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 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

0≤x≤ L/4,

L/4 < x < 3L/4,

3L/4≤ x ≤L