 A vibrating string
is transverse wave case
since the oscillation of the
medium (i.e., the string)
is perpendicular to the propagation direction when the
waves propagatewhich they
do NOT do in
the animation
because the waves are
standing waves.
 The nodes are the unmoving points
and the antinodes are the points where
the wave
amplitude is maximum
or in another meaning the regions between the nodes.
 A wavelength λ is the length of
two antinodes.
This is because a wavelength
is by definition a full spatial cycle of the
wave phenomenon.
Usually a spatial cycle can be characterized as an upanddown cycle.
This definition of
wavelength is the useful one for
analysis of wave phenomenon.
 Image 2 Caption: Note that the
universal symbol for
wavelength is the
small Greek lambda = λ.
 How are the standing waves
produced?
If we drive a vibrating string
with a vibrating machine
at exactly a resonance frequency f_n
(which we explain below),
we get standing waves
with exactly n antinodes.
What if we drive the
vibrating string
off all
resonance frequencies?
We get small
traveling waves that
varied in time.
For off resonance frequencies,
the pulls on the vibrating string
strongly cancel instead of adding up.
Now when you strike a vibrating string,
you simultaneously excite
standing waves
with multiple
superimposed resonance frequencies,
but other
frequency
waves are strongly
suppressed by canceling pulls.
 The allowed resonances
are determined by, among other things) the
boundary conditionswhich in
the case of a vibrating string
are usually fixed endpoints as in the
animation.
 Clearly, the number of antinodes n
for a standing wave
on a vibrating string
must be an integer.
Since an antinode
length is half
a wavelength
(i.e., λ/2), we have the
of number of antinodes formula
L
n =  , where L is the length of the vibrating string.
λ/2
Thus, the wavelength formula is
L
λ =  = 2L/n ,
(n/2)
where the first version makes sense since there are
2
antinodes per
standing wave.
Since we have a general formula
for phase velocity
(the velocity at which
traveling waves propagate:
phase velocity formula fλ=v_phase ,
where v_phase is the
phase velocity
and f is frequency),
we find the resonance frequency
formula for
a vibrating string to be
f_n = n[v_phase/(2L)] .
 For
musical instruments
(like string instruments)
where the vibration of the instrument produces
sound waves,
the resonance frequencies are called
harmonicsthe longest
wavelength
(which is the lowest frequency)
harmonic is the
1st harmonic
or the fundamental
and the others are
higher order harmonics: i.e.,
2nd harmonic,
3rd harmonic,
etc.
In music, one often
uses the alternative terms
fundamental tone (or 1st partial),
2nd tone (or 2nd partial),
3rd tone (or 3rd partial),
etc.
All, the "tones" except the
fundamental tone are
overtones, but
the word
overtone is also used for
noninteger
multiples
of the fundamental
(see Wikipedia: Overtone:
Musical usage term).
 The weighting of
harmonics
in intensity
determines much of the
timbre of the
musical instrument which
makes its sound pleasing.
 Pure frequencies are, in fact, rather boring.
See Wikipedia: Audio frequencies:
Sound file table (scroll down)
and
Wikipedia:
Concert A (standard music reference frequency)
(or Wikipedia:
A440 (pitch standard): Modern practices).
But since the buttons
do NOT always work for on the Wikipedia
sites, see:
To hear
Concert A 440 Hz
with overtones
for a violin
(much more pleasant than the pure
frequency
Concert A),
see
File:A440 violin.mid,
but it's probably just a
synthesizer
violin.

How does one pronounce timbre?
See
Forvo: timbrebut you'll be no wiser
after you've heard all the
pronunciationsanyway
as lumberjacks say
Timber!