PHY-2464 Physical Basis of Music

Transcription

1 Physical Basis of Music Presentation 14 Basics of Flutes Adapted from Sam Matteson s Unit 3 Sessions 26 & 27 Sam Trickey Mar. 1, 2005 Reminder 1: A string with fixed ends has harmonic vibrations given by f n = [n/(2l)] (T/ µ), n= 1, 2, 3, These are standing waves Organ flue pipes, flutes, pan-pipes, pipes, and song whistles are familiar examples of sound in pipes In each of these pipes there is a clear fundamental and a set of apparently harmonic overtones. Standing waves in pipes are formed and sustained by abrupt changes in properties at the pipe ends.

2 Reminder 2: Standing Waves in a Cylindrical Pipe: At a Closed (Stopped) Pipe End the pressure wave reflects without inversion pressure ANTINODE (max) but the displacement wave inverts upon reflection displacement NODE (zero) Reminder 3: Reflection of a Sound Wave in a Stopped Pipe: Pressure anti-node node (max) occurs at the stopped end Pressure Wave Pressure Node Pressure Anti-node node

3 Reminder 4: Reflection of a Sound Wave in a Stopped Pipe: A displacement node forms at the end. Displacement Anti-node node Displacement Wave Displacement Node Reminder 5: Comparison of Pressure Standing Wave in Stopped versus Open Pipes 3 L 4

4 Reminder 6: Stopped Pipes: f stopped stopped = f 2n-1 = v/ λ n = (2n-1) v/ 4L Only odd harmonics of f stopped 1 = v/4l. Open Pipes: f open = f n = v/ λ n = n x v/2l All harmonics of f open 1 = v/2l [= 2 f stopped 1 ] Acoustic impedance goes as 1/S (pipe area) End corrections: add 0.6 pipe radius for each open end to get effective length Modern Transverse Flute (Boehm Flute): Head Joint Body Foot Joint

5 Transverse Flute - a cylindrical open pipe of variable length. Mouthpiece is open Four-component mechanical model The transverse flute is acoustically driven by fluid flow instabilities whose frequency is controlled by the feedback of the resonances of the pipe. Flow Instability Feedback Standing wave frequencies f 1 f 2 f 3 f 4 ~ ~ f n

6 Relationship of flute standing waves and string standing waves. NOTE the animations are DISPLACEMENT amplitudes (not pressure) launc.tased.edu.au/online/sciences/physics/i NSTR.HTM The Transverse Flute is an edge tone instrument The flute is driven by air flow against the edge of the embochure hole. Air flow Embochure hole

7 Edge Tone An air stream striking against an edge produces a fluctuating instability in flow. Air Stream Edge The flow alternates sides. Flow instabilities appear from the small to the very large. Winds in Pacific Ocean Thread in flow

8 Remember this from Presentation 12? Another edge tone; here driving simple harmonic motion of air. (Professor Sam Matteson per form- ing) Turbulence Air mass Oscillation Air of spring air mass Fipple (Amer. Heritage Dict.) A whistlelike mouthpiece for certain wind instruments, such as a recorder or flageolet, that channels the breath toward the sounding edge of a side opening. Fipple

9 Elementary Fipples 1.. Gently blow air from your mouth as in whistling. Below a certain velocity the air flow is stable, but faster than this critical velocity the air flow oscillates. 2. Blow against the edge of a piece of paper. Blow harder and softer. Move the paper closer and farther away from your lips. Notice the change in the sound. Fluid Flow Instability Undulation velocity u = 0.4 v jet v jet

10 Why does the stream oscillate? Short answer: positive feedback. When the stream bends to the left, the stream moves faster on the right side. Bernoulli s Principle: : the faster the flow, the lower the pressure. Therefore, the left-flowing flowing stream will bend back to the right. Bernoulli s s Principle The pressure in a fluid decreases as the velocity increases. Hold a piece of paper by the edge and blow across the top. What do you observe?

11 Other Flutes & Flute-like like Instruments RenaissanceFlutes Penny whistle Fife