# 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 $\left(\lambda =\frac{2\pi }{k}\right).$

What is angular velocity
refer to how fast an object rotates ot revolve to another point
samson
An object undergoes constant acceleration after starting from rest and then travels 5m in the first seconds .determine how far it will go in the next seconds
x=x0+v0t+1/2at^2
Grant
what is the deference between precision and accuracy
ShAmy
In measurement of a set, accuracy refers to closeness of the measurements to a specific value, while precision refers to the closeness of the measurements to each other.
Grant
Thank you Mr..
ShAmy
Are there is a difference between Error and Relative Error...
ShAmy
An aircraft flies 300km due east and 600km due north. determine the magnitude of its displacement
670.8km
iyiola
which formula did you use
Sophy
X^2=300^2+600^2
iyiola
what is Fermat principle.
find the angle of projection at which the horizontal range is twice the maximum height of a projectile
impulse by height fomula
Hello
what is impulse?
impulse is the integral of a force (F),over the interval for which it act
Agbeyangi
What is significance of vector?
magnitude and direction
Chris
what is impulse?
the product of force and time
Robert
find the flow rate of a fluid of viscosity 0.0015N-s/m³ flowing through a pipe of radius 50cm and length 100m at a pressure differential of 200000N/m²
friend ,what is the equation your question
lasitha
What is victor
How are you all?
what is the formula for momentum?
p=mv
Grant
what is the formula for angular displacement ?
lasitha
angular displacement = l/r
anand
friend ,what is the " l "?
lasitha
I have another problem,what is the difference between pure chemistry and applied chemistry ?
lasitha
l stands for linear displacement and l>>r
anand
r stands for radius or position vector of particle
anand
pure chemistry is related to classical and basic concepts of chemistry while applied Chemistry deals with it's application oriented concepts and procedures.
anand
what is the deference between precision and accuracy?
ShAmy
I think in general both are same , more accuracy means more precise
anand
and lessor error
anand
friends,how to find correct applied chemistry notes?
lasitha
what are the main concepts of applied chemistry ?
lasitha
anand,can you give an example for angular displacement ?
lasitha
when any particle rotates or revolves about something , an axis or point ,then angle traversed is angular displacement
anand
like a revolving fan
anand
yeah ,I understood now, thanks!
lasitha
o need this application in French language
Why is it that the definition is not there
Bello
what is physics
the study of matter in relation to energy
Robert
Ampher law
Zia