- #1
Annirak
- 4
- 0
I've been studying the behaviour of speakers and headphones for my own interest. I was able to derive the differential equation governing the motion of the speaker itself, what I've had trouble doing is deriving the pressure created by the speaker.
First the equation I've derived for movement of headphones
[tex]ILB=A(P_F-P_B)+kx+\mu\dot{x}+m\ddot{x}[/tex]
where [tex]I[/tex] is the current through the speaker coil
[tex]L[/tex] is the length of the wire in the speaker
[tex]B[/tex] is the field strength of the magnet
[tex]A[/tex] is the area of the speaker cone
[tex]P_F[/tex] is the pressure at the front of the speaker cone
[tex]P_B[/tex] is the pressure at the back of the speaker cone
[tex]k[/tex] is the spring constant of the speaker cone mounting
[tex]x[/tex] is the displacement of the speaker cone
[tex]\mu[/tex] is the coefficient of friction of the speaker cone mounting
[tex]m[/tex] is the mass of the speaker cone assembly
In a closed space (eg closed-back headphones), [tex]\lambda > a[/tex] where a is the minimum dimension of the cavity, so the pressure can be modeled via the ideal gas law:
[tex]PV=nRT \Rightarrow P=\frac{nRT}{V}[/tex]
Correlating this to displacement via [tex]V=A(x_0+x)[/tex],
[tex]P=\frac{nRT}{A(x_0+x)}[/tex]
[tex]P_F[/tex] and [tex]P_B[/tex] each share this model with different values of [tex]A[/tex] and [tex]x_0[/tex], so that the speaker is essentially a diaphragm mounted part way down a sealed cavity.
Now for an open speaker, I'm not so sure. How is the pressure developed in front of a moving speaker related to the motion of the speaker, assuming that the speaker is not at one end of a closed cavity smaller than the minimum wavelength?
First the equation I've derived for movement of headphones
[tex]ILB=A(P_F-P_B)+kx+\mu\dot{x}+m\ddot{x}[/tex]
where [tex]I[/tex] is the current through the speaker coil
[tex]L[/tex] is the length of the wire in the speaker
[tex]B[/tex] is the field strength of the magnet
[tex]A[/tex] is the area of the speaker cone
[tex]P_F[/tex] is the pressure at the front of the speaker cone
[tex]P_B[/tex] is the pressure at the back of the speaker cone
[tex]k[/tex] is the spring constant of the speaker cone mounting
[tex]x[/tex] is the displacement of the speaker cone
[tex]\mu[/tex] is the coefficient of friction of the speaker cone mounting
[tex]m[/tex] is the mass of the speaker cone assembly
In a closed space (eg closed-back headphones), [tex]\lambda > a[/tex] where a is the minimum dimension of the cavity, so the pressure can be modeled via the ideal gas law:
[tex]PV=nRT \Rightarrow P=\frac{nRT}{V}[/tex]
Correlating this to displacement via [tex]V=A(x_0+x)[/tex],
[tex]P=\frac{nRT}{A(x_0+x)}[/tex]
[tex]P_F[/tex] and [tex]P_B[/tex] each share this model with different values of [tex]A[/tex] and [tex]x_0[/tex], so that the speaker is essentially a diaphragm mounted part way down a sealed cavity.
Now for an open speaker, I'm not so sure. How is the pressure developed in front of a moving speaker related to the motion of the speaker, assuming that the speaker is not at one end of a closed cavity smaller than the minimum wavelength?