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Physics Midterm 2 (Musical Instruments - contains some part/element that…

- - - - doubling the frequency means doubling the velocity in half the time
        
        doubled frequency = quadrupled acceleration
  - - - equilibrium = undisturbed string, restoring force = tension, mass, bending force (add to restoring force) = stiffness of string
        
        modes of vibration, 1st mode being fundamental, freely plucked strings exhibit all of the modes
        
        moving points = anti-nodes, non-moving points = nodes
        
        "nth" mode has "n" times the frequency, i.e. moves of vibration are essentially just harmonics
        
        amt. of specific modes depend on where string is plucked
        
        middle pluck = odd harmonics (hits nodes of even harmonics, therefore no sound)
        
        1/3 of the way pluck = no 3rd, 6th, 9th, etc. harmonics
        
        1/4 of the way pluck = no 4th, 8th, etc. harmonics
        
        near end of string pluck = almost all harmonics --> bright tone
        
        bending force greater for higher frequencies
        
        more bending force for shorter strings/higher notes => inharmonicity gets worse
      - vibrating strings follow Mersenne's law: f = 1/2 sqrt(T/Lm)
        
        frequency doubles when you...half the length, half the diameter, quadruple the tension, decrease density by a quarter
      - notes decay mainly due to friction
        
        air friction - esp. important for thin/hard strings,string loses kinetic E when moving air molecules, decay time depends on frequency as 1/sqrt(f)
        
        internal friction (aka heat) - esp. important for softer strings, decay time depends on frequency as 1/f
  - - - impedance matching - nothing is stopping wave from spreading, eg. string attached to another string
      - mismatched impedance - object has higher impedance than string
        
        important function of soundboard --> prevents strings from losing energy too fast, we want a balance (don't want too much mismatch)
  - - - hammer strikes strings, vibration transmitted to soundboard thru bridge, sound decays, damper stops sound when key is released
    - - strings struck ~10% of way along string --> increases amt. of high harmonics, brighter sound
        
        harder strikes = brighter tones, softer strikes = warmer tones
        
        hammer can't be in contact w string for more than one vibration (otherwise vibration will be dampened), ~5 ms for large hammers and 0.5ms for smallest hammers
        
        hammers smaller/faster at higher end of piano where vibrations are faster
        
        notes don't have as many high harmonics because hammer can't move fast enough
        
        playing notes faster or slower affects harmonics
    - - more pliable soundboard decreases acoustic impedance BUT string needs to provide more energy, therefore we need to increase the velocity with which it vibrates (make hammer faster/heavier) and increase mass (make string heavier)
  - - - main wood resonance = 400 Hz/~G4
      - main air resonance = 260 Hz
      - good violins have separate resonances spread across frequency range of instrument so that every note excites as resonance
        
        in a resonating pipe, wavelength is 2x length of pipe
  - - - 1/2λ = fundamental mode, 1λ = 2nd mode, 3/2λ = 3rd mode, 2λ = 4th mode
    - - 1/4λ = fundamental, 3/4λ = 2nd mode, 5/4λ = 3rd mode, 7/4λ = 4th mode
        
        nth mode has (2n-1)/4 λ in pipe w length L
      - conical pipes - flared, gets wider toward bottom, amplitude decreases with width, all harmonics present
    - - fluid-flow instability - unstable/jet-like air flow interacts w sharp edge that disrupts flow (some air goes into instrument and some goes out)
        
        flute, organ pipe, recorder
        
        driven end = velocity antinode => open pipe
        
        organ: pipes divided into ranks, each rank spans 5 octaves from C2-C7, longest pipe = C2 w frequency 65.4 Hz
        
        recorder: frequency = 365 Hz w all holes covered
        
        flute: instead of mouthpiece that generates edge tone, it is actually generating by blowing thru mouthpiece
        
        frequency of edge tone: air flows back at ~40% of outgoing velocity
        
        time = half of period P (refer to sheet for formula)
        
        overblowing increases frequency by 3x
        
        edge-tone frequency that is close to the natural frequency of the pipe facilitates a mutual relationship in which both will feed each other
      - Reed Instruments
        
        based on Bernoulli effect - faster airflow means lower pressure
        
        reed produces bursts of pressure in mouthpiece => mouthpiece is pressure antinode (open end = pressure node), reed opens bc of pressure and bernoulli effect pulls them closed again when air passes thru
        
        clarinet = closed pipe (closed at one end), pressure vibration at closed end (mouthpiece) drives reed at natural frequency
        
        straight cylindrical cone ==> only odd harmonics, overblowing gives you third harmonic
        
        normal playing uses 1st, 3rd, and 5th harmonics
        
        hard vs soft reeds: hard reeds have sharp resonance and only plays one note. soft reeds have broad resonance and don't have a clearly defined note
        
        hard reed (eg. harmonica)
        
        harmonica - metal reed for each note, sounds depend on whether you blow or draw air
        
        blues harmonica/blues harp plays only major scale (no sharps/flats)
        
        10 holes, 3 octaves - not all notes present, low G occurs twice
        
        chromatic harmonic has slide that allows you to play all 12 notes
        
        soft reed (eg. clarinet)
        
        divided into single and double reeds - not much difference in sound, more for convenience of playing
        
        single reed: saxophone (conical)
        
        flare of pipe ~3 degrees --> wider flare allows for larger tone holes => louder sound
        
        1 more item...
        
        double reed: oboe (concical)
        
        flare of pipe ~1 degree --> narrower flare produces more high harmonics and brighter sound
        
        Brass Instruments = reed instruments where "reed" is player's lips (pressure antinode, closed pipe), usually metal
        
        bugle, trombone, trumpet
        
        change frequency of trombone by sliding pipe --> longer pipe = lower frequency, how far to slide pipe depends on how much more length you need to lower by half-steps
        
        advantages: simple to make, perfect tuning possible
        
        disadvantages: hard to play in tune, hard to play fast/legato, requires cylindrical pipe
        
        problem w cylindrical pipes is that it only produces odd harmonics => less bright sound, fewer basic notes to work off of BUTBUTBUT trombones (and trumpets) give you ALL the harmonics excluding the fundamental
        
        2 more items...
        
        trumpet gets rid of slide in favor of valves --> valves divert air thru extra sections of pipe to make total length longer, plays 2nd-8th harmonics
        
        2nd valve has shortest crook, 1st valve has 2nd-shortest, 3rd valve has longest crook
        
        first and second valves can be pressed together to get third valve BUT note will be slightly sharp --> corrected for by "lipping down" (this is actually preferable to playing 3rd valve alone)
        
        2 more items...
        
        most common # of valves = 3
        
        types of valves: piston and rotary
        
        tuba can play up to 11th harmonic, can play fundamental via extra valve/crook
  - - - muscles around vocal chords allow you to loosen/tighten to make different notes
        
        don't completely close when singing/speaking softly, close and cut off air flow with louder noise --> close for considerable length of time when very loud
- - - - fourier analysis - tells you how much of each harmonic is in a waveform
        
        square and triangle waves - odd harmonics only
        
        sawtooth wave - all harmonics
        
        many high frequencies (aka high harmonics) = bright sound
        
        many low frequencies (aka low harmonics) = "warm sound"
- - - - tensor tympani (attached to hammer) and stapedius (attached to stirrup) activated by autonomic reflexes and prevent ossicles from moving
  - - - basilar membrane - continuum that is excited by different frequencies (excited regions can overlap)
        
        two different theories (read them first) --> place theory: basilar breaks up initial sounds into frequency bands that get sent along different nerves to brain // temporal theory: timing allows brain to make sense of things and perceive pitch more accurately
        
        temporal theory - takes into consideration timing of nerve signals, explains accurate pitch discrimination, pitch identification in complex waveforms, and pitch identification with missing fundamental
        
        timing also plays role in sound localization, sound travels x further to reach one ear (compared to other)
        
        place theory - pitch perception dependent on areas of basilar membrane
        
        notes closer than the critical bandwidth (~minor third or 3 half-steps) excite the same nerve endings, results in dissonance