# 2.2 Energy transmission by waves

 Page 1 / 3

Wave represents distributed energy. Except for standing waves, the wave transports energy along with it from one point to another. In the context of our course, we shall focus our attention to the transverse harmonic wave along a string and investigate energy being transported by it. But the discussion and results would be applicable to other transverse waves as well through a medium. For a base case, we shall consider an ideal case in which there is no loss of energy.

We supply energy by continuously vibrating the free end of taut string. This energy is transmitted by the small vibrating string element to the neighboring element following it. We can see that vibrating small elements (also referred as particle) possess energy as it oscillates (simple harmonic motion for our consideration) in the transverse direction. It can be easily visualized as in the case of SHM that the energy has two components i.e. kinetic energy (arising from motion) and potential energy (arising from position). The potential energy is elastic potential energy like that of spring. The small string element is subjected to tension and thereby periodic elongation and contraction during the cycle of oscillatory motion.

There are few important highlights of energy transport. The most important ones are :

• The energy at the extreme positions along the string corresponds to zero energy.
• The kinetic and elastic potential energies at the mean (equilibrium) position are maximum.

## Kinetic energy

The velocity of string element in transverse direction is greatest at mean position and zero at the extreme positions of waveform. We can find expression of transverse velocity by differentiating displacement with respect to time. Now, the y-displacement is given by :

$y=A\mathrm{sin}\left(kx-\omega t\right)$

Differentiating partially with respect to time, the expression of particle velocity is :

${v}_{p}=\frac{\partial y}{\partial t}=-\omega A\mathrm{cos}\left(kx-\omega t\right)$

In order to calculate kinetic energy, we consider a small string element of length “dx” having mass per unit length “μ”. The kinetic energy of the element is given by :

$dK=\frac{1}{2}dm{v}_{p}^{2}=\frac{1}{2}\mu dx{\omega }^{2}{A}^{2}{\mathrm{cos}}^{2}\left(kx-\omega t\right)$

This is the kinetic energy associated with the element in motion. Since it involves squared cosine function, its value is greatest for a phase of zero (mean position) and zero for a phase of π/2 (maximum displacement). Now, we get kinetic energy per unit length, “ ${K}_{L}$ ”, by dividing this expression with the length of small string considered :

${K}_{L}=\frac{dK}{dx}=\frac{1}{2}\mu {\omega }^{2}{A}^{2}{\mathrm{cos}}^{2}\left(kx-\omega t\right)$

## Rate of transmission of kinetic energy

The rate, at which kinetic energy is transmitted, is obtained by dividing expression of kinetic energy by small time element, “dt” :

$\frac{dK}{dt}=\frac{1}{2}\mu \frac{dx}{dt}{\omega }^{2}{A}^{2}\mathrm{cos}{}^{2}\left(kx-\omega t\right)$

But, wave or phase speed,v, is time rate of position i.e. $\frac{dx}{dt}$ . Hence,

$\frac{dK}{dt}=\frac{1}{2}\mu v{\omega }^{2}{A}^{2}{\mathrm{cos}}^{2}\left(kx-\omega t\right)$

Here kinetic energy is a periodic function. We can obtain average rate of transmission of kinetic energy by integrating the expression for integral wavelengths. Since only ${\mathrm{cos}}^{2}\left(kx-\omega t\right)$ is the varying entity, we need to find average of this quantity only. Its integration over integral wavelengths give a value of “1/2”. Hence, average rate of transmission of kinetic energy is :

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
Got questions? Join the online conversation and get instant answers! By Janet Forrester By OpenStax By Anonymous User By Sarah Warren By Ellie Banfield By Steve Gibbs By Brooke Delaney By Mldelatte By Saylor Foundation By Jessica Collett