Interference of Waves — Question 8

Question 8

Orthogonal transverse components do not cancel each other

Two equal-frequency waves travel along $x$ on an isotropic model string with two transverse coordinates. Their displacement vectors are $A\hat{\mathbf y}\cos\Phi$ and $B\hat{\mathbf z}\cos(\Phi+\delta)$. The quadratic energy uses the squared vector magnitude.

1. Show why the total mean power has no interference cross term for any $\delta$.

2. For $A=B$ and $\delta=\pi$, can the two waves cancel? Contrast with parallel transverse directions.

Question 8 - Solution

Orthogonal transverse components do not cancel each other

Method and mathematical working

Keep displacement as a vector when calculating kinetic and elastic energies. The mixed term is proportional to the dot product of the polarization vectors.$$\begin{aligned}|\mathbf a+\mathbf b|^2&=|\mathbf a|^2+|\mathbf b|^2+2\mathbf a\cdot\mathbf b,\\\mathbf e_y\cdot\mathbf e_z&=0.\\[4pt]\mathbf u_y\cdot\mathbf u_z&=0\Rightarrow P_{\mathrm{cross}}=0,\qquad \overline P=\overline P_y+\overline P_z.\end{aligned}$$

(a) Calculation and conclusion

The cross term is proportional to $\hat{\mathbf y}\cdot\hat{\mathbf z}=0$. Thus mean power is the sum of the separate powers, independent of their relative phase, in this isotropic model.

(b) Interpretation and limits

No. At opposite phase the vector displacement is $A(\hat{\mathbf y}-\hat{\mathbf z})\cos\Phi$, not zero. Only components along the same physical displacement direction can cancel directly. Scalar amplitude addition silently assumes a common polarization direction.

Physical check

An opposite scalar phase cannot cancel two perpendicular vectors. A measurement after projection onto a common analyzer direction would be a different observable and requires its own calculation.