# 16.2 Mathematics of waves  (Page 2/11)

 Page 2 / 11

To construct our model of the wave using a periodic function, consider the ratio of the angle and the position,

$\begin{array}{ccc}\hfill \frac{\theta }{x}& =\hfill & \frac{2\pi }{\lambda },\hfill \\ \hfill \theta & =\hfill & \frac{2\pi }{\lambda }x.\hfill \end{array}$

Using $\theta =\frac{2\pi }{\lambda }x$ and multiplying the sine function by the amplitude A , we can now model the y -position of the string as a function of the position x :

$y\left(x\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{\lambda }x\right).$

The wave on the string travels in the positive x -direction with a constant velocity v , and moves a distance vt in a time t . The wave function can now be defined by

$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{\lambda }\left(x-vt\right)\right).$

It is often convenient to rewrite this wave function in a more compact form. Multiplying through by the ratio $\frac{2\pi }{\lambda }$ leads to the equation

$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi }{\lambda }x-\frac{2\pi }{\lambda }vt\right).$

The value $\frac{2\pi }{\lambda }$ is defined as the wave number    . The symbol for the wave number is k and has units of inverse meters, ${\text{m}}^{-1}:$

$k\equiv \frac{2\pi }{\lambda }$

Recall from Oscillations that the angular frequency    is defined as $\omega \equiv \frac{2\pi }{T}.$ The second term of the wave function becomes

$\frac{2\pi }{\lambda }vt=\frac{2\pi }{\lambda }\left(\frac{\lambda }{T}\right)t=\frac{2\pi }{T}t=\omega t.$

The wave function for a simple harmonic wave on a string reduces to

$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx\mp \omega t\right),$

where A is the amplitude, $k=\frac{2\pi }{\lambda }$ is the wave number, $\omega =\frac{2\pi }{T}$ is the angular frequency, the minus sign is for waves moving in the positive x -direction, and the plus sign is for waves moving in the negative x -direction. The velocity of the wave is equal to

$v=\frac{\lambda }{T}=\frac{\lambda }{T}\left(\frac{2\pi }{2\pi }\right)=\frac{\omega }{k}.$

Think back to our discussion of a mass on a spring, when the position of the mass was modeled as $x\left(t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\omega t+\varphi \right).$ The angle $\varphi$ is a phase shift, added to allow for the fact that the mass may have initial conditions other than $x=\text{+}A$ and $v=0.$ For similar reasons, the initial phase is added to the wave function. The wave function modeling a sinusoidal wave, allowing for an initial phase shift $\varphi ,$ is

$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx\mp \omega t+\varphi \right)$

The value

$\left(kx\mp \omega t+\varphi \right)$

is known as the phase of the wave , where $\varphi$ is the initial phase of the wave function. Whether the temporal term $\omega t$ is negative or positive depends on the direction of the wave. First consider the minus sign for a wave with an initial phase equal to zero $\left(\varphi =0\right).$ The phase of the wave would be $\left(kx-\omega t\right).$ Consider following a point on a wave, such as a crest. A crest will occur when $\text{sin}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right)=1.00$ , that is, when $kx-\omega t=n\pi +\frac{\pi }{2},$ for any integral value of n . For instance, one particular crest occurs at $kx-\omega t=\frac{\pi }{2}.$ As the wave moves, time increases and x must also increase to keep the phase equal to $\frac{\pi }{2}.$ Therefore, the minus sign is for a wave moving in the positive x -direction. Using the plus sign, $kx+\omega t=\frac{\pi }{2}.$ As time increases, x must decrease to keep the phase equal to $\frac{\pi }{2}.$ The plus sign is used for waves moving in the negative x -direction. In summary, $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right)$ models a wave moving in the positive x -direction and $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx+\omega t+\varphi \right)$ models a wave moving in the negative x -direction.

[link] is known as a simple harmonic wave function. A wave function is any function such that $f\left(x,t\right)=f\left(x-vt\right).$ Later in this chapter, we will see that it is a solution to the linear wave equation. Note that $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(kx+\omega t+\varphi \text{′}\right)$ works equally well because it corresponds to a different phase shift $\varphi \text{′}=\varphi -\frac{\pi }{2}.$

## Problem-solving strategy: finding the characteristics of a sinusoidal wave

1. To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right).$
2. The amplitude can be read straight from the equation and is equal to A .
3. The period of the wave can be derived from the angular frequency $\left(T=\frac{2\pi }{\omega }\right).$
4. The frequency can be found using $f=\frac{1}{T}.$
5. The wavelength can be found using the wave number $\left(\lambda =\frac{2\pi }{k}\right).$

#### Questions & Answers

how can I become confident in physics
practice more and more question ,,,,if u get trouble in any one question infact do that again and again.
jyotirmayee
is theory important or numericals
arshdeep
both ,,,,because without understanding theory ,,,,u can't able to solve problemsss or numericals
jyotirmayee
which books I can follow for IIT jee prep for mcq and which for mcq
arshdeep
what can make me good in physics
lovet
how can I solve qsns in physics cuz I understand most of the theories but facing problem during solving the qsns.
Rohan
Can someone please tell what really happened to planet Pluto
what are the first 20 elements in periodic table.
H,O...
Arzoodan
what happened to the rest now, or have you forgotten?
Ejiba
who discover periodic table?
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Ritik
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
Dame
what is the Greek name for calcium
Greek word for calcium is asvestio while Latin name is calf meaning lime or limestone
Ejiba
different types of wave
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
want is meant by the term solar system
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
motion in straight line is kinematic's part
Ritik
yea
Manoj
this defination isn't correct
Arzoodan
motion in one dimension
Anil
what is Lorentz force?
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
what answer fir this
Akshay
what's the working difference between a dynamo and a pump?
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
waaaww... you re awesome
Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
Ritik
i need solutions of unuversity pbysucs volume 1
me too
Nirupam
Help us if anyone knows
Nirupam
bring questions
john
actually if u wanted whole book solution then u should buy the solution book
jyotirmayee
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Nirupam
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
jyotirmayee
Kota Rajasthan
Nirupam
so strange,,,,,r u preparing for pmt,,,?
jyotirmayee
for IIT
Nirupam
then concern near book store
jyotirmayee
in Rajasthan
jyotirmayee
or ask questions here
jyotirmayee
thanks for your help
Nirupam
its fine,,most welcome
jyotirmayee