Longitudinal Waves. Reading: Chapter 17, Sections 17-7 to Sources of Musical Sound. Pipe. Closed end: node Open end: antinode

Transcription

1 Longitudinal Waes Reading: Chapter 7, Sections 7-7 to 7-0 Sources o Musical Sound Pipe Closed end: node Open end: antinode Standing wae pattern: Fundamental or irst harmonic: nodes at the ends, antinode in the middle L, L Resonant requency: L In general, or harmonic number n, L n, L n, or n =,, 3, Resonant requency: n, or n =,, 3, L For pipes with only one open end,

2 L 3 5 7,,, In general, 4L 4L 4L 4L,,, L n, or n =, 3, 5, Resonant requencies: n, or n =, 3, 5, 4 L In general, when a musical instrument produces a tone, the undamental as well as higher harmonics are generated simultaneously This gies rise to the dierent waeorms generated by dierent instruments Hence dierent instruments hae dierent sounds

3 Example 7-6 Weak background noises rom a room set up the undamental standing wae in a cardboard tube o length L = 670 cm with two open ends Assume that the speed o sound in the air within the tube is 343 ms - (a) What requency do you hear rom the tube? (b) I you jam your ear against one end o the tube, what undamental requency do you hear rom the tube? (a) Two open ends, L L L = / L Hz (ans) (067) (b) One ixed end and one open end, L 4 4L L = /4 4L Hz (ans) 4(067) 3

4 Beats Consider two sound waes with slightly dierent requencies: s m Resultant displacement: s cos t and s sm cost s s s sm(cost cost) Using the trigonometric identity we obtain cos cos = cos ( ) cos ( ), m s s cos ( ) t cos ( ) t 4

5 Conclusion: s [ sm cos' t]cost, where ' ( ) and ( ) Since and are nearly equal, >> Hence the resultant displacement consists o an oscillation with angular requency and a slowly changing amplitude with angular requency The amplitude s m cos t is maximum when cos t =, ie times in each repetition o the cosine unction Hence the beat requency is: beat ' beat Musicians use the beat phenomenon in tuning their instruments See Animation Beats 5

6 The oppler Eect See Youtube Fire Engine siren demonstrates the oppler Eect and Example o oppler Shit using car horn etector Moing; Source Stationary In time t, the waeronts moe a distance t, the detector moes a distance t, the range o waes intercepted by the detector = t+ t, the number o waeronts intercepted by the detector = (t+ t)/ 6

7 7 The requency obsered by the detector: ) / ( ' t t t Since =/, / ' Similarly, i the detector moes away rom the source, ' Summarizing, ' Source Moing; etector Stationary

8 In a period T, the distance moed by the waeront W = T, the distance moed by the source = S T, the distance between the waeronts W and W = T S T The requency obsered by the detector: ' ' T S T / S / S I the source moes away rom the detector, ' S Summarizing, ' S General oppler Eect Equation When both the source and detector are moing, ' S S = 0 reduces to the equation or stationary source = 0 reduces to the equation or stationary detector 8

9 Example 7-8 Bats naigate and search out prey by emitting, and then detecting relections o, ultrasonic waes, which are sound waes with requencies greater than what can be heard by a human Suppose a bat emits ultrasound at requency be = 85 khz while lying with elocity b = 900 ms It chases a moth that lies with elocity m = 800 ms (a) What requency md does the moth detect? (b) What requency bd does the bat detect in the returning echo rom the moth? b m be bd m m 9

10 Supersonic Speeds; Shock Waes See Youtube Sonic Boom When approaches S, becomes ininity since ' S When exceeds S, the oppler eect equation does not apply All waeronts bunch along a V-shaped enelope This is called the Mach cone A shock wae is produced Note that the enelope touches the circular waeronts Thereore the radius ending at the tangent point is normal to the Mach cone Mach cone angle: t sin St S S / is called the Mach number 0