# 2.1 Transverse harmonic waves  (Page 3/6)

 Page 3 / 6

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

$\mathrm{sin}\frac{\pi }{3}=\mathrm{sin}\left(\pi -\frac{\pi }{3}\right)=\mathrm{sin}\left(\frac{2\pi }{3}\right)=\frac{1}{2}$

We can determine the slope of the waveform by partially differentiating y-function with respect to "x" (considering "t" constant) :

$⇒\frac{\partial y}{\partial t}=\frac{\partial }{\partial t}A\mathrm{sin}\left(kx-\omega t+\phi \right)=kA\mathrm{cos}\left(kx-wt+\phi \right)$

$\text{At}\phantom{\rule{1em}{0ex}}x=0,t=0,\phi =\pi /3$

$⇒\text{Slope}=kA\mathrm{cos}\phi =kA\mathrm{cos}\frac{\pi }{3}=kAX\frac{\sqrt{3}}{2}=\text{a positive number}$

$\text{At}\phantom{\rule{1em}{0ex}}x=0,t=0,\phi =2\pi /3$

$⇒\text{Slope}=kA\mathrm{cos}\phi =kA\mathrm{cos}\frac{2\pi }{3}=-kAX\frac{\sqrt{3}}{2}=\text{a negative number}$

The smaller of the two (π/3) in first quadrant indicates that wave form has positive slope and is increasing as we move along x-axis. The greater of the two (2π/3) similarly indicates that wave form has negative slope and is decreasing as we move along x-axis. The two initial wave forms corresponding to two initial phase angles in first and second quadrants are shown in the figure below.

We can interpret initial phase angle in the third and fourth quadrants in the same fashion. The sine values of angles in third and fourth quadrants are negative. There is a pair of two angles for which sine has equal negative values. The angle in third quarter like 4π/3 indicates that wave form has negative slope and is further decreasing (more negative) as we move along x-axis. On the other hand, corresponding angle in fourth quadrant for which magnitude is same is 5π/3.

$\mathrm{sin}\frac{4\pi }{3}=\mathrm{sin}\frac{5\pi }{3}=-\frac{1}{2}$

For initial phase angle of 5π/3 in fourth quadrant, the wave form has positive slope and is increasing (less negative) as we move along x-axis. The two initial wave forms corresponding to two initial phase angles in third and fourth quadrants are shown in the figure below.

We can also denote initial phase angles in third and fourth quadrants (angles greater than “π”) as negative angles, measured clockwise from the reference direction. The equivalent negative angles for the example here are :

$-\left(2\pi -\frac{4\pi }{3}\right)=\frac{4\pi }{3}-2\pi =-\frac{2\pi }{3}$

and

$-\left(2\pi -\frac{5\pi }{3}\right)=\frac{5\pi }{3}-2\pi =-\frac{\pi }{3}$

## Particle velocity and acceleration

Particle velocity at a given position x=x is obtained by differentiating wave function with respect to time “t”. We need to differentiate equation by treating “x” as constant. The partial differentiation yields particle velocity as :

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

We can use the property of cosine function to find the maximum velocity. We obtain maximum speed when cosine function evaluates to “-1” :

$⇒{v}_{p\text{max}}=\omega A$

The acceleration of the particle is obtained by differentiating expression of velocity partially with respect to time :

$⇒{a}_{p}=\frac{\partial }{\partial t}{v}_{p}=\frac{\partial }{\partial t}\left\{-\omega A\mathrm{cos}\left(kx-\omega t\right)\right\}=-{\omega }^{2}A\mathrm{sin}\left(kx-\omega t\right)=-{\omega }^{2}y$

Again the maximum value of the acceleration can be obtained using property of sine function :

$⇒{a}_{p\text{max}}={\omega }^{2}A$

## Relation between particle velocity and wave (phase) speed

We have seen that particle velocity at position “x” and time “t” is obtained by differentiating wave equation with respect to “t”, while keeping time “x” constant :

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

On the other hand, differentiating wave equation with respect to “x”, while keeping “t”, we have :

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

The partial differentiation gives the slope of wave form. Knowing that speed of the wave is equal to ratio of angular frequency and wave number, we divide first equation by second :

#### Questions & Answers

there are 3 trimester in human pregnancy
ROHIN Reply
I don't know answer of this question can u help me
ROHIN
yes
Bisa
what is a cell
Fatima Reply
what is genetic
Janet Reply
I join
Janet
what are the branchas of biology
Prisca Reply
zoology, ecology
Millicent
biochemistry,cytology,herpetology...etc
R0se
genetics, microbiology,botany and embryology
Muhammad
what is a cell
Kulunbawi Reply
cell is smallest unit of life. cells are often cell the building blocks of life...
Muhammad
the first twenty element
Orapinega Reply
what are the characteristics of living things?
R0se
growth,respiration,nutrition,sensitivity, movement,irritability, excretion,death.
Obinna
What is the difference between adaptation and competition in animals
Adeyemi Reply
What is biology
Adeyemi
it is a natural science stadey about living things
Zamiil
Biology is the bronch of science which deals with the study of life is called biology
Aziz
what is the x in 300 stands for?
Ogbudu Reply
the properties of life
Clarinda Reply
response to the environment, reproduction, homeostasis, growth,energy processing etc.....
Pushpam
hello.
Daniela
hi
MacPeter
Good
Thomas
what is reproduction
Tims
Reproduction is a fundamental feature of all known life,each individual organism exist as a result of re production.....or else Multiplying...
R0se
a complete virus particle known as
Darlington Reply
These are formed from identical protein subunitscalled capsomeres.
Pushpam
fabace family plant name
Pushpam Reply
in eukaryotes ...protein channel name which transport protein ...
Pushpam Reply
in bacteria ...chromosomal dna duplicate structure called
Pushpam
what is a prokaryotic cell and a eukaryotic cell
Matilda Reply
There are two types of cells. Eukaryotic and Prokaryotic cells. Prokaryotic cells don't have a nucleus or membrane enclosed organelles (little organs within that cell). They do however carry genetic material but it's not maintained in the nucleus. Prokaryotic cells are also one celled.
juanita
Prokaryotic cells are one celled (single celled).
juanita
Prokaryotic cells are Bacteria and Archea
juanita
Prokaryotic cells are smaller than Eukaryotic cells.
juanita
Eukaryotic cells are more complex. They are much bigger than Prokaryotic cells.
juanita
Eukaryotic cells have a nucleus and membrane bound organelles.
juanita
Eukaryotic cells are animals cells which also includes us.
juanita
Eukaryotic cells are also multicellular.
juanita
nice explaination
Amna
eukaryotic cells are individual cells .. but eukaryotes are multicellular organisms which consist of many different types of eukaryotic cells
Will
also eukaryotic cells have mitochondria. prokaryotic cells do not
Will
Good
John
in prokaryotes only ribosomes are present... in eukaryotes mitochondria ...glogi bodies ..epidermis .....prokaryotes one envelop but eukaryotes compartment envelop....envelop mean membrane bound organelles......
Pushpam
prokaryotic cell are cells dat have no true nuclei i.e no cell membrane while eukaryotic cell are cell dat have true nuclei i.e have cell membrane
Divine
grt
Thomas
we have 46 pair of somatic cell and 23 pair of chromosomes in our body, pls can someone explain it to me. pls
Matilda Reply
we have 22 pairs of somatic chromosomes and one pair of sex chromosome
Amna
thanks
Matilda
we have 23 pairs of chromosomes,22 pairs of somatic and one pair of sex chromosomes
Amna
23 chromosomes from dad & 23 chromosomes from mom 23 +23=46 total chromosomes
juanita
X & Y chromosomes are called sex cells, the very presence of a Y chromosome means the person is Male.
juanita
XX Female XY Male
juanita
If a Karyotype has more than 46 Chromosomes then nondisjunction occured. For example, having an extra chromosome 21 will cause Down Syndrome.
juanita
in mammal state the different vertebrae and their location in the body
Igbinigie Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### 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?

 By By Rhodes By By By By Mariah Hauptman By