Inverse Square Relationship

1. ISNS 3371 - Phenomena of Nature Inverse Square Relationship Intensity decreases with increasing distance from the source. the wave is spreading out over a circular (2 dimensions) or spherical (3 dimensions) surface and thus the energy of the sound wave is being distributed over a greater surface area. Since energy is conserved and the area through which this energy is transported is increasing, the power (being a quantity which is measured on a per area basis) must decrease. Just as for gravity and light, sound intensity shows an inverse square relationship - the intensity varies inversely with the square of the distance from the source.

2. ISNS 3371 - Phenomena of Nature Forced Vibration Take a guitar string and stretch it to a given length and a given tightness and pluck it - you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar. Attach the string to the sound box of the guitar - the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a forced vibration. In the case of the guitar string mounted to the sound box, the fact that the surface area of the sound box is greater than the surface area of the string, means that more surrounding air particles will be forced into vibration. This causes an increase in the amplitude and thus loudness of the sound.

3. ISNS 3371 - Phenomena of Nature • Resonance • The string, guitar, and enclosed air vibrate at same frequency - they are in resonance. • Objects which vibrate at the same frequency are in resonance. • Important concept in musical instruments • broad resonance: amplify wide range of frequencies • sharp resonance: amplify narrow range of frequencies. • Note: Resonator does not create energy

4. ISNS 3371 - Phenomena of Nature Production of a Standing Wave in an Air Column

5. ISNS 3371 - Phenomena of Nature Reflection of a Wave at the Open End of a Pipe • pulse of air reaches the end of the tube and its momentum carries it out into the open air, where it spreads out in all directions. • pressure falls very quickly to nearly atmospheric pressure (the air outside is at atmospheric pressure). However, the air still has the momentum to travel away from the end of the pipe. • consequently, it creates a little suction: the air following behind it in the tube is sucked out (a little like the air that is sucked behind a speeding truck). • suction at the end of the tube draws air from further up the tube, and that in turn draws air from further up the tube and so on. • result is that a pulse of high pressure air travelling down the tube is reflected as a pulse of low pressure air travelling up the tube. • the pressure wave has been reflected at the open end, with a change in phase of 180｡.

6. ISNS 3371 - Phenomena of Nature Open and Closed Pipes

7. ISNS 3371 - Phenomena of Nature • Resonance in Closed and Open Pipes - Production of a Standing Wave in an Air Column • Closed pipe • Displacement node at closed end. • Displacement antinode at open end. • Pipe length is 1/4 wave length of fundamental. • Open pipe • Displacement antinodes at both ends. • Pipe length is 1/2 wave length of fundamental. • Node - a point in a standing wave that remains stationary. • Pressure node - no pressure increase • Displacement node - no displacement from normal position

8. ISNS 3371 - Phenomena of Nature Open and Closed Pipe Resonance States/Harmonics Natural frequency dependent on length of pipe fundamental frequency fo 1st harmonic fundamental frequency fo 1st harmonic 2nd harmonic f1 = 2fo 3rd harmonic f1 = 3fo 3th harmonic f2 = 3fo 5th harmonic f2 = 5fo For closed pipe - no "even harmonics”: fundamental frequency is a half-loop or ¼ L. Since every harmonic represents the addition of a complete loop, which contains two half-loops, we can never add just one more half-loop. Thus, we cannot generate even harmonics.

9. ISNS 3371 - Phenomena of Nature Wind Instruments In wind instruments, waves are generated by: lips - brass instruments - trumpets, trombones, etc… (“lip reeds”) reeds - woodwinds - clarinet, oboe, etc… air flow over hole - flute, piccolo, etc…

10. ISNS 3371 - Phenomena of Nature Brass Instruments Sound waves generated by lips - close your mouth, pull your lips back in a strange smile, and blow. The result may be anywhere between a low pitched ‘raspberry' or a high pitched musical note, depending on the tension in your lips- tension on the lips changes frequency of vibration The more tension you apply to your lips the more quickly they spring back into position. If the whole cycle takes a time T (period), then there are (one second)/T cycles per second. So the frequency, f = 1/T. All else equal, high lip tension gives high frequency and so high pitch.

11. ISNS 3371 - Phenomena of Nature Wind Instruments Open/closed pipes: Resonant notes/harmonics selected by lip/reed vibration (or airflow across embouchure hole opening in a flute) - dependent on length and shape of pipe Simple cylindrical pipe - harmonics determined by length

12. ISNS 3371 - Phenomena of Nature In a cylindrical pipe , notes are too far apart to be musically useful and not loud enough. In a flared pipe with a bell, frequency of harmonics raised and closer together - as radius of pipe is increased, sound waves spread out - pressure reduced - raises the frequencies of the standing waves, and raises the frequencies of the low pitched resonances most of all. Flute Clarinet Oboe

13. ISNS 3371 - Phenomena of Nature Bell also contributes - long waves (with the low pitches) are least able to follow the curve of the bell and so are effectively reflected earlier than are the shorter waves - their wavelengths are very much longer than the radius of curvature of the bell - the short waves, on the other hand, are better able to travel into the rapidly widening bell. The further they go into the bell, the easier it is for them to escape into the outside air. So the higher frequency waves are more efficiently radiated as sound outside the instrument. This is a characteristic of the bright sound of brass instruments: the bell radiates several of the higher harmonics well, so power of the harmonics does not decrease with frequency as strongly as it does in the woodwinds. This also makes brass instruments loud, because these strongly radiated high frequencies begin to fall into the range where our ears are most sensitive

14. ISNS 3371 - Phenomena of Nature Woodwind instruments: Vibrations generated by reed - has its own resonance Different harmonics generated by holes in pipe Sound waves reflect off of any discontinuity in the air column - such as a hole in the pipe - so the effective length of a pipe and its resonant frequencies can be changed by drilling holes in the pipe Flute: Open pipe Vibrations generated by airflow across embouchure hole opening - the air jets across hole vibrate Air flows in and out of embouchure hole as sound compression/rarefaction moves back and forth in tube Speed of air jets matched to frequency of note being played - faster air flow means higher harmonic - holes change resonant frequencies

15. ISNS 3371 - Phenomena of Nature Beats • Beats: periodic variations in intensity of sound at a point due to the coexistence of two wave trains having slightly different frequencies. • The two waves interfere constructively and destructively - where depends on difference in frequency. Two waves traveling at the same speed with slightly different frequencies add up - interfere - such that resulting wave travels in the same direction and with the same speed as the two component waves. A “beat” wave is created. The beat wave oscillates with the average frequency, and its amplitude envelope varies according to the difference in frequency. The number of beats per second is equal to the difference in the frequencies of the two primary sounds.

16. ISNS 3371 - Phenomena of Nature Beats To tune a piano, a piano tuner listens for beats produced between a tuning fork and a particular piano string. When the frequencies are identical the beats disappear.

17. ISNS 3371 - Phenomena of Nature The Doppler Effect - Wavelength Shift Due to Motion. Each circle represents the crests of sound waves going in all directions from the train whistle. The circles represent wave crests coming from the train at different times, say, 1/10 second apart. If the train is moving, each set of waves comes from a different location. Thus, the waves appear bunched up in the direction of motion and stretched out in the opposite direction.

18. ISNS 3371 - Phenomena of Nature Hearing the Doppler Effect Animation

19. ISNS 3371 - Phenomena of Nature Doppler Shift vs Velocity Animation