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
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String and Air Instruments Review – Standing waves in String Instruments Examples – String Instruments Longitudinal Waves in Air Standing Waves in Air.
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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
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.