Java Applet for Plucking stringshttp://www.falstad.com/loadedstring/

Use log scaling for harmonics, right + left display

You can see the missing harmonics when you pluck.You can see the sum of the left-moving and right-moving waveYou can see how plucking near the end adds high harmonicsYou can see the sum of the natural modes

A response function for a really bad violin is shown.If the A440 is bowed, How loud (SIL) is the 2nd harmonic, compared with the fundamental?Recall that bowing puts energy in the string natural modes, FALLING 6 dB per octave.

A] the 2nd harmonic is 2 dB higher SILB] the 2nd harmonic is 8 db higherC] the 2nd harmonic is 50 dB higherD] the 2nd harmonic is 58 dB higherE] the 2nd harmonic is absent

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

repeatedly.

There 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…)

The string (or block, in the demo) starts out stationary. When the bow (or desk) applies enough force, the block slips.

The key point is this: when it starts to slide, the friction force goes down, so it slides even faster.

It slides until it is no longer moving with respect to the bow (or desk).

Both of these remarkable facts arise from the physics of bowing

First: sticking & slipping.When the bow is moving fast (relative to the string) the friction is low.

When it is stopped or moving very slowly (relative to the string) the friction is high.

The physics of bowing, continued…

When the stretched string slips, a traveling wave is launched. (OK, yes, it’s also a sum of standing waves, but for now let’s use the yang rather than the yin…)

In fact, traveling waves are launched in both directions. My cartoon (on chalkboard for reference):

Note: the actual waves are different. But the ideas and the result we will get from my cartoons are correct.

Bridge Nut

What happens when the launched traveling waves reflect off the fixed ends of the string? Let’s try it!

A downward pulse on a string reflecting from a fixed end

A] reflects as an upward pulseB] reflects as a downward pulseC] doesn’t reflect at all

At fixed ends, pulses invert, up for down.Here’s what the string looks like after the leftward moving pulse has reflected off the “bridge”.

Now, of course, it’s moving to the right. Here’s the hardest part of our analysis. Consider what happens to a part of the string as the wave passes by.

In particular, what happens to the part of the string touching the bow as the tail end of the wave passes by from left to right?

Bridge Nut

In particular, what happens to the part of the string touching the bow as the tail end of the wave passes by from left to right? (Ignore the bow)

A] It jumps downward (in the direction of the bow motion)B] It jumps upward (opposite to the bow motion)

Bridge Nut

Answer: it jumps downward, in the direction of the bow motion.

The motion of the string downward slows (or stops) its relative motion with respect to the bow, which helps it STICK.

So: the pulse wave reflected ONCE

(off the bridge, or off the nut)

helps to bow to stick to the string again. It will actually stick when the first wave arrives, i.e. the one off the bridge.

When the leftward moving upward pulse hits the nut, it will reflect as a downward pulse.

Both pulses will reach the bowing point (for the second time) at the same time, one having traveled “clockwise” and the other “counterclockwise”

As the tails of these pulses move past, the wave motion of the string is upward

Bridge Nut

The upward string motion encourages another SLIP.

Time between slips = 2L/v.

The frequency of slips = v/(2L) = fundamental frequency of the string!

Any disturbance on a string will “recreate” itself after a time T. The disturbance of slipping recreates itself, and thereby synchronizes slipping with the string fundamental vibration.

The bowed string includes all harmonics, including those with nodes at the bowing point.

The actual string motion is not what I showed in my cartoons.. In fact, the waves act to create a kink that moves around the string.

Because this is a traveling kink, the bowing point is not special.

Timber depends only slightly on bowing point, but strongly on plucking point. http://www.oberlin.edu/faculty/brichard/Apples/StringsPage.html

Move bow toward center, watch force!

When a string is bowed, the place on the string that gets moved the farthest from its rest position is:

A] the place where the bow pulls on the string

B] the center of the string

C] all places (except the ends) get equal displacement, but at different times.

Answer B

For plucking, the harmonics drop about 6 dB/octave. Bowing is a little like plucking everywhere (the kink moves and is anywhere you like at some time), so the energy in the harmonics also falls at 6 dB/octave.

But again… NO MISSING HARMONICS

Acoustic Standing Waves in PipesSonic tube o’ fire.

Open Ends must have P = Patm

Closed Ends must have y = 0 (no displacement)

This is a little like the boundary conditions for a wave on a string (the ends can’t move), but we have two different definitions for “can’t move”… no motion of air, or no “motion” (change) of pressure.

http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-

waves.html

Let’s sketch the displacement and pressure for an open tube. What are the wavelengths of the modes?

Let’s use f = v to find the frequencies.

The frequencies of the natural modes of the air vibrations in an open tube give the harmonic series.

The frequencies of the natural modes of the vibration of the tube body do NOT

give a harmonic series (as we saw when we studied percussion.)

Let’s sketch the displacement and pressure for a tube closed at one end. What are the wavelengths of the modes?

Let’s use f = v to find the frequencies.

The airflow actually extends a little past the end of the pipe.

Essentially, the sound in the pipe diffracts out the end of the tube. For low frequencies, replace L by L + 0.3d, where d = diameter.

For high frequencies (d), there is so little diffraction that the sound doesn’t even “know” it’s at the end of the tube. These frequencies don’t resonate.

For an open pipe, 2L/n, so there are no harmonics for n > L/d.

NEXT THURSDAY APRIL 14We will meet at the Cathedral of St. John in Downtown ABQFor a tour & demonstration of the largest pipe organ in NMhttp://www.stjohnsabq.org/organ.php318 Silver Ave. SW

They will serve us lunch! I need to know if you are coming!Show of hands….

Which could be displacement for a standing wave in an open-ended tube (or choose D, none)?

Ans. A What harmonic is this?

a) 2ndb) 3rdc) 4thd) 6th

Which could be displacement for a standing wave in an tube with one closed end (or choose D, none)?

CThis tube is closed at the left end.

Which could be pressure for a standing wave in an tube with one closed end (or choose D, none)?

C(of course, this tube is closed at the right end!)

Displacement (+ = right) What harmonic is this?

a) 1st (fundamental)b) 2ndc) 3rdd) Its up a fifth from

the fundamental, so not harmonic

If A is the displacement, what curve shows the pressure in pipe at this instant in time?

Displacement + = rightward, - = leftward

Since displacement nodes are pressure antinodes, the answer has to be C or D.

At the point circled, the molecules to BOTH the left & right are displaced closer. So this is a pressure peak! C

Displacement + = rightward, - = leftward

Part of a LASER is a resonant cavity for light, which is a wave having both electric and magnetic fields.

The shapes of the electric and magnetic fields in the laser are exact parallels of the the pressure and displacement in acoustics.

Magnetic Field

Electric Field

Why are we so sensitive to 3-4 kHz sounds?

The auditory canal is a closed pipe with a length of ~2.5 cm.

Fundamental resonance has = 4L = 10 cm.

Fundamental frequency, using f = v = 344 m/s, is 3.4 kHz.

We saw that with violin bowing, stick-slip synchronized with the string fundamental frequency.

This happened because of a feedback mechanism. The “slip” wave recreated itself after time T, encouraging another slip.

With fluid flow over edges, we get another feedback mechanism (recall the Tacoma Narrows bridge!) This leads to a regular oscillation called an EDGE TONE.

Demo with paper

There are many other interesting flow instability phenomena

Singing of power linesFlapping of flagsBlowing of noses (demo!)Wind driven water waves(& Perhaps) Pangea, Gondwana, Rodinia, Kenorland, Nuna, Vaalbara

How long does it take the air to move around this (very approximate) path? That time should be half the period of the oscillation.

€

d = b+ πb /2 = 2.57b

T /2 = d /v j = 2.57b /v j ≈ b /(0.4v j )

So the period = T = b/(0.2vj)

and the frequency fI = 1/T= 0.2vj/b

Jet of 5 m/s with an edge 1 mm away gives 1000 Hz.

So: airstream oscillations at this frequency get reinforced by the “feedback” after 1/2 oscillation.

But if there were 1 1/2 oscillations in the time it takes the disturbance to return to the nozzle, that would also be reinforced. After 1 1/2 oscillations, the stream should also flow in the opposite direction.

If there were 2 1/2 oscillations, that would also be reinforced.

The edge can give oscillations at fII = 3fI, fIII = 5fI

What frequency is produced depends on how fast the air moves and how big the gap b is.

In fact, we don’t use NAKED edgetones to make music at all.

When the edge tone oscillation sends a puff of air into the pipe, that high pressure wave travels to the end of the pipe and reflects.

(Remember, waves reflect whenever they encounter a difference in their propagation properties.)

With a violin string, we saw (experimentally) that the reflection inverts. An upward pulse is reflected downward.

How do pressure pulses reflect off open and closed pipe ends?

Rather than answer experimentally, let’s think about how

the reflection guarantees the boundary conditions

(In fact, a mathematician would say that’s WHY there is a reflection!)

In the following simulation, note how the total displacement of the string (the forward-moving red wave plus the reflected blue wave) is zero at the end.

http://www.walter-fendt.de/ph14e/stwaverefl.htm

What are the boundary conditions for the closed and open pipes?

At an open end of a pipe, the boundary condition is

A] the pressure must not deviate (much) from atmospheric pressure

B] the displacement must be zero

Answer A. The pressure must be nearly atmospheric.As a consequence, pressure waves reflected from an open end:

A] are inverted, low pressure for high pressureB] are uninverted

Answer A. They are inverted, so that the SUM of the incident and reflected waves give no pressure change.

At a closed end of a pipe, the boundary condition is

A] the pressure must not deviate (much) from atmospheric pressure

B] the displacement must be zero

Answer B. Molecules can’t move through a wall. As a consequence, pressure waves hitting a hard wall

A] are inverted, low pressure for high pressureB] are uninverted

Answer B. They are uninverted, giving a pressure antinode (and therefore a displacement NODE) at the wall

In a closed-end organ pipe, when the pressure wave reflection reaches the mouth, it changes the airstream from flowing IN to flowing OUTSIDE the pipe.

That’s a half cycle. So half the period of oscillation is

And the frequency is

Which is just the first natural mode for a half-closed pipe!

€

T

2=2L

v

€

f =v

4L

In an open-end organ pipe, when the pressure wave reflection reaches the mouth, it sucks in MORE. (Remember, it’s INVERTED, and is now a low pressure wave.)

That’s a full cycle. So the period of oscillation is

And the frequency is

Which is just the first natural mode for an open pipe!

€

T =2L

v

€

f =v

2L

The traveling wave picture shows how the pipe provides feedback to the edgetone.

(Just as the traveling wave picture on a bowed string showed how the string provided feedback on the bow slipping.)

Naked Edgetone

“Overblowing”

Other frequencies (overtones) will also give positive feedback. The timbre of the organ (or flute) will include these harmonics.

Recall also: wavelengths Shorter than 2d don’t resonate.

Which spectrum is from a very thin organ pipe(compared with its length)?

Answer C. Which spectrum is from a pipe with a closed end?

Recorders

• At the open end of a resonant tube, the pressure is (nearly) atmospheric.

• Uncovering a small hole does not quite force the pressure at the hole to be atmospheric (not enough air can move through the hole)

But it does reduce the pressure wave there.

Uncovering a small hole higher up has the same effect as uncovering a larger hole farther down.

Demo pitch sliding

Because holes don’t create an absolute NODE in the pressure pattern, we can develop “cross fingerings” or “forked fingerings” to get different notes.

Consider D vs C#. How does closing holes 4-5-6 lower the pitch when hole 3 is open?

In blown pipes, you can get higher notes by exciting higher modes, through overblowing.

(Higher v, or smaller b)

The Transverse Flute

Large tone holes, with a mechanism for closing

Cross fingerings won’t work. Lower holes don’t matter.

Embouchure is adjustable. So the player can hit higher modes by changing the edge tone!

To make a higher frequency edge tone, should the player move his lips closer to edge, or farther?

A] closer B] farther

Closer to edge

Farther from edge