Finite Length Cylindrical Pipes

• In a pipe of finite length, propagating wave components will experience discontinuities at both ends.

• A longitudinal wave component which encounters a discontinuous and finite load impedance $Z_{L}$ at one end of the tube will be partly reflected back into the tube and partly transmitted into the discontinuous medium.

• Wave variables in a finite length tube are then composed of superposed right- and left-going traveling waves. In this way, sinusoidal pressure in the pipe at position $x$ is given by
    $$P(x,t) = \left[C^{+}e^{-jkx} + C^{-}e^{jkx}\right] e^{j\omega t},$$ (22)
  where $C^{+}$ and $C^{-}$ are complex amplitudes.

• From Eq. (19), the corresponding volume velocity is found to be
  

  $$U(x,t) = \left(\frac{A}{\rho c}\right)\left[C^{+}e^{-jkx} - C^{-}e^{jkx}\right] e^{j\omega t}$$

  $$= \frac{1}{Z_{c}}\left[C^{+}e^{-jkx} - C^{-}e^{jkx}\right] e^{j\omega t}$$ (23)

• At any particular position $x$ and time $t$, the pressure and volume velocity traveling-wave components are related by
    $$P^{+} = Z_{c}\,U^{+} \hspace{0.5in} P^{-} = -Z_{c}\,U^{-}$$ (24)
  with
    $$P = P^{+} + P^{-} \hspace{0.5in} U = U^{+} + U^{-}$$ (25)

• The plus $(+)$ superscripts indicate wave components traveling in the positive $x$-direction or to the right, while negative $(-)$ superscripts indicate travel in the negative $x$-direction or to the left.

• The characteristic wave impedance $Z_{c}$ is a frequency-domain parameter, though for plane waves of sound it is purely real and independent of position. Therefore, these relationships are equally valid for both frequency- and time-domain analyses of pressure and volume velocity traveling-wave components.

• Traveling waves of sound are typically reflected at an acoustic discontinuity in a frequency-dependent manner.

• A frequency-dependent reflection coefficient, or reflectance, characterizes this behavior and indicates the ratio of incident to reflected complex amplitudes at a particular frequency.

• Similarly, the ratio of incident to transmitted complex amplitudes at a particular frequency is characterized by a frequency-dependent transmission coefficient, or transmittance.

• For a pipe which extends from $x=0$ to $x=L$ and is terminated at $x=L$ by the load impedance $Z_{L}$, the pressure wave reflectance is
    $$\frac{C^{-}}{C^{+}} = e^{-2jkL}\left[\frac{Z_{L} - Z_{c}}{Z_{L} + Z_{c}}\right]$$ (26)
  and the transmittance is
    $$\frac{P(L,t)}{C^{+}} = e^{-jkL}\left[\frac{2 Z_{L}}{Z_{L} + Z_{c}}\right] e^{j\omega t}.$$ (27)

• The phase shift term $e^{-2jkL}$ in Eq. (26) appears as a result of wave propagation from $x=0$ to $x=L$ and back and has unity magnitude.

• The load impedance $Z_{L}$ characterizes sound reflection and radiation at the end of the pipe.

• For low-frequency sound waves, the open end of a tube can be approximated by $Z_{L} = 0.$ In this limit, the bracketed term of the reflectance becomes negative one, indicating that pressure traveling-wave components are reflected from the open end of a cylindrical tube with an inversion (or a $180^{\circ}$ phase shift). There is no transmission of incident pressure into the new medium when $Z_{L} = 0.$

• If the pipe is rigidly terminated at $x=L,$ an appropriate load impedance approximation is $Z_{L} = \infty,$ corresponding to $U(L,t)=0$ for all time. The bracketed term in Eq. (26) is then equal to one, which implies that pressure traveling waves reflect from a rigid barrier with no phase shift and no attenuation. The pressure “transmittance” (a bit of a misnomer in this case), has a magnitude of two at the rigid barrier.

• The impedance at $x=0,$ or the input impedance of the cylindrical tube, is given by
  

  $$Z_{in} = \frac{P(0,t)}{U(0,t)} = Z_{c}\left[\frac{C^{+} + C^{-}}{C^{+} - C^{-}}\right]$$ (28)

  $$= Z_{c}\left[\frac{Z_{L} \cos (kL) + j Z_{c} \sin(kL)}{jZ_{L} \sin(kL) + Z_{c} \cos(kL)}\right].$$ (29)

• The input impedance of finite length bores can be estimated using the low-frequency approximation $Z_{L} = 0$ for an open end and $Z_{L} = \infty$ for a closed end. In this case, Equation (29) reduces to
    $$Z_{in} = jZ_{c}\tan(kL)$$ (30)
  for the ideally open pipe and
    $$Z_{in} = -jZ_{c}\cot(kL)$$ (31)
  for the rigidly terminated pipe.

• In the low-frequency limit, $\tan(kL)$ is approximated by $kL$ and the input impedance of the open pipe reduces to $j \omega \rho L / A.$ This is the expression for the impedance of a short open tube, or an acoustic inertance.

• Making a similar approximation for $\cot(kL),$ the input impedance of the rigidly terminated pipe reduces to $-(j/\omega) (\rho c^{2}/L A),$ which is equivalent to the impedance of a cavity in the low-frequency limit.

• By equating an open pipe end at $x=0$ with a value of $Z_{in} = 0$ in the previous expressions, the resonance frequencies of the open-closed (o-c) pipe and the open-open (o-o) pipe are given for $n = 1, 2, \ldots$ by
  

  $$f^{(\mbox{o-c})} = \frac{(2 n - 1) c}{4 L}$$ (32)

  $$f^{(\mbox{o-o})} = \frac{n c}{2 L},$$ (33)

  respectively.

• The open-closed pipe is seen to have a fundamental wavelength equal to four times its length and higher natural frequencies that occur at odd integer multiples of the fundamental frequency. The following link provides some animations of longitudinal standing-wave patterns in pipes.

• The open-open pipe has a fundamental wavelength equal to two times its length and higher natural frequencies that occur at all integer multiples of the fundamental frequency.

• The input impedance of an acoustic structure provides valuable information regarding its natural modes of vibration. Various methods for measuring and/or calculating the input impedance of musical instrument bores have been reported (Caussé et al., 1984; Benade, 1959; Plitnik and Strong, 1979; Backus, 1974).