Vibrating cords and sound pipes
Vibrating cords
It is an elastic wire, tended between two fixed points, and succeptible to emit a musical sound by its vibrations.
Such a cord, presumedly cylindrical and homogeneous, can vibrate longitudinally or transversely if one draws aside it from his balance position . In music we will consider only transverse vibrations.
The cords can be out of steel (piano) or bowel of sheep (violin); one can weigh down them by surrounding them of a silver or copper wire: one then obtains plaited catgut strings (serious notes of the piano or the G of the violin).
To draw aside the cord of his position of balance, one can grip it with the finger (grip), a pick (guitarre, mandoline) or with a feather or spine controlled by the keys of a keyboard (harpsichord). The cord can be struck by a hammer furnished with felt (piano) or even scraped by a wheel (hurdy-gurdy). Lastly, in the violin and the instruments of the same type it is attacked by a bow consisted a of tended hairs and rosin coatings to increase their adherence with the cord.
The bow actuates the cord by friction until the moment when the elasticity of the cord overrides the forces of friction and brings back this one to its of balance. The same phenomenon reproduces a great number of times per second and it is that the frequency of this phenomenon is that of the vibration of the cord, thanks to the phenomenon of resonance.
A cord fixed at its two ends always presents a node of vibration at its ends and a certain number of intermediate nodes. This system of standing waves appears by an integer of spindles distributed along the cord. If k spindles is seen, the length of each one of them being l / 2, the total length of the cord L is given by the expression:
L = k * l / 2
g is the frequency and the v speed of the transverse waves one obtains since l = v / g
L = k * v / 2*g
but v = Ö (F / m) from where L = k /2g * Ö (F / m)
or else g = k/2L * Ö (F / m)
g is in hertz (Hz or 1/s)
F is in newton (N)
L is in meters (m)
v is in meter per second (m/s)
k is an integer (k Î |N)
m m is the linear density of the cord, in grams per meter (g/m)
EXERCISE
A 1m cord and total mass 5g vibrating in only one spindle gives a sound of frequency 130.5Hz. Calculate its tension F.
F = 4*m*L²*g² / k² from where (k = 1) F = 4*m*L²*g²
F = 4.5.10exp(-3)*(130.5)² = 345N
F = 35kg
One uses the sonometer to also check the laws of the vibrating cords qualitatively and quantitatively.
To check the law of the harmonics of a cord let us touch very slightly for example this cord in its medium while it vibrates; we remove the fundamental sound to which corresponded a belly in this place, but the cord continues to vibrate while returning the octave of the preceding note, Thus the violonists, by grazing the cord of their violin in a suitable place make return to this cord such or such harmonic.
Sound pipes
It is a tube out of wooden or of metal in general cylindrical of circular or rectangular section, inside whose the air in vibration presents a system of standing waves corresponding to an audio frequency. This vibratory phenomenon is characterized by bellies of vibration (nodes of pressure) and nodes of vibration (bellies of pressures), when they are plane waves, which is the case for a cylindrical pipe. The source of vibration is at the edge of the pipe: mouth of flute or one a sheer actuated by a draught. This source emits a sound badly definite, complex, in which is the suitable frequency to produce the system of standing waves in a given pipe. The vibrating pipe reacts then on the source; the different vibrations than that which reinforces the pipe are very quickly damped out.
Two ends where reign a constant pressure are nodes of pressure thus bellies of vibration.
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V N V
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V N V N V
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<----------l / 2---------->
If there are only these two bellies, the pipe makes the fundamental sound . There can be others corresponding to the various partial sounds.
A pipe is closed when the end opposed to the mouth is provided with a bottom. There is thus a node of vibration on this bottom, but a belly remains with the mouth.
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| N V
| _________________________________________________
<---------------------------L------------------------------------>
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| N V N V
| _________________________________________________
<-------l / 4------>
For the open pipes (length L)
L = k * l / 2
If v is the speed of sound in the gas which fills the pipe l = v / g
g = k* v / 2L
k = 1 corresponds to the fundamental sound, k = 2, 3..... various partial. The partial possible ones are harmonics of the fundamental one.
EXERCISE
Speed of sound in the air being of 340 m/s, calculate the length of the pipe which emits an A3 (440Hz) like its fundamental.
L = k* v / 2g = v / 2g = 340/ 880 = 0.39m
For the closed pipes
L = (2k + 1) * l / 4
g = (2k +1)* v / 4L
k = 0 corresponds to the fundamental sound. k = 1, 2..... various partial, whose frequency is an odd multiple of the frequency fundamental.
Example
panpipes: 180, 160, 144, 135, 120, 108, 96, 90
It will be noticed that a closed pipe gives the same note as an open pipe double length (bumblebee of the organ).