Φ_tt.

5.12 Exercises 159

(d)u_tt−c²u_xx= 0, u(x,0) = cosx, u_t(x,0) =e⁻¹. (e) u_tt−c²u_xx= 0, u(x,0) = log

1 +x² , u_t(x,0) = 2.

(f) u_tt−c²u_xx= 0, u(x,0) =x, u_t(x,0) = sinx.

2. Determine the solution of each of the following initial-value problems:

(a) u_tt−c²u_xx=x, u(x,0) = 0, u_t(x,0) = 3.

(b) utt−c²uxx=x+ct, u(x,0) =x, ut(x,0) = sinx.

(c) utt−c²uxx=e^x, u(x,0) = 5, ut(x,0) =x². (d) u_tt−c²u_xx= sinx, u(x,0) = cosx, u_t(x,0) = 1 +x.

(e) u_tt−c²u_xx=xe^t, u(x,0) = sinx, u_t(x,0) = 0.

(f) u_tt−c²u_xx= 2, u(x,0) =x², u_t(x,0) = cosx.

3. A gas which is contained in a sphere of radiusRis at rest initially, and the initial condensation is given bys₀inside the sphere and zero outside the sphere. The condensation is related to the velocity potential by

s(t) =

1/c² u_t,

at all times, and the velocity potential satisfies the wave equation u_tt=∇²u.

Determine the condensation s(t) for all t >0.

4. Solve the initial-value problem u_xx+ 2u_xy−3u_yy = 0,

u(x,0) = sinx, u_y(x,0) =x.

5. Find the longitudinal oscillation of a rod subject to the initial conditions u(x,0) = sinx,

ut(x,0) =x.

6. By using the Riemann method, solve the following problems:

(a) sin²µ φ_xx−cos²µ φ_yy−

λ²sin²µ cos²µ φ= 0, φ(0, y) =f₁(y), φ(x,0) =g₁(x), φ_x(0, y) =f₂(y), φ_y(x,0) =g₂(x).

(b) x²uxx−t²utt= 0,

u(x, t₁) =f(x), u_t(x, t₂) =g(x). 7. Determine the solution of the initial boundary-value problem

utt= 4uxx, 0< x <∞, t >0, u(x,0) =x⁴, 0≤x <∞,

u_t(x,0) = 0, 0≤x <∞,

u(0, t) = 0, t≥0.

8. Determine the solution of the initial boundary-value problem u_tt= 9u_xx, 0< x <∞, t >0, u(x,0) = 0, 0≤x <∞,

u_t(x,0) =x³, 0≤x <∞,

u_x(0, t) = 0, t≥0.

9. Determine the solution of the initial boundary-value problem utt = 16uxx, 0< x <∞, t >0, u(x,0) = sinx, 0≤x <∞,

ut(x,0) =x², 0≤x <∞,

u(0, t) = 0, t≥0.

10. In the initial boundary-value problem

u_tt =c²u_xx, 0< x < l, t >0, u(x,0) =f(x), 0≤x≤l,

u_t(x,0) =g(x), 0≤x≤l,

u(0, t) = 0, t≥0,

iff andgare extended as odd functions, show thatu(x, t) is given by the solution (5.4.5) forx > ctand solution (5.4.6) forx < ct.

11. In the initial boundary-value problem

u_tt =c²u_xx, 0< x < l, t >0, u(x,0) =f(x), 0≤x≤l,

u_t(x,0) =g(x), 0≤x≤l,

ux(0, t) = 0, t≥0,

iff andgare extended as even functions, show thatu(x, t) is given by solution (5.4.8) forx > ct, and solution (5.4.9) forx < ct.

5.12 Exercises 161 12. Determine the solution of the initial boundary-value problem

u_tt=c²u_xx, 0< x <∞, t >0, u(x,0) =f(x), 0≤x <∞,

u_t(x,0) = 0, 0≤x <∞,

u_x(0, t) +h u(0, t) = 0, t≥0, h= constant.

State the compatibility condition off. 13. Find the solution of the problem

u_tt=c²u_xx, at < x <∞, t >0, u(x,0) =f(x), 0< x <∞,

u_t(x,0) = 0, 0< x <∞,

u(at, t) = 0, t >0,

wheref(0) = 0 andais constant.

14. Find the solution of the initial boundary-value problem u_tt =u_xx, 0< x <2, t >0, u(x,0) = sin (πx/2), 0≤x≤2,

ut(x,0) = 0, 0≤x≤2, u(0, t) = 0, u(2, t) = 0, t≥0.

15. Find the solution of the initial boundary-value problem u_tt= 4u_xx, 0< x <1, t >0, u(x,0) = 0, 0≤x≤1,

ut(x,0) =x(1−x), 0≤x≤1, u(0, t) = 0, u(1, t) = 0, t≥0.

16. Determine the solution of the initial boundary-value problem u_tt=c²u_xx, 0< x < l, t >0, u(x,0) =f(x), 0≤x≤l,

u_t(x,0) =g(x), 0≤x≤l, u_x(0, t) = 0, u_x(l, t) = 0, t≥0,

by extendingf andg as even functions aboutx= 0 and x=l.

17. Determine the solution of the initial boundary-value problem u_tt=c²u_xx, 0< x < l, t >0, u(x,0) =f(x), 0≤x≤l,

ut(x,0) =g(x), 0≤x≤l, u(0, t) =p(t), u(l, t) =q(t), t≥0.

18. Determine the solution of the initial boundary-value problem u_tt=c²u_xx, 0< x < l, t >0, u(x,0) =f(x), 0≤x≤l,

ut(x,0) =g(x), 0≤x≤l, u_x(0, t) =p(t), u_x(l, t) =q(t), t≥0.

19. Solve the characteristic initial-value problem

xy³u_xx−x³y u_yy−y³u_x+x³u_y= 0, u(x, y) =f(x) on y²−x²= 8 for 0≤x≤2, u(x, y) =g(x) on y²+x²= 16 for 2≤x≤4, withf(2) =g(2).

20. Solve the Goursat problem

xy³u_xx−x³y u_yy−y³u_x+x³u_y = 0, u(x, y) =f(x) on y²+x²= 16 for 0≤x≤4, u(x, y) =g(y) on x= 0 for 0≤y≤4, wheref(0) =g(4).

21. Solve

u_tt=c²u_xx,

u(x, t) =f(x) on t=t(x), u(x, t) =g(x) on x+ct= 0, wheref(0) =g(0).

22. Solve the characteristic initial-value problem xuxx−x³uyy−ux= 0, x= 0, u(x, y) =f(y) on y−x²

2 = 0 for 0≤y≤2, u(x, y) =g(y) on y+x²

2 = 4 for 2≤y≤4, wheref(2) =g(2).

23. Solve

uxx+ 10uxy+ 9uyy = 0, u(x,0) =f(x), u_y(x,0) =g(x).

5.12 Exercises 163 24. Solve

4u_xx+ 5u_xy+u_yy+u_x+u_y= 2, u(x,0) =f(x), u_y(x,0) =g(x). 25. Solve

3u_xx+ 10u_xy+ 3u_yy = 0,

u(x,0) =f(x), uy(x,0) =g(x). 26. Solve

u_xx−3u_xy+ 2u_yy = 0,

u(x,0) =f(x), u_y(x,0) =g(x). 27. Solve

x²u_xx−t²u_tt= 0 x >0, t >0, u(x,1) =f(x),

u_t(x,1) =g(x).

28. Consider the initial boundary-value problem for a string of length l under the action of an external force q(x, t) per unit length. The dis- placementu(x, t) satisfies the wave equation

ρ u_tt =T u_xx+ρ q(x, t),

where ρis the line density of the string and T is the constant tension of the string. The initial and boundary conditions of the problem are

u(x,0) =f(x), u_t(x,0) =g(x), 0≤x≤l, u(0, t) =u(l, t) = 0, t >0.

Show that the energy equation is dE

dt = [T u_xu_t]^l₀+ l

0

ρ q u_tdx, whereE represents the energy integral

E(t) = 1 2

l 0

ρ u²_t+T u²_x dx.

Explain the physical significance of the energy equation.

Hence or otherwise, derive the principle of conservation of energy, that is, that the total energy is constant for allt≥0 provided that the string has free or fixed ends and there are no external forces.

29. Show that the solution of the signaling problem governed by the wave equation

utt=c²uxx, x >0, t >0, u(x,0) =u_t(x,0) = 0, x >0,

u(0, t) =U(t), t >0, is

u(x, t) =U

t−x c

H

t−x c

, whereH is the Heaviside unit step function.

30. Obtain the solution of the initial-value problem of the homogeneous wave equation

u_tt−c²u_xx= sin (kx−ωt), −∞< x <∞, t >0, u(x,0) = 0 =ut(x,0), for all x∈R,

wherec,kandω are constants.

Discuss the non-resonance case,ω=ckand the resonance case,ω=ck.

31. In each of the following Cauchy problems, obtain the solution of the system

u_tt−c²u_xx= 0, x∈R, t >0,

u(x,0) =f(x) and u_t(x,0) =g(x) for x∈R, for the givenc,f(x) andg(x):

(a) c= 3, f(x) = cosx, g(x) = sin 2x.

(b) c= 1, f(x) = sin 3x, g(x) = cos 3x.

(c) c= 7, f(x) = cos 3x, g(x) =x.

(d) c= 2, f(x) = coshx, g(x) = 2x.

(e) c= 3, f(x) =x³, g(x) =xcosx.

(f) c= 4, f(x) = cosx, g(x) =xe^−x.

32. Ifu(x, t) is the solution of the nonhomogeneous Cauchy problem u_tt−c²u_xx=p(x, t), for x∈R, t >0,

u(x,0) = 0 =u_t(x,0), for x∈R,

5.12 Exercises 165 and ifv(x, t, τ) is the solution of the nonhomogeneous Cauchy problem

v_tt−c²v_xx= 0, for x∈R, t >0,

v(x,0;τ) = 0, vt(x,0;τ) =p(x, τ), x∈R, show that

u(x, t) = t

0

v(x, t;τ)dτ.

This is known as theDuhamel principlefor the wave equation.

33. Show that the solution of the nonhomogeneous diffusion equation with homogeneous boundary and initial data

ut=κuxx+p(x, t), 0< x < l, t >0, u(0, t) = 0 =u(l, t), t >0, u(x,0) = 0, 0< x < l,

is

u(x, t) = t

0

v(x, t;τ)dτ,

where v =v(x, t;τ) satisfies the homogeneous diffusion equation with nonhomogeneous boundary and initial data

v_tt=κv_xx+p(x, t), 0< x < l, t >0, v(0, t;τ) = 0 =v(l, t;τ), t >0, v(x, τ;τ) =p(x, τ).

This is known as theDuhamel principlefor the diffusion equation.

34. Use the Duhamel principle to solve the nonhomogeneous diffusion equa- tion

u_t=κu_xx+e^−tsinπx, 0< x < l, t >0, with the homogeneous boundary and initial data

u(0, t) = 0, u(1, t) = 0, t >0, u(x,0) = 0, 0≤x≤1.

35. (a) Verify that

u_n(x, y) = exp ny−√

n sinnx, is the solution of the Laplace equation

uxx+uyy = 0, x∈R, y >0, u(x,0) = 0, u_y(x,0) =nexp

−√

n sinnx, wherenis a positive integer.

(b) Show that this Cauchy problem is not well posed.

36. Show that the following Cauchy problems are not well posed:

(a) ut = uxx, x∈R, t >0, u(0, t) = ₂

n sin

2n²t , ux(0, t) = 0, t >0.

(b) u_xx+u_yy = 0, x∈R, t >0,

u_n(x,0)→0, (u_n)_y(x,0)→0, as n→ ∞.

6