- 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

(18) *C*^{+}and*C*^{-}are complex amplitudes. - From Eq. (15), the corresponding volume velocity is found to be

*U*(*x*,*t*)= = (19)

- At any particular position
*x*and time*t*, the pressure and volume velocity traveling-wave components are related by

(20)

(21) - 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

and the transmittance is

(23) - 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
*Z*_{L}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
*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 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 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
*Z*_{L}= 0 for an open end and for a closed end. In this case, Equation (25) reduces to

(26)

(27) - 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*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 by

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 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 (Benade, 1959; Backus, 1974; Caussé et al., 1984; Plitnik and Strong, 1979).

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