The Resonant Ice Cave · AI Generator #GEN-19508D8F

Physics

Problem

A spelunker discovers a narrow ice cave that produces a haunting musical tone when wind blows across its entrance. The cave can be modeled as a cylindrical tube of length

L = 12.0\;\text{m}

that is closed at one end by solid ice and open at the other end where wind flows. The spelunker notices that the fundamental resonant frequency changes as the temperature drops. At temperature

T_1 = 273\;\text{K}

(melting point), the fundamental frequency is

f_1 = 14.3\;\text{Hz}

. When the temperature drops to

T_2 = 243\;\text{K}

(typical winter cave temperature), the frequency becomes

f_2

. However, there's a twist: as the temperature drops, a thin layer of ice forms on the inner walls of the cave, effectively reducing the cave's cross-sectional area and thus its length by

\Delta L = 0.15\;\text{m}

. The spelunker wants to determine whether the frequency change is dominated by the temperature effect on sound speed or by the geometric change. (a) Calculate the fundamental frequency

f_2

at the lower temperature, accounting for both effects. (b) What would the frequency be if only the temperature effect occurred (no change in length)? (c) What would the frequency be if only the geometric effect occurred (no temperature change)? Assume air behaves as an ideal gas and use

\gamma = 1.4

for air.

Intuitive explanation

This problem asks you to separate two competing effects on resonance: how sound speed changes with temperature, and how the physical dimensions shrink. For a tube closed at one end, the fundamental frequency depends on $f = v/(4L)$ , where $v$ is sound speed and $L$ is length—so both the numerator and denominator shift as conditions change. The key insight is to calculate $v$ at each temperature using the ideal gas relationship ( $v \propto \sqrt{T}$ ), then compare three scenarios: both effects together, temperature alone, and geometry alone. This reveals which effect dominates the frequency shift the spelunker hears.

Intuitive explanation

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