Adding mass will increase natural frequency by a ratio of the square root of masses.

How much mass does one add for it to be accurate? The amount of mass the pipe has to displace? While not terrible, how do we quantify that?

With regards to tension, it only serves to increase natural frequency.

As motive force is applied, it bends the pipe to the tune of Sb=My/I.

However, if you imagine if instead of bending a volume, you're actually compressing one side and tensioning the other Sa=+/-F/A. And further, if you think of Hooke's law applying... F=kx, applying tension means it takes more energy to bend the pipe, and therefore the natural frequency will increase.

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