Bending wave speed grows as the square root of frequency

Evidence

A major assumption behind Equation 1.11 is that the shear contribution to the lateral displacement is small. This is likely to be true if the radius of the bend is large with respect to the thickness of the plate, that is, at long wavelengths. However, when the radius of the bend is of a similar size to the thickness, this condition is no longer satisfied, and the wave propagated asymptotically approaches that of a transverse shear wave. This gives an upper limit on the phase velocity of a bending wave, which is equal to that of the transverse shear wave in the material. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 129 (#qualifies, via linked claim)

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

• Bending waves disperse because their wavenumber grows only with the square root of frequency (similarity 0.91)
• Bending wave velocity tops out at the material’s transverse shear wave speed (similarity 0.91)
• Dispersive media split wave speed into phase and group velocity (similarity 0.91)
• Bending waves disperse with frequency, so their waveshape is not preserved (similarity 0.90)
• Dispersive waves in stiff bars produce inharmonic modal frequencies (similarity 0.88)
• Dispersion causes bending waves to lose their original waveshape (similarity 0.88)
• Applied forces in plates generate bending waves rather than pure shear waves (similarity 0.87)
• Longitudinal and shear waves preserve waveshape because phase velocity is frequency-independent (similarity 0.86)
• Shear waves stay non-dispersive while bending waves disperse with frequency (similarity 0.85)
• The wave equation fλ equals c applies to any wave in any medium (similarity 0.85)
• Stiffness makes transversal waves dispersive and their modal tones inharmonic (similarity 0.82)
• Wave propagation speeds depend on shape beyond simple strings (similarity 0.81)
• Denser materials slow sound while stiffer bonds speed it up (similarity 0.80)
• Materials with movable boundaries support wave types beyond longitudinal waves. (similarity 0.80)
• Shape-dependent spring constants make vibration speeds complex in solid objects (similarity 0.80)

Broader Connections

• Bending wave velocity tops out at the material’s transverse shear wave speed #qualifies (0.90) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 129
• Bending waves disperse with frequency, so their waveshape is not preserved #expands (0.85) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 128
• Dispersive waves in stiff bars produce inharmonic modal frequencies #expands (0.80)