Solution Found!
A string (of tension T and density ) with xed ends at x =
Chapter 5, Problem 5.10(choose chapter or problem)
. A string (of tension T and density \(\rho\)) with fixed ends at x = 0 and x = l is hit by a hammer so that u(x, 0) = 0, and \(\partial u / \partial t(x, 0)=V\) in \(\left[-\delta+\frac{1}{2} l, \delta+\frac{1}{2} l\right]\) and \(\partial u / \partial t(x, 0)=0\) elsewhere. Find the solution explicitly in series form. Find the energy
\(E_{n}(h)=\frac{1}{2} \int_{0}^{l}\left[\rho\left(\frac{\partial h}{\partial t}\right)^{2}+T\left(\frac{\partial h}{\partial x}\right)^{2}\right] d x\)
of the nth harmonic \(h=h_{n}\). Conclude that if \(\delta\) is small (a concentrated blow), each of the first few overtones has almost as much energy as the fundamental. We could say that the tone is saturated with overtones.
Questions & Answers
QUESTION:
. A string (of tension T and density \(\rho\)) with fixed ends at x = 0 and x = l is hit by a hammer so that u(x, 0) = 0, and \(\partial u / \partial t(x, 0)=V\) in \(\left[-\delta+\frac{1}{2} l, \delta+\frac{1}{2} l\right]\) and \(\partial u / \partial t(x, 0)=0\) elsewhere. Find the solution explicitly in series form. Find the energy
\(E_{n}(h)=\frac{1}{2} \int_{0}^{l}\left[\rho\left(\frac{\partial h}{\partial t}\right)^{2}+T\left(\frac{\partial h}{\partial x}\right)^{2}\right] d x\)
of the nth harmonic \(h=h_{n}\). Conclude that if \(\delta\) is small (a concentrated blow), each of the first few overtones has almost as much energy as the fundamental. We could say that the tone is saturated with overtones.
ANSWER:Step 1 of 4
Given conditions are:
Boundary conditions are homogenous and linear and therefore, one can solve it using the method of separation of variables.