physics of sound at pipe

[alejandrito20]

in a pipe , open and closed at the ends, the frecuency is

[math]f=(2n-1)\frac{v}{4L}[/math],

if [math]L[/math] decreases, then [math]f[/math] increasses.

 

I don't understand why in a text of physics of music (in spanish) says:

 

"in a pipe, closed in the vocal fold, and open in lips....in the lips there is antinodes of velocity...

if the lips widen, then frecuency increasses"

 

someone tell me the mathematics asociated to this afirmation?

[baxtrom]

in a pipe , open and closed at the ends, the frecuency is

[math]f=(2n-1)\frac{v}{4L}[/math],

if [math]L[/math] decreases, then [math]f[/math] increasses.

 

I don't understand why in a text of physics of music (in spanish) says:

 

"in a pipe, closed in the vocal fold, and open in lips....in the lips there is antinodes of velocity...

if the lips widen, then frecuency increasses"

 

someone tell me the mathematics asociated to this afirmation?

 

 

In a pipe which is closed in one end (x = 0) and open in the other end (x = L), the boundary conditions become

 

[math] u = 0 [/math] at [math] x = 0 [/math], and

 

[math] p = 0 [/math] at [math] x = L [/math],

 

where [math]u[/math] is the particle velocity and [math]p[/math] the acoustic pressure. The first condition is natural since the wall prohibits motion. The second condition is perhaps less intuitive but can be seen as "grounding" the acoustic pressure. The pressure wave is so to say short circuited into the void. And, as it happens the physics of wave propagation imply that if at [math] x=L [/math] there is a pressure node, then there must be a velocity antinode at the same location. This follows from the fact that the pressure in a channel can be written as the composition of one wave traveling in the +x direction and another traveling in the -x direction. Using complex notation,

 

[math] p = e^{i \omega t} \Big[ A e^{-i k x} + B e^{i k x} \Big] [/math],

 

where [math] \omega [/math] is the frequency and [math] k [/math] is the "wave number". The particle velocity becomes

 

[math] u = \frac{1}{\rho c} e^{i \omega t} \Big[ A e^{-i k x} - B e^{i k x} \Big] [/math],

 

where [math] \rho [/math] is the density and [math] c [/math] the speed of sound. Note the change of sign for the B-wave, which is due to the fact that it is traveling in the -x direction. Note also that if [math] u = 0 [/math] then also the derivative of [math] p [/math] is zero and thus the amplitude of [math] p [/math] is an extreme.

 

The boundary condition at [math] x=L [/math] is actually only approximate and more complicated expressions are found in the litterature.