Solution Found!
In Section 4.2 it was shown that the displacement u of a nonuniform string satisfies 0
Chapter 4, Problem 4.4.2(choose chapter or problem)
In Section 4.2 it was shown that the displacement u of a nonuniform string satisfies
\(\rho_{0} \frac{\partial^{2} u}{\partial t^{2}}=T_{0} \frac{\partial^{2} u}{\partial x^{2}}+Q\),
where Q represents the vertical component of the body force per unit length. If Q = 0, the partial differential equation is homogeneous. A slightly different homogeneous equation occurs if \(Q=\alpha u\).
(a) Show that if \(\alpha<0\), the body force is restoring (toward u = 0). Show that if \(\alpha>0\), the body force tends to push the string further away from its unperturbed position u = 0.
(b) Separate variables if \(\rho_{0}(x)\) and \(\alpha(x)\) but \(T_{0}\) is constant for physical reasons. Analyze the time-dependent ordinary differential equation.
(c) Specialize part (b) to the constant coefficient case. Solve the initial value problem if \(\alpha<0\):
\(\begin{aligned}
u(0, t)=0, & u(x, 0)=0 \\
u(L, t)=0, & \frac{\partial u}{\partial t}(x, 0)=f(x) .
\end{aligned}\)
What are the frequencies of vibration?
Questions & Answers
QUESTION:
In Section 4.2 it was shown that the displacement u of a nonuniform string satisfies
\(\rho_{0} \frac{\partial^{2} u}{\partial t^{2}}=T_{0} \frac{\partial^{2} u}{\partial x^{2}}+Q\),
where Q represents the vertical component of the body force per unit length. If Q = 0, the partial differential equation is homogeneous. A slightly different homogeneous equation occurs if \(Q=\alpha u\).
(a) Show that if \(\alpha<0\), the body force is restoring (toward u = 0). Show that if \(\alpha>0\), the body force tends to push the string further away from its unperturbed position u = 0.
(b) Separate variables if \(\rho_{0}(x)\) and \(\alpha(x)\) but \(T_{0}\) is constant for physical reasons. Analyze the time-dependent ordinary differential equation.
(c) Specialize part (b) to the constant coefficient case. Solve the initial value problem if \(\alpha<0\):
\(\begin{aligned}
u(0, t)=0, & u(x, 0)=0 \\
u(L, t)=0, & \frac{\partial u}{\partial t}(x, 0)=f(x) .
\end{aligned}\)
What are the frequencies of vibration?
ANSWER:Step 1 of 6
The given PDE of displacement 
Here, 