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9. Assuming that u(x,t) = X(x)T(t) and substituting for u in Eq.(1) leads to the pair of O.D.E. X" + OX = 0,
T" + a2OT = 0. Applying the B.C. u(0,t) = 0 and
ux(L,t) = 0 to u(x,t) we see that we must have X(0) = 0
and X'(L) = 0. By considering the three cases O < 0,
O = 0, and O > 0 it can be shown that nontrivial solutions of the problem X" + OX = 0, X(0) = 0, X'(L) = 0 are possible if
and only if O = (2n-1) 2n 2/4L2, n = 1,2,... and the
corresponding solutions for X(x) are proportional to sin[(2n-1)nx/2L]. Using these values for O we find that T(t) is a linear combination of sin[(2n-1)nat/2L] and cos[(2n-1)nat/2L]. Now, the I.C. ut(x,0) implies that T'(0) = 0 and thus functions of the form
un(x,t) = sin[(2n-1)nx/2L]cos[(2n-1)nat/2L], n = 1,2,... satsify the P.D.E. (1), the B.C. u(0,t) = 0, ux(L,t) = 0,
and the I.C. ut(x,0) = 0. We now seek a superposition of
these fundamental solutions un that also satisfies the
I.C. u(x,0) = f(x). Thus we assume that
u(x,t) = cnsin[(2n-1)nx/2L]cos[(2n-1)nat/2L]. The
I.C. now implies that we must have
f(x) = y cnsin[(2n-1)nx/2L]. From Problem 39 of Section
10.4 we see that f(x) can be represented by such a series and that
cn = (2/L) [Lf(x)sin[(2n-1) nx/2L]dx, n = 1,2,... .
Substituting these values into the above series for u(x,t) yields the desired solution.
10a. From Problem 9 we have
[cos(-------------------) - cos(
(2n-1)n(L-2) )] 4L
(2n-1)n _ (2n-1)n
trigonometric relations for cos(A ± B). Substituting this value of cn into u(x,t) in Problem 9 yields the desired solution.
13. Using the chain rule we obtain ux = u^x + u^Hx =
u^ + u^ since ?x = nx = 1. Differentiating a second time
gives uxx = u^ + 2u^n + u^n. In a similar way we obtain ut = u^t + unnt = -au^ + au^, since ?t = -a, nt = a. Thus utt = a2 (u^ - 2u^n + unn). Substituting for uxx and utt in the wave equation, we obtain u^ = 0. Integrating both sides of u^n = 0 with respect to n yields u^(?,n) = y(^) where Y is an arbitrary function of ?.
Integrating both sides of u^(?,n) = Y(?) with respect to ? yields u(?,n) = Jy (^)d^ + ^(n) = ^(?) + V(n) where y(n) is an arbitrary function of n and lY(?)d? is some function of ? denoted by ^(?). Thus u(x,t) = u(?(x,t),n (x,t)) = ^(x - at) + y(x + at).
14. The graph of y = sin(x-at) for the various values of t is
indicated in the figure on the next page. Note that the
graph of y = sinx is displaced to the right by the
distance "at" for each value of t.
Similarly, the graph of y = ^(x + at) would be displaced to the left by a distance "at" for each t. Thus ^(x + at) represents a wave moving to the left.
16. Write the equation as a2uxx = utt + a 2u and assume u(x,t) = X(x)T(t). This gives a2X"T = XT" + a 2XT,
X" 1 T" 2
or — = — (— + a2) = a. The separation constant a is X a2 T
—X2 using the same arguments as in the text and earlier problems. Thus X" + X 2X = 0, X(0) = 0, X(L) = 0 and T" + (a2 + z2X2)T = 0, T'(0) = 0. If we let pn = X^a2+a2,
we then have un(x,t) = cospntsin-, where X n = -.
Using superposition we obtain u(x,t) = cncospntsin
n = 1 J_1
and thus u(x,0) = > cnsin = f(x). Hence cn are given
n = 1
by Eq. (22).
17a. We have u(x,t) = ^(x-at) + y(x+at) and thus ut(x,t) = -a^'(x-at) + a%'(x+at). Hence u(x,0) = ^(x) + y(x) = f(x) and
ut(x,0) = -a^'(x) + ay'(x) = 0. Dividing the last
equation by a yields the desired result.
17b. Using the hint and the first equation obtained in part (a) leads to ^(x) + y(x) = 2^(x) + c = f(x) so ^(x) = (1/2)f(x) - c/2 and y(x) = (1/2)f(x) + c/2. Hence
u(x,t) = ^ (x - at) + y(x + at) = (1/2)[f(x - at) - c] + (1/2)[f(x + at) + c] = (1/2)[f(x - at) + f(x + at)].
f(x + at)
17c. Substituting x + at for x in f(x) yields |2 -1 < x + at < 1
Subtracting "at" from both sides of the inequality then yields
I 2 -1 - at < x < 1 - at
f(x + at) = -j .
[ 0 otherwise
18a. As in Problem 17a, we have u(x,0) = ^(x) + y(x) = 0 and ut(x,0) = -a^'(x) + ay(x) = g(x).
18b. From part (a) we have y(x) = -^(x) which yields
-2a^(x) = g(x) from the second equation in part a.
Integration then yields $(x) - $(x0) = —I g(?)d? and
2 a *^0
y(x) = (1/2a) fxg(^)d?, - ^(x0). x0
18c. u(x,t) = ^(x-at) + y(x+at)
= -(1/2a) [xatg(^)d^ + ^(x0) + (1/2a) fx+atg(^)d?, - ^(x0) x0 x0
(1/2a)[[x+atg(^)d^ - [x_atg(^)d^] x0 x0
(1/2a)[| x+atg(^)d^ + [x° g(?)d?]
24. Substituting u(r,0,t) = R(r)0(0)T(t) into the P.D.E.
yields R"0T + R'0T/r + R0"T/r2 = R0T"/a2 or equivalently R"/R + R'/rR + 0"/0r2 = T"/a2T. In order for this
equation to be valid for 0 < r < r0, 0 < 0 < 2n, t > 0,
it is necessary that both sides of the equation be equal to the same constant -O. Otherwise, by keeping r and 0 fixed and varying t, one side would remain unchanged while the other side varied. Thus we arrive at the two equations T" + Oa2T = 0 and r2R"/R + rR'/R + Or2 = -0"/0.