Library / Structural dynamics and vibration testing / SDY-14
Structural dynamics and vibration testing
Flexural wave packet group speed
Speed at which the energy of a narrow-band bending wave packet travels along a uniform beam, from its wavenumber κ.
- Ref
- SDY-14
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- cg [m/s]
For a narrow-band packet on a uniform Euler–Bernoulli beam the group speed is cg = dω/dκ = 2κ√(EI/μ). Energy and the envelope of the packet travel at this speed, which is twice the phase speed. It rises in proportion to the wavenumber, so short waves carry energy fastest.
Speed at which the energy of a narrow-band bending wave packet travels along a uniform beam, from its wavenumber κ. It is twice the phase speed at the same frequency.
Calculator
- Wavelength, 2π/κ3.142 m
- Angular frequency, κ²√(EI/μ)40 rad/s
- Phase speed, cg/220 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A packet with wavenumber 2 rad/m (wavelength 3.14 m) travels at 40 m/s.
- Doubling the wavenumber doubles the group speed, so an echo of a short-wavelength burst returns sooner than a long-wavelength one.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| κ Wavenumber | 0.1 to 50 rad/m | 2 | 1,000 | 998 | +10% | +10 % |
| μ Mass per metre | 0.1 to 20 kg/m | 12.65 | 178.9 | 166.2 | +10% | −4.65 % |
| EI Flexural rigidity | 10 to 2,000 N·m² | 8.944 | 126.5 | 117.5 | +10% | +4.88 % |
Sweep of Wavenumber
| Wavenumber (rad/m) | Group speed (m/s) |
|---|---|
| 0.1 | 2 |
| 6.337 | 126.7 |
| 12.57 | 251.5 |
| 18.81 | 376.3 |
| 25.05 | 501 |
| 31.29 | 625.8 |
| 37.52 | 750.5 |
| 43.76 | 875.3 |
| 50 | 1,000 |
Sweep one variable
Variables and how each one contributes
Group speed
What the equation gives you. Everything else feeds it.
Wavenumber
The wavenumber, in rad/m: 2π divided by the wavelength. Shorter waves travel faster on a beam. Raising it by 10 % (to 2.2) moves the result by +10 %, from 40 to 44 m/s.
Flexural rigidity
The flexural rigidity, in N·m²: Young’s modulus times the second moment of area. A stiffer section raises the speed or frequency. Raising it by 10 % (to 220) moves the result by +4.88 %, from 40 to 41.95 m/s.
Mass per metre
The mass per metre of the beam, in kg/m. A heavier beam lowers the speed or frequency. Raising it by 10 % (to 2.2) moves the result by −4.65 %, from 40 to 38.14 m/s.
Worked example with the values above
- Stiffness-to-mass term: √(EI/μ) = √(200 / 2) = 10.
- Multiply by 2κ = 4.
- cg = 40 m/s.
Assumptions and limits
- A uniform Euler–Bernoulli beam: slender, constant section, small deflections, no rotary inertia or shear deformation.
- For a trial shape, it satisfies the geometric support conditions.
Common mistakes
- Using the formula for thick or short beams, where shear and rotary inertia lower the frequencies.
- Using the wrong dimensionless root for the supports.
- Mixing the mass per metre with the total mass.
Statistics and uncertainty
Frequency scales with the square root of EI/μ and inversely with L², so a 2 % error in length gives a 4 % error in frequency. Section properties and the true support conditions are the dominant uncertainties in practice.
Where it is used
- Benchmarking a finite-element beam model against a closed-form result.
- Estimating bending frequencies and wave speeds for bars, shafts, pipes and cantilevers.
Maths you need
- Second moment of area: the section property that sets bending stiffness
- Mode shape: the pattern of motion at one natural frequency
- Wavenumber: rad/m, 2π over wavelength