Reflection from a Radius Mismatch

1. Download the pd patch vir_tube2.pd, and open it in pd. Figure 2 shows a sample screen capture of the patch.

2. The patch simulates an acoustic tube driven at one end by the input stimulus signal, stim. The length of this tube may be varied using the first horizontal slider on the patch. The other end of the tube is joined to a second cylindrical acoustic tube, whose radius may be adjusted by the second slider on the patch. While the radius of the first tube is set to a constant $10$mm, the radius of the second tube may be adjusted to a value between $0$mm and $100$mm. The second tube is assumed to extend infinitely far to the right. Finally, the temperature may be adjusted using the third slider on the patch.

3. Adjust the first tube length to its minimum possible value. Next, adjust the temperature to the value at which sound should propagate at approximately $1$ft/ms.

4. Using the middle slider, set the second tube radius to $0$mm, its minimum possible value. This is one way to simulate closure of the end of the first tube. What is the reflection coefficient for this radius mismatch?

5. Click the large circular button on the patch. You should see a response similar to that of Figure 2. Does the reflected signal agree with your prediction?

6. Next adjust the second tube radius to $100$mm, its maximum possible value. Repeat the previous two steps for this new radius mismatch. What situation does this radius mismatch resemble?

7. Adjust the second tube radius to an arbitrary value between the minimum and maximum possible values, and repeat the two steps prior to the previous step.

8. What value for the second tube radius should result in negligible reflection of the stimulus signal? Adjust the second tube radius to this value, and click the large circular button to launch a wave towards the tube junction. Do you achieve negligible reflection as desired?

Figure 2: Screen capture showing the cascaded acoustic tube patch with variable first tube length, variable second tube radius, and variable air temperature.