# 2.1 Transverse harmonic waves  (Page 4/6)

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$⇒\frac{{v}_{p}}{\frac{\partial y}{\partial x}}=-\frac{\omega }{k}=-v$

$⇒{v}_{p}=-v\frac{\partial y}{\partial x}$

At a given position “x” and time “t”, the particle velocity is related to wave speed by this equation. Note that direction of particle velocity is determined by the sign of slope as wave speed is a positive quantity. We can interpret direction of motion of the particles on the string by observing “y-x” plot of a wave form. We know that “y-x” plot is a description of wave form at a particular time instant. It is important to emphasize that a wave like representation does not show the motion of wave. An arrow showing the direction of wave motion gives the sense of motion. The wave form is a snapshot (hence stationary) at a particular instant. We can, however, assess the direction of particle velocity by just assessing the slope at any position x=x. See the plot shown in the figure below :

The slope at “A” is positive and hence particle velocity is negative. It means that particle at this position - at the instant waveform is captured in the figure - is moving towards mean (or equilibrium) position. The slope at “B” is negative and hence particle velocity is positive. It means that particle at this position - at the instant waveform is captured in the figure - is moving towards positive extreme position. The slope at “C” is negative and hence particle velocity is positive. It means that particle at this position - at the instant waveform is captured in the figure - is moving towards mean (or equilibrium) position. The slope at “D” is positive and hence particle velocity is negative. It means that particle at this position - at the instant waveform is captured in the figure - is moving towards negative extreme position.

We can crosscheck or collaborate the deductions drawn as above by drawing wave form at another close instant t = t+∆t. We can visualize the direction of velocity by assessing the direction in which the particle at a position has moved in the small time interval considered.

## Different forms of wave function

Different forms give rise to a bit of confusion about the form of wave function. The forms used for describing waves are :

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

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

Which of the two forms is correct? In fact, both are correct so long we are in a position to accurately interpret the equation. Starting with the first equation and using trigonometric identity :

$\mathrm{sin}\theta =\mathrm{sin}\left(\pi -\theta \right)$

We have,

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

Thus we see that two forms represent waves moving at the same speed ( $v=\omega /k$ ). They differ, however, in phase. There is phase difference of “π”. This has implication on the waveform and the manner particle oscillates at any given time instant and position. Let us consider two waveforms at x=0, t=0. The slopes of the waveforms are :

$\frac{\partial }{\partial x}y\left(x,t\right)=kA\mathrm{cos}\left(kx-\omega t\right)=kA=\text{a positive number}$

and

$\frac{\partial }{\partial x}y\left(x,t\right)=-kA\mathrm{cos}\left(\omega t-kx\right)=-kA=\text{a negative number}$

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Almas
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yeah
Joseph
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no can't
Lohitha
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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
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da
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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
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ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
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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
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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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