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# INTRODUCTION

The Doppler effect, also known as Doppler shift, is a phenomenon that occurs when the source of waves moves in relation to an observer. A common practical demonstration of the Doppler Effect is an ambulance crossing you with its siren blaring!

![amblance](//images.ctfassets.net/4yflszkpcwkt/1M5oLmBnrsN7C21xCCvQeP/673c0e2055a1ada52d1e7f0fe320fc80/thumb-2021-09-09.png)
[Source](https://pressprogress.ca/calgary-and-edmonton-keep-running-out-of-ambulances/)

## DOPPLER EFFECT FOR LIGHT

When a moving source emits light waves, the Doppler effect results in a __red or blue shift in frequency__. The behavior of sound waves and other types of waves is similar (even if they are not identical).

The relative velocity u of the source and observer for motion towards each other determines the __Doppler shift for light__.

## DOPPLER EFFECT FORMULA

![formula](//images.ctfassets.net/4yflszkpcwkt/1CbLuH2AwKgItcPCd9uaSm/a9494519e3cdd10c129e9162ba12e0ea/doppler_effect_equation.jpg)
[Source](https://study.com/cimages/multimages/16/doppler_effect_equation.jpg)

## APPLICATIONS OF DOPPLER'S EFFECT

__1. Sirens:__ 

A passing emergency vehicle's siren will begin higher than its stationary pitch, slide down as it passes, and then continue lower as it recedes from the observer.

2. Apart from these, Radar, Medical imaging and blood flow measurement, Flow measurement, Velocity profile measurement, Satellite communication, Audio, Vibration measurement, Developmental biology, and so on all utilize Doppler's effect.

## LIMITATIONS OF THE DOPPLER EFFECT

1. The velocity of the source of sound must be less than that of the velocity of sound, i.e., Vs&lt;V.
2. The velocity of the observer must be less than the velocity of sound, i.e., Vo&lt;V.
3. If the velocity of sound of the source is greater than that of the velocity of sound, then due to shock waves, the wavefront gets distorted. Consequently, the change in frequency will not be observed by the observer.

## EXAMPLE OF DOPPLER EFFECT IN DAILY LIFE 

- A small __speed gun__ can accurately assess your vehicle's speed and is based on the Doppler Effect! 
- A speed gun is a device that police officers use to keep an eye on fast automobiles. 
- It consists mostly of a radio transmitter and receiver. 
- After bouncing off the target item, a speed gun broadcasts a radio wave signal and gets the same signal back. 
- The frequency of the return waves is higher than the frequency of the sent waves while the object is approaching the radar, and it is lower when the item is moving away. 
- The radar speed gun can calculate the speed of the object from which the waves have been bounced back based on this difference.

![speed gun](//images.ctfassets.net/4yflszkpcwkt/2JnIJfJvJODcxNIbzj8rSG/8c00ea3b73fe39b0d405c74df88de661/police-rada.jpg)
[Source](https://studiousguy.com/wp-content/uploads/2020/10/police-rada.jpg) 

## SUMMARY

* Doppler Effect is the phenomenon of motion-related frequency change.
* You might have heard the changing pitch of the siren of an ambulance that passes with its siren blaring: When an ambulance approaches you, the siren’s pitch sounds higher than when it moves away from you. This change is a common Doppler principle.
* The Doppler effect is applicable for both light and sound. 

## FAQs

**Q. Who discovered the Doppler effect?**

Christian Andreas Doppler is best known for his groundbreaking idea of the Doppler effect.

**Q. How can the Doppler effect be applied to everyday life?**

The Doppler effect has a variety of practical applications. Meteorologists, for example, utilize the Doppler effect to track storms in addition to police radar. The Doppler effect was even employed by doctors to diagnose cardiac issues.

**Q. How does the Doppler effect prove that the universe is expanding?**

This is due to the fact that when the object moves away, the waves spread out wider. As the waves get farther apart, the wavelength lengthens, pushing the spectrum toward the red end of the spectrum. The Doppler effect explains the redshift, which indicates that the cosmos is expanding. 

_We hope you enjoyed studying this lesson and learned something cool about the **Doppler Effect!** Join our Discord community to get any questions you may have answered and to engage with other students just like you! We promise, it makes studying much more fun!😎_

## REFERENCE
1. Applications and Limitations of Doppler Effect: [https://www.toppr.com/ask/content/concept/applications-and-limitations-of-doppler-effect-209165/](https://www.toppr.com/ask/content/concept/applications-and-limitations-of-doppler-effect-209165/) Accessed 14th April 2022.
2. Doppler Effect: [https://byjus.com/physics/doppler-effect/#Defining-Doppler-Effect](https://byjus.com/physics/doppler-effect/#Defining-Doppler-Effect) Accessed 14th April 2022.
3. Doppler Effect: [https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DopplerEffect.htm](https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DopplerEffect.htm) Accessed 14th April 2022.
4. Doppler Effect Examples: [https://studiousguy.com/doppler-effect-examples/](https://studiousguy.com/doppler-effect-examples/) Accessed 14th April 2022.

The Doppler effect, also known as the Doppler shift, is the change in frequency of a wave as a result of an observer moving away from the wave source. It was named after Christian Doppler, an Austrian physicist who first described the phenomena in 1842.


Doppler Effect Study Guide

# INTRODUCTION

The term "ultrasound" refers to sound waves having frequencies higher than the human hearing limit. The top capacity varies per person, but it is often approximately 20,000 Hertz. Ultrasound has the same physical properties as audible sound.

## ULTRASOUND DEFINITION

Ultrasound is a type of non-ionizing radiation that images the body using high-frequency sound waves. It is a real-time examination that allows for the evaluation of moving structures as well as the measurement of blood flow velocity and directionality within a vessel.

Bats, for example, detect their prey through a variety of ultrasonic range (echolocation) techniques. They have the ability to detect frequencies above 100 kHz and possibly up to 200 kHz.

![ultra](//images.ctfassets.net/4yflszkpcwkt/7ny2rC6oGzmGm44eh1aOoq/486bb15bdeb12f492c011e189c0c69c2/applications-of-ultrasound.png)
[Source](https://www.aplustopper.com/wp-content/uploads/2016/09/applications-of-ultrasound.png)

## USES OF ULTRASOUND
* Ultrasounds are used in a variety of applications such as navigation, medical, imaging, cleaning, and communication.
* During pregnancy, it is used to examine the uterus and ovaries and keep an eye on the developing baby's health.

![ultrass](//images.ctfassets.net/4yflszkpcwkt/1P6gWrlVY186fOMeY2H9xq/2003731bf816fb14bf7157ebcb92e367/ultrasound-1.jpg)
[Source](https://images.emedicinehealth.com/images/article/main_image/ultrasound-1.jpg)

* __Sonar__ - Submarines used this technology to navigate by passing high-frequency waves beneath the water. These sound waves may easily pass over water and are employed for undersea communication and detection.
* __Echocardiography__ (ECG)
* __Detecting Cracks in metal__ -  Waves are commonly used to identify cracks or fissures in high-rise buildings and substantial metal components.

Apart from these, ultrasound waves are also used in many places.

## PROPERTIES OF ULTRASOUND

Ultrasonic waves are sound waves with a frequency greater than 20 kHz. The human ear cannot hear these sound waves. Ultrasound waves with a very high amplitude are referred to as sonic.

Some important properties of Ultrasound:

* Ultrasonic waves have a frequency that is __higher__ than what humans can hear (20 kilohertz).
* Their wavelengths are __shorter__. As a result, they have a lot of penetrating power.
* They are __unable__ to travel in a vacuum.
* They are reflected, refracted, and absorbed.
* They contain a __lot of energy__. They can be transmitted across long distances with little energy loss.
* When they flow through objects, they produce a lot of __heat__.

## APPLICATIONS OF ULTRASOUND

* It's utilized in security systems to detect even the tiniest movement within a defined region.
* Acoustic microscopy is used in industry to analyze the uniformity and cleanliness of liquids and solids using ultrasound.
* Ultrasound waves vibrate a metal sheet in humidifiers, causing the water to spray as a fine mist.
* To weld polymers, ultrasonic welding is utilized to generate heat.
* Ultrasonic cleaning is effective for cleaning delicate jewelry, watches, and glasses.

# SUMMARY
* The term "ultrasound" refers to sound waves having frequencies higher than the human hearing limit. 
* The top capacity varies per person, but it is often approximately 20,000 Hertz. 
* Ultrasound has the same physical properties as audible sound.
* Ultrasounds are used in a variety of applications such as navigation, medical, imaging, cleaning, and communication.

## FAQs

**Q. What are the three properties of ultrasound?**

1. Ultrasonic waves have a frequency that is higher than what humans can hear (20 kilohertz).
2. Their wavelengths are shorter. As a result, they have a lot of penetrating power.
3. They are unable to travel in a vacuum.

**Q. How is physics used in ultrasound?**

Materials with a piezoelectric property can generate ultrasound waves. The piezoelectric effect is a phenomenon that occurs when a mechanical force (squeeze or stretch) is applied to certain materials, resulting in the creation of an electric charge.

**Q. Explain the uses of ultrasound.**

Ultrasounds are used in a variety of applications such as navigation, medical, imaging, cleaning, and communication.

_We hope you enjoyed studying this lesson and learned something cool about **Ultrasound**! Join our Discord community to get any questions you may have answered and to engage with other students just like you! We promise, it makes studying much more fun!😎_

## REFERENCE 
1. Properties of Ultrasonic Waves: [https://byjus.com/jee/properties-of-ultrasonic-waves/](https://byjus.com/jee/properties-of-ultrasonic-waves/)** **Accessed 14th April 2022.
2. Ultrasound: [https://www.vedantu.com/physics/ultrasound](https://www.vedantu.com/physics/ultrasound) Accessed 14th April 2022.
3. Applications of Ultrasound: [https://www.slideshare.net/abhijitcool18/application-of-ultrasound-in-separation-process](https://www.slideshare.net/abhijitcool18/application-of-ultrasound-in-separation-process) Accessed 14th April 2022.


Ultrasound refers to sound waves having frequencies higher than the human hearing limit. Ultrasound has the same physical qualities as "regular" sound, with the exception that humans cannot hear it. In healthy young people, this limit varies from person to person and is around 20 kilohertz.


 Ultrasound Study Guide

# INTRODUCTION 
![pitch](//images.ctfassets.net/4yflszkpcwkt/5duKB2paiqDhPnxfXHNH5e/3a9d02f79e8a94c03b901df2efc8b31b/pitch.jpg)
[Source](https://www.toppr.com/guides/physics/sound/characteristics-of-sound/)

The pitch of sound is determined by the frequency of vibration of the sound waves that produce them. A high frequency (e.g., 880 Hz) is seen as a high pitch, while a low frequency (e.g., 55 Hz) is regarded as a low pitch.

Low-frequency sounds include a bass drum, thunder, and a man's deep voice. High-frequency sounds include a high-pitched whistle, squeak, and a child's voice. The loudness or softness of a sound is measured in intensity.

## HOW ARE PITCH AND FREQUENCY RELATED?

The frequency of a sound wave is linked to our perception of the pitch of that sound. Sound waves with a high frequency are perceived as high-pitched, while sound waves with a low frequency are seen as low-pitched.

### FREQUENCY OF SOUND

- Sound waves are measured in __hertz (Hz)__, which is the number of waves that pass through a particular place in a second. 
- Sounds with a frequency of __20 Hz to 20,000 Hz__ are generally audible to humans. 
- __Infrasound__ refers to sounds having frequencies less than 20 hertz. 
- Humans cannot hear infrasound because it is too low-pitched. 
- __Ultrasound__ refers to sounds with frequencies greater than 20,000 hertz. Humans cannot hear ultrasound because it is too high-pitched.

- Other species are capable of hearing noises in the ultrasonic spectrum. 
- Dogs, for example, can detect sounds with frequencies of up to 50,000 Hz.
- You may have seen special whistles that dogs can hear but not humans. 
- Whistles make noises that are too high in frequency for the human ear to detect. 
- Even higher-frequency sounds are audible to other animals. 
Bats can detect sounds with frequencies exceeding 100,000 Hz.

![waves](//images.ctfassets.net/4yflszkpcwkt/1NY19ekfNszZANHsDHKLb4/e99c0c6214959a251fa6f4dff99f8335/u11l2a2.gif)
[Source](http://www.physicsclassroom.com/Class/sound/u11l2a2.gif)

### PITCH OF SOUND

- Pitch is the __quality of sound__ that distinguishes an acute (or strident) note from a grave or flat note. 
- In music, the phrase 'pitch' is the frequently used. 
- It is determined by the sound wave's __frequency__. 
- A higher frequency note has a higher pitch than a lower frequency note.

* This is determined by the frequency of the waves' __vibrations__.
* The sound is harsh and has a high pitch if the frequency of vibration is higher. 
* On the other hand, if a sound is described to have a lower pitch, it means that it vibrates at a lower frequency.
* Pitch is influenced by __changes in frequency__. The shrillness of the sound increases as the frequency rises.

![pitch](//images.ctfassets.net/4yflszkpcwkt/5RMMo7HGKVhQsvCzMxAyti/9d1926b462f969478297f1d323c06a32/a80bf9f958f926a160067488787a42d6--high-pitch-oval-windows.jpg)
[Source](https://i.pinimg.com/474x/a8/0b/f9/a80bf9f958f926a160067488787a42d6--high-pitch-oval-windows.jpg)

## SUMMARY
* The pitch of sounds is determined by the frequency of vibration of the sound waves that produce them.
* Sound waves with a high frequency produce high-pitched noises, whereas sound waves with a low frequency produce low-pitched sounds.
* Pitch is a sound property that distinguishes between harsh and flat sounds.
* Pitch is unaffected by the amount of energy received by the ear per unit of time.

## FAQs

**Q. What is the frequency and pitch in sound?**

A frequency's sensation is generally referred to as a sound's pitch. A high-frequency sound wave corresponds to a high pitch sound, whereas a low-frequency sound wave corresponds to a low pitch sound.

**Q. How frequency and pitch are related?**

The frequency of a sound wave is linked to our perception of the pitch of that sound.

**Q. How do frequency and pitch affect the sound?**

The higher the pitch of the sound we hear, the higher the frequency waves oscillate.

**Q. Explain the term 'pitch' of a sound. On what factor does the 'pitch' of a sound depend?**

The frequency of a sound determines its pitch. Because the pitch of a sound is directly proportional to its frequency, low-frequency sounds are described as having a low pitch, while high-frequency sounds are described as having a high pitch.

_We hope you enjoyed studying this lesson and learned something cool about the **Frequency and Pitch of Sound**! Join our Discord community to get any questions you may have answered and to engage with other students just like you! We promise, it makes studying much more fun!😎_

## REFERENCE  
1. Pitch and Loudness of Sound: [https://byjus.com/physics/loudness-of-sound/](https://byjus.com/physics/loudness-of-sound/)** **Accessed 14th April 2022.
2. Frequency and Pitch of Sound: [https://flexbooks.ck12.org/cbook/ck-12-physics-flexbook-2.0/section/12.4/primary/lesson/frequency-and-pitch-of-sound-ms-ps/](https://flexbooks.ck12.org/cbook/ck-12-physics-flexbook-2.0/section/12.4/primary/lesson/frequency-and-pitch-of-sound-ms-ps/) Accessed 14th April 2022.

A frequency's sensation is generally referred to as a sound's pitch. A high-frequency sound wave corresponds to a high pitch sound, whereas a low-frequency sound wave corresponds to a low pitch sound.

Frequency And Pitch Of Sound Study Guide

# INTRODUCTION

![whisper](//images.ctfassets.net/4yflszkpcwkt/5YCMpxld6GsxMVPz3lhhXM/4c61a4fab2f93a4a2d89ae14c1cc9f6a/PRI_80043681.jpg)
[Source](https://metro.co.uk/2019/08/13/fancy-getting-paid-30-an-hour-to-whisper-into-the-ears-of-stressed-out-people-10562170/)

When a friend speaks to you in hushed tones, you may need to lean in close to hear what they’re saying. Similarly, if your friend calls out to you from across the gymnasium. Even though they’re several meters away, their voice is now loud enough for you to plainly hear them! It is safe to say that the volume of sounds can fluctuate.

A listener's perception of how loud or soft a sound is referred to as loudness. The strength of the sound waves determines the loudness of the sound. The quantity of energy in sound waves is measured by intensity. A decibel is a measurement of intensity (dB).

## DECIBEL LEVELS

The graph below depicts the decibel levels of various sounds. Sound waves get more intense, and noises become louder as decibel levels rise. Loudness increases by ten times for every ten-decibel increase in sound intensity. As a result, a 30-decibel "silent" room is ten times louder than a 20-decibel whisper, and a 40-decibel gentle rainfall is 100 times louder. 

Remember: Sounds with a high decibel level are hazardous. They can harm the ears and cause hearing loss.

![decibels](//images.ctfassets.net/4yflszkpcwkt/1KkgdVCySpx95Ql0qpYtgg/c430ee986b53bc06c50a9483113677b2/f-d_456f837f13d56d784512929f6135733a22d77f6aa75e5fcdf95ee874_IMAGE_THUMB_POSTCARD_TINY_IMAGE_THUMB_POSTCARD_TINY.1)
[Source](https://flexbooks.ck12.org/cbook/ck-12-physics-flexbook-2.0/section/12.3/primary/lesson/intensity-and-loudness-of-sound-ms-ps/) 

## DIFFERENCE BETWEEN LOUDNESS AND INTENSITY 

<table>
 <tr>
 <td>Loudness
 </td>
 <td>Intensity
 </td>
 </tr>
 <tr>
 <td>The ear's response to sound is measured by its loudness.
 </td>
 <td>The sound power per unit area is known as intensity.
 </td>
 </tr>
 <tr>
 <td>Decibels are used to measure loudness.
 </td>
 <td>Watt per meter square is the unit of measurement for intensity.
 </td>
 </tr>
 <tr>
 <td>The sensitivity of the human ears determines how loud something is.
 </td>
 <td>The sensitivity of the human ears has no bearing on the intensity.
 </td>
 </tr>
 <tr>
 <td>Loudness is a personal preference.
 </td>
 <td>The term "intensity" refers to a numerical value that can be measured.
 </td>
 </tr>
</table>

# SUMMARY
* A listener's perception of how loud or soft a sound is is referred to as loudness. The intensity, or quantity of energy, in sound waves determines the loudness of the sound. The decibel is a measurement of intensity (dB).
* Sound waves get more intense, and noises become louder as decibel levels rise. Loudness increases by ten times for every ten-decibel increase in sound intensity.
* The amplitude of the sound waves and the distance they have traveled from the source of the sound determines the intensity of the sound.

## FAQs

**Q. What does the loudness of a sound depend on?**

The loudness of a sound wave is determined by its amplitude. A loud sound has a bigger amplitude, while a soft sound has a lesser amplitude.

**Q. Does sound intensity depend on frequency?**

The energy per second that a wave can send to a unit area of a surface is measured by its intensity. (Watts per square meter is the unit of measurement for intensity.) The vibration rate equals the frequency of a sound wave, and the intensity is determined by the amplitude.

**Q. Does intensity change when a sound gets louder?**

The intensity of the sound increases as the amplitude of the sound wave increases.

_We hope you enjoyed studying this lesson and learned something cool about the **Intensity and Loudness of Sound**! Join our Discord community to get any questions you may have answered and to engage with other students just like you! We promise, it makes studying much more fun!😎_

## REFERENCE 
1. Distinguish Between Loudness and Intensity of Sound: [https://byjus.com/questions/distinguish-between-loudness-and-intensity-of-sound/](https://byjus.com/questions/distinguish-between-loudness-and-intensity-of-sound/) Accessed 14th April 2022.
2. Intensity and Loudness of Sound: [https://flexbooks.ck12.org/cbook/ck-12-physics-flexbook-2.0/section/12.3/primary/lesson/intensity-and-loudness-of-sound-ms-ps/](https://flexbooks.ck12.org/cbook/ck-12-physics-flexbook-2.0/section/12.3/primary/lesson/intensity-and-loudness-of-sound-ms-ps/) Accessed 14th April 2022.

The amount of sound energy traveling through a unit area per second is the intensity of a sound wave. Loudness is a measurement of the ear's response to sound. Its amplitude determines how loud a sound is. The intensity of a sound is determined by its amplitude, which is perceived as loud by the ear.


Intensity And Loudness Of Sound Study Guide

# INTRODUCTION

![palne](//images.ctfassets.net/4yflszkpcwkt/4yB7SJNtHn4XsgOXu3qNz/b977ff12042e509dc0253f85aa4e4826/speed.jpg)
[Source](https://blog.soundcloud.com/2011/12/13/found-sounds-the-speed-of-sound/)

Sound speed is the rate at which sound waves travel through various materials. The contemporary value for the speed of sound in dry air at a temperature of 0 °C (32 °F) is 331.29 meters (1,086.9 feet) per second.

## SOUND SPEED IN DIFFERENT MEDIUMS

- Sound’s speed in colloquial speech refers to the speed of sound waves in the air. 
- Sound travels at different speeds in different substances.
- Gases are the slowest, liquids are faster, and solids are the fastest. 
- While sound travels at 343 meters per second in air, it travels at 1,481 meters per second in water (almost 4.3 times faster) and 5,120 meters per second in iron (almost 15 times as fast). 
- Sound travels at approximately 12,000 m/s through a very hard solid like a diamond, nearly 35 times its speed in air and the fastest it can travel under normal conditions.

## FACTORS AFFECTING THE SPEED OF SOUND 

The speed of sound is affected by the density and temperature of the material through which it travels.

![speed](//images.ctfassets.net/4yflszkpcwkt/ORnzmVlDZ8Qgn6u9YP31G/6fbe7626113e45082ac8af2acfbfe4f5/sound.gif)
[Source](https://www.grc.nasa.gov/www/BGH/Images/sound.gif)

### THE MEDIUM'S DENSITY

When the medium is dense, the molecules are packed tightly together, allowing sound to travel faster. As a result, as the density of the medium increases, so does the speed of sound.

### THE MEDIUM'S TEMPERATURE

The temperature has a direct relationship with the speed of sound. As a result, the speed of sound increases as the temperature rises.

## SPEED OF SOUND FORMULA

Because the speed of sound is defined as the distance traveled by a sound wave in a given amount of time, it can be calculated using the following formula:

**v = λ f**

Where v denotes velocity,  λ  denotes sound wave wavelength, and f denotes frequency.

For all waves, the relationship between the speed of sound, its frequency, and wavelength is the same. The distance between neighboring compressions or rarefactions determines the wavelength of a sound. The quantity of waves that pass a location per unit of time has the same frequency as the source.

**Example**

How long does a sound wave with a frequency of 2 kHz and a wavelength of 35 cm take to traverse 1.5 kilometers?

Solution:

We know that the formula for calculating sound speed is:

__v = λ v__

We get by substituting the numbers in the equation.

v = 0.35 m × 2000 Hz = 700 m/s

The time it takes a sound wave to traverse 1.5 kilometers can be calculated as follows:

__Time = Velocity/Distance Traveled__

We get this by substituting the numbers in the equation.

Time = 1500 meters / 700 meters per second = 2.1 seconds

# SUMMARY

* The speed of "sound" is actually the rate at which a minor disturbance is transmitted across a medium.
* The contemporary value for the speed of sound in dry air at a temperature of 0 °C (32 °F) is 331.29 meters (1,086.9 feet) per second.
* Sound travels at different speeds in different substances: sound travels the slowest in gases, the fastest in liquids, and the fastest in solids.

## FAQs

**Q. What can affect the speed of sound?**

The speed of sound is affected by the density and temperature of the material through which it travels.

**Q. What limits the speed of sound?**

Because sound is a wave that propagates by causing surrounding particles to interact, its speed is determined by the density of a material and the way its atoms are bonded together. Atoms can only travel so fast, and that movement limits the speed of sound.

**Q. Does temperature affect the speed of sound?**

Yes, the temperature of the material affects the sound’s speed.

**Q. Does pressure affect the speed of sound?**

The speed of sound is unaffected by gas pressure at a constant temperature.

_We hope you enjoyed studying this lesson and learned something cool about **Speed of Sound**! Join our Discord community to get any questions you may have answered and to engage with other students just like you! We promise, it makes studying much more fun!😎_

## REFERENCE 
1. Speed of Sound: [https://byjus.com/physics/speed-of-sound-propagation/](https://byjus.com/physics/speed-of-sound-propagation/) Accessed 14th April 2022.
2. Speed of Sound: [https://en.wikipedia.org/wiki/Speed_of_sound](https://en.wikipedia.org/wiki/Speed_of_sound)  Accessed 14th April 2022.

The speed of "sound" is actually the rate at which a minor disturbance is transmitted across a medium.

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 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 sin(x) 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! 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.


The Fourier series is used to represent a complex function as an infinite summation of sine and cosine functions. In this article, you will see how it is done with a simple example. 


## 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 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

## INTRODUCTION 

You’re on a picnic with your family at the beach. It’s sunny, the water is great, and people are surfing the high waves. You decide to venture a little further into the water until it is chest-deep. A big wave passes over you, picks you up, and sets your right back down where you were standing. Every other wave does the same. So why don’t the waves carry you to the shore although they are moving towards it from the ocean? Let’s find out! 

![Waves](//images.ctfassets.net/4yflszkpcwkt/6LJu8vMYU71O9p274bOgd6/9aa58375316400c36c17a586155afcce/1030_oceanwaves-1028x579.png)
[Source](https://www.sciencenewsforstudents.org/article/explainer-understanding-waves-and-wavelengths) 

### WHAT ARE WAVES?

Everybody has an idea what a wave is. But how would you characterize what defines a wave? In simple terms, a wave can be described as a disturbance that propagates in a medium. The ocean, for example, is a medium through which a wave travels. Waves can travel in air as the medium as well; it is what we call sound! Waves can also travel in all kinds of different materials. 

A disturbance is the displacement of the particles in the medium which creates the wave. For example, when you throw a rock into clear water, it creates a disturbance that disperses out as tiny little waves or ripples. Disturbance in the air can be a sudden change of pressure that our ears perceive as sound. 

### PROPERTIES OF WAVES

To understand waves, it is essential to explore the physical properties of waves. 

### 1. WAVE TYPE

Based on the direction of the displacement of the particles, there are two types of waves: longitudinal and transverse waves. 

![Transverse and Longitudinal Waves](//images.ctfassets.net/4yflszkpcwkt/3BoT5Fcvc2l69E8ngBOIib/d548099c0d974347c1ba5949729c95f2/longitudinal-transverse-waves.png.pagespeed.ce.kWtYTjwnEP.png)
[Source](https://www.onlinemathlearning.com/transverse-longitudinal-wave.html)

- In a longitudinal wave, the particles of the medium are displaced in the same direction as the wave's motion. Example: Waves on slinky and sound waves.
- In a transverse wave, the particles are displaced perpendicular to the direction of wave travel. Examples: Ocean waves. 

### 2. FREQUENCY AND WAVELENGTH

![Ocean Waves](//images.ctfassets.net/4yflszkpcwkt/2DESPTvpso7WDBorWNYKcY/61ca9c4428cb3115b0261524efc58919/wave-illustration.gif)
[Source](https://www.boatus.com/app/views/201812/ocean-wave-basics.asp)

Waves always travel in crests and troughs. The crest is the highest part of the wave, and the trough is the lowest part of the wave. Frequency is the number of crests passing through a point every second. The distance between successive crests or subsequent troughs is called the wavelength of the wave. The height of a crest or the depth of a trough is called the amplitude of the wave. 

### PROPAGATION OF WAVES

From the types of waves discussed above, we see that a wave is a disturbance in a medium that transports energy through the medium and not matter. That means that the energy traveling through the medium only displaces the particles in the medium from a rest position; it does not cause the migration of the particles to a new location. This is the reason why the waves at the beach lift you and drop you back in the same place as it passes. The wave doesn’t carry you with it to the shore. 

## CONCLUSION

- Waves are a disturbance in a medium that transports energy through the medium.
- Depending on the displacement of the particles, there are two types of waves: longitudinal and transverse waves. 
- Waves only transport energy and not matter. 

### FAQs:

__1. What is wave definition?__

A wave is a disturbance in a medium that travels from one point to another. 

__2. What are 4 types of waves?__

The four types of waves include transverse waves, longitudinal waves, and surface waves (mechanical waves). The 4th type of wave is an electromagnetic wave such as light.  

*We hope you enjoyed studying this lesson and learned something cool about __Waves__! Join our Discord community to get any questions you may have answered and to engage with other students just like you! We promise, it makes studying much more fun!* 😎

## SOURCES: 
1. Types of Waves. https://www.ck12.org/physics/types-of-waves/lesson/Types-of-Waves-PPC/?referrer=concept_details. Accessed 18 Feb 2022.
2. Vibrations and Waves - Lesson 1 - The Nature of a Wave. https://www.physicsclassroom.com/class/waves/Lesson-1/What-is-a-Wave. Accessed 18 Feb 2022.  
3. Waves. https://courses.lumenlearning.com/boundless-physics/chapter/waves/. Accessed 18 Feb 2022.  

Primarily, there are two types of waves: Longitudinal and transverse waves. In this article, you will learn about their properties and how they differ. 

Waves Study Guide

## INTRODUCTION 

It’s New Year’s Eve, and you’re off to the fireworks display in the city with your family. The clock hits midnight, and the whole crowd erupts in a cheer as they reach the end of their count down. The fireworks start with aerial shells shooting into the air, and as always, you see the flash and sparkly stars before hearing the “thuds”! Have you ever stopped to nerd out and ponder why it is that the sound always arrives a few seconds after the flash? If you’re thinking sound waves, then you are definitely headed in the right direction!

![Fireworks](//images.ctfassets.net/4yflszkpcwkt/1NWRHp1ZqK2wUqPii0l1bV/cd60bf228c40246fa679c51ae3b9afc4/fireworks-july4th-1-1024x446.jpg)
[Source](https://avanti.in/science-of-fireworks/)

### WHAT IS SOUND?

In physics, a sound wave is a vibration that propagates as mechanical waves through a medium. Sound can travel in solids, liquids, and gasses. It travels the fastest in solids and slowest in liquids. Sound waves are not too different from the ripples that form on a lake when you drop a stone into it. The energy travels outward in the form of a wave getting weaker the further it goes. Sound waves are similar, except they are 3-dimensional and travel outward like a bubble. The human ear is designed to detect the pressure of this bubble, and it is what we perceive as sound. 

### WHAT ARE SOUND WAVES?

![Sound waves](//images.ctfassets.net/4yflszkpcwkt/DVrkaUIbIDldwEBnWhD4F/2aeeefe57e09995748a5c591cf8d36bb/uu5vc9wnypgui6xwi0ds.png)
[Source](https://www.tuttee.co/blog/phys-sound-waves)

A sound wave is a type of mechanical wave called a longitudinal wave. It is characterized by the wave's energy propagating parallel to the direction of the vibrating particles. In the air, sound moves as a wave that generates continuous compression and rarefaction of gas molecules from the source to our ears. 

When the tuning fork in the figure rings, it continuously transmits the vibration to the air molecules in its surroundings. This vibration gets passed on to adjacent air molecules in all directions as a wave with alternating high (compression) and low pressure (rarefaction). Our ears perceive sound by detecting the low and high-pressure periods that transmit the vibration to our eardrums.

![Sound](//images.ctfassets.net/4yflszkpcwkt/6SoaD98C0dZcAOHwOks5UI/f91d733aebff44d1f4493ec72e40e702/Sound.png)
[Source](https://www.uoanbar.edu.iq/eStoreImages/Bank/13622.pdf)

Longitudinal waves which make up sound are different from transverse waves. In transverse waves, the propagation of energy is perpendicular to the direction of vibration of the particles. Waves on a stretched string and waves on the water are examples of transverse waves. 

Since sound requires a medium to travel, it cannot travel in a vacuum such as space. 

### SPEED OF SOUND

Sound waves being longitudinal depend on the density and elastic property of the medium to travel faster or slower. Sound travels fastest in solids and slowest in gas. The molecules are tightly packed together in solids and transmit sound faster than liquids and gasses. 

The speed of sound in air at sea level is about 346 m/s, whereas the speed of sound in Aluminum is 6,320 m/s! Going back to the fireworks display, sound travels a lot slower than light. Therefore, you always hear the sound after a few seconds of the flash. 

## CONCLUSION:

- A sound wave is a vibration that propagates as mechanical waves through a medium.
- Sound waves are longitudinal waves that travel as compression and rarefaction of air molecules. 
- The speed of sound is different in different mediums. 

### FAQs:

__1. What is sound?__

Going by the sound definition in physics, it is a pressure wave generated by a vibrating object perceived by our ears. 

__2. What are the three types of sound?__

Sound waves can be classified as longitudinal, mechanical, and pressure waves.

__3. What is an example of sound?__

Lightning, everyday sounds are coming from vehicles, birds, and speech.

*We hope you enjoyed studying this lesson and learned something cool about __Sound__! 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. 17.1 Sound Wave. https://flexbooks.ck12.org/cbook/ck-12-physics-flexbook-2.0/section/12.1/primary/lesson/sound-waves/. Accessed 28 Jan 2022.  
2. Types of Waves. https://byjus.com/physics/types-of-waves/.  Accessed 28 Jan 2022.  
3. Are sound waves longitudinal waves or transverse waves?. https://www.tuttee.co/blog/phys-sound-waves.  Accessed 28 Jan 2022.  

Sound waves are mechanical waves that require a medium for propagation. The speed of sound is different in different mediums. 


Sound Study Guide

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!