# 13.1 An introduction to the analysis of brain waves  (Page 2/8)

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## 1.

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

$sin\left(2,\pi ,\omega ,t\right)$

Aside : The wave described by $sin\left(t\right)$ has frequency $1/2\pi$ . If we instead write $sin\left(2\pi t\right)$ , 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 $\ell$ is wavelength, we write this in the following way:

$\ell =\frac{1}{\omega }$

A wave with wavelength $\ell$ is written as $sin\left(\frac{2\pi t}{\ell }\right)$ .

## 2.

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

$a·sin\left(2\pi t\right)$

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.

## 3.

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

$sin\left(2\pi \omega t+p\right)$

Since a sin wave repeats every $2\pi$ , the following is always true (that is, for any $\varphi$ ):

$sin\left(\varphi \right)=sin\left(\varphi +2\pi \right)$

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

$cos\left(\varphi \right)=sin\left(\varphi +\pi /2\right)$

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

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.

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

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 $\frac{1}{6}$ ) 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.

## 1.1

[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.

## 1.2

Use the identities $cos\left(\varphi \right)=sin\left(\varphi +\pi /2\right)$ and $sin\left(\varphi \right)=sin\left(\varphi +2\pi \right)$ 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:

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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