<< Chapter < Page Chapter >> Page >

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 sin k x ω t

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

v p = y t = - ω A cos k x ω t

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 :

d K = 1 2 m v p 2 = 1 2 μ x ω 2 A 2 cos 2 k x ω t

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 = K x = 1 2 μ ω 2 A 2 cos 2 k x ω t

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” :

K t = 1 2 μ x t ω 2 A 2 cos 2 k x ω t

But, wave or phase speed,v, is time rate of position i.e. x t . Hence,

K t = 1 2 μ v ω 2 A 2 cos 2 k x ω t

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 cos 2 k x ω t 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 :

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit 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
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Oscillation and wave motion' conversation and receive update notifications?