String and Air Instruments

• Review – Standing waves in String Instruments

• Examples – String Instruments

• Longitudinal Waves in Air

• Standing Waves in Air Instruments (open-open)

• Standing Waves in Air Instruments (open-closed)

• Summary Air Instruments (open-open, open-closed)

• Examples – String and Air Instruments

Standing waves in String Instruments• String anchored between 2 points and velocity fixed

• Allowed opening widths

• In general

• Allowed wavelengths– ,2,3….

• Allowed frequencies Velocity is–

Example 12-7

• Frequency of highest note

• Frequency of lowest note

• Ratio

Example 12-8

• Allowed wavelengths in string,2,3….

•Frequency in AirSame as string v = 440 Hz.

• Wavelength in AirDifferent because of different velocity

Longitudinal Waves in Air• Traveling sound wave

https://sites.google.com/site/physicsflash/home/sound

• Pressure and Displacement Nodes/Antinodes

sound.swf

Standing waves in Air – open/open end (1)• Display as transverse wave (easier to see)• Pressure wave node at both ends

• Result:– Pressure wave node at both ends

– Pipe length must be some multiple of ½ wavelength!

Standing waves in Air – open/open end (2)

• Animation – Pressure wave node at both endshttp://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta2fix.html

• Result:– Pressure wave node at both ends

– Pipe length is some multiple of ½ wavelength!

sta2fix.swf

Standing waves in Air – open/open end (3)• Allowed widths

• In general

• Allowed wavelengths– ,2,3….

• Allowed frequencies Velocity is–

Standing waves in Air – open/closed end (1)• Display as transverse wave (easier to see)• Pressure wave node at one end, antinode at other

• Result– Pressure wave node at one end, antinode at other

– Pipe length is some odd multiple of ¼ wavelength

Standing waves in Air – open/closed end (2)

• Animation – Pressure wave node at end, antinode at otherhttp://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta1fix.html

• Result– Pressure wave node at end, antinode at other

– Pipe length is some odd multiple of ¼ wavelength

sta1fix.swf

Standing waves in Air – open/closed end (3)• Allowed widths

• In general

• Allowed wavelengths– ,3,5….

• Allowed frequencies Velocity is–

Comparison of waves on string and air

• Both have– Wavelength – distance between peaks at fixed time– Frequency – rate of repetitions at fixed position (like your ear)– Wave velocity

• Differences– String wave velocity varies with tension and mass/length

– String has ½- wavelength harmonics

– Air wave velocity set at 343 m/s (at 20° C)

– Air has ½- or ¼- wavelength harmonics

Examples of String and Air Instruments

• String Instruments– Guitar– Violin– Piano

• Air Instruments– Flute– “Trombone”– Soda bottle

Examples• Examples– Problem 25 – Open & closed, 1st 3 harmonics– Problem 26 – Coke bottle– Problem 27 – Range of human hearing, pipe lengths– Problem 28 – Guitar sounds with fret– Problem 29 – Guitar sounds with fret– Problem 30 – Length of organ pipe– Problem 32 – Flute– Problem 34 – Pipe multiple harmonics

Problem 25 – Organ Pipe

• Open at both ends

Closed at one end

<<skip even harmonics

Problem 26 – Coke bottle

• Open/closed fundamental

•Closed 1/3 way up

Problem 27 – Full-range Pipe Organ

• Open/open fundamental

• Lowest frequency

• Highest frequency

Problem 28 – Guitar

• Original frequency of 3rd harmonic (on string)

• Fingered frequency of fundamental

• Ratio

Problem 29 – Guitar (1)

• Unfingered frequency of fundamental (on string)

• Fingered frequency of fundamental

• Ratio

Problem 29 – Guitar (2)

• wavelength of 440 Hz fundamental in string

• frequency in air

440 Hz

• Wavelength in air different because of air

Problem 30 – Organ Pipe

• Corrected velocity to 21°C

• Allowed frequencies

• Length is

• Wavelength

same inside and outside tube

Problem 32 - Flute

• Flute open at both ends (open-open)

• Allowed frequencies

• Length is

Problem 34 – Assume open-open <?>

• Write n and (n+1) harmonics in terms of fundamental

• Subtract

• So the difference of any 2 harmonics should be the fundamental.

????!!

Problem 34 – Assume open-closed <?>

• Write n and (n+2) odd harmonics in terms of fundamental

• Subtract

• So the difference of any 2 harmonics should be twice fundamental.

success!!