Determine eigenvalue-problem for steel pole

[schniefen]

Homework Statement

Consider a steel pole of length $L=1$ m, which is not connected (free ends on both sides). For simplicity, we study only motions in the direction along the pole. Determine the eigenvalue-problem you need to solve.

Relevant Equations

The wave equation: $\frac{\partial^2 \psi(x,t)}{\partial x^2}=\frac{\rho}{E}\frac{\partial^2 \psi(x,t)}{\partial t^2}$, where $\rho$ denotes density and $E$ pressure. Also, since the ends are free, $\psi(0,t)=0$ and $\psi(L,t)=0$.

If we assume that $\psi$ has a Fourier transform $\hat{\psi}$, so that $\psi(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\psi}(x,\omega)e^{i\omega t}\mathrm{d}\omega$, then the wave equation reduces to $-\rho\omega^2\hat{\psi}(x,\omega)=E\frac{\partial^2 \hat{\psi}(x,\omega)}{\partial x^2}$, since the integrands need to equal. Also, we have $\hat{\psi}(0,\omega)=0$ and $\hat{\psi}(L,\omega)=0$.

This already reveals the eigenvalue-problem, however, the answer given is $-\lambda f(x)=\frac{\partial^2 f(x)}{\partial x^2}$ with $f(0)=f(L)=0$. How can $\hat{\psi}(x,\omega)$ be rewritten in terms of a function only depending on $x$ and why the partial derivatives notation if the function only does depend on a single variable?

 

[haruspex]

Are those the right boundary conditions for free ends? Looks like fixed ends to me.

 

[schniefen]

You are right, I was wrong. The boundary conditions should read $\frac{\partial \psi(x,t)}{\partial x}\bigg\rvert_{x=0}=0$ and $\frac{\partial \psi(x,t)}{\partial x}\bigg\rvert_{x=L}=0$. And these are the same for $\hat{\psi}(x,\omega)$.

 

[schniefen]

A follow-up to the question is to determine the vibrational frequencies. We have

$-\frac{\rho\omega^2}{E}\hat{\psi}(x,\omega)=\frac{\partial^2 \hat{\psi}(x,\omega)}{\partial x^2},$​

and define $\lambda=\frac{\rho\omega^2}{E}$. Then $\hat{\psi}(x,\omega)=\cos{(\sqrt{\lambda}x)}$ satisfies the above equation as well as $\frac{\partial \hat{\psi}(x,t)}{\partial x}\bigg\rvert_{x=0}=0$. The second boundary condition gives

$\frac{\partial \hat{\psi}(x,t)}{\partial x}\bigg\rvert_{x=L}=-\sqrt{\lambda}\sin{(\sqrt{\lambda}L)}=0.$​

This implies either $\lambda=0$ or

$\lambda_j=\left(\frac{(j+1)\pi}{L}\right)^2$
​

and thus

$\omega^2=\omega_j^2=\frac{E}{\rho}\left(\frac{(j+1)\pi}{L}\right)^2.$​

The frequencies are then simply $f_j=\frac{\omega_j}{2\pi}=\sqrt{\frac{E}{\rho}}\frac{j+1}{2L}$. I am a little uncertain about the range of $j$ though. There should be a frequency that is $0$, which would imply $j=-1,0,1,2, ...$.

 

[haruspex]

This implies either λ=0 or

Why break it into two cases? Doesn't $\lambda_j=(\frac{j\pi}L)^2$, $j=0, 1, 2,..$ cover it?