The Ideal Waveguide

First we address the scalar case. For an ideal acoustic tube, we have the following wave equation [27]:

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

where $p(x,t)$ denotes (scalar) pressure in the tube at the point $x$ along the tube at time $t$ in seconds. If the length of the tube is $L_R$, then $x$ is taken to lie between $0$ and $L_R$. We adopt the convention that $x$ increases ``to the right'' so that waves traveling in the direction of increasing $x$ are referred to as ``right-going.'' The constant $c$ is the speed of sound propagation in the tube, given by $c=\sqrt{K/ \mu }$, where $K$ is the ``tension'' of the gas in the tube, and $\mu$ is the mass per unit volume of the tube. The dual variable, volume velocity $u$, also obeys (1) with $p$ replaced by $u$. The wave equation (1) also holds for an ideal string, if $p$ represents the transverse displacement, $K$ is the tension of the string, and $\mu$ is its linear mass density.

The wave equation (1) follows from the more physically meaningful equations [28, p. 243]:

$$\displaystyle -\frac{\partial p(x,t)}{\partial x} = \mu \frac{\partial u(x,t)}{\partial t}$$ (2)

$$\displaystyle % needed for each line -\frac{\partial u(x,t)}{\partial x} = K^{-1} \frac{\partial p(x,t)}{\partial t} \,.$$ (3)

Equation (2) follows immediately from Newton's second law of motion, while (3) follows from conservation of mass and properties of an ideal gas.

The general traveling-wave solution to (1), or (2) and (3), was given by D'Alembert as [27]

$$\begin{array}{l} p(x,t)=p^+(x-c t)+p^-(x+c t) \\ u(x,t)=u^+(x-c t)+u^-(x+c t) \,, \end{array}$$ (4)

where $p^+, p^-, u^+, u^-$ are the right- and left-going wave components of pressure and velocity, respectively, and are referred to as wave variables. This solution form is interpreted as the sum of two fixed wave-shapes traveling in opposite directions along the tube. The specific waveshapes are determined by the initial pressure $p(x,0)$ and velocity $u(x,0)$ throughout the tube $x\in [0,L_R]$.