Interference of Waves — Question 1

Question 1

Partial constructive interference need not exceed one component amplitude

Two waves travel together on the same string: $y_1=A\cos\Phi$ and $y_2=A\cos(\Phi+2\pi/3)$, with $\Phi=kx-\omega t$.

1. Find the resultant amplitude and phase.

2. Compare its mean power with the power of one wave and with the sum of the two individual powers.

Question 1 - Solution

Partial constructive interference need not exceed one component amplitude

Method and mathematical working

Add the two equal phasors by resolving them along the angle bisector. Squaring the resultant amplitude gives the power ratio for the same traveling-wave impedance.$$\begin{aligned}A_{\mathrm{res}}&=2A\cos(\phi/2),\qquad \phi=2\pi/3,\\\overline P_{\mathrm{res}}&=\overline P_0(A_{\mathrm{res}}/A)^2.\\[4pt]y&=2A\cos(\pi/3)\cos(\Phi+\pi/3)=A\cos(\Phi+\pi/3),\\\overline P&=\overline P_0,\qquad\overline P_1+\overline P_2=2\overline P_0.\end{aligned}$$

(a) Calculation and conclusion

The sum-to-product identity gives $y=2A\cos(\pi/3)\cos(\Phi+\pi/3)=A\cos(\Phi+\pi/3)$. Its amplitude is $A$, not $2A$.

(b) Interpretation and limits

At fixed string parameters and frequency, power is proportional to amplitude squared. The resultant power equals the power $P_0$ of one component, while the individual powers sum to $2P_0$. The interference cross term is negative here, reducing the coherent total relative to that sum.

Physical check

The amplitude lies between zero and $2A$. At this phase the cross term is negative, so the resultant power is smaller than the sum of the powers assigned to isolated contributions.