3.3 Waves and Stuff

Science of Music2007

Last Time

Dr. Koons talked about consonance and beats.

Let’s take a quick look & listen at what this means ….

Listen First

Recall what a single frequency tone sounds like

Play on Sound Generator A=440 Hz. The Graph:

0.005 0.01 0.015 0.02

-1

-0.5

0.5

1

440 Hz. and 450 Hz. Compared

0.005 0.01 0.015 0.02

-1

-0.5

0.5

1

0.005 0.01 0.015 0.02

-1

-0.5

0.5

1

440 Hz.

450 Hz.

440 Hz. and 450 Hz. Together

0.005 0.01 0.015 0.02

-1

-0.5

0.5

1

440 Hz. and 450 Hz. Togetherfor a longer time (0.1 second)

0.02 0.04 0.06 0.08 0.1

-1

-0.5

0.5

1

440 Hz. and 450 Hz. Togetherfor an even longer time (0.5 second)

0.1 0.2 0.3 0.4 0.5

-1

-0.5

0.5

1Tbeat

From this graph we see that Tbeat = 0.1 seconds. fbeat = 10 Hz..

Why??

0.02 0.04 0.06 0.08 0.1

-1

-0.5

0.5

1

In phase out of phase in phase again

The Concept Two sounds start together (in

phase). After Tbeat seconds they get

back together again. One of the waves must have

gone through N cycles while the other went through (N+1) complete cycles.

They are therefore together again!

0.02 0.04 0.06 0.08 0.1

-1

-0.5

0.5

1

The *(*@)$# math

121221

21

21

211

21

211

21

21

2211

2111

11

)(

)1(

ffTTTT

TTf

TT

TTT

TT

TTNT

TT

TN

TTTN

TNTN

beat

beat

The beat frequency between two simultaneous tones is equal toThe difference between the frequencies of the two tones!

Back to

The frequencies

Lm

T

LL

vvf

vf

L

/2

1

2

2

11

11

1

These are the frequencies at which theString RESONATES.

RESONANCE

STRINGS HAVE MORE THAN ONE RESONANT FREQUENCY

The RESONANT frequencies

L

vf

L

2

2

12

1

2

2

ff

SOL

f

BUT

Standing Wave Produced by TWO waves traveling along a

string in opposite directions. Each wave reflects at the end of the string

and then goes the other way. Both waves travel with the same velocity

over the same length of string. Many pairs of waves may travel along the

string at the same time. More than one set of standing waves is

possible on the string at the same time.

actual string shape

fundamental

first overtone, second harmonic,second octave

second overtone, third harmonic,fifth above second octave

third overtone, fourth harmonic,second octave.

etc.

Resonance of Strings

Multiple Frequencies Determine the Timbre

What does sound look like?

Repeats

Music

For short periods of time, a musical sound is PERIODIC.

It has a weird shape. How do we produce a strange looking

but periodic shape? Answer: FOURIER

Fourier’s Theorem Any periodic signal can be broken

down into a sum of simple sine waves at different frequencies and sizes (amplitudes).

This theorem allows us to understand why different instruments sound different even when playing what we perceive as the same tone.

Square Wave

How do we create weird periodic shapes??

FOURIER THEOREM

For Example

The FOURIER spectra for each of these consists of asingle frequency.

When you strike the string All standing wave modes are excited

at the same time. “Non-Standing” waves die out quickly. This is a Fourier thing. If we could initially shape the string

exactly to one of these modes, then this would be the only one that would be excited.

But we can’t do this .. can we???

We can! Sorta..

Two different plucks would require different sets of harmonics to create the shape.

These will produce somewhat different instrument sounds.

Modes

Place finger near thecenter of the stringand the strike it. Theodd overtones should be suppressed.

Plucked atMidpoint

Touched atMidpoint

So – How does the guitar work?

Each string that is plucked will vibrate in one of its fundamental modes. The shape of the initial string stretch determines which modes will be excited.

Each mode is established by waves bouncing back and forth along the instrument.

More ..

The sound from the instrument depends on how and where along the string it is plucked.

The strings by themselves emit little sound. The connection to the bridge causes the sound box to move and resonate, a topic we will discuss later.

So now you know!