Superposition Interference and Standing Waves In the first part of Chap 14 we considered the motion of a single wave in space and time What if there are two waves present simultaneously in the same place and time Let the first wave have 1 and T1 while the second wave has 2 and T2 The two waves or more can be added to give a resultant wave this is the Principle of Linear Superposition Consider the simplest example 1 2 Since both waves travel in the same medium the wave speeds are the same then T1 T2 We make the additional condition that the waves have the same phase i e they start at the same time Constructive Interference The waves have A1 1 and A2 2 Here the sum of the amplitudes Asum A1 A2 3 y y1 y2 A2 A1 Sum If the waves 1 2 and T1 T2 are exactly out of phase i e one starts a half cycle later than the other Destructive Interference If A1 A2 we have complete cancellation Asum 0 y y1 y1 0 A1 sum A2 These are special cases Waves may have different wavelengths periods and amplitudes and may have some fractional phase difference Here are a few more examples exactly out of phase but different amplitudes Same amplitudes but out of phase by 2 Example Problem Speakers A and B are vibrating in phase They are directed facing each other are 7 80 m apart and are each playing a 73 0 Hz tone The speed of sound is 343 m s On a line between the speakers there are three points where constructive interference occurs What are the distances of these three points from speaker A Solution Given fA fB 73 0 Hz L 7 80 m v 343 m s v 343 m s vT 4 70 m 73 0 Hz L 2 x x is the distance to the first constructive interference point L The next point node is half a x wave length away 2 2 L Where n 0 1 2 3 for all nodes x n 2 2 7 8 4 70 n 0 x 0 3 9 m 2 2 7 8 4 70 n 1 x 1 1 55 m 2 2 Behind 7 8 4 70 n 2 x 2 0 8 m speaker A 2 2 7 8 4 70 n 1 x 1 6 25 m 2 2 Speaker A A1 A2 sum x x B Beats Different waves usually don t have the same frequency The frequencies may be much different or only slightly different If the frequencies are only slightly different an interesting effect results the beat frequency Useful for tuning musical instruments If a guitar and piano both play the same note same frequency f1 f2 constructive interference If f1 and f2 are only slightly different constructive and destructive interference occurs The beat frequency is f b f1 f 2 or 1 1 1 In terms of periods Tb T1 T2 as f 2 f1 f b 0 The frequencies become tuned Example Problem When a guitar string is sounded along with a 440Hz tuning fork a beat frequency of 5 Hz is heard When the same string is sounded along with a 436Hz tuning fork the beat frequency is 9 Hz What is the frequency of the string Solution Given fT1 440 Hz fT2 436 Hz fb1 5 Hz fb2 9 Hz But we don t know if frequency of the string fs is greater than fT1 and or fT2 Assume it is f b1 f s f T 1 and f b 2 f s f T 2 f s f b1 f T 1 5 440 445 Hz f s f b 2 f T 2 9 436 445 Hz If we chose fs smaller f b1 f T 1 f s and f b 2 f T 2 f s f s f T 1 f b1 440 5 435 Hz f s f T 2 f b 2 436 9 427 Hz Standing Waves A standing wave is an interference effect due to two overlapping waves transverse wave on guitar string violin longitudinal sound wave in a flute pipe organ other wind instruments The length dictated by some physical constraint of the wave is some multiple of the wavelength You saw this in lab a few weeks ago Consider a transverse wave f1 T1 on a string of length L fixed at both ends If the speed of the wave is v not the speed of sound in air the time for the wave to travel from one end to the other and back is 2L v If this time is equal to the period of the wave T1 then the wave is a standing wave 1 2L v v T1 f1 1 2 L f1 v 2 L 1 Therefore the length of the wave is half of a wavelength or a half cycle is contained between the end points We can also have a full cycle contained between v v end points 2 L f 2 2 L f2 Or three half cycles v v 3v 3 L f 3 2 f3 3 3 L 2 L 2 3 Or n half cycles Some notation v f n n n 1 2 3 4 2 L For a string fixed at both ends f1 1st harmonic or fundamental f 2 2 f1 2nd 1st overtone f 3 3 f1 3rd 2nd overtone f 4 4 f1 4th 3rd overtone The zero amplitude points are called nodes the maximum amplitude points are the antinodes Longitudinal Standing Waves Consider a tube with both ends opened If we produce a sound of frequency f1 at one end the air molecules at that end are free to vibrate and they vibrate with f1 The amplitude of the wave is the amplitude of the vibrational motion SHM of the air molecule changes in air density Similar to the transverse wave on a string a standing wave occurs if the length of the tube is a half multiple of the wavelength of the wave For the first harmonic fundamental only half of a cycle is contained in the tube v f1 2L Following the same reasoning as for the transverse standing wave all of the harmonic frequencies are v f n n n 1 2 3 2 L Open open tube Identical to transverse wave except number of nodes is different nodes n 1 string nodes n Open open tube An example is a flute It is a tube which is open at both ends v fa 2 La v fb fa 2 Lb x mouthpiece x La Lb We can also have a tube which is closed at one end and opened at the other open closed At the closed end the air molecules can not vibrate the closed end must be a node The open end must be an anti node The distance between a node and the next adjacent anti node is 1 4 of a wavelength Therefore the fundamental frequency of the openclosed tube is v f1 since L 4 …
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