# Connexions

You are here: Home » Content » Fourier Series

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Waves and Optics"

"This book covers second year Physics at Rice University."

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

# Fourier Series

Module by: Paul Padley. E-mail the author

Summary: A brief introduction to Fourier Series starting from the normal modes of an oscillating string. The concept is then extended to Fourier's integral theorem.

## Fourier Analysis

### Fourier Series

Lets go back to the case of a string fixed at 0 0 and L L , its n t h n t h harmonic is y n ( x , t ) = A n sin ( n π x L ) cos ( ω n t δ n ) y n ( x , t ) = A n sin ( n π x L ) cos ( ω n t δ n ) In fact all the modes could be permitted, and so any possible motion of the string can be completely specified by: y ( x , t ) = n = 1 A n sin ( n π x L ) cos ( ω n t δ n ) . y ( x , t ) = n = 1 A n sin ( n π x L ) cos ( ω n t δ n ) . This has been rigorously shown by mathematicians but the complete proof is beyond our scope in this course. Lets accept the mathematicians word on this. We could take a snapshot of this function at a time t = t 0 t = t 0 . Then we could write y ( x ) = n = 1 B n sin ( n π x L ) y ( x ) = n = 1 B n sin ( n π x L ) where B n = A n cos ( ω n t 0 δ n ) . B n = A n cos ( ω n t 0 δ n ) . Likewise we could look at one point at space and look at the oscillations as a function of time. In that case we would get. y ( t ) = n = 1 C n cos ( ω n t δ n ) y ( t ) = n = 1 C n cos ( ω n t δ n ) Lets work with the time snapshot, y ( x ) = n = 1 B n sin ( n π x L ) y ( x ) = n = 1 B n sin ( n π x L ) We need to figure out what the B n B n factors are and this is what Fourier figured out. We can multiply both sides by the sin sin of a particular harmonic y ( x ) s i n ( n i π x L ) = n = 1 B n sin ( n π x L ) s i n ( n i π x L ) y ( x ) s i n ( n i π x L ) = n = 1 B n sin ( n π x L ) s i n ( n i π x L ) and now we can integrate both sides Recall cos ( θ φ ) = cos θ cos φ + sin θ sin φ cos ( θ φ ) = cos θ cos φ + sin θ sin φ cos ( θ + φ ) = cos θ cos φ sin θ sin φ cos ( θ + φ ) = cos θ cos φ sin θ sin φ So sin θ sin φ = 1 2 [ c o s ( θ φ ) cos ( θ + φ ) ] sin θ sin φ = 1 2 [ c o s ( θ φ ) cos ( θ + φ ) ] Thus This is equal to zero at the limits 0 , L 0 , L except for the particular case when n = n i n = n i . In that case sin ( n π x L ) sin ( n i π x L ) x = sin 2 ( n π x L ) x sin ( n π x L ) sin ( n i π x L ) x = sin 2 ( n π x L ) x So you get After all that we should see that for each term in the sum is zero, except the case where n i = n n i = n . Thus we can simplify the equation: 0 L y ( x ) sin ( n π x L ) x = L 2 B n . 0 L y ( x ) sin ( n π x L ) x = L 2 B n . or B n = 2 L 0 L y ( x ) sin ( n π x L ) x B n = 2 L 0 L y ( x ) sin ( n π x L ) x The above is a very specific form of the Fourier Series for a function spanning an interval from 0 0 to L L and passing through zero at x = 0 x = 0 .

#### More General Case

One could write a more general case for the Fourier Series which applies to an interval spanning L L to L L and not constrained to pass through zero. In that case one can write y ( x ) = a 0 2 + n = 1 [ a n cos ( n π x L ) + b n sin ( n π x L ) ] y ( x ) = a 0 2 + n = 1 [ a n cos ( n π x L ) + b n sin ( n π x L ) ] where A n = 1 L L L y ( x ) cos ( n π x L ) x    n = 0 , 1 , 2 , 3 , A n = 1 L L L y ( x ) cos ( n π x L ) x    n = 0 , 1 , 2 , 3 , and B n = 1 L L L y ( x ) sin ( n π x L ) x    n = 1 , 2 , 3 , B n = 1 L L L y ( x ) sin ( n π x L ) x    n = 1 , 2 , 3 , You can then look at the symmetry of the problem and see if just sin sin or cos cos can be used. For example if y ( x ) = y ( x ) y ( x ) = y ( x ) then use cosines. If y ( x ) = y ( x ) y ( x ) = y ( x ) use the sines.

### Fourier Integral Theorem

In fact Fourier's theorem can be taken to a next step. This is Fourier's integral theorem. That is any function (even if it is not periodic) can be represented by f ( x ) = 1 π 0 [ A ( k ) cos ( k x ) + B ( k ) s i n ( k x ) ] d k f ( x ) = 1 π 0 [ A ( k ) cos ( k x ) + B ( k ) s i n ( k x ) ] d k where A ( k ) = f ( x ) cos ( k x ) x A ( k ) = f ( x ) cos ( k x ) x B ( k ) = f ( x ) sin ( k x ) x B ( k ) = f ( x ) sin ( k x ) x A A and B B are called the Fourier transforms of f ( x ) f ( x ) Lets look at an example.

f ( x ) = E o | x | < L / 2 f ( x ) = 0 | x | > L / 2 f ( x ) = E o | x | < L / 2 f ( x ) = 0 | x | > L / 2 right away you can set B ( x ) = 0 B ( x ) = 0 from symmetry arguments A ( k ) = f ( x ) cos ( k x ) x = L / 2 L / 2 E 0 cos ( k x ) x = E o k sin ( k x ) | L / 2 L / 2 = E o k [ sin ( k L 2 ) sin ( k L 2 ) ] = 2 E o k sin ( k L 2 ) = E 0 L sin ( k L 2 ) k L 2 A ( k ) = f ( x ) cos ( k x ) x = L / 2 L / 2 E 0 cos ( k x ) x = E o k sin ( k x ) | L / 2 L / 2 = E o k [ sin ( k L 2 ) sin ( k L 2 ) ] = 2 E o k sin ( k L 2 ) = E 0 L sin ( k L 2 ) k L 2

### Closing word

Up until now in the course we have been dealing with very simple waves. It turns out that any complicated wave that can possibly exist can be constructed from simple harmonic waves. So while it may seem that an harmonic wave is an over simplification, it can be used in even the most complex cases.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

### Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks