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Doppler Effect Study Guide

Acoustics

Is sound and noise the same thing? Maybe yes, maybe no. Let's learn more with these study guides, interactive 3D models and more.

 Ultrasound Study Guide

Frequency And Pitch Of Sound Study Guide

Intensity And Loudness Of Sound Study Guide

Speed Of Sound Study Guide

## INTRODUCTION 

In the domain of science and engineering, there are several phenomenons that are periodic in nature. For instance, sound waves, vibrations in structures, as well as current and voltage in AC circuits are examples. To study these things, it is essential to break them down into functions that can be analyzed. Fourier analysis is the process used to understand the constituent components of fundamental and harmonic frequencies in these systems. 

![Fourier analysis](//images.ctfassets.net/4yflszkpcwkt/22PJ9Qyda7imUBP1zTuE6D/15def9e470df9e373a192638fa05d05f/1_hMuADytSQyWNEGds41uogg.png)
[Source](https://levelup.gitconnected.com/representing-a-square-wave-with-a-fourier-series-and-python-6d43beb19442)

### WHAT IS THE FOURIER SERIES? 

A Fourier series is used to represent a periodic function f(x) as an infinite sum of sine and cosine functions. The study of the Fourier series is known as harmonic analysis. It is a highly useful tool to analyze arbitrary periodic functions by breaking them up into simpler terms that can be solved individually and then combined to solve the original problem. 

### PERIODIC FUNCTIONS

A periodic function is a unique relationship between members of two different sets. A periodic function is a relationship that repeats itself over a set of values. Consider the plot of the periodic function. 
F(x) = f(x np) where p is the period and n is a set of whole numbers n = 1, 2, … and so on. Here the graph shows a sine function with the least period of 2π as well as the periods -2π, 4π, 6π, etc. 

![Periodic Function](//images.ctfassets.net/4yflszkpcwkt/4bBRuqX3DyUqabj7boofL5/54308322647e27f4a299fc59d5feb12a/Periodic_Function.png)
[Source](https://mathworld.wolfram.com/PeriodicFunction.html) 

Consider the functions sin(x) and sin(2x). The figure below shows each function as a waveform plotted on a graph. When you add the two, you make a new wave. 

![wave-superposition](//images.ctfassets.net/4yflszkpcwkt/30N4mkfYashCDBqeE8I89g/ab750a11446290789d950e62e6b0ee63/wave-superposition.svg)
[Source](https://www.mathsisfun.com/physics/waves-introduction.html)

Complicated waves in many systems can be analyzed by breaking them down into smaller functions that can be solved individually and added up later. Consider the square wave shown below, for example: 

![fourier-square](//images.ctfassets.net/4yflszkpcwkt/1C2Hw9jN4Hp7QmRvVDOBRb/a830b34b0334a1daf3b6282cdb3cd239/fourier-square.svg)
[Source](https://www.mathsisfun.com/calculus/fourier-series.html)

It is possible to use sine waves to make a square wave. To that, we start with a <span id="fourier-select-1"></span> function: 

![fourier-square-sine1](//images.ctfassets.net/4yflszkpcwkt/152fmv8jXRPDbX7GLpiDvP/08045cfd90c9031496c6ad331ebe3309/fourier-square-sine1.svg)
[Source](https://www.mathsisfun.com/calculus/fourier-series.html) 

Now take a new function sin(3x)/3 : 

![fourier-square-sine3](//images.ctfassets.net/4yflszkpcwkt/7s0cFEI7nAh26CYygO6DGY/4e5f50d546b4bd569783526a4ba5d615/fourier-square-sine3.svg)
[Source](https://www.mathsisfun.com/calculus/fourier-series.html) 

Add the two functions sin(x) sin(3x)/3 to generate this wave: 

![fourier-square-sine1-3](//images.ctfassets.net/4yflszkpcwkt/3WIukejPP4NYXlwPFE8VUD/ec2806c883a2f23a3822a74c3f1077b5/fourier-square-sine1-3.svg)
[Source](https://www.mathsisfun.com/calculus/fourier-series.html) 

Now consider the wave of the function sin(5x)/5: 

![fourier-square-sine5](//images.ctfassets.net/4yflszkpcwkt/44tuQtgcfUFhBx24xfVx7q/c6ff1797cd67aa8057a3f37cc4dabc9d/fourier-square-sine5.svg)
[Source](https://www.mathsisfun.com/calculus/fourier-series.html)

Add the above functions sin(x) sin(3x)/3 sin(5x)/5 to get: 

![fourier-square-sine1-5](//images.ctfassets.net/4yflszkpcwkt/6OFLvc29tV375Xhxy93bLC/0d4829891f3a26ddded2d00ac3cc479d/fourier-square-sine1-5.svg)
[Source](https://www.mathsisfun.com/calculus/fourier-series.html) 

Similarly if you add 20 more sine waves sin(x) sin(3x)/3 sin(5x)/5 ... sin(39x)/39 we get a wave that’s close: 

If this process continues infinitely, we will arrive at a square wave, which is the principle behind the Fourier series. The same process can be used to approximate a lot of different functions like the ones shown below: 

![FourierSeries](//images.ctfassets.net/4yflszkpcwkt/3O9OBgmyrFiV1Vam8aHKFu/936d3d4e4d49b9bcb8b9b744fd6d7042/FourierSeriesExamples_1200.svg)
[Source](https://mathworld.wolfram.com/FourierSeries.html) 

### THE FOURIER SERIES FORMULA

The Fourier series formula for a function f(x) (such as the square wave in the above example) between the interval [-L, L] is given by: 

![Fourier-series-formula](//images.ctfassets.net/4yflszkpcwkt/2iOMJnBPG5ZwzgFauZqIyk/33e4caca2d3d6c521212d4c03b7e14a7/Fourier-series-formula.png)
[Source](https://byjus.com/maths/fourier-series/) 

### HARMONIC ANALYSIS 

Fourier series is a branch of harmonic analysis. In music, if a note has a frequency of (f) which is called the fundamental frequency, its integer multiples such as 2f, 3f, 4f so on are called harmonics. These harmonics are where the instrument playing the note has its peak frequencies. For example, if the fundamental frequency of the first harmonic is 50 Hz, the second harmonic will be 100 Hz, and the third harmonic will be 150 Hz. 

![harmonics](//images.ctfassets.net/4yflszkpcwkt/30uGHkogMXHz56esffzjyb/ac04515f9b46c0a524b568ae91e021ba/harmonics.svg)
[Source](https://www.teachmeaudio.com/recording/sound-reproduction/fundamental-harmonic-frequencies) 

## CONCLUSION:

- Science and engineering problems often encounter phenomenons that repeat periodically. 
- To analyze complex periodic functions, it is essential to break them down into smaller functions.
- A Fourier series is used to represent a periodic function f(x) as an infinite sum of sine and cosine functions. 
- Fourier series is a branch of harmonic analysis. 

*We hope you enjoyed studying this lesson and learned something cool about The __Fourier Series__! Join our Discord community to get any questions you may have answered and to engage with other students just like you! Don't forget to download our app to experience our fun VR classrooms - we promise it makes studying much more fun!* 😎

## SOURCES: 

1. Fourier Series: 
2. https://mathworld.wolfram.com/FourierSeries.html#:~:text=A%20Fourier%20series%20is%20an,the%20sine%20and%20cosine%20functions. Accessed 25 Feb 2022.
3. Harmonic Analysis: https://mathworld.wolfram.com/HarmonicAnalysis.html. Accessed 25 Feb 2022. 
4. Periodic Function: https://mathworld.wolfram.com/PeriodicFunction.html. Accessed 25 Feb 2022. 
5. Fourier Series: https://byjus.com/maths/fourier-series/. Accessed 25 Feb 2022. 
6. Fourier Series: https://www.mathsisfun.com/calculus/fourier-series.html. Accessed 25 Feb 2022. 
7. Fundamental and Harmonic Frequencies: https://www.teachmeaudio.com/recording/sound-reproduction/fundamental-harmonic-frequencies. Accessed 25 Feb 2022.
8. Sound Spectrum: https://newt.phys.unsw.edu.au/jw/sound.spectrum.html Accessed 25 Feb 2022.


Fourier - Making Waves Study Guide

Waves Study Guide

Sound Study Guide

TL;DR Fourier - Making Waves

TL;DR Ultrasound

TL;DR Doppler Effect

TL;DR Frequency And Pitch Of Sound

TL;DR Intensity And Loudness Of Sound

TL;DR Speed Of Sound

TL;DR Waves

All the sounds that we perceive around and about us come from vibrations and travels in a wave much the same as a wave in water. Let's get to know these waves quickly!