Wave Frequency in Piston: Pressure Variation & Adiabatic Compression

[WiFO215]

When you have a wave in sound inside a piston of length L, A.P. French says that the fundamenal frequency $$\nu$$ of an oscillation is given by

$$\nu$$ = 1/4L $$\sqrt{(\gamma$$ p / $$\rho)}$$

Where p is the pressure, $$\gamma$$ is the factor that accounts for adiabatic compression of the gas, and $$\rho$$ is it's density.

My question is this : doesn't the pressure p vary as you compress the gas? How can you assume it to be constant?

 

[WiFO215]

Hold up. I think I got it. That's not ANY pressure that you plug into that formula, it's the pressure when the gas is at equilibrium.

 

[charng]

I can understand your confusion about the assumption of constant pressure in this equation. However, it is important to note that this equation is derived under certain ideal conditions, and in reality, the pressure may indeed vary as the gas is compressed. This equation serves as a simplified model to understand the relationship between wave frequency and piston length, and it can still provide valuable insights and predictions in certain scenarios.

To address your concern, we must first understand the concept of adiabatic compression. Adiabatic compression refers to the process of compressing a gas without any heat exchange with the surroundings. In this process, the gas is compressed quickly enough that there is not enough time for heat transfer to occur. This results in an increase in pressure and temperature of the gas.

Now, let's consider the equation provided by A.P. French. The term \gamma represents the ratio of specific heats for the gas, which takes into account the change in temperature during adiabatic compression. This means that the equation already accounts for the variation in pressure due to the change in temperature.

Furthermore, the assumption of constant pressure is often made in ideal gas laws and thermodynamic equations as a simplification to make calculations and predictions easier. In reality, the pressure may vary, but the overall relationship between wave frequency and piston length still holds true.

In conclusion, while the pressure may indeed vary during adiabatic compression, the equation provided by A.P. French takes into account this variation through the use of the \gamma factor. This equation serves as a useful tool to understand the relationship between wave frequency and piston length, and it can still provide reliable predictions in certain scenarios. As scientists, it is important to understand the assumptions and limitations of equations and models, but they can still be valuable in helping us understand complex phenomena.