Calculator
Use the method selector to switch between core phase-velocity relations, string waves, acoustic waves, and electromagnetic propagation models.
Formula used
vₚ = f × λThis is the direct phase-velocity definition. Multiply frequency by wavelength when both are known in the same medium.
vₚ = ω / kUse angular frequency and wave number when the wave is already represented in sinusoidal form.
λ = vₚ / fRearrange the phase-velocity relation to recover wavelength from known speed and frequency.
f = vₚ / λRearrange the same relation to solve frequency from phase velocity and wavelength.
vₚ = √(T / μ)For a stretched string, phase velocity grows with tension and falls with linear mass density.
vₚ = √(K / ρ)For acoustic waves in a fluid-like medium, speed increases with bulk modulus and decreases with density.
vₚ = c / nFor electromagnetic waves, phase velocity in a medium equals the speed of light in vacuum divided by refractive index.
How to use this calculator
- Select the calculation method that matches the data you already know.
- Enter values in the visible fields and choose the correct engineering or scientific units.
- Press Calculate to place the result block above the form.
- Review phase velocity, wavelength, frequency, period, angular frequency, and wave number together.
- Use the Plotly graph to inspect how the chosen variable affects the result.
- Download a CSV or PDF report when you need to save or share the output.
Example data table
| Scenario | Known inputs | Method | Phase velocity |
|---|---|---|---|
| Air sound wave | 50 Hz, 6.8 m | f and λ | 340 m/s |
| Guitar string | 120 N, 0.006 kg/m | T and μ | 141.421 m/s |
| Water acoustic wave | 2.2 GPa, 1000 kg/m³ | K and ρ | 1,483.24 m/s |
| Glass optical wave | n = 1.50 | c and n | 1.9986e8 m/s |
| Sinusoidal model | ω = 314 rad/s, k = 2 rad/m | ω and k | 157 m/s |
FAQs
1. What is phase velocity?
Phase velocity is the speed at which a constant phase point, such as a crest, moves through space. It is commonly written as vₚ and depends on frequency, wavelength, and medium properties.
2. Is phase velocity the same as group velocity?
No. Phase velocity tracks a phase point, while group velocity tracks the envelope or energy packet. In dispersive media, the two speeds can differ significantly.
3. Why do unit selections matter here?
Wave calculations are simple in form but sensitive to unit mistakes. Converting everything to SI units first prevents mismatched scales and keeps the derived speed, wavelength, or frequency physically consistent.
4. When should I use ω and k instead of f and λ?
Use ω and k when your wave is written in sinusoidal notation, such as y = A sin(kx − ωt). That form directly gives the phase velocity relation vₚ = ω/k.
5. Can this calculator handle string waves?
Yes. Select the string method and enter tension plus linear density. The tool applies vₚ = √(T/μ), then derives frequency or wavelength when one optional value is also supplied.
6. How is refractive index used for optical waves?
The optical method assumes phase velocity equals c/n. A higher refractive index lowers phase velocity. If you also enter frequency or wavelength, the companion quantity is derived for that medium.
7. Why might my optional frequency and wavelength disagree?
If both optional values imply a speed that differs from the medium-based speed, the inputs are inconsistent. This often happens from incorrect units, rounded values, or properties copied from different conditions.
8. What does the graph show after calculation?
The Plotly graph changes with the selected method. It shows how the main result varies as one controlling input changes, while the other relevant quantity remains constant.