Listening to Liberty: How Modal Analysis Sheds Light on the Sound of the Liberty Bell

Sean Collier – smc604@psu.edu

The Pennsylvania State University Applied Research Lab, University Park, PENNSYLVANIA, 16804, United States

Jonathan Young
Aaron Stearns

Popular version of 4aMU5 – Modal Analysis of a Liberty Bell Replica
Presented at the 190th ASA Meeting
Read the abstract at https://eppro01.ativ.me/web/planner.php?id=ASASPRING2026

–The research described in this Acoustics Lay Language Paper may not have yet been peer reviewed–

More often than not, what crosses your mind when hearing “Liberty Bell” is not the chime or musicality of the bell; rather, it is likely the infamous crack through its left-hand side. On this 250th anniversary, it is worthwhile to pause in its silence, and consider how it might have sounded when newly cast in 1751.

Predicting the sound of a bell feels, at first pass, relatively straightforward. The bell has a simple geometry, which is important for understanding how its vibration patterns radiate sound to your ears. The bell is made of bronze, which is essential for predicting the exact pitches at which the bell will sound, as well as how long it rings. These parameters, considered together, define this action of vibrations along the surface pushing air into oscillation, leading to the radiation of sound and the bell’s characteristic chime. The physics are well understood, but the Liberty Bell, in particular, is not. That is, surprisingly little exists in the way of geometry and material composition for the bell, which makes it a particularly challenging example.

Figure 1. Photo of the Liberty Bell replica at Penn State Behrend

Simply using the meager information available online produces, frankly, a terrible sounding bell that in presence does not match the silhouette of the iconic bell. Rather than toy with parameters until things seemed “right,” we sought out a replica to measure its geometry and vibroacoustic response, so to calibrate our prediction. Modal Analysis, the act of exciting a structure to understand its vibration patterns – called modes – as well as its frequencies at which the modes vibrate, is a common tool used to isolate this information in a meaning and practical way. Some results from this modal analysis are compared in Figure 2, showing the measured vibration patterns for the replica to classical results from Rossing and Perrin [1].

Figure 2. Comparison of the first few vibration patterns between the replica and theory

Knowing the modes and frequencies was only half the effort, though, as we noted that the geometry defines so much of the sound that we eventually perceive. Indeed, small changes to the geometry could alter the prediction considerably. To have the model be as close to truth as possible, a 3D scan of the replica was done to produce a geometry – making it likely the most accurately modeled cast bell to ever exist! Once the geometry, frequencies, and modes are in place, the prediction could be tuned so to back out the bell’s material properties – “Bell Bronze”, intrinsic to the distinct ring of bells.

Figure 3. 3-dimensional scan of the replica to define the cross-section and model geometry

Through modal analysis of this replica, we were able to tune a predictive model of the bell to match the measured vibroacoustic response. Beyond the pretty shapes, the analysis tells us how pitches in the chime relate in strength and in time, illuminating the evolution of the sound over time and adding scientific context to something so often overlooked in the story of the Liberty Bell.

References:

[1] Rossing, Thomas D., and Robert Perrin. “Vibrations of bells.” Applied Acoustics 20.1 (1987): 41-70