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

≤

16(L − x)

≤ x

≤ L

(a) Find the displacement

u(x, t)

for the given initial position

f(x).

(Use a to represent an arbitrary constant.)
u(x, t) =

∞

n = 1

Solutions

Expert Solution

genius_generous answered 1 year ago

Next > < Previous

Related Solutions

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.

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 point mass M is suspended in the middle of a string whose ends are fixed...

A point mass M is suspended in the middle of a string whose ends are fixed at the same level distance 2ℓ apart; the mass is distance h below the ends of the string. The total mass of the string is m ≪ M. (a) Find the speed of sound waves along the string and (b) the principal frequency of string vibrations on either side of the mass. (c) Do you think this is the smallest vibration frequency in the...

A thin string of length L = 3m is pinned at the two ends. Based on...

A thin string of length L = 3m is pinned at the two ends. Based on the material linear density ρ and on the traction force T, it can be assumed that √ T/ρ = 2 m/s. The string is initially straight when, at time t = 0, it is tapped in two narrow intervals (1 − 1/8, 1 + 1/8) and (2 − 1/8, 2 + 1/8) with opposite velocities +1m/s and −1m/s, respectively. i) Derive the field equation...

A string 4.40 m long is held fixed at both ends. If a sharp blow is...

A string 4.40 m long is held fixed at both ends. If a sharp blow is applied to the string at its center, it takes 0.0400 s for the pulses to travel to the ends of the string and return to the middle. a)What is the lowest standing wave frequency for this string? b)What is the second lowest standing wave frequency for this string?

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?

(a) A string of length L is fixed at one end (in other words, y(0) =...

(a) A string of length L is fixed at one end (in other words, y(0) = 0) and free at the other end (y 0 (L) = 0). Consider the position-dependent wave equation for the string. What are f, g, w, α1, β1, α2 and β2 of this Sturm-Liouville system? Find the eigenvalues and eigenvectors of this system. How may these be useful for solving problems? (Aside: this is known as the one-dimensional Helmholtz equation.) (b) Consider a string that...

. If a string fixed at both ends resonates in its fundamental mode with a frequency...

. If a string fixed at both ends resonates in its fundamental mode with a frequency of 150 Hz, at which of the following frequencies will it not resonate?

For a string fixed at both ends with a constant tension, if you double the frequency...

For a string fixed at both ends with a constant tension, if you double the frequency what happens to the wavelength? For a string fixed at both ends with a constant tension, if you triple the harmonic number what happens to the frequency? Consider the 20th harmonic on a string fixed at both ends. Determine the total KE & net force on each node. For a string fixed at both ends, if you quadruple the tension what happens to the...

Subjects