Modeling Pressure in a Sound Wave Instrument with External Force

[Josiaah]

Hello everyone,

I’m working on a theoretical model for a sound wave instrument that involves a pipe. Traditionally, sound wave instruments utilize an open tube, but I’m interested in exploring what happens when I apply a force F to the pipe.

How can I theoretically model the relationship between the pressure inside the pipe and the applied force F? Given that this modification might invalidate the standard D'Alembert equation, what alternative approaches should I consider? Additionally, how can I relate this setup energetically through equations?

Any insights or references would be greatly appreciated!

Thank you!

 

[Baluncore]

Welcome to PF.

How can I theoretically model the relationship between the pressure inside the pipe and the applied force F?

Do you use an internal piston in the tube, to change the length and pressure?
We need a diagram that shows the way the force is applied to the closed pipe.

The speed of sound is not pressure dependent, but rapid changes to internal pressure, that momentarily heat or cool the internal gas, will change the speed of sound inside the pipe momentarily.

Applying a force to the pipe will elastically deform the tube material, but that will probably not alter the wall elasticity or pipe volume.

 

[Josiaah]

Thanks for the response! To clarify, I'm not using an internal piston, but instead applying a force to the pipe using an external paddle (as shown in the diagram below). The tube is open at one end, and the paddle strikes the tube externally, applying a force F to the pipe wall. The idea is that this external impact causes pressure waves inside the pipe.

 

[Baluncore]

You are striking the wall of the tube, at the open end of the tube. I will assume it is a quick strike, and that the paddle is not still closing the tube, when a reflected wave returns. I also assume that there is no bending sideways wave generated in the tube wall.

Two things will be initiated.

1. A compression wave travels quickly away from the paddle, along the wall of the tube. When that reaches the closed end of the tube, it will move the back wall, to generate a reflection that will travel back towards the paddle through the wall. At the same time, a depressive wave will travel backwards in the air in the tube, returning to the open end, where it will be reflected again.

2. A slower air compression wave, followed by an immediate depression, will be launched by the paddle, to travel at the speed of sound in air, from the paddle, towards the closed end of the tube, where it will be reflected, returning to the open end of the tube, to be reflected again from the open end, by then without the paddle.

The sum of those waves will be a complex sound with two resonant frequencies. Just what it will sound like, will be determined by the length of the tube, and by the speed of sound in the tube wall material.

There will be critical pipe lengths where the resonant tube-wall wave and the resonant wave in the air will coincide in time, to reinforce or cancel.

What is the wall material?
What is the approximate length of the tube?

 

[Josiaah]

Thank you for the detailed explanation!

I’d like to clarify a couple of points: the tube I’m working with is open on both ends, and only one end is struck by the paddle. The paddle doesn't close the tube; it just applies a quick force to the open end. The tube is made of PVC, and it’s relatively short.

I know the formula relating the tube’s length to its resonant frequency, and I’m aware that the amplitude of the sound wave increases with the strength of the hit — the harder you strike, the louder the sound. So there must be a relationship between the applied force and the amplitude of the sound wave. However, I’m struggling to fully model this relationship in terms of equations.

I was also wondering if approaching it energetically might simplify the modeling. By looking at the energy transferred from the paddle to the pressure waves in the air, perhaps I can derive a clearer expression for this relationship. Once I have a working model, I’d like to validate it experimentally by varying parameters such as force and material.

Thanks again for your help!

 

[Baluncore]

Thank you for the detailed explanation! I’d like to clarify a couple of points: the tube I’m working with is open on both ends, and only one end is struck by the paddle. The paddle doesn't close the tube; it just applies a quick force to the open end. The tube is made of PVC, and it’s relatively short.

Speed of sound in PVC = 2395 m/s.
Speed of sound in air = 343 m/s.

With both ends of the tube open, the wave in the wall of the tube will not be important to the sound, but it will consume significant energy from the paddle. The energy lost to the tube will be determined by the paddle material and how well energy is coupled into the tube material. It will come down to reflection and transfer coefficients for the collision. That will not be easy to model theoretically with equations. I expect you will need to model it numerically, or measure it while experimenting.

You are simply exciting an organ pipe, open at both ends, by slapping one end with a paddle, in effect bouncing the paddle off an end. Apprentice plumbers often play the PVC pipe, with the palm of their hand.