Superposition: Question 7

Worked solution

Part (a): Conditions for constructive and destructive interference

Constructive interference occurs at a point where the path difference between the two waves is a whole number of wavelengths: $\text{path difference} = n\lambda, \qquad n = 0, 1, 2, \dots$

Destructive interference occurs at a point where the path difference is an odd number of half-wavelengths: $\text{path difference} = \left(n+\tfrac12\right)\lambda, \qquad n = 0, 1, 2, \dots$

Part (b): Conditions for a stable interference pattern

For $S_1$ and $S_2$ to produce a stable, observable interference pattern, they must be coherent, which requires:

• the same frequency (and hence the same wavelength), and
• a constant (non-varying) phase difference between them.

(In this case, being driven in phase from the same oscillator ensures both conditions are met.)

Part (c): Path difference at $P$

The path difference is the difference between the two distances from the sources to $P$: $\Delta = |S_1P - S_2P| = |25.2 - 19.2| = 6.0\text{ cm}$

Check (recompute independently): starting from $19.2\text{ cm}$ and adding $6.0\text{ cm}$ gives $19.2+6.0=25.2\text{ cm}$, matching the given distance from $S_1$, consistent.

So the path difference is $\Delta = \boxed{6.0\text{ cm}}$.

Part (d): Type of interference at $P$

Express the path difference as a number of wavelengths, using $\lambda = 2.4\text{ cm}$: $\frac{\Delta}{\lambda} = \frac{6.0}{2.4} = 2.5$

Check (recompute independently): $2.4 \times 2.5 = 2.4\times2 + 2.4\times0.5 = 4.8+1.2=6.0$, confirms $6.0/2.4=2.5$ exactly.

Since $2.5 = 2 + \tfrac12$, the path difference is $\left(2+\tfrac12\right)\lambda$, an odd number of half-wavelengths. By the condition stated in part (a), this means the two waves arrive at $P$ exactly out of phase, so the interference at $P$ is destructive.

Final answers

• (a) Constructive: path difference $=n\lambda$; destructive: path difference $=(n+\tfrac12)\lambda$
• (b) Same frequency/wavelength; constant phase difference (coherent)
• (c) Path difference $= \boxed{6.0\text{ cm}}$
• (d) $6.0\text{ cm} = 2.5\lambda = (2+\tfrac12)\lambda$, so the interference at $P$ is destructive