Results

Figure 7.4 shows the object reflection data for a test object consisting of stepped cylinders and is a longer version of the test object used in chapter 5. The first section of the object is a 50 mm long section with a radius of 5mm (the same as the source tube), the second section is 650 mm long with an internal radius of 6.25 mm and the third section is 650 mm long with an internal radius of 9.4 mm. The object reflections are not long enough in this example to overlap with the source reflections, which can clearly be seen starting at around 60 ms.

Figure 7.4: Reflections from long stepped cylinder test object including source reflections

Figure 7.5 displays the second half of the object reflection data, labelled $R_j$. Also shown is $R_j^+$, the result of applying the source reflection filter to give the forward travelling component of $R_j$, due to source reflections. $R_j^-$ is the result of subtracting $R_j^+$ from $R_j$ to give the backward travelling component, due to any remaining object reflections, with the source reflections cancelled. As we expect for this example, the data in the region shown consists entirely of source reflections with $R_j^-$ equal to zero, give or take a small amount of noise. Figure 7.6 shows the full object reflection data, before and after the cancellation of the source reflections. The success of the post-processing subtraction of the source reflections can be clearly seen.

Figure 7.5: Second half of long stepped cylinder test object reflections

Figure 7.6: Long stepped cylinder test object reflections

Figure 7.7 on the other hand shows the reflections from a French horn. The object reflections are too long to be separated in the time domain from the source reflections. Subtracting the source reflections is therefore vital if the response of the instrument is to be measured.

Figure 7.7: French horn object reflections

Figure 7.8 and figure 7.9 show bore reconstructions of the French Horn. Both use the virtual dc tube method. The line labelled ``Old method'' shows the reconstructed bore of a French horn using the pressure signal shown in figure 7.7 truncated just before the source reflections arrive. There is an upward trend in the bore reconstruction of a section of the pipe which is, in reality, cylindrical. The reconstruction is not accurate because the object reflections have not completely died out at this point. Also shown is a line labelled ``Crook,'' which shows the reconstructed bore of the detached crook (the section of tubing between the mouthpiece and the rest of the instrument). Since the crook section is short, the object and source reflections are separated in the pressure recording and, in this case, truncation of the signal gives an accurate reconstruction. The ``New method'' result uses the post processed version of the figure 7.7 object reflections with the source reflections removed.

The source reflection cancellation reconstruction of the French horn agrees with the reconstruction of the crook section showing that the technique enables longer objects to be measured accurately. While we have not verified that the remainder of the horn is reconstructed correctly (in fact there will still be errors due to higher mode propagation at the bell), it is clear that deviation has been greatly reduced.

Figure 7.8: French horn bore reconstruction, first 3.5 metres

Figure 7.9: French horn bore reconstruction, first 4.3 metres