String and Air Instruments

1. 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

2. 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

3. Example 12-7 • Frequency of highest note • Frequency of lowest note • Ratio

4. Example 12-8 • Allowed wavelengths in string ,2,3…. • Frequency in Air Same as string v = 440 Hz. • Wavelength in Air Different because of different velocity

5. Longitudinal Waves in Air • Traveling sound wave https://sites.google.com/site/physicsflash/home/sound • Pressure and Displacement Nodes/Antinodes

6. 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!

7. Standing waves in Air – open/open end (2) • Animation – Pressure wave node at both ends http://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!

8. Standing waves in Air – open/open end (3) • Allowed widths • In general • Allowed wavelengths • ,2,3…. • Allowed frequencies Velocity is

9. 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

10. Standing waves in Air – open/closed end (2) • Animation – Pressure wave node at end, antinode at other http://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

11. Standing waves in Air – open/closed end (3) • Allowed widths • In general • Allowed wavelengths • ,3,5…. • Allowed frequencies Velocity is

12. 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

13. Examples of String and Air Instruments • String Instruments • Guitar • Violin • Piano • Air Instruments • Flute • “Trombone” • Soda bottle

14. 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

15. Problem 25 – Organ Pipe • Open at both ends Closed at one end <<skip even harmonics

16. Problem 26 – Coke bottle • Open/closed fundamental • Closed 1/3 way up

17. Problem 27 – Full-range Pipe Organ • Open/open fundamental • Lowest frequency • Highest frequency

18. Problem 28 – Guitar • Original frequency of 3rd harmonic (on string) • Fingered frequency of fundamental • Ratio

19. Problem 29 – Guitar (1) • Unfingered frequency of fundamental (on string) • Fingered frequency of fundamental • Ratio

20. Problem 29 – Guitar (2) • wavelength of 440 Hz fundamental in string • frequency in air 440 Hz • Wavelength in air different because of air

21. Problem 30 – Organ Pipe • Corrected velocity to 21°C • Allowed frequencies • Length is • Wavelength same inside and outside tube

22. Problem 32 - Flute • Flute open at both ends (open-open) • Allowed frequencies • Length is

23. 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. ????!!

24. 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!!