1Superposition and Standing Waves• Superposition• Constructive and destructive interference• Standing waves• Harmonies and tone• Interference from two sources• Beats2Principle of SuperpositionWhen two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacement due to each individual wave.3Constructive and Destructive Interference•Constructive –amplitude of the 2 waves is of the same sign•Destructive –amplitude of the 2 waves is of the opposite sign4Standing WaveTwo waves traveling in the opposite directions with the same amplitudeThe two waves interfere and create a standing wave5Nodes and Antinodes• Nodes –displacement does not change• Antinodes –displacement changes with maximum amplitude6Nodes and Antinodes –longitudinal wavesNodes and antinodes can be defined as pressure or velocity. Text book defines as pressure – other sources define as velocity of particles in the medium7ReflectionsWhen a wave meets a boundary it is reflected.A hard boundary will invert the reflection, a soft boundary will keep the original senseAnimation courtesy of Dr. Dan Russell, Kettering University http://paws.kettering.edu/~drussell8Reflection at a discontinuity• At a discontinuity in the medium – e.g. passing from higher to lower density, we get partial transmission and partial reflection.• From low to high density we also get an inversion at the reflection9Modes• Certain wavelengths will fit on a fixed length of medium.• These are called modes• The number of antinodes gives us the mode number10ModesThe wavelengths of the modes for a medium length L, can be described by,...4,3,2,12mmLm11ModesThe frequencies of the modes for a medium length L, can be described by,...4,3,2,12mLmvvfmm12Special ModesWhen m=1 we get the lowest frequency, called the fundamental frequency...4,3,2,1211mmffLvfm13Applications• Stringed instruments – we know the velocity of the wave in the string is:• To keep the tension on stringed instruments the same, the strings linear density, μ is changedssTLfTv21114LasersThe laser has a full reflector and partial reflector. The light produced in the cavity is leaked at one end by a mirror that is only 99% efficient.15Standing sound waves in pipes• A closed end pipe will reflect the wave• An open end pipe will partially transmit and partially reflect the sound wave – it is a discontinuity in the medium16Sound waves in a pipe• The open end of a pipe will be a pressure node – the pressure will constant• A closed end of the pipe will be a pressure antinode – the pressure fluctuates from minimum to maximum value17Sound wave modes in a pipeRepresentation of longitudinal waves in open-open, closed-closed and open-closed pipes18Standing waves in an open-closed pipeWe can get one quarter wavelengths in an open-closed pipe:...7,5,3,144mLmvfmLmm19Physics of the human earSound travels into the ear, vibrates the ear drum, which amplifies the sound, and sends it down the cochlea20Physics of the human earThe sound resonates hair cells in the cochlea (0.5nm) to fire neurons21Shape of soundA guitar string will have many higher frequencies, or harmonics. They add to the tone quality, or timbre.22InterferenceTwo wave sources operating at the same frequency will add (constructively and destructively) and lead to interference patterns.23Constructive interference• Amplitudes will add when the waves are in phase• This happens when the path length difference is a whole number of wavelengths.,...3,2,1,0mmd24Destructive interference• Amplitudes will cancel when the waves are out of phase• This happens when the path length difference is a half wavelength off.,...3,2,1,021mmd25Destructive interference in head phones• Active noise reduction is when the incoming sound is inverted and rebroadcast• Commonly used on air flights.• Selective frequency response.26Beats• Consider two waves of slightly different frequency• The amplitudes add and cancel and give rise to beats.27BeatsThe time between the beats is dependent on the difference between the two frequencies28BeatsThere are 2 new frequencies, the frequency of the oscillation, fosc, and the beat frequency, fbeat 212121ffffffbeatosc29Summary• Superposition• Constructive and destructive interference• Standing waves• Harmonies and tone• Interference from two sources• Beats30Homework problemsChapter 16 Problems41, 54, 56, 61, 62,
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