10.5 There and back again - an exploration of the liouville  (Page 2/10)

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The remainder of this document is divided as follows. In "Problem Setup" , we provide a short introduction to the forward and inverse problems under consideration. In "The Transformations" , we analyze indetail the transformation used to turn the wave equation into the Sturm-Liouville equation and describe a method for inverting it. In "Numerical Results" , we present the results of the main collection of numerical experiments we conducted, which use a particular Sturm-Liouville potentialfunction from [link] . A discussion of the errors in these results is presented in "Error Analysis: Recovering Eigenvalues" . "Corresponding q(t) and ρ(x) Functions in Closed Form" contains results for other functions in closed form we studied. Finally, we provide a brief description of the numerical methodswe used in "Numerical Methods" .

The forward problem

We begin by modeling the behavior of a string of length $L$ with spatially-varying mass density $\rho \left(x\right)>0$ for $x\in \left[0,L\right]$ . For a long string experiencing small vibration, let $u\left(x,t\right)$ be the vertical displacement of the string at time $t\ge 0$ . This $u\left(x,t\right)$ satisfies the 1-D wave equation

$\rho \left(x\right){u}_{tt}\left(x,t\right)={u}_{xx}\left(x,t\right),\phantom{\rule{2.em}{0ex}}0\le x\le L,$

subject to the Dirichlet boundary conditions

$u\left(0,t\right)=u\left(L,t\right)=0,\phantom{\rule{2.em}{0ex}}t\ge 0$

where ${u}_{tt}\left(x,t\right)=\frac{{\partial }^{2}u}{\partial {t}^{2}}$ . Assuming the variables can be separated, we write $u\left(x,t\right)=\psi \left(t\right)y\left(x\right)$ and after making the appropriate substitutions, obtain

$\rho \left(x\right){\psi }^{\text{'}\text{'}}\left(t\right)y\left(x\right)=\psi \left(t\right){y}^{\text{'}\text{'}}\left(x\right),$

hence

$\frac{1}{\rho \left(x\right)}\frac{{y}^{\text{'}\text{'}}\left(x\right)}{y\left(x\right)}=\frac{{\psi }^{\text{'}\text{'}}\left(t\right)}{\psi \left(t\right)}.$

As the left side depends only on $x$ and the right side depends only on $t$ , we conclude that both sides must be equal to the same constant, which we denote by $-\lambda$ . Equating the left side with $-\lambda$ and rearranging yields

$-{y}^{\text{'}\text{'}}\left(x\right)=\lambda \rho \left(x\right)y\left(x\right)$

with boundary conditions $y\left(0\right)=y\left(L\right)=0$ . The constant $\lambda$ is said to be an eigenvalue of the boundary-value problem [link] - [link] , and $y\left(x\right)$ is its corresponding eigenfunction . Finding all $\lambda$ such that [link] has a solution satisfying the boundary conditions is called the forward eigenvalue problem , and the collection of all such $\lambda$ is called the problem's spectrum . Physically, the forward problem asks one to determine a string's natural frequencies from a description of its physicalcomposition.

The inverse problem will do the opposite, asking for information about the string from its eigenvalues.

The inverse problem

In contrast to the forward problem, the inverse eigenvalue problem asks if one can recover information about a string's physical composition fromknowledge of its frequencies. Mathematically, this amounts to asking if we can determine the mass density function $\rho \left(x\right)$ given the spectrum of the problem [link] - [link] .

In determining physical properties of objects, the inverse eigenvalue problem is involved in answering various questions. For a vibrating string one question is, "What is the isospectral set , the mass density of all strings of a given length with a given set of frequencies?" The inverse eigenvalue problem could be used in designing a musical instrument that will produce a desired sound. Also, when a particular object is being designed and it is known that the object will be experiencing vibrations of certain frequencies, the inverse problem can be used to ensure the object will not resonate at these frequencies. Additionally, the inverse problem is involved in structural composition analysis. By collecting acoustic data for a structure, a bridge or building, for example, it can be determined if there are "cracks" or "faults" in the object's components. This allows for analysis of a structure without dismantling the structure.

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