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The reflection and transmission of mechanical waves is presented.

Reflection and transmission

Four possible cases for the reflection and transmission of a wave on a string.

The first figure shows the 4 possible cases for reflection and transmission at an interface. Lets solve the problem, which is shown in the next figure

since μ 1 μ 2 it must be that v 1 v 2 .

Note that we are assuming that Young's Modulus is constant across the boundary.
So we get y i n c = A cos ( k 1 x ω t ) y r e f = B cos ( k 1 x + ω t ) y t r a n s = C cos ( k 2 x ω t ) (note the reflected wave goes the other direction).

On the left side of the junction we have y l = y i n c + y r e f = A cos ( k 1 x ω t ) + B cos ( k 1 x + ω t ) and on the right side of the junction we have y r = y t r a n s = C cos ( k 2 x ω t ) . At the boundary x = 0 the wave must continuous, that is there are no kinks in it. Thus we must have y l ( 0 , t ) = y r ( 0 , t ) y l ( x , t ) x | x = 0 = y r ( x , t ) x | x = 0

So from the first equation A cos ( ω t ) + B cos ( ω t ) = C cos ( ω t ) A + B = C y l ( x , t ) x | x = 0 = y r ( x , t ) x | x = 0 A k 1 sin ( ω t ) k 1 B sin ( ω t ) = k 2 C sin ( ω t ) ( A B ) k 1 sin ω t = C k 2 sin ω t A B = k 2 k 1 C now solve for B and C A + B = C A B = k 2 k 1 C 2 A = ( 1 + k 2 k 1 ) C Thus we can define the transmission coefficient t r C / A = 2 k 1 k 1 + k 2 and the refection coefficient r B / A = C A 1 = k 1 k 2 k 1 + k 2 note how the amplitudes can change at the boundary

If μ 2 < μ 1 then we must have k 2 < k 1 since v = ω / k = T / μ and ω and T must be fixed. We see that k μ In this case we see that the amplitude of the wave gets bigger when it moves into a less dense medium. We have probably all experienced this in reallife. As waves come ashore they become bigger. This is because shallower water is effectively less dense. A tsumami in open ocean may have animperceptable amplitude but when it comes ashore it can be tremendous. This seems almost counter intuitive, but in any closed system the energy and powerare conserved but there is no rule saying amplitude has to be conserved.

Lets look at the reflected and transmitted power. Recall Power: P = 1 2 μ ω 2 A 2 v For the incident and reflected waves μ and ν are the same so the reflected power coefficient (reflected power / incident power) P R = ( B / A ) 2 = ( k 1 k 2 k 1 + k 2 ) 2 To do transmitted power lets first rewrite the power equation. Recall v = ν λ = 2 π ν k = ω k Also v = T μ so μ = T v 2 = T ( ω k ) 2 = T k 2 ω 2 so now P = 1 2 μ ω 2 A 2 v becomes P = 1 2 T k 2 ω 2 ω 2 A 2 ω k or P = 1 2 T k ω A 2 .

Watch out, in the above lines A was used to denote amplitude in general and in the following line it specifically refers to the incoming wave.
The transmission power coefficient is thus: P T = 1 2 T k 2 ω C 2 1 2 T k 1 ω A 2 Note that ω and T are the same for both waves P T = k 2 C 2 k 1 A 2 earlier we showed C / A = 2 k 1 k 1 + k 2 so P T = ( 2 k 1 k 1 + k 2 ) 2 k 2 k 1 P T = 4 k 1 k 2 ( k 1 + k 2 ) 2 Note that P R + P T = 1 which means that energy is conserved.

Now lets look at the 4 specific cases we have:

Rigid wall

μ so k 2 r = k 1 k 2 k 1 + k 2 = k 1 k 2 1 k 1 k 2 + 1 r 1 Also P R + 1 P T 0 So wave is reflected and inverted, but has same power

Free end

μ 0 so k 2 0 r = k 1 k 2 k 1 + k 2 = k 1 k 1 r + 1 Also P R + 1 P T 0 So wave is reflected and has same power

Moving to higher density

μ 2 > μ 1 k 2 > k 1 so r < 0 t r > 0

Moving to lower density

μ 2 < μ 1 k 2 < k 1 so r > 0 t r > 1 Note the transmitted wave's amplitude is larger than the original.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
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Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
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