Exploring Fourier methods with beer bottles

Abstract

As anyone who has blown across the mouth of a beer bottle knows, beer bottles have a well-defined fundamental frequency. This paper shows how a beer bottle's acoustical resonance can be modeled as a one-dimensional driven-damped oscillator and includes enough detail to be useful in undergraduate laboratory experiments. While the frequency-domain Green's function of the bottle can be extracted through sequential pure-tone measurements, sufficient data to fit the model's parameters can be collected in just a few seconds when Fourier methods are used.

Figures (6)

• Figure 1: Experimental setup. A microphone (A) measures the signal from a speaker (B), which is driven by an amplifier (C). (The stack of books boosts the speaker up to the microphone height.) Time-series data from the amplifier voltage and the microphone signal are measured and fed into the computer (D), which may also monitor the room temperature via an optional thermometer (E). Each measurement is taken both with the bottle below the microphone, and without the bottle.
• Figure 2: Left: Real and imaginary parts of $G(\omega)$. Right top: Magnitude of $G(\omega)$. Right bottom: Phase of $G(\omega)$. An unrealistically large $\beta = \omega_0/5$ has been used for plotting; typically, $\beta \ll \omega_0$, making the peaks in $G(\omega)$ much narrower.
• Figure 3: Measurements obtained from pure tones at different angular frequencies. Top: Normalized amplitude data $P_B/P_S$ and $P_M/P_S$, along with a fit to our model (Eq. \ref{['eq:driven-damped amplitude']}). Bottom: Measured phase $\delta_B$ (Eq. \ref{['eq:delta_B']}) vs. the fit to our model (Eq. \ref{['eq:G(omega) phase shift prediction']}, with the same parameters as for the upper panel).
• Figure 4: Measurements corresponding to a 20.0 s chirp signal sweeping from 100-300 Hz sent by the speaker. Top: The microphone signal $p_M(t)$ with the bottle underneath it. Bottom: The microphone signal $p_S(t)$ without the bottle below it. For an ideal speaker, the $p_S(t)$ amplitude would be even across all frequencies.
• Figure 5: Top: Spectra of $p_S$ and $p_M$, close to the resonance frequency. Bottom: The function $R(\omega)$, with the frequencies $\omega_1$ and $\omega_2$ labeled by vertical dashed lines ($\omega_1 < \omega_2$), and the value $R_1$ labeled by a horizontal dashed line.