# 6. Line Spectrum

**Recall** from earlier trigonometry that we can express the sum of a sine term and a cosine term, with the same period, as follows:

`a\ cos θ + b\ sin θ = R\ cos ( θ − α)`

where

`R=sqrt(a^2+b^2)`

and

`alpha=arctan(b/a)`

Likewise, the Fourier series

`f(t)=a_0/2` `+sum_(n=1)^oo(a_n\ cos n omega t` `{:+b_n\ sin n omega t)`

can be written in **harmonics form**

`f(t)=C_0+` `sum_(n=1)^ooC_n\ cos(n omegat-varphi_n)`

where

`C_1 cos (ωt - varphi_1)` is called the

fundamental`C_2 cos (2ωt - varphi_2)`, the

second harmonic`C_3 cos (3ωt - varphi_3)`, the

third harmonic... etc; and`varphi_n` =

phase angle`C_n=`

harmonic amplitude`=sqrt((a_n)^2+(b_n)^2`

We met harmonics before.

A plot showing each of the harmonic amplitudes in the wave is
called the **line spectrum**.

**Note:** Waves
with discontinuities such as the saw tooth and square wave have
spectra with slowly decreasing amplitudes since their series have
strong high harmonics. Their 10th harmonics will often have
amplitudes of significant value compared to the fundamental.

In contrast, the series of waveforms without discontinuities and with a generally smooth appearance will converge rapidly to the function and only a few terms are required to generate the wave.

### Example 1

Plot the first 4 terms of the line spectrum for the Fourier Series:

`f(t)=pi/2` `+sum_(n=1)^oo1/(2n-1)cos nt` `+sum_(n=1)^oo((-1)^n)/(2n)sin nt`

Answer

We can see from the series that

`a_n=1/(2n-1)`

`b_n=((-1)^n)/(2n)`

Now, using `C_n=sqrt((a_n)^2+(b_n)^2)` for each term, we have:

`a_n=1/(2n-1)` | `b_n=((-1)^n)/(2n)` | `C_n=sqrt((a_n)^2+(b_n)^2)` |

`a_1=1` | `b_1=-1/2` | `C_1=sqrt(1^2+(-1/2)^2)=1.118` |

`a_2=1/3` | `b_2=1/4` | `C_2=sqrt((1/3)^2+(1/4)^2)=0.4167` |

`a_3=1/5` | `b_3=-1/6` | `C_3=sqrt((1/5)^2+(-1/6)^2)=0.260` |

`a_4=1/7` | `b_4=1/8` | `C_4=sqrt((1/7)^2+(1/8)^2)=0.190` |

The resulting line spectrum is:

The line spectrum for the function `f(t)`.

The above Fourier series has an interesting graph. (NOTE: The following is **not** the plot of line spectrum asked for in the question, and that we found in the Answer. It's the plot of the above Fourier series.)

Graph of the first 30 terms of `f(t)`.

## Music Examples

### Example 2: Harmonics and Sound

When we listen to different musical instruments playing the
same note, they sound different to us because of the different
combinations of **harmonics** contained in the
note.

For example, if a flute and a violin both play G above middle C, the harmonic spectrum is quite different.

G has a frequency of 392 Hz and the harmonics are all multiples of this fundamental frequency (or about 800 Hz, 1200 Hz, 1600 Hz, etc).

**Harmony** (2 or more notes sounding at the same
time) works because of harmonics (look for the chord GBD
contained within the harmonics of the note G).

You can see the relative values of the harmonics in the following sound spectrum images of a flute and a violin playing G4.

**Flute**

**Violin**

[Images source: University of New South Wales Physics Department]

### Example 3: Java Applet - Fourier Series and Sound

The following link leads to a great Java applet to play with. (It's on an external site.) He is using Fourier Series expressed in the form

`f(t)=C_0+sum_(n=1)^ooC_ncos(n omega t-varphi_n)`

You can change the mean value (*C*_{0}),
amplitude of each harmonic (*C*_{n}) and also
*f*_{n}, which changes the phase shift. You can
hear the sounds that result from adding different harmonics.

While you're there, check out the other applets and lots of interesting stuff at: