What did the Liberty Bell sound like? Science is on the case.

The Liberty Bell, with its famous crack, hasn’t been rung in 180 years, but that hasn’t stopped scientists from pondering what the bell might have sounded like when tolling. In honor of the nation’s 250th anniversary, researchers at Penn State University have modeled its acoustic properties, drawing on data collected from one of the bell’s replicas.

自由钟因其著名的裂纹已经有180年没有敲响了，但这并没有阻止科学家们思考它在鸣响时可能发出的声音。为了纪念美国建国250周年，宾夕法尼亚州立大学的研究人员利用从该钟复制品中收集的数据，对其声学特性进行了建模。

The work was first presented at a meeting of the Acoustical Society of America in Philadelphia earlier this year, followed by an article in Physics Today. Historians aren’t sure exactly when the Liberty Bell acquired its distinctive crack, just that it occurred sometime between 1817 and 1846.

这项研究成果于今年早些时候在费城举行的美国声学学会会议上首次发表，随后刊登在《今日物理》杂志上。历史学家们并不确定自由钟究竟是在何时出现那道标志性裂纹的，只知道它发生在1817年至1846年之间的某个时间。

An 1846 account in the Public Ledger described the ringing of the bell on February 23 that year and mentioned that the bell had developed a crack, which someone had attempted to fix by filing the sides down. But it didn’t work: The February 23 ringing extended the crack even more, so much so that “the old Independence Bell … now hangs in the great steeple irreparably cracked and forever dumb.” Nobody has heard it ring since.

1846年《公共分类账报》的一篇报道描述了当年2月23日钟声敲响的情况，并提到钟上已经出现了一道裂纹，有人曾试图通过打磨侧面来修复它。但这种方法没有奏效：2月23日的鸣响使裂纹进一步扩大，以至于“古老的独立钟……如今悬挂在大尖塔中，裂纹无法修复，从此永远沉默。”此后，再也没有人听到过它的钟声。

This wasn’t the first time the bell had cracked; the rim cracked shortly after it arrived in Philadelphia in 1752, the first time the clapper struck while testing the sound. It was recast by two local foundry workers, John Pass and John Stow, not once but twice, because the first recast produced a sound that one bystander likened to coal scuttles banging together.

这并不是该钟第一次破裂；1752年它刚运抵费城时，在测试声音的首次敲击中，钟沿就出现了裂纹。它由当地两名铸造工人约翰·帕斯和约翰·斯托重新铸造，而且不止一次，因为第一次重铸后的声音被一位旁观者形容为像煤斗撞击在一起的声音。

The second recast’s sound was deemed acceptable, and the bell was typically used to summon the Assembly, among other purposes, until it fell into disuse after the American Revolutionary War. As for why the bell was so prone to cracking, a 1975 study of the bell’s chemical composition revealed that the metals used formed “an extremely brittle alloy” that also made it easier for souvenir collectors to chip off pieces from the rim.

第二次重铸的声音被认为是可以接受的，此后该钟通常被用于召集议会等用途，直到美国独立战争后被废弃。至于为什么该钟如此容易破裂，1975年对其化学成分的一项研究显示，所使用的金属形成了一种“极其脆弱的合金”，这也使得纪念品收藏者更容易从钟沿上敲下碎片。

We do know something about the general acoustics of bells, dating back at least to a seminal 1987 paper by Thomas D. Rossing and Robert Perrin exploring the forms and vibrational modes of church bells, carillon bells, and handbells. A given bell’s sound is determined by relatively simple geometry—which determines the vibration patterns radiating sound to listeners’ ears—and the material of which it is made (bronze), which determines the exact pitch and duration of the ring.

我们确实对钟的一般声学原理有所了解，这至少可以追溯到托马斯·D·罗辛和罗伯特·佩林在1987年发表的一篇开创性论文，该论文探讨了教堂钟、钟琴和手铃的形状及振动模式。特定钟的声音由相对简单的几何形状（决定了向听众耳朵辐射声音的振动模式）以及制造材料（青铜，决定了钟声的确切音高和持续时间）所决定。

Per Sean Collier, co-author of this latest article, relying solely on the meager online information about the Liberty Bell to replicate its sound produces “a terrible sounding bell that in presence does not match the silhouette of the iconic bell.” There are several full-size replicas of the Liberty Bell made in the 20th century, including one housed on the Penn State Behrend campus, and one can tap that bell to produce a ring.

据最新文章的合著者肖恩·科利尔称，仅依靠网上关于自由钟的微薄信息来复制其声音，会产生“一种声音极其糟糕的钟，其外观也与标志性的自由钟轮廓不符”。20世纪制造了几个自由钟的全尺寸复制品，其中包括存放在宾夕法尼亚州立大学贝伦分校的一个，人们可以敲击那个钟来发出响声。

But scientists desire more quantitative information, such as the specific notes, which notes ring the longest, and how the bell deforms as it vibrates. And that requires modal analysis, a technique used by bell manufacturers to assess tuning and make any necessary corrections.

但科学家们需要更定量的研究信息，例如具体的音符、哪些音符持续时间最长，以及钟在振动时如何变形。这需要模态分析，这是钟表制造商用来评估调音并进行必要修正的一种技术。

For instance, Western church bells are typically tuned to the first five normal frequencies, according to the authors—the hum note, fundamental (the note that is most dominant to the human ear), tierce, quint, and nominal—following well-defined ideal ratios. “Musically, the frequencies sound as a minor triad, which is why church bells are commonly perceived as having a somber tone,” they wrote. Further, “The human ear perceives the pitch of a bell as a subjective tone called the strike note,” arising from the combination of how the mode shapes interact, and the listener’s own expectations.

例如，作者指出，西方教堂钟通常根据明确的理想比例，调至前五个标准频率——基音（hum note）、主音（人耳最能感知到的音符）、三度音、五度音和名义音。他们写道：“从音乐上讲，这些频率听起来像是一个小三和弦，这就是为什么教堂钟通常被认为具有忧郁的音调。”此外，“人耳将钟的音高感知为一种被称为‘打击音’的主观音调”，这是由模式形状的相互作用以及听众自身的期望共同产生的。

Collier and his co-authors, Jon Young and Aaron Stearns, used the Penn State Behrend replica to collect data on mode shapes, natural frequencies, and loss factors. They glued accelerometers to different parts of the replica and then tapped various areas on the bell with a hammer equipped with a force gauge. The accelerometers captured the bell’s response to being tapped, and the force gauge measured the force of impact as a function of time.

科利尔及其合著者乔恩·杨和亚伦·斯特恩利用宾夕法尼亚州立大学贝伦分校的复制品收集了关于模式形状、固有频率和损耗因子的数据。他们将加速度计粘在复制品的不同部位，然后用配备力传感器的锤子敲击钟的各个区域。加速度计捕捉到了钟对敲击的反应，而力传感器则测量了随时间变化的冲击力。

The authors then extracted the bell’s natural vibrational frequencies from that data, along with the corresponding mode shapes and loss factors. The results: The replica’s hum note was 174 Hz; the fundamental was 323 Hz; the tierce was 381 Hz; the quint was 510 Hz; and the nominal was 623 Hz. The ratios of those frequencies fell very close to those of an ideally tuned bell. And the mode shapes of the fundamental frequency closely matched those of typical Western church bells.

随后，作者从这些数据中提取了钟的固有振动频率，以及相应的模式形状和损耗因子。结果显示：该复制品的基音为174赫兹；主音为323赫兹；三度音为381赫兹；五度音为510赫兹；名义音为623赫兹。这些频率的比率与理想调音的钟非常接近。此外，其主音的模式形状也与典型的西方教堂钟高度吻合。

Measuring loss factors proved trickiest. Collier et al.’s measured values were within the typical range, but loss factors are impacted by the type of bronze used and casting imperfections. The 20th-century bell-making processes used to cast the replica were different from those in the 18th century used to cast the original Liberty Bell. So the loss factors might have been similar; it’s just harder to predict.

测量损耗因子是最棘手的。科利尔等人的测量值处于典型范围内，但损耗因子会受到所用青铜类型和铸造缺陷的影响。用于铸造该复制品的20世纪制钟工艺与18世纪铸造原始自由钟的工艺有所不同。因此，损耗因子可能相似，只是更难预测。