Interference of Waves — Question 10

Question 10

A coherent power increase requires the source work to be reconsidered

Two same-direction equal-amplitude sinusoidal contributions on one string are maintained by controlled sources. Each contribution alone would carry mean power $P_0$. With both sources active, the resulting field amplitudes are held fixed while the relative phase is varied.

1. Find the total mean power for equal phase and opposite phase.

2. Explain why comparing the equal-phase result with $2P_0$ does not demonstrate creation of energy. Which assumption about source work would be invalid?

Question 10 - Solution

A coherent power increase requires the source work to be reconsidered

Method and mathematical working

Apply the quadratic traveling-wave power law to the total amplitude. Then use energy conservation for the complete driven system, including source work.$$\begin{aligned}A_{\mathrm{res}}(0)&=2A,\qquad A_{\mathrm{res}}(\pi)=0,\\\overline P(0)&=4\overline P_0,\qquad \overline P(\pi)=0.\\[4pt]P(\phi)&=P_0|1+e^{i\phi}|^2=2P_0(1+\cos\phi),\\P(0)&=4P_0,\qquad P(\pi)=0.\end{aligned}$$

(a) Calculation and conclusion

Equal phase gives amplitude $2A$ and power $4P_0$. Opposite phase gives zero resultant field and zero net carried power for the ideal coincident modes. The cross term changes the total.

(b) Interpretation and limits

One cannot assume that each source continues doing exactly its isolated work when both act on the same coupled field. The total displacement and velocity at the drives alter their work, including possible energy exchange between sources. Energy conservation applies to the complete sources-plus-string system, not to isolated-source power values reused without checking interaction.

Physical check

The work needed to maintain prescribed source motions depends on the combined force they experience. Assigning each driven source an immutable isolated-source power would omit this change in work.