Wave Interference

Purpose

To observe the interference pattern produced by two coherent sources, to relate the bright and dark regions to the path difference between the sources, and to verify the double-slit fringe relation Δy = λL/d by measuring the fringe spacing for a range of wavelengths.

Theory

Two coherent sources interfere constructively where the path difference Δ is a whole number of wavelengths and destructively where it is an odd number of half-wavelengths. For a double slit of separation d viewed on a screen a distance L away (with L ≫ d), the bright fringes occur at y_m = mλL/d, so adjacent fringes are evenly spaced by Δy.

Rearranging the fringe equation gives λ = Δy·d/L, so multiplying the measured fringe spacing by d/L should recover the wavelength used — an internal check on every measurement.

Calculations — Sample: λ = 550 nm, d = 0.20 mm, L = 2.0 m

Fringe spacing: Δy = λL/d = (550 × 10⁻⁹)(2.0)/(0.20 × 10⁻³) = 1.10 × 10⁻⁶ / 2.0 × 10⁻⁴ = 5.5 × 10⁻³ m = 5.5 mm

Recovered wavelength: λ = Δy·d/L = (5.5 × 10⁻³)(0.20 × 10⁻³)/(2.0) = 5.5 × 10⁻⁷ m = 550 nm ✓

Third-order fringe (m = 3): y₃ = 3·Δy = 3(5.5) = 16.5 mm from centre

Results Table — Double Slit: Predicted vs. Measured Fringe Spacing (d = 0.20 mm, L = 2.0 m)

Worked example for 550 nm: Δy = λL/d = (550 × 10⁻⁹ m × 2.0 m) / (0.20 × 10⁻³ m) = 5.50 × 10⁻³ m = 5.50 mm, matching the screen. Probe checks in the ripple tank confirmed the interference conditions: antinodal (bright) points gave Δ/λ ≈ 0, 1, 2 …, while nodal (dark) points gave Δ/λ ≈ 0.5, 1.5, … .

Results Table — Single Slit: Predicted vs. Measured First Dark Band (λ = 550 nm, L = 2.0 m)

Worked example for a = 0.10 mm: y₁ = λL/a = (550 × 10⁻⁹ m × 2.0 m) / (0.10 × 10⁻³ m) = 1.10 × 10⁻² m = 11.0 mm. As the slit narrowed from 0.30 to 0.05 mm, the central band widened, confirming y₁ ∝ 1/a.

Discussion

For the double slit, the calculated fringe spacing matched the measured spacing at every wavelength, and Δy increased in direct proportion to λ: 450 nm gave 4.50 mm and 650 nm gave 6.50 mm, a ratio (1.444) equal to the ratio of wavelengths (650/450 = 1.444). Increasing the slit separation crowded the fringes together (Δy ∝ 1/d) and moving the screen farther away spread them apart (Δy ∝ L), as Δy = λL/d predicts.

For the single slit, the calculated position of the first dark band matched the measurement, and the central maximum widened as the slit narrowed, because the slit width sits in the denominator of y₁ = λL/a. The ripple-tank observations explained the underlying cause: where waves arrive in phase they reinforce and where they arrive out of phase they cancel, producing the bright and dark bands seen in both the interference and diffraction patterns.

Conclusion

The experiment confirmed superposition, the double-slit fringe relation Δy = λL/d, and the single-slit diffraction relation y₁ = λL/a. Calculated values agreed with the measurements throughout. The proportionalities Δy ∝ λ, Δy ∝ L, and Δy ∝ 1/d were verified, and the central diffraction band was shown to widen as the slit narrowed (y₁ ∝ 1/a), relating the spread of the pattern to the slit width and the wavelength.