# 12.2 Boundary conditions  (Page 3/5)

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For example, if we consider a rope that can move in a pipe such that it can have

• both ends free to move (Case 1)
• one end free and one end fixed (Case 2)
• both ends fixed (Case 3).

Each of these cases is slightly different because the free or fixed end determines whether a node or anti-node will form when a standingwave is created in the rope. These are the main restrictions when we determine the wavelengths of potential standing waves. These restrictions are known as boundary conditions and must be met.

In the diagram below you can see the three different cases. It is possible to create standing waves with different frequencies andwavelengths as long as the end criteria are met.

The longer the wavelength the less the number of anti-nodes in the standing waves. We cannot have a standing wave with no anti-nodes becausethen there would be no oscillations. We use $n$ to number the anti-nodes. If all of the tubes have a length $L$ and we know the end constraints we can find the wavelength, $\lambda$ , for a specific number of anti-nodes.

## One node

Let's work out the longest wavelength we can have in each tube, i.e. the case for $n=1$ .

Case 1 : In the first tube, both ends must be anti-nodes, so we must place onenode in the middle of the tube. We know the distance from one anti-node to another is $\frac{1}{2}\lambda$ and we also know this distance is L. So we can equate the two and solve for the wavelength:

$\begin{array}{ccc}\hfill \frac{1}{2}\lambda & =& L\hfill \\ \hfill \lambda & =& 2L\hfill \end{array}$

Case 2 : In the second tube, one end must be a node and the other must be ananti-node. Since we are looking at the case with one node, we are forced to have it at the end. We know the distance from onenode to another is $\frac{1}{2}\lambda$ but we only have half this distance contained in the tube. So :

$\begin{array}{ccc}\hfill \frac{1}{2}\left(\frac{1}{2},\lambda \right)& =& L\hfill \\ \hfill \lambda & =& 4L\hfill \end{array}$

Case 3 : Here both ends are closed and so we must have two nodes so it isimpossible to construct a case with only one node.

## Two nodes

Next we determine which wavelengths could be formed if we had two nodes. Remember that we are dividing the tubeup into smaller and smaller segments by having more nodes so we expect the wavelengths toget shorter.

Case 1 : Both ends are open and so they must be anti-nodes. We can have twonodes inside the tube only if we have one anti-node contained inside the tube and one on each end. This means we have 3 anti-nodes in thetube. The distance between any two anti-nodes is half a wavelength. This means there is half wavelength between the left sideand the middle and another half wavelength between the middle and the right side so there must be one wavelength inside the tube. The safestthing to do is work out how many half wavelengths there are and equate this to the length of the tube L and then solve for $\lambda$ .

$\begin{array}{ccc}\hfill 2\left(\frac{1}{2}\lambda \right)& =& L\hfill \\ \hfill \lambda & =& L\hfill \end{array}$

Case 2 : We want to have two nodes inside the tube. The left end must be anode and the right end must be an anti-node. We can have one node inside the tube as drawn above. Again we can count the number ofdistances between adjacent nodes or anti-nodes. If we start from the left end we have one half wavelength between the end and the nodeinside the tube. The distance from the node inside the tube to the right end which is an anti-node is half of the distance to anothernode. So it is half of half a wavelength. Together these add up to the length of the tube:

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
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