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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.


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.

Depiction of a beaded string with variable names

The tension is given by σ . The quantities 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 k ˜ and 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 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:

σ k ( v k ( t ) - v k + 1 ( t ) ) + σ k - 1 ( v k ( t ) - v k - 1 ( t ) ) + m k v k ' ' = 0 , k = 1 , 2 , , n 1

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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