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u(x, t) = 1[f (x — at) + f (x + at)].
This form of the solution was obtained by D’Alembert in 1746.
Hint: Note that the equation f'(x) = 0(x) is solved by choosing f(x) = 0(x) + c.
r( ) _ |2, —1 < x < 1,
f(x ) = 0, otherwise.
2, —1 + at < x < 1 + at,
f(x — at) =
Also determine f(x + at).
(d) Sketch the solution found in part (b)at t = 0, t = 1/2a, t = 1/a, and t = 2/a, obtaining the results shown in Figure 10.7.7. Observe that an initial displacement produces two waves moving in opposite directions away from the original location; each wave consists of one-half of the initial displacement.
18. Consider the wave equation
a2uxx = utt
in an infinite one-dimensional medium subject to the initial conditions
u(x, 0) = 0, ut(x, 0) = g(x), —to < x < to.
(a) Using the form of the solution obtained in Problem 13, show that
0(x) + f(x) = 0,
—a0'(x) + af' (x) = g(x).
(b) Use the first equation of part (a) to show that f'(x) = —0'(x). Then use the second equation to show that —2a0(x) = g(x), and therefore that
0(x) = — — g(%) + 0(x0),
where x0 is arbitrary. Finally, determine f(x).
10.7 The Wave Equation: Vibrations of an Elastic String
FIGURE 10.7.7 Propagation of initial disturbance in an infinite one-dimensional medium. (c) Show that
u(x, t) = — g(%) d
19. By combining the results of Problems 17 and 18 show that the solution of the problem
a2uxx = utt,
u(x, 0) = f (x), ut(x, 0) = g(x), — to < x < to
is given by
u(x, t) = -[f (x — at) + f (x + at)] + — g(%) d
2 2a Jx-at
Problems 20 and 21 indicate how the formal solution (20), (22) of Eqs. (1), (3), and (9) can be
shown to constitute the actual solution of that problem.
20. By using the trigonometric identity sin A cos B = i[sin(A + B) + sin(A — B)] showthat the solution (20) of the problem of Eqs. (1), (3), and (9) can be written in the form (28).
21. Let h(i~) represent the initial displacement in [0, L], extended into (—L, 0) as an odd function and extended elsewhere as a periodic function of period 2L. Assuming that h, h', and h" are all continuous, show by direct differentiation that u(x, t) as given in
Chapter 10. Partial Differential Equations and Fourier Series
Eq. (28) satisfies the wave equation (1) and also the initial conditions (9). Note also that since Eq. (20) clearly satisfies the boundary conditions (3), the same is true of Eq. (28). Comparing Eq. (28) with the solution of the corresponding problem for the infinite string (Problem 17), we see that they have the same form provided that the initial data for the finite string, defined originally only on the interval 0 < x < L, are extended in the given manner over the entire x-axis. If this is done, the solution for the infinite string is also applicable to the finite one.
22. The motion of a circular elastic membrane, such as a drumhead, is governed by the twodimensional wave equation in polar coordinates
urr + (1/r )ur + (1/r 2)uee = a~2utt.
Assuming that u(r, 9, t) = R(r)®(9)T(t), find ordinary differential equations satisfied by R(r), ®(9), and T(t).
23. The total energy E (t) of the vibrating string is given as a function of time by
E(t) = j ^2pu2(x, t) + 1 Tu2(x, f)J dx; (i)
the first term is the kinetic energy due to the motion of the string, and the second term is the potential energy created by the displacement of the string away from its equilibrium position.
For the displacement u(x, t) given by Eq. (20), that is, for the solution of the string problem with zero initial velocity, show that
_ 2 t ^
E (t) = -4^7 ' (ii)
Note that the right side of Eq. (ii) does not depend on t. Thus the total energy E is a constant, and therefore is conserved during the motion of the string.
Hint: Use Parseval’s equation (Problem 37 of Section 10.4 and Problem 17 of Section 10.3), and recall that a2 = T/p.
10.8 Laplace’s Equation
One of the most important of all partial differential equations occurring in applied
mathematics is that associated with the name of Laplace8: in two dimensions
uxx + uyy = 0, (1)
and in three dimensions
uxx + uyy + uzz = 0 (2)
For example, in a two-dimensional heat conduction problem, the temperature u (x, y, t) must satisfy the differential equation
a2(uxx + uyy) = ut, (3)
8Laplace’s equation is named for Pierre-Simon de Laplace, who, beginning in 1782, studied its solutions extensively while investigating the gravitational attraction of arbitrary bodies in space. However, the equation first appeared in 1752 in a paper by Euler on hydrodynamics.
10.8 Laplace’s Equation
where a2 is the thermal diffusivity. If a steady-state exists, u is a function of x and y only, and the time derivative vanishes; in this case Eq. (3) reduces to Eq. (1). Similarly, for the steady-state heat conduction problem in three dimensions, the temperature must satisfy the three-dimensional form of Laplace’s equation. Equations (1) and (2) also occur in other branches of mathematical physics. In the consideration of electrostatic fields the electric potential function in a dielectric medium containing no electric charges must satisfy either Eq. (1) or Eq. (2), depending on the number of space dimensions involved. Similarly, the potential function of a particle in free space acted on only by gravitational forces satisfies the same equations. Consequently, Laplace’s equation is often referred to as the potential equation. Another example arises in the study of the steady (time-independent), two-dimensional, inviscid, irrotational motion of an incompressible fluid, which centers about two functions, known as the velocity potential function and the stream function, both of which satisfy Eq. (1). In elasticity the displacements that occur when a perfectly elastic bar is twisted are described in terms of the so-called warping function, which also satisfies Eq. (1).