Vibrating StringsA standing waves = A left-going wave + A right-going wavehttp://www.walter-fendt.de/ph14e/stwaverefl.htm

Consider a wave pulse sent down a string. (Demo)This traveling wave pulse

A] is a sum of harmonic standing waves, with appropriate phases

B] cannot be a sum of harmonic standing waves, since it moves and they all stay in the “same place”

Any free vibration of an object is a sum of natural modes. Answer A.

Traveling waves & Standing waves are simply two different, but equally correct, ways of describing the excitations of a string.

We can use whichever description works best for the question at hand.

Vibration Recipes for Plucked or Hammered Strings

Plucking at a node of a mode DOES NOT excite that mode.

This is the same result as hitting a percussion instrument exactly at a node.

Note that these are subtly different “initial conditions”… in one case (percussion), we impart a velocity to a part of the object. In the other case (plucking), we impart a displacement to a part of the object.

Mode 1

Mode 2

Mode 3

Mode 4

Plucking: Dirichlet Initial ConditionsPercussion: Neumann Initial Conditions

Vibration Recipes for Plucked or Hammered Strings

Although the vibration recipe IS the same, plucking DOES displace other parts of the string, far away from the plucking point.

So it’s not obvious why a pluck at a node can’t excite that mode.

But it can’t.

Mode 4

The reason for this has to do with how much two different functions (string shapes) “look like” each other… a concept that we can define precisely using calculus, following a recipe set down by Fourier. The shape of the plucked string shown doesn’t “look like” mode 4 AT ALL… mathematically we say these functions have “no overlap (integral)”

Another plucking result from Fourier analysis: the overall ‘envelope’ of the spectrum of modes drops 6 dB/octave (energy per mode) - dashed line

Mode n has a node at L/n from the end of the string… and another one twice as far, 3x, 4x, etc.

So if you pluck at L/5, you get no 5th, 10th, 15th… mode

Mode 1

Mode 2

Mode 3

Mode 4

Just because an object has a particular distribution of energy in its various modes, does NOT mean that the sound from it will have the same distribution of energy in the overtones.

We have to consider how the motion of the vibration is converted into sound.

Easiest case: electric guitar. “Pickups” measure the motion of the string where they are positioned.

Mode 1

Mode 2

Mode 3

Mode 4

Suppose you pluck a guitar string at L/4This will put energy into modes 1,2,3,5,6,7,9,10,11… etc. falling 6 dB per octave. Note that modes 4,8,12 etc are missing.

If your electric guitar has a pickup at L/2, what modes will you HEAR from the amp?

A] only odd modesB] only even modes C] only even modes except multiples of 4

This electric guitar was plucked at L/4. Where is the pickup?

A] L/2 B] L/8 C] L/10 D] L/12

We’ll talk more about how acoustic stringed instruments convert string vibration into sound next week.

1) Velocity of hammer >> velocity of key

2) Hammer is free of key when striking

3) Damper is held off string when key pressed

Most piano notes are triple strung. That makes them louder, of course, but also has other interesting consequences!

Vibration Recipes for Piano

A high frequency natural mode of a piano string is shown. The string is struck with a soft hammer, shown as the blue oval. When the hammer hits the string, it compresses to the extent shown.

Can this hammer excite this mode efficiently?

A] yes, if it is not centered on a nodeB] no, no matter where it is

No. Striking over an area produces the same effect as striking at each point and adding… but in this mode some places struck would be moving up while other places would be moving down.

Vibration Recipes for Piano

Suppose we use a hammer with a hard, narrow inner core, shown in dark blue.

Can this hammer excite this mode efficiently? Assume it is not centered on a node.

A] yes, no matter how hard it hitsB] yes, if it hits hard enough that the

hard part hits the string B] no

Vibration Recipes for Piano

Piano hammers are harder in the middle. So the timbre changes when you hit the keys harder… more higher harmonics… brighter sound.

You can also get a very bright sound by putting tacks in the hammers! (demo?)

Vibration Recipes for Piano

Striking at a node discriminates against BUT DOES NOT ELIMINATE vibration of that mode.

This is different from plucked strings. The initial condition of the string (after the hammer has struck) includes both displacement and velocity…

Another viewpoint: to some extent, the hammer creates a traveling wave. The traveling wave travels to places where the mode of interest does NOT have a node.

Vibration Recipes for Piano

Not only do we need to worry about whether the hammer is too big to excite a high order mode, we also need to worry about whether it is pressing on the string too long to excite a high order mode.

Recall: if the pressing time > T/2, the mode won’t be excited.

Mostly for this reason, high order modes are suppressed more in piano than in guitaror harpsichord (6db/octave).

What mode (harmonic) of C2 is the same frequency as C6 ?

Four octaves

= 2 x 2 x 2 x 2 = 16

The sustain & decay of piano notes

Rapid initial decay in intensity, but followed by a long sustain.

http://www.youtube.com/watch?v=P-Q9D4dcYng 4:20

This is surprising. Most decays are “simple exponentials”, with half the amplitude lost in t1/2 (though, of course, different overtones decay with different t1/2)

Here, the same tone changes its decay rate!!

The properties of the piano note decay arise from the fact that (for most notes) there are 3 strings.

We need to understand in detail why this is so.

Recall that work = energy flow, and work = force x distance.

When the hammer hits 3 strings, they all begin to vibrate together. They all push on the bridge on the soundboard. With 3 strings pushing on the bridge, it wiggles (roughly) 3 times as far as it would for one string.

Each string is now exerting its FORCE through 3x distance. So the rate of energy flow out of the string and into the bridge (and soundboard) is 3x larger PER STRING.

(Since there are 3 strings, the total rate of energy flow (power) is 9 times that of a single string.)

Three strings vibrating in phase lose energy much faster than a

Single string.

So: piano notes have a rapid INITIAL decay

The 3 strings are never perfectly “in tune”. For example, the A440 might have strings at 439.8 Hz, 440 Hz, 440.1 Hz

(Recall a JND is at least half a Hz)

After a few seconds, the strings will NOT be vibrating IN PHASE.

Recall what you know about SILs & interference… when sounds are IN PHASE (constructive interference), the amplitudes ADD. When sounds are randomly phased, the intensities ADD.

Intensity = Energy per unit area per unit time.

We are interested in how much energy per unit time is delivered to the bridge. Just as with sound intensities, if the sounds are randomly phased, the average energy per unit time is just the sum of the energy per unit time for each string.

So: when the 3 strings are out of phase, they will lose energy at 3 times the rate of a single string.

Of course, they have 3 times the energy of a single string, so they lose the same fraction of energy per unit time as a single string.

“Polarization”

Another cause of long sustain is the polarization of the string wave.

If the string is vibrating in a vertical plane, it transfers its energy quickly to the bridge & soundboard, giving a rapid decay.

If the string is vibrating in the horizontal plane, it only couples weakly. It is a quieter, but LONGER LASTING note.

In any real piano, the hammer causes some of both horizontal and vertical string vibrations.

Una Corda (“Quiet”) Pedal - Literally “One String” (but actually 2 of the 3)

Because the strings can push on the bridge, and the bridge can push on other strings, the strings are vibrationally coupled (like with the coupled pendulums.) The coupling is very weak, so it doesn’t shift the frequencies perceptibly.

Just as with the coupled pendulums, there are natural modes of the 3 strings. Here are the 3 natural modes corresponding to the lowest longitudinal mode of 3 strings. (nb book shows some modes, but they are not the natural modes, as you can tell by analogy with the pendulums.)

Just as the horizontal “polarization” of a piano string vibration does not efficiently transfer energy to the bridge, a motion of the 3 strings that tends to rock the bridge left to right will not transfer energy well. Only motion where all 3 strings pull UP & DOWN on the bridge together will transfer energy well.

Which natural mode below will transfer energy to the bridge best?

A B C

Which natural mode below will give the loudest sound on a piano (if the amplitudes are the same) ?

Mode A Mode B Mode C

Which natural mode below will give the shortest-lasting note (with the dampers off)?

Mode A Mode B Mode C

In normal operation, the hammer is designed to hit all three strings, exciting mode A.

With the una corda pedal pressed, the hammer hits only two strings.

Mode A Mode B Mode C

Since any motion is a sum of natural modes, the motion of the two right strings must be a sum of the motions shown here. It is!

KEY POINTS: 1. Much of the energy is in modes B & C, which are quiet but long-lastingThis is very different from hitting all three strings, which puts all the energy in mode A.

2. Do not think that the left string NEVER moves. These modes have slightly different frequencies, and so (just as with coupled pendulums) the energy will shift to the left string after a while. Mode A Mode B Mode C

Bottom line, musically: with the “soft pedal” pressed, the sound is not just quieter… it is smoother, more sustained, and less percussive.

Inharmonicity

We have ignored the bending stiffness of the strings.The pitches of the harmonics are affected… especially for THICK strings.

Why is this a problem? Let’s consider:If we keep the tension and the string density and diameter the same, suppose the A3 (220 Hz) is a meter long. How long does the A0 string have to be? (Three octaves down in pitch)?

A] 3 metersB] 4 metersC] 8 meters

Well, we can’t have an 8 m piano. So we can lower the pitch by using a heavier string(increasing )

(Decreasing tension too much gives a floppy string thatdoesn’t have enough energyto drive the soundboard.)

But using a heavier, thicker string usually increases the stiffness and inharmonicity.

Solution: wrap the strings. Adds mass to slow the vibration, but doesn’t change bending stiffness much.

€

fn =n

2L

T

μ

The Bowed String - Violins (& Guitars if named Jimmy)

The violin bow is made sticky with rosin. As it is pulled across a string, it sticks and slips,

repeatedly.DemoThere are two remarkable features of this process:

1. The slipping is synchronized with the fundamental frequency of the string!

(For example, if a violinist bows an A 440, the bow slips exactly 440 times per second!)

2. The bowing of a string excites ALL harmonics, INCLUDING THOSE WHICH HAVE NODES AT THE BOWING POINT!

(This means that bowing is somehow different from repeated plucking at the bowing point…)

We’ll address bowing in more detail in the next lecture.

•When the string slips off the bow, it overshoots (inertia!)

•The force on the bridge tracks the motion of the string: when the string is above the rest position, it pulls the bridge upward in the diagram, when below, it pulls the bridge downward.

This is a rocking motion.

T = period of string fundamental

The spectrum of this motion is shown below. Note the absence of any missing harmonics! All harmonics are present, even those with nodes at the bowing point!

T = period of string fundamental

Rocking the bridge vibrates the violin. The vibrating violin (not the string) moves the air.

End view Side view

Baroque violins used gut strings with less tension (modern use mostly steel at higher tension); A was tuned to < 420 Hz, rather than 440 Hz;Bridges were lower (so there was less down-bearing).

So modern violins are louder and more “brilliant”

The bowed violin string is not the source of sound from a violin.Rather, it is an oscillator that drives a resonant object (the body):It is the body that makes sound.

That’s why a Stradivarius ≠ “Crescent Brand Student Violin”

Resonance

When an object is driven at the frequency of one of itsnatural modes, it oscillates strongly (i.e. with a large amplitude).

Let’s do some experiments with the resonant mass & spring.

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Red line - positionBlue line - force

If the driving force is well below the natural mode frequency, what graph best shows the relationship of force (in blue) to position (in red)?

A

B

C

Answer A. The mass just tracks the applied force. This motion is “stiffness” (or compliance) - limited

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Red line - positionBlue line - force

If the driving force is well above the natural mode frequency, what graph best shows the relationship of force (in blue) to position (in red)?

A

B

C

Answer C. The inertia of the mass prevents it from following the driving force. The amplitude is inertia-limited

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Red line - positionBlue line - force

If the driving force is at the resonant frequency, what curve shows the relationship of force to position?

A

B

C

Answer B. The mass is lagging behind the driving force by a quarter of a cycle.

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Red line - positionBlue line - force

Between the pink arrows, the force is pulling up (it’s positive) on the mass.

During this time

A] the mass moves in the direction of the applied force

B] the mass moves opposite to the direction of the applied force

C] the mass has no net motion

B - at resonance

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Red line - positionBlue line - force

Between the pink arrows, the force is pulling up

and the mass is moving up.

So the force does work on the mass.

Work (force x distance) is a transfer of energy

Energy is transferred to the mass.

B - at resonance

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Red line - positionBlue line - force

In the other half cycle, the force is down and the motion is down. So the force is still doing work on the mass, adding energy.

So what limits the amplitude of the motion? If we keep adding energy, the amplitude should grow forever, OR THE ENERGY MUST LEAVE SOMEHOW

B - at resonance

The added energy is lost to friction or radiation. This motion is DISSIPATION-LIMITED

SUMMARY SO FAR:

“Resonance” happens when we drive a system at one of its natural mode frequencies.

When driving at that frequency, we transfer energy into the system at a high rate.

The system will oscillate strongly, with an amplitude that is only limited by losses due to friction or radiation.

1.0

0.5

0.0

-0.5

-1.0

8006004002000time

1.0

0.5

0.0

-0.5

-1.0

8006004002000

1.0

0.5

0.0

-0.5

-1.0

8006004002000

Driving below resonance frequency

Driving above resonance frequency

What happens when we are far off the resonance frequency?

In the half cycle where the force is positive, the mass has no net motion (it ends up exactly where it started), so there is NO TRANSFER OF ENERGY!

If we increase the damping (or dissipation) in the system, the amplitude of all motion is reduced.

HOWEVER, the resonant amplitude is reduced most. Recall that we said the resonant amplitude was DISSIPATION-LIMITED.

The overall effect is to give a much broader, but weaker response.

Am

plit

ude o

f D

riven M

oti

on

frequency

Note that adding friction doesn’t affect the amplitude here much. That’s because here we are compliance-limited

Note that adding friction doesn’t affect the amplitude here much. That’s because here we are inertia-limited

Let’s experiment with using a tuning fork to drive different “systems”

Metal Gong - Weak dissipation, sharp resonance

Boomwacker tube - Resonance of the air column!

Wood frame - Strong dissipation (tone doesn’t last) … Broad resonance (all forks work equally)

You may have noticed that the frequency the fork sounds with the boomwacker tube is an octave below its sound with the wood frame.

The motion of the base of the fork is an octave higher than the prongs!

Back to violins & other acoustic strings…

The violin body (and the air inside) has many resonant natural modes. We could plot the amplitude of motion for each input frequency, but that’s very hard to do (and not what

we really want, anyway…) Instead, we can plot the sound output (in dB) as a function of the frequency of a standard sinusoidal input at the bridge.

Mass & Spring

Violin

Recall that we can also find natural modes by hitting the violin. With a priceless Stradivarius, we use an IMPACT HAMMER

Violin

Note that this is the acoustic output for a given energy input.

If we used 10x lower energy input, the output would be 10 dB lower.

The shape of the curve would be exactly the same!

The combination of many resonances, some sharp and some broad, gives the overall response function.

Violin

What the bowed string provides as input(energies)

Each input must be multiplied by the response function at that frequency, to get the final sound spectrum

A B

If Output = A x B

Then log (Output) = log(A) + log(B)

We need to add to each line in A the length indicated in B

A B

Two sounds have the spectra shown.

What is true?

A] Sound A has the same timbre as B, but is louderB] Sound A has the same timbre as B, but is quieterC] Sounds A & B have different timbres

Ans. A Every harmonic is down by the same factor, so the waveform is the same, just smaller

log(Input energy from bow)

+

log(Response)

=

log(Sound Output)

You should appreciate that we can move any of these curves UP or DOWN… that won’t change the timbre of the note!

Any shift just changes the loudness of the final note.

By the way… the complexity of the response function means that, during vibrato, the timbre changes as the pitch changes!

It’s hard to do experiments with impact hammers. Sometimes we just play the violin as loudly as possible on every note and measure the output.

Open circle = air resonance

Higher frequency filled circle = wood resonance

Lower frequency filled circle = “wood prime”, not a genuine resonance, but the tendency of notes an octave below the wood resonance to be loud.