Phasors — Question 7

Question 7

Spatially counterpropagating phasors cannot share one translation factor

Use $y(x,t)=\operatorname{Re}[Z(x)e^{-i\omega t}]$ with $Z(x)=Ae^{ikx}+Be^{-ikx}$, where $A,B$ are positive real amplitudes and $k>0$.

1. Find $|Z(x)|^2$ and determine when it can reach zero.

2. Why cannot the two terms be replaced by a single constant amplitude times $e^{ikx}$ when $B\ne0$?

Question 7 - Solution

Spatially counterpropagating phasors cannot share one translation factor

Method and mathematical working

At each position add the two spatial phasors. Multiply the result by its complex conjugate to obtain the local squared amplitude.$$\begin{aligned}Z(x)&=Ae^{ikx}+Be^{-ikx},\\|Z(x)|^2&=A^2+B^2+2AB\cos(2kx).\\[4pt]|Z|_{\min}&=|A-B|,\quad A=B:\quad kx=(m+1/2)\pi\Rightarrow Z=0.\end{aligned}$$

(a) Calculation and conclusion

$|Z|^2=A^2+B^2+2AB\cos(2kx)$. Its minimum is $(A-B)^2$, so zeros exist only for $A=B$, at $kx=(n+1/2)\pi$.

(b) Interpretation and limits

The ratio $Z(x)/e^{ikx}=A+Be^{-2ikx}$ varies with $x$. Factoring out the common time dependence is valid, but the two opposite spatial phases cannot both be absorbed into one constant rightward phasor. This spatially varying amplitude is the basis of standing-wave structure.

Physical check

The minimum is $(A-B)^2$, so a true zero requires equal component amplitudes. The position dependence cannot generally be absorbed into one constant phasor for the entire string.