Chapter 17 The Principle of Linear Superposition and Interference Phenomena 17 1 The Principle of Linear Superposition When the pulses merge the Slinky assumes a shape that is the sum of the shapes of the individual pulses 17 1 The Principle of Linear Superposition When the pulses merge the Slinky assumes a shape that is the sum of the shapes of the individual pulses 17 1 The Principle of Linear Superposition THE PRINCIPLE OF LINEAR SUPERPOSITION When two or more waves are present simultaneously at the same place the resultant disturbance is the sum of the disturbances from the individual waves 17 2 Constructive and Destructive Interference of Sound Waves When two waves always meet condensation to condensation and rarefaction to rarefaction they are said to be exactly in phase and to exhibit constructive interference 17 2 Constructive and Destructive Interference of Sound Waves When two waves always meet condensation to rarefaction they are said to be exactly out of phase and to exhibit destructive interference 17 2 Constructive and Destructive Interference of Sound Waves 17 2 Constructive and Destructive Interference of Sound Waves If the wave patters do not shift relative to one another as time passes the sources are said to be coherent For two wave sources vibrating in phase a difference in path lengths that is zero or an integer number 1 2 3 of wavelengths leads to constructive interference a difference in path lengths that is a half integer number 1 2 of wavelengths leads to destructive interference 17 2 Constructive and Destructive Interference of Sound Waves Example 1 What Does a Listener Hear Two in phase loudspeakers A and B are separated by 3 20 m A listener is stationed at C which is 2 40 m in front of speaker B Both speakers are playing identical 214 Hz tones and the speed of sound is 343 m s Does the listener hear a loud sound or no sound 17 2 Constructive and Destructive Interference of Sound Waves Calculate the path length difference 3 20 m 2 2 40 m 2 2 40 m 1 60 m Calculate the wavelength v 343 m s 1 60 m f 214 Hz Because the path length difference is equal to an integer 1 number of wavelengths there is constructive interference which means there is a loud sound 17 2 Constructive and Destructive Interference of Sound Waves Conceptual Example 2 Out Of Phase Speakers To make a speaker operate two wires must be connected between the speaker and the amplifier To ensure that the diaphragms of the two speakers vibrate in phase it is necessary to make these connections in exactly the same way If the wires for one speaker are not connected just as they are for the other the diaphragms will vibrate out of phase Suppose in the figures next slide the connections are made so that the speaker diaphragms vibrate out of phase everything else remaining the same In each case what kind of interference would result in the overlap point 17 2 Constructive and Destructive Interference of Sound Waves 17 3 Diffraction The bending of a wave around an obstacle or the edges of an opening is called diffraction 17 3 Diffraction single slit first minimum sin D 17 3 Diffraction Circular opening first minimum sin 1 22 D 17 4 Beats Two overlapping waves with slightly different frequencies gives rise to the phenomena of beats 17 4 Beats The beat frequency is the difference between the two sound frequencies 17 5 Transverse Standing Waves Transverse standing wave patters 17 5 Transverse Standing Waves In reflecting from the wall a forward traveling half cycle becomes a backward traveling half cycle that is inverted Unless the timing is right the newly formed and reflected cycles tend to offset one another Repeated reinforcement between newly created and reflected cycles causes a large amplitude standing wave to develop 17 5 Transverse Standing Waves String fixed at both ends v f n n 2L n 1 2 3 4 17 5 Transverse Standing Waves v f n n 2L n 1 2 3 4 17 5 Transverse Standing Waves Conceptual Example 5 The Frets on a Guitar Frets allow a the player to produce a complete sequence of musical notes on a single string Starting with the fret at the top of the neck each successive fret shows where the player should press to get the next note in the sequence Musicians call the sequence the chromatic scale and every thirteenth note in it corresponds to one octave or a doubling of the sound frequency The spacing between the frets is greatest at the top of the neck and decreases with each additional fret further on down Why does the spacing decrease going down the neck 17 6 Longitudinal Standing Waves A longitudinal standing wave pattern on a slinky 17 6 Longitudinal Standing Waves Tube open at both ends v f n n 2L n 1 2 3 4 17 6 Longitudinal Standing Waves Example 6 Playing a Flute When all the holes are closed on one type of flute the lowest note it can sound is middle C 261 6 Hz If the speed of sound is 343 m s and the flute is assumed to be a cylinder open at both ends determine the distance L 17 6 Longitudinal Standing Waves v f n n 2L L n 1 2 3 4 nv 1 343 m s 0 656 m 2 f n 2 261 6 Hz 17 6 Longitudinal Standing Waves Tube open at one end v f n n 4L n 1 3 5 17 7 Complex Sound Waves 17 7 Complex Sound Waves