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We will use the theory of plane wave propagation as set out at the start
of chapter 2 for the analysis which follows in this chapter.
The possibility of including higher mode effects in the analysis will be
discussed in chapter 7.
The pressure at the input to a tubular object will be the sum of the forward
and backward going waves from equations (5.1) and
(5.2):

(5.3) 
Similarly, the volume velocity follows from equation (2.7), giving

(5.4) 
The input impedance is defined as the ratio of the pressure and volume velocity
at the input plane. So far we have obtained a time domain expression for
the pressure and volume velocity if the input impulse response is known.
The input impedance is generally frequency dependent however, so
we must take the Fourier transform of the signals and divide the frequency
components to get the input impedance at a particular frequency.
The Fourier transform of an impulse is 1 for all frequencies. We define
as the Fourier transform of .
The Fourier transform of the pressure is then

(5.5) 
and the volume velocity is

(5.6) 
Dividing in the frequency domain gives the input impedance as

(5.7) 
This equation lets us easily calculate the input impedance of an
object once the input impulse response is obtained by measurement.
Since it is impossible to produce a perfect impulse, measurement of the
input impulse response is not a straight forward task and will be discussed
later in the
chapter. For now we proceed with the background theory to pulse reflectometry.
Back to Kemp Acoustics Home
Next: Layer peeling bore reconstruction
Up: Acoustic pulse reflectometry
Previous: Input impulse response
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Jonathan Kemp
20030324