Library / Foundations: Physics / FPH-30
Foundations: Physics
Phase and path difference
Phase angle between two waves from their path difference.
- Ref
- FPH-30
- Rev
- v1
- Field
- Foundations: Physics
- Result
- Δφ [rad]
For coherent waves initially in phase in the same medium, path difference Δx = mλ gives constructive interference for integer m.
Waves whose paths differ by one wavelength arrive in step; a half wavelength puts them exactly out of step.
Calculator
rad
cm
cm
- Phase difference in degrees90 °
- Path difference in wavelengths0.25 λ
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.
Phase difference (rad) current value 1.571 rad
ΔxPath differenceKey driver
0.10510.5
λWavelength
0.9424.71
- A half-wavelength difference (180°) gives destructive interference, the basis of noise cancelling.
- A whole number of wavelengths gives constructive interference.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Δx Path difference | 1 to 100 cm | 0.1047 | 10.47 | 10.37 | +10% | +10 % |
| λ Wavelength | 20 to 100 cm | 0.9425 | 4.712 | 3.77 | +10% | −9.09 % |
Sweep of Path difference
| Path difference (cm) | Phase difference (rad) |
|---|---|
| 1 | 0.1047 |
| 1.778 | 0.1862 |
| 3.162 | 0.3312 |
| 5.623 | 0.5889 |
| 10 | 1.047 |
| 17.78 | 1.862 |
| 31.62 | 3.312 |
| 56.23 | 5.889 |
| 100 | 10.47 |
Sweep one variable
Phase difference (rad)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
Δφ
Phase difference
What the equation gives you. Everything else feeds it.
Δx
Path difference
Difference in distance travelled. Directly proportional.
λ
Wavelength
Wavelength. Inversely proportional.
Worked example with the values above
- Path difference in wavelengths: 15 / 60 = 0.25.
- Each wavelength is 2π rad.
- Δφ = 1.571 rad (90°).
Assumptions and limits
- Coherent sources.
- Same medium, same speed.
Common mistakes
- Mixing units of Δx and λ.
- Confusing phase in degrees and radians.
Statistics and uncertainty
Path differences are small compared with the paths, so measurement errors in the paths themselves are amplified when finding Δx.
Where it is used
- Radar and sonar array beamforming.
- Interference patterns.
Maths you need
- Wavelength: metres per cycle
- Radians: 2π per cycle