Chapter 7

The Plucked String

Combinations of Modes

• when a system is struck and left to its own devices, any possible motion is made up of a collection of the natural frequencies.

Normal Modes of a Two Mass Chain

Two Mass Normal Modes

Mode 1

Mode 2

Normal Mode Notes

• If we start the system in a normal mode, it persists.

• Let us get to a new configuration in steps

New Starting ConfigurationMethod 1

Move to normal mode 1 first, followed by mode two.

New Starting ConfigurationMethod 2

Move to normal mode 2 first, followed by mode one.

Resulting Motion

Original shape not maintained

Conclusions

• If the initial shape agrees with a normal mode, the system will retain its shape.

• If the initial shape is not one of the normal modes, the system will not retain its shape.

• By using various amounts of the normal modes, we can construct any initial pattern we like.

The Plucked String

• Now the model is increased to a large number of balls (like in Chapter 6).

• The first three normal modes are…

Mode 1

Mode 2

Mode 3

Standing Waves

• When two sets of waves of equal amplitude and wavelength pass through each other in opposite directions, it is possible to create an interference pattern that looks like a wave that is “standing still.”– Waves on a violin string reflect off of the

bridge. The reflected wave acts as the second wave moving in the opposite direction from the original.

Standing WavesClick inside the box

Notes on Standing Waves

• There is no vibration at a node.

• There is maximum vibration at an antinode.

is twice the distance between successive nodes or successive antinodes.

Mode Excitation of a Plucked String

• The position at which the string is plucked determines which of the partials (normal modes or standing waves) will be excited.

Place the Plectrum Here

Notice that each mode has some amplitude at the dotted line

Mode 1

Mode 2

Mode 3

Change Plectrum Position

• Modes one and three are at maximum here• Mode two is at a zero.

o Can’t expect to excite a mode that has no amplitude

Notes

• Notice that modes one and three are symmetrical about the mid-point

• Mode two is antisymmetrical

• Plucking a string at the node of any mode will not excite that mode.

• Plucking a string at the antinode of a mode gives the strongest excitaiton.

Generalize• In general, the excitation of a mode is

proportional to the amplitude of the mode at the plucking point. Notice that the previous statements are included here.

Harmonic Amplitudesfor Plucked Strings

2n 1a (1/n ) a

Thus even if we pluck a string at the best possible point to excite mode 4, the initial amplitude will be 1/16 of a1

The amplitudes of the higher harmonics are related to the amplitude of the fundamental

Harmonic AmplitudesPlucked Strings

Harmonic Amplitude

1 a1

2 ¼ a1

3 a1

4 a1

5 a1

91

161

251

String Plucking “Recipe”

• Consider a string that is plucked ¼ of the way from one end

• The first four modes are shown next, where the amplitudes are the same for simplicity

First Four Normal Modes

a1

a2

a3

a4

with a little trig…

Amplitudes• a1 = 0.707

• a2 = 1.00

• a3 = 0.707

• a4 = 0.00

• First four modes set the pattern for the others• Every fourth mode has initial amplitude of zero• Every other even mode has amplitude one• All odd modes have amplitude 0.707

Amplitudes of First Eight Modes of a Plucked String (1/4 point)

Mode Number 1 2 3 4 5 6 7 8

Normalized Mode Amplitude (a)

0.707 1.000 0.707 0.000 0.707 1.000 0.707 0.000

Mode NumberSquared (b)

1 4 9 16 25 36 49 64

Initial Amplitude(a/b)

0.707 0.250 0.079 0.000 0.028 0.028 0.014 0.000

Normalized Amplitude(Initial Amplitude/0.707)

1.000 0.353 0.111 0.000 0.040 0.039 0.020 0.000

Notes

• The absence of multiples of 4 is the result of the plucking point. Had it been at the ⅓ position, then modes that are multiples of three are missing.

• To remove the nth mode and its multiples, pluck at the 1/nth position.

String½⅓¼

More Notes

• Plucking near one of these positions weakens the corresponding modes.

• High order modes are weak because of the 1/n2 dependence.

Striking a String

• Plucking gave a 1/n2 dependence of the amplitudes of the normal modes

• Striking gives a 1/n dependenceo The higher modes have more relative amplitude

Amplitudes of First Eight Modes of a Struck String (¼ point)

Mode Number 1 2 3 4 5 6 7 8

Normalized Mode Amplitude (a)

0.707 1.000 0.707 0.000 0.707 1.000 0.707 0.000

Mode Number (b)

1 2 3 4 5 6 7 8

Initial Amplitude(a/b)

0.707 0.500 0.236 0.000 0.141 0.167 0.101 0.000

Normalized Amplitude(Initial Amplitude/0.707)

1.000 0.707 0.333 0.000 0.200 0.236 0.143 0.000

Comparison of Amplitudes

Amplitudes of the Normal Modes

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Mode Number

Re

lati

ve

Am

plit

ud

e

Struck

Plucked

Tuning Forks

• Striking a tuning fork at a point ¼ to ½ of the way from the end excites only mode 1

• The end of the tuning fork is not fixed as in the examples aboveo This puts an antinode at that end

• For a tuning fork struck at the ¼ point, normal modes might look like…

Tuning Fork Normal Modes

FixedEnd

OpenEnd

Observations

• Mode 1 is near its max

• Mode 2 is close to a node

• Mode 3 is close to a node

• Mode 4 is close to a maxo Higher modes have rather low amplitude

(Mode 4 would have only 1/16th the initial amplitude of Mode 1).

Guitar Pick-up Locations

• Stimulated by large amplitudes of the string

• Modes that are amplified are those whose amplitudes are near maximum at the location of the pickup

• Place the pickup at the ¼ point and plucked the string at the center.

Normal modes of the Guitar String

Observations

• Only the odd modes are excited and they are at their antinode.

• Even modes are at their nodes

Amplitudes of First Eight Modesof a Plucked String (1/2 point)

Mode Number 1 2 3 4 5 6 7 8

Normalized Mode Amplitude (a)

1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00

Mode NumberSquared (b)

1 4 9 16 25 36 49 64

Initial Amplitude(a/b)

1.000 0.000 0.111 0.000 0.040 0.000 0.020 0.000

Normalized Amplitude

1.000 0.000 0.111 0.000 0.040 0.000 0.020 0.000

Now to the Pickup• We have to now modify the amplitude table

to get the amplitude that the excited modes have at the pickup position.

Normalized 1/4 Point Resultant Amplitude Amplitude

1.000 0.707 0.707

0.000 1.000 0.000

0.111 0.707 0.078

0.000 0.000 0.000

0.040 0.707 0.028

0.000 1.000 0.000

0.020 0.707 0.014

0.000 0.000 0.000

Moving the pickup

• We change the amplitude of the excited modes and therefore the mix of the normal modes.

• Also changing notes on the same string means using different frets, changing the length of the string

Comparing Plucking Positions

Amplitude Comparison

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9

Normal Mode

Am

plit

ud

e

1/4 Point

Mid-Point