Dispersive waves in stiff bars produce inharmonic modal frequencies
Evidence
Equation 1.11 is significantly different from that for quasi-longitudinal and transverse shear waves. In particular, the velocity is now frequency dependent and increases with frequency. This results in dispersive propagation of waves with different frequencies traveling at different velocities. Therefore waveshape is not preserved in bending wave propagation. One can hear the effect of this if one listens to the “chirp” sound emitted by ice covering a pond when hit by a thrown rock. The dispersion and the fourth root arise because, unlike quasi-longitudinal and transverse shear waves, the spatial derivatives in the wave equation are fourth order instead of second order because the bending wave is an amalgam of longitudinal and transverse waves. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 128 (
#expands, via linked claim)
Categorization
Category: Causal Relationships
Related
- Stiffness makes transversal waves dispersive and their modal tones inharmonic (similarity 0.91)
- Dispersive media split wave speed into phase and group velocity (similarity 0.88)
- Bending wave speed grows as the square root of frequency (similarity 0.88)
- Bending stiffness makes bar vibrations dispersive and inharmonic. (similarity 0.88)
- Bending waves disperse with frequency, so their waveshape is not preserved (similarity 0.87)
- Longitudinal and shear waves preserve waveshape because phase velocity is frequency-independent (similarity 0.85)
- Dispersion causes bending waves to lose their original waveshape (similarity 0.84)
- Real string stiffness slightly raises wave velocity with frequency, affecting timbre. (similarity 0.84)
- Bending waves disperse because their wavenumber grows only with the square root of frequency (similarity 0.83)
- Instrument stiffness versus tension determines how widely its modes are spaced (similarity 0.83)
- A tiny extra mass ruins a string’s harmonic spacing and makes it sound sour (similarity 0.82)
- String stiffness makes higher frequencies travel faster, changing timbre (similarity 0.82)
- The wave equation fλ equals c applies to any wave in any medium (similarity 0.82)
- A string’s stiffness raises its higher-frequency modes above the ideal values (similarity 0.81)
- Higher modes of a lumped bead string are not equally spaced, ruining periodicity (similarity 0.81)
Broader Connections
- String stiffness raises component mode frequencies above the harmonic series, creating inharmonicity
#corroborates(0.95) - Bending waves disperse with frequency, so their waveshape is not preserved
#expands(0.90) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 128