Phys 1240 Fa 05, SJP 6-1
Chapter 10.1: Natural vibrations of a string.
We skipped ahead to this section because it contains an essential piece of physics for understanding
"harmonics" and timbre. We've already talked about it some, but not in a lot of detail. It's the idea of a
standing wave on a string, like we discussed back in Chapter 4.
If you wiggle a string, of course a traveling (transverse) wave will propagate. It will have a certain
SPEED which is given by the formula v= Tension
(mass per length) .
We haven't derived this formula (although it looks a little like the formula for the frequency of
oscillation of a mass on a spring!) Let's see if we can make sense of it. It says that if the tension gets
bigger, then waves travel faster. Seems reasonable - we saw that in class with a demo of rubber tubing,
and with a slinky. More tension!allows the wiggling material to "send the message" to the next piece
of material that much easier and faster. The signal travels faster! The fact that mass is in the
downstairs also seems reasonable - heavy materials will be sluggish, waves should travel more slowly.
The square root is NOT very obvious, but if you think carefully about the units, you'll see that you
need it. (I leave that as a puzzle for the ambitious - tension has units of force (which is mass *
acceleration. Can you see that the units really do work out, AFTER taking the square root, to be
meters/second?)
If the string is "clamped" at both ends (like a string on any musical instrument), then when you send
waves down, they will bounce off the end, come back, and superpose with the waves going the other
way. We saw (with simulations) that when you superpose two traveling waves going in opposite
directions, you get a standing wave. Why is it called "standing"? Well, it's not obvious moving left or
right any more. If you stare at it, you just see string bobbing up and down, there is no longer any sense
of "left or right" motion. You can UNDERSTAND it as the superposition of a left-going and a rightgoing wave, those back and forth waves add up to build a standing wave.
In the end, the result is just a standing wave!
In general, if you wiggle the string at some random frequency f, you will generate waves with a very
specific wavelength, given by my favorite wave formula, f*λ=v.
Here, "v" is the speed of waves on this string [v = square root of tension/(mass per length)]
It's a little funny, because we have standing waves that don't (apparently) move anywhere. So you
might think v should be zero, but no, you just use the "v" of the traveling waves that BUILD the
standing wave. This formula says the wavelength, λ, is going to be determined... if you wiggle at
some frequency f, then λ comes out how it comes out, given by v/f. (v is fixed, unless you change the
tension, or change the material itself, it's some particular, constant number. It is NOT, in general, 344
m/s! It's the speed of a wave along the string, NOT the speed of sound in air) Now, if λ is just some
random number, try to picture what this standing wave will look like. Waves are traveling back and
forth, back and forth - sometimes adding constructively, sometimes destructively. It won't look like
much of anything, just a random, chaotic wiggling of string. There is no "resonance", nothing special
happens. You wiggle, the string responds, no obvious patterns appear, no buildup of energy...
But what if you wiggle at JUST the right (magic, special!) frequency...
such that the WAVELENGTH that you create "fits" perfectly into the
length of the string. This is a very different story. Consider, e.g., a
SINGLE wavelength just "fitting" perfectly. That would look like this:
As this wave bounces, and reflects, you will get the same shape
superposing on top of itself over and over.
Phys 1240 Fa 05, SJP 6-2
Except, when you bounce off a rigid wall, a wave will "flip" upside
down. That means that the superposing waves (going right and left, back
and forth) sometimes adds up, but at other times completely cancel out
(but just for an instant!) The wave changes with time. In this next picture,
I show in dashes and dots the string at different times. In ONE period, the
standing wave smoothly goes from the solid, through the dash, to the
dots, and back again to where it started.
Try to picture it. It's a "standing wave". There are parts of the wave that NEVER move at all. (The two
ends at the walls, of course, because the string is attached there. But also in this picture there's another
spot, right at the center. It's called a NODE. Or, more specifically, an INTERIOR NODE. (to
distinguish it from the two ends).
There are also spots that wiggle the most. Those are called ANTI-NODES. In this figure, there are two
antinodes, one at the ¼ of the way across point, the other at the ¾ of the way across point.
This wave is "happy", if I may anthropomorphize. You must feed energy in to build it up, but the wave
just builds and builds on itself. It is a RESONANT wave, and can build up to a large amplitude. It has
a definite wavelength (in the case shown, λ is exactly L) and therefore ONE particular definite
frequency. If you change the wavelength a little, it won't "fit" on the string, and the reflected waves
don't "match", you get a jumble, not a resonance.
There is one longer wavelength that fits perfectly. This may be a surprise
(since the picture at the top of the page has ONE full wavelength, how can
you get fewer than that?) The answer is that waves pass through zero in
the middle, too, so you can actually fit a HALF wavelength on this same
string. Here's what the standing wave would look like:
Stare at the picture, convince yourself that this is a different wavelength
than the previous picture - it's exactly TWICE the wavelength above, in
fact! This wave would fit perfectly in a length 2L, in other words, this wave has λ = 2L. (Another way
to think of it is that exactly one "half wavelength" fits, or mathematically,(λ/2)= L)
This is the LONGEST POSSIBLE wavelength that fits in this length string. Anything longer will not
yet have reached zero at the side walls.
Since λ f = v, the largest wavelength means the smallest frequency.
We call this smallest frequency the FUNDAMENTAL FREQUENCY, or f1, of this string.
So the fundamental frequency, f1 = v/(2L).
If you shake a string at this frequency, you have a resonance. It absorbs lots of energy from the shake,
and builds up a nice big amplitude, like a kid on a swing being pushed at just the right frequency. The
fundamental wave looks like this picture: it bobs "up and down". Can you see that there is ONE
antinode (right in the middle). And there are no interior nodes. (Only nodes are at the walls, and I
generally don't count them because they're always there)
So f1 = v/(2L), and λ1 = 2L.
Phys 1240 Fa 05, SJP 6-3
Looking back at the picture on the top of the previous page, you can see that the NEXT SHORTER
allowed wavelength is L. We give this new wavlength a new name - since it's the next allowed wave,
we call it #2, or λ2 = L. Another way to think of it is that now two "half wavelengths" just fit.
What will the frequency be for this wave? Well, as always, f*λ=v, which means
So f2 = v/(L), and λ2 = L. Let me re-write this in a more suggestive way:
f2 = 2 v/(2L), and λ2 = (2L)/2
Why did I do that? Because I'm thinking about the pattern as we go up to smaller waves that will also
fit. E.g, the NEXT one that fits is one where THREE "half wavelengths"
just fit. Think about that for a second, look at the picture. Can you SEE
the three "half waves" there? Count 'em!
In English, I would say "3 half-wavlengths = L". In math, that says
3 (λ/2) = L, which means λ = (2L)/3.
And the corresponding frequency is f = v/λ = v/(2L/3) = 3 v / (2L)
You should check that last little step of algebra. It's important, and sets the stage for the pattern.
Remember, this last example is the "third pattern", the "third allowed wave on this string", also
referred to as the "third harmonic wave".
People call these different allowed patterns the harmonics of the string. They are also called "modes".
So the fundamental (which we indicate with a little "1" subscript) is also the FIRST harmonic, or the
FIRST mode. (It has ONE antinode, in the middle. Look back at the picture, bottom of previous page)
The next higher mode (with a little "2" subscript) is the SECOND harmonic.
It has TWO antinodes and ONE node.
The next higher one, the one we just did, has a "3" subscript. It's the THIRD harmonic.
Look at the picture again - it has (count them!) THREE antinodes and TWO nodes.
The pattern continues.
f1 = v/(2L), and λ1 = (2L)
f2 = 2 v/(2L), and λ2 = (2L)/2
f3 = 3 v/(2L), and λ3 = (2L)/3
f4 = 4 v/(2L), and λ4 = (2L)/4
If your eyes just glazed and you "skimmed" over those formulas 'cause they're mathy, go back and
stare. I need you to see the pattern. What is the 5th frequency? What's the fifth wavelength?
How about the 99th? If you see the pattern, you can just write it down!
Notice also that
f2 = 2 f1
f3 = 3 f1
f4 = 4 f1
etc. (Look! Convince yourself! Do you see this?)
So the "allowed" or special or harmonic or resonant (all synonyms!) frequencies (or modes, there are
lots of words we use to say the same thing, sorry!) are all just INTEGER multiples of the fundamental.
So if the fundamental vibrates at, say, 110 Hz, then you'll also have resonances of the SAME string at
220, 330, 440, 550, 660, (etc.) Hz. These are called the allowed "harmonics"
Phys 1240 Fa 05, SJP 6-4
Remember when we talked about octaves? Doubling the frequency is an octave.
That means that f2 is exactly one octave above f1. But f3 is NOT two octaves higher than f1(!!)
Let's think about that - if you go from 110 to 220 Hz, that's one octave. The next octave is DOUBLE
that, i.e. 440 Hz. That's f4. But wait, what about f3? There's an EXTRA resonant frequency in between,
f3 at 330 Hz, which is NOT a perfect octave above f1!
Indeed, going from 440 to 880 Hz (that's the NEXT octave higher), there are now THREE extra tones
which our string can play, in between (at 550, 660, and 770 Hz) which are not perfect octaves of f1.
If you look at the table in the back flyleaf of your book, you can identify which notes those
"harmonics" are. They sound very nice together - when you write music, it's great to know about
harmonics and make use of them! They "fit" nicely, and sound quite consonant together.
I picked f1=110 Hz totally arbitrarily. You can start with a string of any length, and adjust the tension
(or pick the mass) so that you can get any fundamental you want. But once you have that, then if you
don't make any changes in the nature of the string itself, you can ONLY get resonances at these
harmonics, i.e. at any integer times the fundamental.
What we've said so far is that, given a particular string, it will have a particular fundamental
frequency that it resonates at ("likes to vibrate at"), as well as an infinite number of others, but still a
very specific set, only and exactly those frequencies that are an integer (whole number) times greater
than the fundamental. If you excite the string at one of these harmonic frequencies, it "hums", and if
you try to excite it at any OTHER frequency, it "dies".
If you play a real string instrument, and pluck it, what will happen? In a sense, plucking is complicated
and you might think of this as "feeding energy in" at many different frequencies. But like any resonant
system, MOST of those frequencies die out quickly - only the particular (special) resonant frequencies
will "build up" and leave the string "humming". Generally, the lowest (fundamental) frequency is the
one you will hear loudest. You will excite the LOWEST modes more than the higher ones. But ALL
the higher frequencies, in principle, have been excited (at least a little) too. The one string is humming
at many different frequencies (all the harmonics) at the SAME TIME. Isn't that odd? You can hear it,
pluck a string and listen!
So if a string has a fundamental frequency f1=110 Hz, and you pluck it, you will hear mostly this tone
(called A2, which is 2 A's below middle C on the piano). But if you listen carefully, you will always
hear a little bit of f2=220 Hz (which is A3, just below middle C on the piano) and a little bit of f3=330
Hz (which is E4, the E just above middle C, and for those of you who are musicians, a perfect fifth
above A3!) and a little bit of the higher ones, generally fainter and fainter but all playing at the SAME
time. That's the difference between a real string, with this mix of harmonic frequencies, and our purely
electronic perfect frequency sin wave. The frequency generator makes a SINGLE frequency, which I
generally find sounds a little annoying. A violin or guitar playing what is (nominally) the exact same
note sounds more pleasant, precisely because it has this richer structure - there is MORE than one note
playing at the same time on the same string!
We'll come back to this idea in the next chapter - (which will be Chapter 8, we're going to skip around
a little now) - called "harmonic analysis" - we will analyze the harmonics that coexist in a given string.
Each instrument has its own characteristic "strengths" of the higher harmonics, giving rise to the
unique sound (timbre) of that instrument, even when it plays the same note as another instrument...
Phys 1240 Fa 05, SJP 6-5
Bottom line summary of harmonics: For a string with speed v, you have only special allowed modes
of vibration
1) The fundamental. The first harmonic. The first mode.
It has the longest wavelength possible: λ1 = 2L
It has the smallest frequency possible, f1 = v/ λ1 = v/(2L)
It has no internal nodes, and one antinode. (Just look at the picture!)
2) The second mode, the second harmonic. (Sometimes called the "first
overtone" by musicians, 'cause it's the first tone OVER the fundamental.)
It has a SHORTER wavelength, λ2 = (2L)/2.
(Note, it's half the wavelength of the fundamental)
It has a higher frequency, f2 = 2 f1
It has one node (in the middle) and two antinodes. (Look, convince yourself!)
3) The third mode, the third harmonic. (the "second overtone")
It has λ3 = (2L)/3
It has , f3 = 3 f1
It has two interior nodes, and three antinodes.
...
n) The n'th mode, the n'th harmonic. (the "n-1"th overtone)
It has λn = (2L)/n ,
fn = n f1
It has (n-1) interior nodes, and n antinodes.
(What is "n" for the figure shown?)