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 ZL 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
where C+ and C- are complex amplitudes.
From Eq. (15), the corresponding volume velocity is found to be
At any particular position x and time t, the pressure and volume velocity traveling-wave components are related by
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 Zc 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 ZL, the pressure wave reflectance is
and the transmittance is
The phase shift term e-2jkL in Eq. (22) appears as a result of wave propagation from x=0 to x=L and back and has unity magnitude.
The load impedance ZL characterizes sound radiation at the open end of the pipe.
For low-frequency sound waves, the open end of a tube can be approximated by ZL = 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 phase shift). There is no transmission of incident pressure into the new medium when ZL=0.
If the pipe is rigidly terminated at x=L, an appropriate load impedance approximation is
corresponding to U(L,t)=0 for all time. The bracketed term in Eq. (22) 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
The input impedance of finite length bores can be estimated using the low-frequency approximation ZL = 0 for an open end and
for a closed end. In this case, Equation (25) reduces to
for the ideally open pipe and
for the rigidly terminated pipe.
In the low-frequency limit, is approximated by kL and the input impedance of the open pipe reduces to
This is the expression for the impedance of a short open tube, or an acoustic inertance.
Making a similar approximation for the input impedance of the rigidly terminated pipe reduces to
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 Zin = 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
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 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.