# 10.1 Experiments with the three-spectral inverse problem for a

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This report summarizes work done as part of the Physics of Strings PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module describes the three-spectral inverse problem for a beaded string and presents experimental results of its application.

## Introduction

How well can we predict a string's mass distribution by simply listening to its vibration? While considering this question, previous experiments have been limited in the types of strings that could be studied. When only considering two spectra (fixed-fixed and fixed-flat), acquiring the necessary data required us to force the beaded strings to be symmetric about the midpoint. This condition has severely limited the possible experiments. However, recent theoretical developments by Boyko and Pivovarchik have expanded the regime of experimental work with beaded strings. Here we consider three fixed-fixed spectra (whole string, clamped left section, and clamped right section), and show that the information contained in these three spectra may be written as two sets of two spectra problems. Thus, for an arbitrary beaded string, it is possible to measure the frequencies of vibration of three sections of the string. It is then possible to convert these spectra into two separate inverse problems with well known solutions. An algorithm for the recovery of the length and mass information of the string is given by Cox, et. al. . Here is presented the theoretical framework and an experimental setup to predict the masses and lengths of any arbitrary beaded string as long as the string meets our much shorter list of requirements.

## The three-spectral forward problem

We begin by considering a beaded string with at least two beads. The string is artificially separated at an interior point into a left part and a right part, with each part containing at least one mass. The two parts join to form a continuous string. This string vibrates with particular characteristic frequencies depending on the tension in the string, the masses of the beads, and the lengths between them. The forward problem is concerned with finding the spectra given a beaded string's properties.

The tension is given by $\sigma$ . The quantities ${\ell }_{k}$ and ${m}_{k}$ represent the lengths between the beads and the masses of the beads for the left part of the string. The quantities $\stackrel{˜}{{\ell }_{k}}$ and ${\stackrel{˜}{m}}_{k}$ represent those respective properties for the right part. There are ${n}_{1}$ masses on the left and ${n}_{2}$ masses on the right. These properties describe a uniquely determined beaded string. Let ${v}_{k}$ and ${\stackrel{˜}{v}}_{k}$ represent the displacements of the masses in the vertical direction. The equations of motion for this system are governed by the following system of ODE's:

$\frac{\sigma }{{\ell }_{k}}\left({v}_{k}\left(t\right)-{v}_{k+1}\left(t\right)\right)+\frac{\sigma }{{\ell }_{k-1}}\left({v}_{k}\left(t\right)-{v}_{k-1}\left(t\right)\right)+{m}_{k}{v}_{k}^{\text{'}\text{'}}=0,\phantom{\rule{1.em}{0ex}}k=1,2,\cdots ,{n}_{1}$

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20/(×-6^2)
Salomon
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Salomon
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Salomon
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Salomon
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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