1. Vibrations in Strings and Pipes – Learning Outcomes  Describe stationary waves in strings and pipes, particularly the relationship between frequency and length.  Use string and wind instruments.  Discuss string and woodwind sections in orchestras.  HL: Describe harmonics in strings and pipes.  HL: Solve problems about harmonics in strings and pipes. 1

2. Harmonics  In addition to the fundamental frequency, recall that instruments often produce overtones.  Overtones are frequencies above the fundamental frequency that instruments produce.  In physics, we discuss the similar idea of harmonics – integer multiples of the fundamental frequency.  The 1st harmonic is the fundamental frequency, f.  The 2nd harmonic is 2f.  The 3rd harmonic is 3f, etc. 2

3. Stationary Waves in Pipes  Three rules for stationary waves in pipes:  Closed ends have a node,  Open ends have an anti-node,  There must be a node between two adjacent anti- nodes and an anti-node between two adjacent nodes.  We discuss two types of pipes, those with two open ends, and those with one open and one closed end.  Examples of open-end pipes are flutes, piccolos.  Examples of closed-end pipes are panpipes, clarinet. 3

4. Open Pipes  Open pipes have anti-nodes at both ends.  The fundamental frequency of an open pipe has two anti-nodes at the ends with one node between them. 4  e.g. if the above pipe is 1 m long, what is the wavelength of the standing wave in the pipe?  e.g. if the speed of sound in air is 340 m s -1, what is the fundamental frequency / 1st harmonic of the pipe?

5. HL: Harmonics in Open Pipes  Recalling the rules for open pipes, what does the 2nd harmonic look like? 5  What does the 3rd harmonic look like?

6. HL: Open Pipes  e.g. draw the first three harmonics produced in an open pipe of length 0.75 m.  Calculate the frequency of each and compare them to the fundamental frequency.  e.g. In the open pipe instrument below, the note A4 is produced with the illustrated fingering. If A4 is at 440 Hz, what is the distance between the mouthpiece and the open hole? 6

7. Closed Pipes  Closed pipes have an anti-node at the open end and a node at the closed end.  The fundamental frequency of a closed pipe has one node and one anti-node: 7  e.g. if the pipe above is 1 m long, what is the wavelength of the standing wave in the pipe?  e.g. if the speed of sound in air is 340 m s -1, what is the fundamental frequency / 1st harmonic of the pipe?

8. HL: Harmonics in Closed Pipes  Recalling the rules for closed pipes, what does next harmonic look like? 8  What about the harmonic after that?

9. HL: Closed Pipes  e.g. Draw the first three harmonics that appear in a closed pipe of length 0.90 m.  Calculate the frequency of each and compare them to the fundamental frequency. 9 Open PipeClosed Pipe HarmonicOvertoneHarmonicOvertone 1st- - 2nd1st3rd1st 3rd2nd5th2nd 4th3rd7th3rd 5th4th9th4th

10. Stationary Waves in Strings  Two rules for stationary waves in strings:  Nodes at both ends,  Anti-node between adjacent nodes and node between adjacent anti-nodes.  The fundamental frequency has a node at both ends and one anti-node between them: 10  If the string above is 1 m long, what is the wavelength of the wave produced within the string?

11. HL: Harmonics in Strings  Recalling the rules for stretched strings, what does the second harmonic look like? 11  What does the third harmonic look like?

12. Strings  Strings work slightly differently to pipes because the sound is produced in a different medium (pipes produce sound in air) which travels into air.  When waves travel between media, they refract, which changes their speed and wavelength. We can’t use just wavelength and the speed of sound in air to determine frequency as we did with pipes.  When waves refract, their frequency is preserved, so the frequency of a wave within the string is the same as the frequency after it enters air. We need the frequency produced in the string. 12

13. HL: Strings 13

14. HL: Strings  e.g. A wire of length 3 m and mass 0.6 kg is stretched between two points so that the tension in the wire is 200 N. Calculate its fundamental frequency.  e.g. When the tension in a stretched string is 40 N, its fundamental frequency is 260 Hz. Find its fundamental frequency if its tension is increased to 160 N. 14

15. Orchestras  Orchestras are made up of many different instruments.  Brass and woodwind sections use air columns in pipes to create their sound. Trumpets, trombones and various horns in the brass section use cylindrical pipes. Flutes and clarinets in the woodwind section use them also.  The string section uses stretched strings to create music. Violins and cellos vary length to create different notes. Harps and pianos use separate strings for each note. 15