solve the wave equation utt=4uxx for a string of length 3 with both ends kept free...

solve the wave equation utt=4uxx for a string of length 3 with both ends kept free for all time (zero Neumann boundary conditions) if initial position of the string is given as x(3-x), and initial velocity is zero.

Expert Solution

Colby Messinger answered 2 years ago

A string of length L = 1 is held fixed at both ends. The string is...

A string of length L = 1 is held fixed at both ends. The string is initially deformed into a shape given by u(x, t = 0) = sin^2(πx) and released. Assume a value of c2 = 1. Find the solution u(x, t) for the vibration of the string by separation of variables.

Solve the following wave equation using Fourier Series a2uxx = utt, 0 < x < 1,...

Solve the following wave equation using Fourier Series a2uxx = utt, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x2, ut(x,0) = 0

Solve the following wave equation using Fourier Series a2uxx = utt, 0 < x < pi,...

Solve the following wave equation using Fourier Series a2uxx = utt, 0 < x < pi, t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinx, ut(x,0) = pi - x

1. A string is fixed at both ends. If only the length (L) increases, then the...

1. A string is fixed at both ends. If only the length (L) increases, then the fundamental frequency goes: (circle one) up down 2. If only the tension (T) increases, the fundamental goes: (circle one) up down 3. If only the linear mass density increases, the fundamental goes: (circle one) up down 4. The tension (T) in a string fixed across a 1 m length (L) is 400 Newtons, and the linear mass density of the string is 0.01 kg/m....

A taut string of density p and length b is fixed at both ends. At a...

A taut string of density p and length b is fixed at both ends. At a distance 3b/7 from the origin it is pulled up by an amount h, and at a distance 4b/7 it is pulled down by the same amount. Both points are released simultaneously. Describe the motion and the amplitudes of the normal modes (So, which modes are not excited and why not)

A string with a length of 35 cm is fixed at both ends. Waves travel along...

A string with a length of 35 cm is fixed at both ends. Waves travel along it at a speed of 4 m/s. What is the frequency of its lowest mode of standing waves? At what distance from the end of the string is the first node if the string is vibrating at four times its fundamental frequency?

If friction is present, the wave equation takes the form utt − c2 uxx = −r...

If friction is present, the wave equation takes the form utt − c2 uxx = −r ut, where the resistance r > 0 is a constant. Consider a periodic source at one end: u(0, t) = 0, u(l, t) = Aeiωt . (a) Show that the PDE and the BC are satisfied by u(x, t) = Aeiωt sin βx sin βl , where β2c2 = ω2 − irω. (b) No matter what the IC, u(x, 0) and ut(x, 0), are,...

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0)...

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0) = 0 ut(x, 0) = 0.1x(π − x) u(0,t) = u(π,t) = 0 (a) Solve the problem using the separation of variables. (b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).

Consider an elastic string of length L whose ends are held fixed. The string is set...

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). Let L = 10 and a = 1 in parts (b) and (c). (A computer algebra system is recommended.) f(x) = 16x L , 0 ≤ x ≤ L 4 , 4, L 4 < x < 3L 4 , 16(L − x) L , 3L 4...

6. Consider the one dimensional wave equation with boundary conditions and initial conditions: PDE : utt...

6. Consider the one dimensional wave equation with boundary conditions and initial conditions: PDE : utt = c 2 uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) = f(x), ut(x, 0) = g(x) a) Suppose c = 1, L = 1, f(x) = 180x 2 (1 − x), and g(x) = 0. Using the first 10 terms in the series, plot the solution surface and enough time snapshots to display the dynamics of the...

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