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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
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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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