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The Physics of Musical Instruments

Class Notes – 24

Pipes with holes; Recorders, flutes

The Physics of Musical Instruments – Fall 2017 – October 24 – 1 Topics

• More on acoustic impedance • The end correction for a flute or recorder • The effect of a side hole • The flute – a open at both ends with side holes

The Physics of Musical Instruments – Fall 2017 – October 24 – 2 More Complex Pipes

• We are now beginning to consider more complex pipes and air – Bottles, blown across the top (fun stuff) but more seriously… – Bells at the end (why?) – Pipes with finger holes – And, eventually, pipes with non-cylindrical bores • The concept of acoustic impedance is very helpful here, even without getting highly quantitative.

The Physics of Musical Instruments – Fall 2017 – October 24 – 3 Energy Transfer

• When the impedance changes along an acoustic pipe, there can be a reflection

• If one of the impedances is exactly zero, ALL OF THE ENERGY IS REFLECTED. • We used this idea to analyze the open end of an ideal pipe

The Physics of Musical Instruments – Fall 2017 – October 24 – 4 A Finite Discontinuity

• Consider a discontinuity in pipe

• Energy reflection in this case is 1/9 or 11%

The Physics of Musical Instruments – Fall 2017 – October 24 – 5 Add a Transition Piece

• With a transition segment added, the impedance discontinuity is reduced, and there is less reflection • Only 6% is reflected

The Physics of Musical Instruments – Fall 2017 – October 24 – 6 A Tapered Transition

• The impedance is lowered gradually left to right, but because the impedance is changing, some of the energy is always reflected

• The lower example will look more “abrupt” than the upper one, especially when the wavelength becomes greater than the length of the taper

The Physics of Musical Instruments – Fall 2017 – October 24 – 7 Transition to Open Air

• Bells of and wind instruments are tapered and provide transitions between the in-pipe impedance and the external air – We will return to this later in the semester • For now, consider the abrupt end of a flute or a recorder – The small makes an abrupt transition to some small but finite acoustic impedance of the air – The reflector is imperfect – some of the pressure variation extends into the air

The Physics of Musical Instruments – Fall 2017 – October 24 – 8 Topics

• More on acoustic impedance • The end correction for a flute or recorder • The effect of a side hole • The flute – a pipe open at both ends with side holes

The Physics of Musical Instruments – Fall 2017 – October 24 – 9 How Big is the Effect?

• Johnston presents a sketch for the case of a closed pipe

• The pipe behaves as if it were slightly longer than the physic pipe because of the imperfect reflection at the impedance discontinuity at the open end

The Physics of Musical Instruments – Fall 2017 – October 24 – 10 Lowest Pitch of a Recorder

• We measure pitch of two different recorders and calculate the corresponding wavelengths • We recall that for an ideal pipe open at both ends, the wavelength will be twice the pipe length

The Physics of Musical Instruments – Fall 2017 – October 24 – 11 Recorder Length

• – – ~520 Hz – Wavelength is 2.1 feet, twice the pipe length – Effective pipe length is 1.05 feet or 13 inches – Length of recorder is at most 12 inches • – F – ~350 Hz – Wavelength is 3.1 feet, twice the pipe length – Effective pipe length is 1.55 feet or 19 inches – Length of recorder is at most 17 inches • The effective pipe length is longer than for an ideal pipe – the acoustic impedance looking out from the end of an open pipe is not exactly zero!

The Physics of Musical Instruments – Fall 2017 – October 24 – 12 Truth in Lending

• Johnston’s diagram can be misleading.

• The pressure variation does not “go to zero” some distance outside the end. • There is a new acoustic radiation field starting at the pipe end, but the effect of the reflection on the mode inside the pipe is “as if” it did.

The Physics of Musical Instruments – Fall 2017 – October 24 – 13 Topics

• More on acoustic impedance • The end correction for a flute or recorder • The effect of a side hole • The flute – a pipe open at both ends with side holes

The Physics of Musical Instruments – Fall 2017 – October 24 – 14 A Side Hole Shortens the Pipe • A tiny hole doesn’t change the impedance enough to affect the • When the hole is large enough, the effective pipe length Le is shortened to the hole position

The Physics of Musical Instruments – Fall 2017 – October 24 – 15 Impedance Again

Tiny discontinuity Big discontinuity in impedance; in impedance; weak reflection strong reflection

Resonance Resonance determined by determined by pipe length hole position

The Physics of Musical Instruments – Fall 2017 – October 24 – 16 Side Holes add Reflections

• The side holes add small cavities that makes the no longer a simple ideal pipe

Added volume and , like Benade’s assembly of bottles

Impedance “bumps” even when the hole is closed

The Physics of Musical Instruments – Fall 2017 – October 24 – 17 Making a Scale

Recorder fingering chart • If the holes are large enough, they permit the playing of a scale • All instruments have “fingering charts”

The Physics of Musical Instruments – Fall 2017 – October 24 – 18 It’s Not Always Obvious

• Effects of closing non-adjacent holes can be important

The Physics of Musical Instruments – Fall 2017 – October 24 – 19 A More Careful Look

The Physics of Musical Instruments – Fall 2017 – October 24 – 20 Getting an

• So far, we have been looking at the fundamental mode of the resonant pipe • How can we excite next higher ? • Three options: – Overblow – Put a hole somewhere to suppress the fundamental (usually for the thumb) – Combinations of the two

The Physics of Musical Instruments – Fall 2017 – October 24 – 21 Back to the Recorder

Half Hole

The Physics of Musical Instruments – Fall 2017 – October 24 – 22 Topics

• More on acoustic impedance • The end correction for a flute or recorder • The effect of a side hole • The flute – a pipe open at both ends with side holes

The Physics of Musical Instruments – Fall 2017 – October 24 – 23 The Flute

• A cylindrical pipe, open at both ends, with an array of side holes of similar size • Holes on the flute can be open or covered with pads

The Physics of Musical Instruments – Fall 2017 – October 24 – 24 Looking at the Flute

• We’ll look quickly into the article posted under “additional resources” – http://newt.phys.unsw.edu.au/jw/fluteacoustics.html

• We will also look at a brief video – https://www.youtube.com/watch?v=zLLlBNUZmfQ

The Physics of Musical Instruments – Fall 2017 – October 24 – 25