Dispersive media split wave speed into phase and group velocity

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

• Longitudinal and shear waves preserve waveshape because phase velocity is frequency-independent (similarity 0.91)
• Bending wave speed grows as the square root of frequency (similarity 0.91)
• Dispersion causes bending waves to lose their original waveshape (similarity 0.89)
• Dispersive waves in stiff bars produce inharmonic modal frequencies (similarity 0.88)
• Bending waves disperse because their wavenumber grows only with the square root of frequency (similarity 0.88)
• Bending waves disperse with frequency, so their waveshape is not preserved (similarity 0.88)
• The wave equation fλ equals c applies to any wave in any medium (similarity 0.88)
• Shear waves stay non-dispersive while bending waves disperse with frequency (similarity 0.83)
• Phase velocity measures single-frequency speed while group velocity tracks the signal envelope (similarity 0.81)
• Waves always refract toward regions of slower wave speed (similarity 0.81)
• Stiffness makes transversal waves dispersive and their modal tones inharmonic (similarity 0.81)
• Phase velocity and group velocity differ in dispersive media (similarity 0.81)
• Frequency stays constant across media, so slow speed shortens wavelength (similarity 0.80)
• Wave propagation speeds depend on shape beyond simple strings (similarity 0.79)
• Bending wave velocity tops out at the material’s transverse shear wave speed (similarity 0.78)

Broader Connections

• Bending wave speed grows as the square root of frequency #exemplifies (0.90)
• Bending waves disperse with frequency, so their waveshape is not preserved #expands (0.90) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 128
• Dispersive waves in stiff bars produce inharmonic modal frequencies #exemplifies (0.85)