Subject: Speech Processing

Topic: Speech Production, Acoustic Phonetics and Auditory Perception

Difficulty: High

(i) There will be some energy lost due to the friction between air and walls of the tube and also due to vibration of the tube.

(ii) Consider the vibration of the walls of the vocal tract. When the air pressure inside the tube varies the walls will experience varying force.

(iii) The cross-sectional area of the tube will change depending on the pressure inside, if the walls are elastic.

(iv) If the walls are reacting locally, then we can say that the area A(x,t) will be a function of pressure P(x,t).

(v) Since pressure variations are negligible, the changes in the cross-sectional area can be treated as small disturbances in the normal area and

$ \therefore A(x,t) = A_0(x,t) + \delta A(x,t) $

where, A$_0$(x,t) - nominal area and $\delta$A(x,t) is the small disturbance.

(vi) Considering the mass and the elastic nature of the wall, the relation between pressure changes P(x,t) and the disturbance $\delta$A(x,t) can be represented as:

$ m_w \frac{d^2 (\delta A)}{dt^2} + b_w \frac{d (\delta A)}{dt} + K_w (\delta A) = P(x,t) $

where,

m$_w$ is mass of vocal tract

b$_w$ is damping of vocal tract

K$_w$ is the stiffness of vocal tract

Now neglecting all the second and higher order terms in s/A and PA, we can write,

$ \frac{- \delta s}{\partial x} = \frac{1}{PC^2} \frac{\delta (PA_0)}{\delta t} + \frac{\delta A_0}{\delta t} + \frac{\delta (\delta A)}{\delta t} $ (Sound propogation through a tube having soft wall similar to vocal tract)