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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

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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