Revisiting Koehler's experiment of measuring the ratio of the specific heats of air by self-sustained oscillations: a more concise theoretical interpretation

TL;DR

The paper addresses why Koehler's self-sustained oscillations in the Koehler apparatus have a period close to the Rüchardt frequency $ω$. It develops a transparent theoretical framework by deriving a dimensionless, piecewise-linear differential-system model (PWLD) with a separation surface at $y=0$, and then reduces the 3D PWLD to a tractable 2D PWLD to study periodic solutions. The analysis shows that, near each half-space, the dynamics are governed by stable focus behavior with decaying harmonic perturbations of frequency near unity in scaled time, and that the overall period remains near $2π$ (the reduced Rüchardt period) when the ball passes the hole once in each direction per cycle. The key finding is that, if a periodic solution exists with a hole-cross per half-cycle, the oscillator’s frequency indeed closely matches the Rüchardt frequency $ω$, providing an accessible, geometry-based explanation that supports teaching and student exploration of nonlinear self-sustained oscillations.

Abstract

We revisit Koehler's experiment, a clever modification of Ruchardt's experiment designed to measure the ratio of specific heats of gas. The theory of self-sustained oscillations in Koehler's experiment was provided by Koehler (1950). However, its lengthy and dense analysis may pose challenges to readers due to the complexity of the calculations. Following Koehler's approximation for pressure changes, we explicitly present the model equations as piecewise linear differential systems and qualitatively analyze the periodic solutions from a geometric perspective. This concise and transparent approach addresses a fundamental question about Koehler's experiment: why is the oscillation frequency nearly equal to the Ruchardt frequency? Our analysis avoids intricate calculations and will be particularly helpful for teachers and students who encounter Koehler's experiment in general physics laboratory classes.

Figures (7)

• Figure 1: (a) and (b) Schematic diagrams of Rüchardt apparatus and Koehler apparatus. (c) An image of experimental apparatus.
• Figure 2: (a) and (b) The phase portrait and transition map of 2D system with separation line $\{p=0\}$. The parameters of numerical simulation are $G_{+}=0.3$, $G_{-}=0.075$, and $J=1.0628\times10^{-3}$. The red solid circle and blue hollow circle are two foci, while the hollow upward triangles and downward triangles are the intersections of the trajectory and the separation line. (c)-(f) The oscillations of $u$ and $p$, respectively. (d), (f) correspond to the enlarged display of shadow region in (c), (e) respectively.
• Figure 3: (a) and (b) The phase portrait and transition map of 2D system with separation line $\{p=p_1\neq 0\}$. The parameters of numerical simulation are $G_{+}=0.3$, $G_{-}=0.075$, and $J=1.0628\times10^{-3}$. The red solid circle and blue hollow circle are two foci, while the hollow upward triangles and downward triangles are the intersections of the trajectory and the separation line. (c) The oscillations of $u$. (d) The enlarged display of shadow region in (c).
• Figure 4: (a) and (b) The phase portrait and transition map of 3D system with separation plane $\{y=0\}$. The parameters of numerical simulation are $G_{+}=0.3$, $G_{-}=0.075$, and $J=1.0628\times10^{-3}$. Under the setting of these parameters, the oscillation of displacement $y$ is symmetrical with respect to plane $\{y=0\}$. (c)-(h) The oscillations of $y$, $u$ and $p$, respectively. (d), (f) and (h) correspond to the enlarged display of shadow region in (c), (e) and (g) respectively.
• Figure 5: The phase portrait, transition map and $y$, $u$ oscillation of 3D system with separation plane $\{y=0\}$. The simulation parameters corresponding to the figures in the left column are $G_{+}=0.3$, $G_{-}=0.075$, and $J=1.2\times10^{-3}$. The simulation parameters corresponding to the figures in the right column are $G_{+}=0.3$, $G_{-}=0.075$, and $J=0.9\times10^{-3}$. under the setting of these parameters, the oscillation of displacement $y$ is asymmetrical with respect to plane $\{y=0\}$.