# 12.2 Boundary conditions  (Page 4/5)

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$\begin{array}{ccc}\hfill \frac{1}{2}\lambda +\frac{1}{2}\left(\frac{1}{2}\lambda \right)& =& L\hfill \\ \hfill \frac{2}{4}\lambda +\frac{1}{4}\lambda & =& L\hfill \\ \hfill \frac{3}{4}\lambda & =& L\hfill \\ \hfill \lambda & =& \frac{4}{3}L\hfill \end{array}$

Case 3 : In this case both ends have to be nodes. This means that the length ofthe tube is one half wavelength: So we can equate the two and solve for the wavelength:

$\begin{array}{ccc}\hfill \frac{1}{2}\lambda & =& L\hfill \\ \hfill \lambda & =& 2L\hfill \end{array}$
If you ever calculate a longer wavelength for more nodes you have made a mistake. Remember to check if your answers make sense!

## Three nodes

To see the complete pattern for all cases we need to check what the next step for case 3 is when we have an additional node. Below is the diagram for the casewhere $n=3$ .

Case 1 : Both ends are open and so they must be anti-nodes. We can have threenodes inside the tube only if we have two anti-nodes contained inside the tube and one on each end. This means we have 4 anti-nodes in thetube. The distance between any two anti-nodes is half a wavelength. This means there is half wavelength between every adjacent pairof anti-nodes. We count how many gaps there are between adjacent anti-nodes to determine how many half wavelengths there are and equatethis to the length of the tube L and then solve for $\lambda$ .

$\begin{array}{ccc}\hfill 3\left(\frac{1}{2}\lambda \right)& =& L\hfill \\ \hfill \lambda & =& \frac{2}{3}L\hfill \end{array}$

Case 2 : We want to have three nodes inside the tube. The left end must be anode and the right end must be an anti-node, so there will be two nodes between the ends of the tube. Again we can count the number ofdistances between adjacent nodes or anti-nodes, together these add up to the length of the tube. Remember that the distance between the node and anadjacent anti-node is only half the distance between adjacent nodes. So starting from the left end we count 3 nodes, so 2 half wavelength intervals and then only anode to anti-node distance:

$\begin{array}{ccc}\hfill 2\left(\frac{1}{2}\lambda \right)+\frac{1}{2}\left(\frac{1}{2}\lambda \right)& =& L\hfill \\ \hfill \lambda +\frac{1}{4}\lambda & =& L\hfill \\ \hfill \frac{5}{4}\lambda & =& L\hfill \\ \hfill \lambda & =& \frac{4}{5}L\hfill \end{array}$

Case 3 : In this case both ends have to be nodes. With one node in between there aretwo sets of adjacent nodes. This means that the length of the tube consists of two half wavelength sections:

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

## Superposition and interference

If two waves meet interesting things can happen. Waves are basicallycollective motion of particles. So when two waves meet they both try to impose their collective motion on the particles. This can havequite different results.

If two identical (same wavelength, amplitude and frequency) waves are both trying to form a peak then they are able to achieve the sum oftheir efforts. The resulting motion will be a peak which has a height which is the sum of the heights of the two waves. If two waves areboth trying to form a trough in the same place then a deeper trough is formed, the depth of which is the sum of the depths of the twowaves. Now in this case, the two waves have been trying to do the same thing, and so add together constructively. This is called constructive interference .

If one wave is trying to form a peak and the other is trying to form a trough, then they are competing to do different things. Inthis case, they can cancel out. The amplitude of the resulting wave will depend on the amplitudes of the two waves that are interfering. If the depth ofthe trough is the same as the height of the peak nothing will happen. If the height of the peak is bigger than the depth of thetrough, a smaller peak will appear. And if the trough is deeper then a less deep trough will appear. This is destructive interference .

how can chip be made from sand
is this allso about nanoscale material
Almas
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 is the latest information on a no technology how can I find it
William
currently
William
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
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