consider an elastic string of length l whose ends are held fixed the s

Question

Consider an elastic string of length L whose ends are held fixed. The
string is set in motion with no initial velocity from an initial position
u(x,0) = f(x). In each of Problems 1 through 4, carry out the
following steps. Let L = 10 and a = 1 in parts (b) through (d).
a. Find the displacement u(x, t) for the given initial position
f(x).
Gb. Plot u(x, t) versus x for 0≤x≤ 10 and for several values
oft between t = 0 and t = 20.
G c. Plot u(x, t) versus t for 0 ≤ ≤ 20 and for several values
of x.
Gd. Construct an animation of the solution in time for at least
one period.
e. Describe the motion of the string in a few sentences.
0≤x≤L/2,
1. f(x) =
(2x/L,
2(L-x)/L,
(4x/L,
2. f(x) = 1,
4. f(x) =
3. f(x) = 8x(L- x)²/L³
(4(L-X)/L, 3L/4≤x≤L
L/2 < x < L
0 0≤x≤L/2-1
1,
L/2-1<x<L/2+1 (assume L> 2),
0, L/2+1 ≤ x ≤1
S4.x/L,
0≤x≤ L/4,
L/4< x < 3L/4,
Consider an elastic string of length L whose ends are held fixed. The
string is set in motion from its equilibrium position with an initial
velocity u(x, 0) = g(x). In each of Problems 5 through 8, carry out
the following steps. Let L = 10 and a = 1 in parts (b) through (d).
a. Find the displacement u(x, t) for the given g(x).
G b. Plot u(x, t) versus x for 0 ≤ x ≤ 10 and for several values
oft between t = 0 and 1 = 20.
Gc. Plot u(x, f) versus t for 0 << 20 and for several values
of x.
Gd. Construct an animation of the solution in time for at least
one period.
e. Describe the motion of the string in a few sentences.
0≤x≤L/2,
5. g(x) =
(2x/L,
[2(L-x)/L,
6. g(x) = ¹,
4(L-x)/L,
7. g(x) = 8x(L-x)²/L³
L/2<x<L
0≤x≤ L/4,
L/4 < x < 3L/4,
3L/4≤ x ≤L