# The network wave equation  (Page 2/11)

 Page 2 / 11

## The wave equation

The vibration of a string in one dimension can be understood through the standard wave equation, given by

$\frac{{\partial }^{2}u}{\partial {t}^{2}}={c}^{2}\frac{{\partial }^{2}u}{\partial {x}^{2}},\phantom{\rule{2.em}{0ex}}u\left(0,t\right)=u\left(\ell ,t\right)=0$

where $u$ describes string displacement, $c$ is a constant describing wave speed and $\ell$ is the length of the string. The string is fixed at displacement 0 at the endpoints and assume without loss of generality $c=1$ . This equation is derived in much more detail in . This second order partial differential equation can likewise be rewritten as a system of two ordinary differential equations in time

$\begin{array}{ccc}\hfill \frac{\partial u}{\partial t}& =& v\hfill \\ \hfill \frac{\partial v}{\partial t}& =& {c}^{2}\frac{{\partial }^{2}u}{\partial {x}^{2}}\hfill \end{array}$

or equivalently, the first order matrix equation

$\frac{\partial }{\partial t}\left[\begin{array}{c}u\\ v\end{array}\right]=\left[\begin{array}{cc}0& I\\ {c}^{2}\frac{{\partial }^{2}}{\partial {x}^{2}}& 0\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$

## Eigenvalues, eigenfunctions, and their significance

We are especially interested in the eigenvalues $\lambda$ and associated eigenfunctions of the wave equation, such that

$\left[\begin{array}{cc}0& I\\ \frac{{\partial }^{2}}{\partial {x}^{2}}& 0\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]=\lambda \left[\begin{array}{c}u\\ v\end{array}\right],\phantom{\rule{2.em}{0ex}}\frac{{\partial }^{2}u}{\partial {x}^{2}}={\lambda }^{2}u$

Since only trigonometric functions satisfy both our equation and our boundary conditions, our eigenfunctions take the form $u\left(x\right)=Asin\left(\lambda x\right)+Bcos\left(\lambda x\right)$ . Applying our boundary condition at $x=0$ to $u\left(x\right)$ reveals that $B=0$ . Since we can then set $A$ as an arbitrary scaling factor, our eigenfunction $u\left(x\right)$ is simply $sin\left(\lambda ,x\right)$ . By applying our second boundary condition at $x=\ell$ , we can see that $\lambda$ is of the form $\frac{i\pi n}{\ell }$ for any nonzero integer $n$ . We then get the eigenpairs

${\lambda }_{n}=\frac{i\pi n}{\ell },\phantom{\rule{2.em}{0ex}}{u}_{n}\left(x\right)=sin\left({\lambda }_{n}x\right)$

These eigenfunctions constitute an infinite-dimensional basis for any solution to the wave equation, with ${u}_{i}\left(x\right)$ orthogonal to ${u}_{j}\left(x\right)$ for $i\ne j$ with respect to the inner product

$⟨{u}_{i},{u}_{j}⟩\equiv {\int }_{0}^{\ell }{u}_{i}\left(x,t\right){u}_{j}\left(x,t\right)\phantom{\rule{0.166667em}{0ex}}dx$

Intuitively, these correspond to the fundamental modes of a string - any vibration of the string can be decomposed into a linear combination of the fundamentals. The magnitude of each eigenvalue, likewise, is related to the frequency at which the corresponding fundamental mode vibrates - in other words, each eigenvalue is tied to a note in the progression of the Western scale. As we will see, this linear progression of the eigenvalues is lost when a single string is replaced by a network of strings, leading to more of a dissonant sound when a network is plucked.

## Finite element solution method

In this report, we use the finite element method to numerically solve for solutions to the wave equation. The idea behind this method is based on picking a finite-dimensional set of $N$ basis functions ${\phi }_{i}\left(x\right)$ that span the space on which the solution is defined. We then calculate the best approximation

$u\left(x,t\right)=\sum _{j=1}^{N}{c}_{j}\left(t\right){\phi }_{j}\left(x\right)$

to the solution from the span of these basis functions via the solution to a matrix equation $Ac=f$ . Recall the definition of our inner product $⟨{u}_{i},{u}_{j}⟩\equiv {\int }_{0}^{\ell }{u}_{i}\left(x,t\right){u}_{j}\left(x,t\right)\phantom{\rule{0.166667em}{0ex}}dx$ . Then, $A$ is

$A=\left[\begin{array}{cccc}〈{\phi }_{1},,,{\phi }_{1}〉& 〈{\phi }_{1},,,{\phi }_{2}〉& ...& 〈{\phi }_{1},,,{\phi }_{N}〉\\ 〈{\phi }_{2},,,{\phi }_{1}〉& 〈{\phi }_{2},,,{\phi }_{2}〉& ...& 〈{\phi }_{2},,,{\phi }_{N}〉\\ ⋮& & \ddots & \\ 〈{\phi }_{N},,,{\phi }_{1}〉& 〈{\phi }_{n},,,{\phi }_{2}〉& ...& 〈{\phi }_{n},,,{\phi }_{N}〉\end{array}\right],\phantom{\rule{2.em}{0ex}}f=\left[\begin{array}{c}〈f,,,{\phi }_{1}〉\\ 〈f,,,{\phi }_{2}〉\\ ⋮\\ 〈f,,,{\phi }_{N}〉\end{array}\right]$

$A$ is called the Gramian matrix - a matrix whose $ij$ th entry is the inner product between the $i$ and $j$ th basis functions. After solving for the vector $c={\left[{c}_{1},{c}_{2},...,{c}_{N}\right]}^{T}$ , we can reconstruct our best approximation to the solution.

We first rearrange our PDE into a more flexible form. Given a function $v\left(x\right)$ obeying the same boundary conditions as $u$ , multiply both sides of our wave equation by this function and integrate over the interval $\left[0,\ell \right]$

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
what is hormones?
Wellington
Got questions? Join the online conversation and get instant answers!