Derivation of Acoustic Wave Propagation in a Tube and the Cascaded-Tube Section Scattering Relations

An excellent derivation of the acoustic wave equation in a tube may be found in [Cook 1990]. The main points are summarized here:

We require the formulae for point-wise conservation of momentum and mass within the tube:

$$a(x)\frac{\partial p(x,t)}{\partial x} = -\rho \frac{\partial u(x, t)}{\partial t} ,$$ (1)

and

$$\frac{\partial u(x, t)}{\partial x} = -\frac{a(x)}{\rho c^2} \frac{\partial p(x,t)}{\partial t},$$ (2)

where $a(x)$ is the cross-sectional area of the tube at point $x$ (in square meters), $p(x,t)$ is the longitudinal pressure (in Newtons per square meter), $\rho$ is the density of the fluid in the tube (in kilograms per cubic meter), $u(x,t)$ is the volume velocity of the fluid in the tube (in cubic meters per second), and $c$ is the speed of sound in the fluid.

If $a(x)$ is assumed to be a constant, $A$, then combining the two equations yields

$$\frac{\partial^2 u(x,t)}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 u(x,t)}{\partial t^2} ,$$ (3)

or, for pressure,

$$\frac{\partial^2 p(x,t)}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 p(x,t)}{\partial t^2} .$$ (4)

These may each be recognized as dual forms of the homogeneous wave equation, the solution of which allows for arbitrary left- and right-going traveling wave components (these will be hereafter referred to as $u^{+}$ and $u^{-}$ for volume velocity, and $p^{+}$ and $p^{-}$ for pressure). The general solutions to Eq. (3) and Eq. (4) are given by:

$$u(x,t) = u^{+}(t-\frac{x}{c}) + u^{-}(t+\frac{x}{c}),$$ (5)

and

$$p(x,t) = p^{+}(t-\frac{x}{c}) + p^{-}(t+\frac{x}{c}).$$ (6)

Since the forward- and reverse-traveling wave components each satisfy the wave equation separately (this can be shown by direct substitution), Eq. (1) yields

$$A \frac{\partial p^{+}(t-\frac{x}{c})}{\partial x} = - \rho \frac{\partial u^{+}(t-\frac{x}{c})}{\partial t}$$ (7)

$$\Rightarrow \frac{\partial p^{+}(t-\frac{x}{c})}{\partial x} = \frac{\rho c}{A} \frac{\partial u^{+}(t-\frac{x}{c})}{\partial x}$$ (8)

$$\Rightarrow p^{+}(t-\frac{x}{c}) = \frac{\rho c}{A} u^{+}(t-\frac{x}{c}),$$ (9)

and similarly, for the left-traveling wave component,

$$A \frac{\partial p^{-}(t+\frac{x}{c})}{\partial x} = - \rho \frac{\partial u^{-}(t+\frac{x}{c})}{\partial t}$$ (10)

$$\Rightarrow \frac{\partial p^{-}(t+\frac{x}{c})}{\partial x} = - \frac{\rho c}{A} \frac{\partial u^{-}(t+\frac{x}{c})}{\partial x}$$ (11)

$$\Rightarrow p^{-}(t+\frac{x}{c}) = - \frac{\rho c}{A} u^{-}(t+\frac{x}{c}).$$ (12)

Define the characteristic impedance of the tube as

$$R = \frac{\rho c}{A}$$ (13)

$$\Rightarrow p^{+}(t-\frac{x}{c}) = R u^{+}(t-\frac{x}{c}),$$ (14)

$$p^{-}(t+\frac{x}{c}) = - R u^{-}(t+\frac{x}{c}).$$ (15)

Consider a pair of tubes (with characteristic impedances $R_1$ and $R_2$) in cascade. The scattering relation for the junction between these tubes may be derived as follows:

$$0 = u_1-u_2$$ (16)

$$= u_1^{+} + u_1^{-} - u_2^{+} - u_2^{-}$$ (17)

$$= G_1(p_1^{+} - p_1^{-}) + G_2(p_2^{-} - p_2^{+})$$ (18)

$$= G_1(2 p_1^{+} - p_J) + G_2(2 p_2^{-} - p_J)$$ (19)

$$\Rightarrow p_J = \frac{2 p_1^{+} G_1 + 2 p_2^{-} G_2}{G_1 + G_2}$$ (20)

$$\Rightarrow p_1^{-} = p_J - p_1^{+}$$ (21)

$$= \frac{G_1 - G_2}{G_1 + G_2} p_1^{+} + \frac{2 G_2}{G_1 + G_2} p_2^{-}$$ (22)

$$= \frac{R_2 - R_1}{R_1 + R_2} p_1^{+} + \frac{2 R_1}{R_1 + R_2} p_2^{-},$$ (23)

$$\Rightarrow p_2^{+} = p_J - p_2^{-}$$ (24)

$$= \frac{G_2 - G_1}{G_1 + G_2} p_2^{-} + \frac{2 G_1}{G_1 + G_2} p_1^{+}$$ (25)

$$= \frac{R_1 - R_2}{R_1 + R_2} p_2^{-} + \frac{2 R_2}{R_1 + R_2} p_1^{+},$$ (26)

where $u_1$ and $u_2$ are the volume velocities in the two tubes at the junction, $u_1^{+}$ and $u_1^{-}$ are the forward- and reverse-traveling volume velocity wave components in tube 1 at the junction, $u_2^{+}$ and $u_2^{-}$ are the forward- and reverse-traveling volume velocity wave components in tube 2 at the junction, $p_1$ and $p_2$ are the pressures in the two tubes at the junction, $p_1^{+}$ and $p_1^{-}$ are the forward- and reverse-traveling pressure wave components in tube 1 at the junction, $p_2^{+}$ and $p_2^{-}$ are the forward- and reverse-traveling pressure wave components in tube 2 at the junction, $p_J$ is the junction pressure (must be identical to $p_1$ and $p_2$ to avoid a pressure gradient across a zero-width (and hence, zero-mass) section), and $G_1 = \frac{1}{R_1}$ and $G_2 = \frac{1}{R_2}$ are the acoustic conductances in the two tube sections. Note Eq. (16) expresses the conservation of volume velocity, and Eq. (19) uses the aforementioned continuity of pressure across the junction.

The following subsections deal with sample implementations of this vocal synthesis technique.