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Physics of Musical Instrument

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Physics of Musical Instrument. FLUTE. Introduction . There are over millions of musical instruments in the world. Some instruments can produce sounds by air vibration. Vibrating the lips (trumpet) , reed (oboe) and blowing air in the tube (piccolo) are three methods to make the air vibrate.

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  • There are over millions of musical instruments in the world.
  • Some instruments can produce sounds by air vibration.
  • Vibrating the lips (trumpet) , reed (oboe) and blowing air in the tube (piccolo) are three methods to make the air vibrate.
  • Flute is the one producing sound with blowing air into the pipe.
the air jet vibrates
The air jet vibrates
  • In a flute, the air jet from the player\'s lips travels across the embouchure-hole opening and hits the sharp edge of the hole.
  • The cause of the disturbance of the jet is the sound vibration in the flute tube, which causes air to flow into and out of the embouchure hole.

Cross section of the flute at the embouchure


A air jet striking an edge and being alternately deflected up and down

  • If the jet speed is matched to the frequency of the note being played, then the jet will flow into and out of the embouchure hole in just the right phase to amplify the sound and cause the flute to produce a continued note.
To play a high note, the travel time of waves on the jet must match the higher frequency, and this can be done by increasing the blowing pressure (which increases the jet speed) and moving the lips forward to shorten the distance along the jet to the edge of the embouchure hole.

f = 1/T

the flute is an open pipe
The flute is an open pipe
  • The flute is open at both ends. It\'s obvious that it\'s open at the far end. The player\'s lower lip covers only part of the embouchure hole, leaving a large part of the hole open to the atmosphere.

The dotted line indicated the position of lower lip


Assumption :

  • 1. Flute is cylinder in shape.
  • 2. All holes are closed.
  • 3. Embouchure hole is at the end of the flute.
The natural vibrations in the flute are due to the standing waves. The Flute is open to the air at the ends means that the total pressure at the ends of the pipe must be approximately atmospheric pressure (acoustic pressure = 0 ).
  • Acoustic pressure equals zero at either end. These points are called pressure nodes.
  • The pressure needs not be atmospheric inside the flute. The maximum variation in pressure occurs at the middle.
tone holes
tone holes
  • When the the tone holes are opened, starting from the far end, the pressure node moves closer up the pipe .This makes the pipe shorter.
  • Holes can also serve as register holes.
  • The register hole makes the played note (at least) one octave higher, because it is halfway along the working length of the flute and so facilitates the second harmonic of C4(fo).


if you play C4 (fo) and then lift your left thumb, you are opening a hole halfway down the instrument . This encourages the even harmonics, so the flute \'jumps up\' to C5 (2fo).


To play a high note with short wavelength, a register hole can be opened at a different fraction of the length.

open and closed pipe
Open and Closed pipe

Closed pipe

  • The disturbance from the embouchure hole leads the vibration of the air inside the pipe to produce standing wave.
  • The antinode is at the opening of the pipe while the node is at the end of the closed pipe.

Length of the pipe = 1/4 (wavelength)n where n is an odd integer

n = 1, 3, 5, 7…

n = 1, 3, 5, 7…


Open pipe (flute)

  • The antinodes are at both openings of the open pipe.

Length of the pipe = ½ (wavelength)n where n is any integer.

n = 1, 2, 3, 4…

n = 1, 2, 3, 4…

fundamental and overtone
Fundamental and overtone


  • It is the sound with lowest frequency of the standing wave in the pipe (i.e. f = nv/2L ,where n =1)

Overtones (for flute)

  • it is the sound with frequency that is the multiple of the fundamental frequency in open pipe.( i.e. 2fo, 3fo)

Both fundamental and overtone are called harmonic.


There is a limited range of frequency for air vibrating in the pipe. Therefore, the number of overtones is limited.

  • This is called the harmonic series, and notes with these frequencies have the pitches shown below.

Length of OUR flute (L ) = 0.663m

V= 350m/s

for sound in warm, moist air

C4 fo = v/2L



Theoretical frequency of C4=261.63Hz

% error =100(263.95-261.63)/261.63


C5 f =2fo



Theoretical frequency of C5=523.25Hz

% error =100(527.9-523.25)/523.25



Closed pipe

  • When a sound wave hits the closed end, it is reflected.
  • The incident and reflected waves were overlapped along the air column.
  • A longitudinal stationary wave is set up if the length of the air column is such that a node is formed at the closed end and an antinode is formed at the open end.
  • The air at the antinode at the mouth of the tube forces the air outside the tube to vibrate strongly and produces a loud sound of the same frequency
open pipe flute
Open pipe (flute)
  • There is standing wave of air molecules inside the flute.
  • If the frequency of the air blown in is the same as that of the standing wave inside, resonance will occur.
  • With different fingerings, a different frequency of standing wave inside the pipe is set up.
  • By blowing air in, different notes will be produced.

The above calculated frequencies such as fo, 2fo and 3fo are the resonant frequency.


End Correction

  • In the real case, the antinode is not exactly at the end of the opening.
  • The antinode is usually outside the opening at distance c from the opening. This is called the end correction.
  • c is related to the wavelength. The shorter the wavelength ,the smaller the c.
  • This correction can reduce the percentage of error when calculating the frequency of sound.

When the radius of bore be r, c = 0.6r .

For closed pipe,

  • c appears at one end of the pipe.

When n = 1,


λ= 4 (l+ c)


For open pipe (flute),

  • c appears at both ends of the pipe.

When n = 1,


λ= 2 (l+2c)



Here are some waveforms of flute from different books :


y = Asinωt

Tuning fork’s waveform

Flute’s waveform

The sound consists of five harmonics, then

y = sin(x)+0.8sin(2x+2)+0.3cos(3x)-0.4cos(4x)+0.2sin(5x)

From the equation, the frequencies of the five harmonics are ω﹑2ω﹑3ω﹑4ω and 5ω, where ω is the fundamental frequency.

where ψn is a constant.



The flute made of wood with higher density has a better sound.







Density = 2. 0








There was a lot of information in the internet. We had to spend quite a lot of time to read through it. To understand those information, we face many difficulties. Nevertheless, after having this challenge, we have learnt much more than those in the books.

group members
Group Members

Chu Hon Ting (5)

Liu Pik Yin (18)

Siu Lee Lee (23)