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11. String Telephone

Matthias Mller

German Team

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Task

How do the intensity of sound, transmitted along

a string telephone, and the quality ofcommunication between the transmitter andreceiver depend upon the distance, tension in theline and other parameters? Design an optimalsystem.

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Overview

Eperimental Set-up

Theory Quality and Intensity Losses in

Different Parts of the Telephone Transmission into String and Out of it Damping and Dispersion in String

Parametric Optimiziation Materials,Dimensions

Experimental Confirmation

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Properties of Optimal System

High quality of communication low dispersion

low frequency-dependence of transmittedintensity (no resonance in speakingrange)

Only of secondary importance: transmitted power as high as possible

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Shape of Horns

Horns with no resonance within range

Exponential horns (no reflection noresonance)

Short cans Eigenfrequency above 1000 Hz for length

smaller 5 cm

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Experimental Set-up

Lx

can

membrane

string canItrans Irec

transmitter receiver

3 important parts:

Transmission from can into string

Damping and dispersion in the string

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Experimental Set-up

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Intensity of Sound Wave

P = 12cAv2

Abbreviation: Z c, Z ZA

Power: P = 12Zv2

Intensity: I= PA

= 12Zv2

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Transmission through Membrane

Transmission of wave from one media throughmembrane into other media.

Ptrans = mb(Pinc Prefl)

vtrans = vmb = vinc vrefl

Remember P = 12Zv2

Therefore

PtransPinc

= 42mbZtransZinc

(Ztrans + mbZinc)

2

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Oscillation of Membrane

transmitter-Membrane

Equation of motion for small piece of membrane:

2z

t2+ Za

z

t+ Tz+ ekr

2b

(F0 + Dz) = p(t)

r Radial Coordinate of membranez(r,,t) Elongation

T Tension of membrane

F0 + Dz Force excerted by string

p(t) Incoming wave

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Identities of Good Membrane

Good quality for high resonance frequencies

High frequencies for low mass, high tension,

stiffness Membrane should be stiff

Tension in membrane increases with tensionin thread

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Optimization of Membrane

Input: White Noise; tension: 30 N, 40 N, length:8.5 m

200 400 600 800 1000

high tension

low tension

f [Hz]

relative intensity

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Noise without Rush

200 400 600 800 1000

f [Hz]

relative intensity

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Good Membran

Input: White Noise; tension: 40 N, length: 4.5 m

200 400 600 800 1000

f [Hz]

relative intensity

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Minimizing of String-Damping

High tension, thin string to eliminate bends instring

Intensity of damped wave:

I= I0 e2L

= 0 calculated by thermal losses

rubber aluminium iron titanium

0 [106s/m] 110 11 8.2 5.4

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Dispersion in String

Dispersion if phase velocity c depends onfrequency

Occurs due to damping:

pressure wave: p = pe(tx)

c = |()| = |+ 0|

Low damping low dispersion

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Measurement of Dispersion

200 400 600 800 1000

0.5

1

1.5

2

2.5

3

f [Hz]

travelling time [ms]

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Overall Transmission

Irec Ptrans 1

Acyl

4mbAmbZcyl

AstrZstr

2

e20L

Acyl should be small Amb should be large

problem as Amb

at end of can with Acyl

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Pressure Chamber

Acyl Amb

Effect of pressure chamber:

Larger Amb better transmission from can to string

Lower Acyl

intensity I=PA highest at opening

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New Overall Transmission

Including reflection can pressure chambern Amb

Acyl

Irec Ptrans Amb16mbZcyl

AstrZstr2 n3

(n + 1)4 e20L

High receiver-intensity for:

Large crossectional area of membrane

Maximum for n = 3, so Acyl =13Amb

Small wave impedance AstrZstr of the string

Low damping in the string

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Optimal System

System with pressure chamber diameter of membrane e.g. 15cm

diameter of can about 8.7 cm Membrane of stiff material, high tension in it

String of suitable metal (aluminium,titanium,. . . )

Thin string with high tension (for no bends)

Damping and dispersion increases withlength

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