# 2.1 Transverse harmonic waves  (Page 2/6)

 Page 2 / 6

It should be clearly understood that repetition of phase (displacement) is not identified by the magnitude of displacement alone. In the figure shown below, the points “A”, “B” and ”C” have same y-displacements. However, the phase of the waveform at “B” is not same as that at “A”. Note that sense of vibration at “A” and “B” are different (As a matter of fact, the particle at "A" is moving down whereas particle at "B" is moving upward. We shall discuss sense of vibrations at these points later in the module). Since phase is identified by the y-displacement and sense of vibration at a particular position, we need to compare the sense along with the magnitude of y-displacement as well to find the wavelength on a waveform. The wavelength, therefore, is equal to linear distance between positions of points “A” and “C”.

$AC=\lambda$

Mathematically, let us consider y-displacement at a position specified by x=x. At time t = 0, the y-displacement is given by :

$y\left(x=x,t=0\right)=A\mathrm{sin}kx$

According to definition of wavelength, the y-displacement is same at a further distance “λ”. It means that :

$⇒y\left(x=x,t=0\right)=y\left(x=x+\lambda ,t=0\right)$

$⇒A\mathrm{sin}kx=A\mathrm{sin}k\left(x+\lambda \right)=A\mathrm{sin}\left(kx+k\lambda \right)$

The equality given above is valid when :

$⇒k\lambda =2\pi$

$⇒k=\frac{2\pi }{\lambda }$

## Speed of transverse harmonic wave

We can determine speed of the wave by noting that wave travels a linear distance “λ” in one period (T). Thus, speed of wave is given by :

$v=\frac{\lambda }{T}=\nu \lambda =\frac{\frac{2\pi }{k}}{\frac{2\pi }{\omega }}=\frac{\omega }{k}$

It is intuitive to use the nature of phase to determine wave speed. We know that phase corresponding to a particular disturbance is a constant :

$kx-\omega t=\text{constant}$

Speed of wave is equal to time rate at which disturbance move. Hence, differentiating with respect to time,

$⇒k\frac{dx}{dt}-\omega =0$

Rearranging, we have :

$⇒v=\frac{dx}{dt}=\frac{\omega }{k}$

Here, “ω” is a SHM attribute and “k” is a wave attribute. Clearly, wave speed is determined by combination of SHM and wave attributes.

Problem : The equation of harmonic wave is given as :

$y\left(x,t\right)=0.02\mathrm{sin}\left\{\frac{\pi }{6}\left(8x-24t\right)\right\}$

All units are SI units. Determine amplitude, frequency, wavelength and wave speed

Solution : Simplifying given equation :

$y\left(x,t\right)=0.02\mathrm{sin}\left\{\frac{\pi }{3}\left(12x-24t\right)\right\}=0.02\mathrm{sin}\left\{\left(4\pi x-8\pi t\right)\right\}$

Comparing given equation with standard equation :

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

Clearly, we have :

$⇒A=0.02\phantom{\rule{1em}{0ex}}m$

$⇒\omega =2\pi \nu =8\pi$

$⇒\nu =4\phantom{\rule{1em}{0ex}}Hz$

Also,

$⇒k=\frac{2\pi }{\lambda }=4\pi$

$⇒\lambda =0.5\phantom{\rule{1em}{0ex}}m$

Wave speed is given as :

$⇒v=\frac{\omega }{k}=\frac{8\pi }{0.5}=16\pi \phantom{\rule{1em}{0ex}}\frac{m}{s}$

## Initial phase

At x=0 and t =0, the sine function evaluates to zero and as such y-displacement is zero. However, a waveform can be such that y-displacement is not zero at x=0 and t=0. In such case, we need to account for the displacement by introducing an angle like :

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

where “φ” is initial phase. At x=0 and t=0,

$y\left(0,0\right)=A\mathrm{sin}\left(\phi \right)$

The measurement of angle determines following two aspects of wave form at x=0, t=0 : (i) whether the displacement is positive or negative and (ii) whether wave form has positive or negative slope.

For a harmonic wave represented by sine function, there are two values of initial phase angle for which displacement at reference origin (x=0,t=0) is positive and has equal magnitude. We know that the sine values of angles in first and second quadrants are positive. A pair of initial phase angles, say φ = π/3 and 2π/3, correspond to equal positive sine values as :

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!