The crest of a Parasaurolophus dinosaur skull is shaped...

The crest of a Parasaurolophus dinosaur skull is shaped somewhat like a trombone and contains a nasal passage in the form of a long, bent tube open at both ends. The dinosaur may have used the passage to produce sound by setting up the fundamental mode in it. (a) If the nasal passage in a certain Parasaurolophus fossil is 2.0 $\mathrm{m}$ long, what frequency would have been produced? (b) If that dinosaur could be recreated (as in Jurassic Park , would a person with a hearing range of 60 $\mathrm{Hz}$ to 20 $\mathrm{kHz}$ be able to hear that fundamental mode and, if so, would the sound be high or low frequency? Fossil skulls that contain shorter nasal passages are thought to be those of the female Parasaurolophus. (c) Would that make the female's fundamental frequency higher or lower than the male's?

The crest of a Parasaurolophus dinosaur skull is shaped somewhat like a trombone and contains a nasal passage in the
form of a long, bent tube open at both ends. The dinosaur may
have used the passage to produce sound by setting up the fundamental mode in it. (a) If the nasal passage in a certain
Parasaurolophus fossil is 2.0 $\mathrm{m}$ long, what frequency would have
been produced? (b) If that dinosaur could be recreated (as in Jurassic Park , would a person with a hearing range of 60 $\mathrm{Hz}$ to
20 $\mathrm{kHz}$ be able to hear that fundamental mode and, if so, would the
sound be high or low frequency? Fossil skulls that contain shorter nasal passages are thought to be those of the female
Parasaurolophus. (c) Would that make the female's fundamental
frequency higher or lower than the male's?

Key Concepts

Physical Dimensions and Sound Pitch

The length of the resonating structure, such as a tube, is inversely proportional to the pitch of the sound produced. Longer tubes generate lower frequencies (bass sounds), while shorter tubes produce higher frequencies, which is essential when comparing resonant sounds produced by different-sized structures.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system naturally oscillates. In the context of open tubes, it is determined by the length of the tube and the speed of sound, representing the most basic sound that can be produced by the resonating structure.

Hearing Range

The human hearing range typically spans from around 20 Hz to 20 kHz. Understanding this range is important when evaluating whether a particular sound produced by a resonant system is audible to the human ear and whether it is perceived as a high or low pitch.

Resonance in Open Tubes

In acoustics, a tube or pipe that is open at both ends supports standing waves where the fundamental mode has antinodes at both ends. The frequency of this mode is given by the relation f = v/(2L), where v is the speed of sound and L is the length of the tube.

Speed of Sound

This concept is critical because sound waves travel through a medium, and the speed at which they do so affects the wavelength and frequency of the sound produced. The speed of sound in air is a constant factor used when relating the physical dimensions of a resonating tube to the frequency of sound.

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