Physics 1251 The Science and Technology of Musical Sound

1. Physics 1251The Science and Technology of Musical Sound Unit 3 Session 26 MWF Sound in Pipes

2. Physics 1251 Unit 3 Session 26 Sound in Pipes A standing wave on a string (tied at the ends) of length 3.0 m has two other nodes. What is the wavelength of the string wave? There are 6 node-antinode distances. Therefore, L = 6 ‧ λ/4 = 3.0 m; λ =2.0 m.

3. Physics 1251 Unit 3 Session 26 Pipes 1′ Lecture: • Sound in pipes can produce standing waves in the air column. • Standing waves in air columns produce pressure nodes and displacement nodes (and antinodes) at different places. • A change in the acoustic impedance of the air column produces a reflection. • Organ pipes and the flute are examples of open or unstopped pipes.

4. Physics 1251 Unit 3 Session 26 Sound in Pipes Standing Waves: • Strings – reflected waves combine to produce cancellation through destructive interference at nodes and constructive interference at antinodes. String Wave Demonstration ‒redux

5. Physics 1251 Unit 3 Session 26 Sound in Pipes Standing Waves in a Cylindrical Pipe: • A Closed or Stopped Pipe – the pressure wave reflects without inversion, but the displacement wave inverts upon reflection. • Thus, a pressureanti-node will occur at the wall; but, on the other hand, a displacementnode will occur at the same place.

6. Pressure Node Pressure Anti-node + - + - λ/4 λ/4 Physics 1251 Unit 3 Session 26 Sound in Pipes Reflection of a Sound Wave in a Stopped Pipe: A pressure anti-node appears at a wall. Pressure Wave

7. Physics 1251 Unit 3 Session 26 Sound in Pipes Displacement Standing Wave Visualization

8. Physics 1251 Unit 3 Session 26 Sound in Pipes Reflection of a Sound Wave in a Stopped Pipe: A displacement node appears at a wall. Displacement Anti-node Displacement Wave Displacement Node λ/4 λ/4

9. Physics 1251 Unit 3 Session 26 Sound in Pipes Comparison of Pressure and Displacement Standing Wave in a Double Stopped Pipe λ/4 λ/4 Pressure Wave Displacement Wave

10. Physics 1251 Unit 3 Session 26 Sound in Pipes Standing waves in a pipe are an example of the property of Interference of sound waves.

11. Physics 1251 Unit 3 Session 26 Sound in Pipes Recall: • f λ = v • Thus, f = v/ λ; L = Nna λ/4 • So f = Nna v/4L • Nna = 2 n ; fn = 2 n v/4L fn = n v/2L In double stopped pipe.

12. Physics 1251 Unit 3 Session 26 Sound in Pipes Why does the sound wave reflect? Because of an abrupt change in a property of the medium.

13. Physics 1251 Unit 3 Session 26 Sound in Pipes 80/20Acoustic Impedance: Z = p/U Acoustic Impedance is the ratio of the pressure p of a sound wave to the flow U (= u S) that results. For a plane wave in a tube of cross section S (m2) in air the acoustic impedance is: Z = ρv/S = 415/ S rayl

14. Physics 1251 Unit 3 Session 26 Sound in Pipes Acoustic Impedance and Reflection: The pressure that reflects is pout = R ‧ pin R = (Z2 – Z1 )/ (Z2 + Z1) At an immoveable wall U = 0 (no displacement) irrespective of the pressure and, thus, Z → ∞ R ≈ 1

15. Physics 1251 Unit 3 Session 26 Sound in Pipes But what about an open pipe? Does the sound reflect? Yes! Z = p/U p drops suddenly near the end of the pipe, since S → ∞. Thus, Z ≈ 0, R ≈ -1

16. λ/4 3λ/4 λ/4 λ/4 4λ/4 Physics 1251 Unit 3 Session 26 Sound in Pipes Comparison of Pressure Standing Wave in a Single Stopped and an Open Pipe L

17. Physics 1251 Unit 3 Session 26 Sound in Pipes 80/20For Stopped Pipe: Nna = odd number = 2n-1, n=1,2,3,4 … λn = 4L/ Nna = 4L / (2n-1) fstopped = f2n-1 = v/ λn = (2n-1) v/ 4L 80/20Only odd harmonics of fstopped 1 = v/4L.

18. λ/4 3λ/4 λ/4 λ/4 4λ/4 Physics 1251 Unit 3 Session 26 Sound in Pipes Comparison of Pressure Standing Wave in a Single Stopped and an Open Pipe L

19. Physics 1251 Unit 3 Session 26 Sound in Pipes Open versus Stopped Pipe Demonstration

20. Physics 1251 Unit 3 Session 26 Sound in Pipes 80/20For Open Pipe: Nna = even number = 2n, n=1,2,3,4… λn = 4L / Nna = 4L/(2n) = 2L/ n fopen = fn = v/ λn = n ‧ v/2L 80/20All harmonics of fopen1 = v/2L [= 2 fstopped 1 ]

21. Physics 1251 Unit 3 Session 26 Sound in Pipes End Correction for Open Pipe δ ≈ 0.6 a for a ≪λ ; δ ≈ 0 a for a > λ / 4 a Radius L + δ δ

22. Physics 1251 Unit 3 Session 26 Sound in Pipes Organ Pipes Open Pipes

23. Physics 1251 Unit 3 Session 26 Sound in Pipes Organ Pipes Pipe Organs use both open and stopped pipes in different ranks because the timbre is different for each. Open Pipes Stopped Pipes

24. Physics 1251 Unit 3 Session 26 Sound in Pipes Transverse Flute

25. Physics 1251 Unit 3 Session 26 Sound in Pipes Transverse Flute 80/20The transverse flute is a cylindrical open pipe. Mouthpiece is open

26. Physics 1251 Unit 3 Session 26 Sound in Pipes Other Flute-like Instruments Kaen RenaissanceFlute Penny whistle Fife

27. Physics 1251 Unit 3 Session 26 Sound in Pipes Stopped Pipe “Flute-like” Instruments The Pan Pipe: Only odd harmonics = different timbre Lower fundamental = shorter pipe

28. Physics 1251 Unit 3 Session 26 Sound in Pipes Summary: • fopen = fn = n ‧ v/2L • fstopped = f2n-1 = (2n-1) v/ 4L • Stopped and open cylindrical pipes have different timbres. • Impedance: Z = p/U • An abrupt change in Z is responsible for the reflections that lead to standing waves in pipes.