<< Chapter < Page Chapter >> Page >

The wave equation

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

2 u t 2 = c 2 2 u x 2 , u ( 0 , t ) = u ( , t ) = 0

where u describes string displacement, c is a constant describing wave speed and 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

u t = v v t = c 2 2 u x 2

or equivalently, the first order matrix equation

t u v = 0 I c 2 2 x 2 0 u v

Eigenvalues, eigenfunctions, and their significance

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

0 I 2 x 2 0 u v = λ u v , 2 u x 2 = λ 2 u

Since only trigonometric functions satisfy both our equation and our boundary conditions, our eigenfunctions take the form u ( x ) = A sin ( λ x ) + B cos ( λ x ) . Applying our boundary condition at x = 0 to u ( x ) reveals that B = 0 . Since we can then set A as an arbitrary scaling factor, our eigenfunction u ( x ) is simply sin λ x . By applying our second boundary condition at x = , we can see that λ is of the form i π n for any nonzero integer n . We then get the eigenpairs

λ n = i π n , u n ( x ) = sin ( λ n x )

These eigenfunctions constitute an infinite-dimensional basis for any solution to the wave equation, with u i ( x ) orthogonal to u j ( x ) for i j with respect to the inner product

u i , u j 0 u i ( x , t ) u j ( x , t ) d x

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 φ i ( x ) that span the space on which the solution is defined. We then calculate the best approximation

u ( x , t ) = j = 1 N c j ( t ) φ j ( x )

to the solution from the span of these basis functions via the solution to a matrix equation A c = f . Recall the definition of our inner product u i , u j 0 u i ( x , t ) u j ( x , t ) d x . Then, A is

A = φ 1 , φ 1 φ 1 , φ 2 ... φ 1 , φ N φ 2 , φ 1 φ 2 , φ 2 ... φ 2 , φ N φ N , φ 1 φ n , φ 2 ... φ n , φ N , f = f , φ 1 f , φ 2 f , φ N

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

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

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask