Phasors — Question 8

Question 8

Phase unwrapping restores a continuous spatial advance

Three equally spaced sensors report phases $170^\circ,-175^\circ,-160^\circ$ using the interval $(-180^\circ,180^\circ]$. The separation is $d=.020\,\mathrm{m}$. Independent information says phase increases by less than $180^\circ$ per sensor step.

1. Unwrap the phases and infer wavelength.

2. Explain what ambiguity would remain without the independent advance bound.

Question 8 - Solution

Phase unwrapping restores a continuous spatial advance

Method and mathematical working

Add complete turns to preserve the specified positive, sub-$180^\circ$ increment between adjacent sensors. Then convert the unwrapped increment to a spatial gradient.$$\begin{aligned}170^\circ,\,-175^\circ,\,-160^\circ&\longrightarrow170^\circ,185^\circ,200^\circ,\\kd&=15^\circ=\frac\pi{12},\qquad \lambda=\frac{2\pi d}{\pi/12}.\\[4pt]k&=\frac{\pi/12}{0.020}=\frac{25\pi}{6}\,\mathrm{rad/m},\qquad\lambda=\frac{2\pi}{k}=0.480\,\mathrm m.\end{aligned}$$

(a) Calculation and conclusion

Continuous phases can be $170^\circ,185^\circ,200^\circ$, so each step advances $15^\circ$. Thus $d/\lambda=15/360$, giving $\lambda=24d=.480\,\mathrm{m}$.

(b) Interpretation and limits

Additional full $360^\circ$ advances could occur between sensors while leaving the reported principal phases unchanged. The positive-advance bound chooses the small increment. A numerical discontinuity at the reporting branch cut does not represent a physical jump in the wave.

Physical check

Both neighboring intervals independently give the same phase advance. Without the increment bound, additional whole turns could be inserted and the wavelength would not be uniquely determined.