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Pipes and frequencies
To study the physical backgrounds of the acoustical properties of the alphorn we have to go back to the theory of standing sound waves in pipes, in particular conical pipes.
First we repeat 4 well-known models in this theory, Then, in the next section, we will investigate which of these models applies to the acoustics of the alphorn and to what extent.
Model 1:
Open cylindrical pipe, with length L
Vibrating of the air column admits the following frequencies:
f, 2f, 3f, 4f, 5f, ............... where f = v / 2L
v is the speed of sound: v = 331(1 + t / 273) ½ (m/s under t degrees Celsius).
t = 15: v = 340,0
t = 20: v = 342,9
Physical explanation
Based on elementary physics, by studying the possible node and anti-node patterns for the sound wave.
Model 2:
Closed cylindrical pipe (i.e. closed at one end), with length L
Vibrating of the air column admits the following frequencies:
f, 3f, 5f, 7f, ............... where f = v / 4L
Physical explanation
Also elementary, like above.
Model 3:
Conical pipe, with length L
Vibrating of the air column admits the following frequencies:
f, 2f, 3f, 4f, 5f, ............... where f = v / 2L
Physical explanation
Less simple. Can be shown by solving the wave equation.
Model 4:
Truncated conical pipe, with length L and inner diameters d1 and d2
Vibrating of the air column admits the following frequencies:
f1, f2, f3, f4, f5, ............... with
(formula 2 of Neville Fletcher)
where c is the speed of sound and L' = L + 0,3 d2, the so-called acoustic length.
0,3 d2 is the end-correction at the open end op the pipe.
Condition
The apex a = d1L / (d2-d1) is not to small. If a is to small, then model 3 applies.
Physical explanation
Not simple. See: N.H. Fletcher and T.D. Rossing - The Physics of Musical Instruments, 1991.
Particular case: d1 = d2
Then fn = 2 (n-½) c / 4L' = (2n-1) c / 4L'. This is model 2, with L replaced by L'.
The relationship between the models 2, 3 and 4
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If a is to small (a → 0), then model 3 (not a special case of model 4) applies
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If a is not to small, then model 4 applies
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If a → ∞ (i.e. d1 → d2), then model 2 (special case of model 4) applies
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© 2007 J. de Ruiter
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