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Frequency measures how often a wave passes. We can make a wave with frequency ω by writing:

sin 2 π ω t

Aside : The wave described by sin ( t ) has frequency 1 / 2 π . If we instead write sin ( 2 π t ) , we will have a wave with frequency 1, which is easier to work with.

We can express the same information in terms of wavelength. Wavelength is how close neighboring waves are to each other. It is inversely proportional to frequency, which means that the higher the frequency, the smaller the wavelength. If is wavelength, we write this in the following way:

= 1 ω

A wave with wavelength is written as sin ( 2 π t ) .


Amplitude measures how high the wave is. We can make a wave of amplitude a by writing:

a · sin ( 2 π t )

Sometimes a will be negative. In this case, we still say that the wave has amplitude a , but note that the function will be flipped across the x-axis.


Phase describes how far the wave has been shifted from center. To create a wave with phase p , we write:

sin ( 2 π ω t + p )

Since a sin wave repeats every 2 π , the following is always true (that is, for any ϕ ):

sin ( ϕ ) = sin ( ϕ + 2 π )

Aside : A cosine wave is a sine wave shifted back by a π / 2 , a quarter of the standard wave:

cos ( ϕ ) = sin ( ϕ + π / 2 )

Every sine (cosine) wave can be described completely by these characteristics. These are shown in [link] (phase = 0 for simplicity):

a dog on a bed
A sin wave.

To code [link] in MATLAB, use:

>>t = 0:.01:1;>>amp = 3;>>freq = 1;>>phase = 0>>y = amp*sin(freq*2*pi*t + phase);

Hearing sine waves

The sine wave represents a pure tone. To hear one, we use the MATLAB function sound() , which converts a vector into sound. Find the frequency that is specified, and compare to human range of hearing. Should we be able to hear this sound?

>>freq = 1000;>>samp_rate = 1e4;>>duration = 1;>>samples = 0 : (1/samp_rate) : duration;>>sound_wave = sin(2 * pi * samples * freq);>>sound(sound_wave, samp_rate);

Enter the above code into Matlab to hear the sound.

Adding sine waves

We can add together multiple sin waves to accomplish different shapes. For example, if we add the wave [ 4 · sin ( 2 π t ) ] and the wave [ sin ( 6 · ( 2 π t ) ) ] we get:

a dog on a bed
A compound wave

Look at the figure and try to identify the effect of each wave. The first wave has frequency 1 and amplitude 4. This accounts for the large up-and-down motion that only goes through one cycle in the figure. The second wave has frequency 6 (wavelength 1 6 ) and amplitude 1. This accounts for the small wiggles that happen many times in the figure.

Figure 4 is implemented in MATLAB with the following code:

>t = 0:.01:1;>>y = 4*sin(2*pi*t)+sin(6*2*pi*t);>>plot(t,y)

A wave of any shape can be expressed as a sum of sin and cosine waves, although it may take infinitely many. In the end of this module we will find interesting uses for the this fact.



[link] two shows a sum of two sine waves (with phase 0). Try to replicate it, and report what the amplitudes and frequencies of each wave is.

Perro sentado en la cama
The compound wave for exercise 1.1


Use the identities cos ( ϕ ) = sin ( ϕ + π / 2 ) and sin ( ϕ ) = sin ( ϕ + 2 π ) to solve for p in the following equations by making the above substitutions. Check your answer visually against the figures of sine and cosine waves:

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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