# Vibrations on a string

 Page 1 / 1
Vibrations on a string give rise to waves and normal modes

## Vibrations on a string

Consider the forces on a short fragment of string ${F}_{y}=T{\mathrm{sin}}\left(\theta +\Delta \theta \right)-T{\mathrm{sin}}\theta$ ${F}_{x}=T{\mathrm{cos}}\left(\theta +\Delta \theta \right)-T{\mathrm{cos}}\theta$ Assume that the displacement in y is small and $T$ is a constant along the stringthus $\theta$ and $\theta +\Delta \theta$ are smallthen ${F}_{x}\approx 0$ we can see this by expanding the trig functions ${F}_{x}\approx T\left[1-\frac{{\left(\theta +\Delta \theta \right)}^{2}}{2}-1+\frac{{\theta }^{2}}{2}+\dots \right]$ or ${F}_{x}\approx T\theta \Delta \theta$ which is very small.On the other hand ${F}_{y}\approx T\left[\theta +\Delta \theta -\theta +\dots \right]$ or ${F}_{y}\approx T\Delta \theta$ which is not nearly as small. So we will consider the $y$ component of motion, but approximate there is no x component $\begin{array}{c}{F}_{y}=T{\mathrm{sin}}\left(\theta +\Delta \theta \right)-T{\mathrm{sin}}\theta \\ \approx T{\mathrm{tan}}\left(\theta +\Delta \theta \right)-T{\mathrm{tan}}\theta \\ =T\left(\frac{\partial y\left(x+\Delta x\right)}{\partial x}-\frac{\partial y}{\partial x}\right)\\ =T\frac{{\partial }^{2}y}{\partial {x}^{2}}\Delta x\end{array}$ Also we can write: ${F}_{y}=m{a}_{y}$ $m=\mu \Delta x$ where $\mu$ is the mass density ${a}_{y}=\frac{{\partial }^{2}y}{\partial {t}^{2}}$ now have $T\frac{{\partial }^{2}y}{\partial {x}^{2}}\Delta x=\mu \Delta x\frac{{\partial }^{2}y}{\partial {t}^{2}}$ $\frac{{\partial }^{2}y}{\partial {x}^{2}}=\frac{\mu }{T}\frac{{\partial }^{2}y}{\partial {t}^{2}}$ Note dimensions, get a velocity $\frac{T}{\mu }={{\text{v}}}^{2}$ $\frac{{\partial }^{2}y}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}y}{\partial {t}^{2}}$ The second space derivative of a function is equal to the second time derivative of a function multiplied by a constant.

## Normal modes on a string

Before considering traveling waves, we are going to look at a special case solution to the wave equation. This is the case of stationary vibrations of astring.

For example here, lets consider the case where both ends of the string are fixed at $y=0$ . Now we vibrate the string. Every point along the string acts like a littledriven oscillator. So lets assume that every point on string has a time dependence of the form ${\mathrm{cos}}\omega t$ and that the amplitude is a function of distance Assume $y\left(x,t\right)=f\left(x\right){\mathrm{cos}}\omega t$ then $\frac{{\partial }^{2}y}{\partial {t}^{2}}=-{\omega }^{2}f\left(x\right){\mathrm{cos}}\omega t$ $\frac{{\partial }^{2}y}{\partial {x}^{2}}=\frac{{\partial }^{2}f}{\partial {x}^{2}}{\mathrm{cos}}\omega t$ Substitute into wave equation $\frac{{\partial }^{2}y}{\partial {x}^{2}}=\frac{1}{{{\text{v}}}^{2}}\frac{{\partial }^{2}y}{\partial {t}^{2}}$ $\frac{{\partial }^{2}f}{\partial {x}^{2}}{\mathrm{cos}}\omega t=-\frac{{\omega }^{2}}{{{\text{v}}}^{2}}f\left(x\right){\mathrm{cos}}\omega t$ Then every $f\left(x\right)$ that satisfies: $\frac{{\partial }^{2}f}{\partial {x}^{2}}=-\frac{{\omega }^{2}}{{{\text{v}}}^{2}}f$ is a solution of the wave equation

A solution is (requiring $f\left(0\right)=0$ since ends fixed) $f\left(x\right)=A{\mathrm{sin}}\left(\frac{\omega x}{{\text{v}}}\right)$ Another boundary condition is $f\left(L\right)=0$ so get $A{\mathrm{sin}}\left(\frac{\omega L}{{\text{v}}}\right)=0$ Thus $\frac{\omega L}{{\text{v}}}=n\pi$ $\omega =\frac{n\pi {\text{v}}}{L}$

Be careful with the equations above: ${\text{v}}$ is the letter vee and is for velocity. now we introduce the frequency $\nu$ which is the Greek letter nu.

recall $\nu =\omega /2\pi$ so ${\nu }_{n}=\frac{n{\text{v}}}{2L}=\frac{n}{2L}{\left(\frac{T}{\mu }\right)}^{\frac{1}{2}}$ This is a very important feature of wave phenomena. Things can be quantized. This is why a musical instrument will play specific notes. Note, that wemust have an integral number of half sine waves ${\lambda }_{n}=\frac{2L}{n}$ end up with ${f}_{n}\left(x\right)={A}_{n}{\mathrm{sin}}\left(\frac{2\pi x}{{\lambda }_{n}}\right)$ leading to ${y}_{n}\left(x,t\right)={A}_{n}{\mathrm{sin}}\left(\frac{2\pi x}{{\lambda }_{n}}\right){\mathrm{cos}}{\omega }_{n}t$ where ${\omega }_{n}=\frac{n\pi }{L}{\left(\frac{T}{\mu }\right)}^{\frac{1}{2}}=\frac{n\pi }{L}{\text{v}}=n{\omega }_{1}$ ${\omega }_{1}$ is the fundamental frequency

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Madison Christian By By Saylor Foundation By OpenStax By Katy Keilers By Madison Christian By Rohini Ajay By Maureen Miller By Michael Pitt By Jams Kalo