Almost all of my references do not designate in relation to + or (-) waves. Bell's work does. In relation to a wave reflecting of a surface, (ala Bell), do not remember a designation associated to it.
Forgive me if I am repeating something you already know...I just want to make sure we are on the same page. When the exhaust port is opened, a positive pressure wave is produced. If this wave encounters the open end of a pipe, a portion of that wave changes sign, and a portion continues on as positive pressure.
A simple open pipe produces a strong negative wave, but the duration is too short to be useful. Having a cone at the end stretches out the duration of the negative wave, at the expense of the strength.
If the negative wave returns at the right time, it can greatly assist with scavenging when the pressure inside the cylinder is low. Assisting too much draws charge into the exhaust at the same time velocity from the transfers is decreasing, resulting in less charge density in the cylinder.
The solution is to have the remaining positive wave reflect back, which it will do when it hits a constriction in the pipe. Again, a cone can stretch out the duration at the expense of intensity. If this wave returns just prior to the exhaust port closing, it can force charge back into the cylinder. If the exhaust is not long enough, however, the positive wave returns too early, and interferes with scavenging.
From what i have gathered, the resonator (muffler) wants to default to its single resonant frequency. As extra exhaust is pushed into it, and then released, the pressure will drive it out. Since, the incoming exhaust charge is of higher pressure than contained inside the muffler, the exhaust will overshoot out of the muffler exit hole and produce a slight vacuum inside the muffler. At this point the exhaust gasses remaining in the muffler will oscillate into and out of the exit hole at the muffler's frequency.
The exit hole in the exhaust, or a stinger, is not involved in tuning, but rather acts as a resistor to produce back pressure. This back pressure also works to prevent too much charge from being drawn through the exhaust port. The "vacuum" you envision is actually an increase in pressure.
Pressure remains an unsolvable variable at least for me mathematically.
It isn't the only unsolvable variable. The math to determine tuned length only works for pipes. If, say on a motorcycle, you flatten one side to make clearance for the frame, the muffler will be pretty much useless.
Even working with pipes, the math will only be an approximation, and you still will be left with "cut and try".
Working on an visual estimate of how it effects the muffler damping by 3D graphing, 3 mufflers tuned to the same frequency, using the same muffler volume, but with differing area values of exit holes, and stinger pipe lengths. Much like i did for the examples A, B, C earlier in the thread. Maybe there is a pattern trend line when graphed against vs. frequency then again vs RPM.
The differences you will see will likely be due to striking a balance between creating the right amount of back pressure and restricting flow, rather than wave action.
Now that's funny!
