17-3 Interference

17-3 Interference#

Prompts

  • Two sound waves with the same wavelength are emitted in phase from two sources and meet at a point P. How do you find the phase difference \(\phi\) at P from the path length difference \(\Delta L\)?

  • When is interference fully constructive? Fully destructive? Give both the phase condition (\(\phi\)) and the path-length condition (\(\Delta L/\lambda\)).

  • Convert a phase difference of \(\pi/2\) rad to degrees and to number of wavelengths. What type of interference does it give?

  • Two in-phase sources are separated by 1.5\(\lambda\). On the perpendicular bisector, what is \(\Delta L\)? Along the line through the sources, what is \(\Delta L\)? Where do you get constructive vs destructive interference?

Lecture Notes#

Overview#

  • When two sound waves with the same wavelength overlap at a point, the resultant displacement is the sum of the two (superposition, section 16-4).

  • The type of interference—constructive, destructive, or intermediate—depends on the phase difference \(\phi\) at that point.

  • If the waves are emitted in phase and travel in roughly the same direction, \(\phi\) is determined by the path length difference \(\Delta L\): one wave travels farther than the other, so it is shifted in phase.

Phase difference from path length difference#

Two waves with wavelength \(\lambda\) are emitted in phase from sources S\(_1\) and S\(_2\). They reach a common point P after traveling distances \(L_1\) and \(L_2\). The path length difference is

(124)#\[ \Delta L = |L_2 - L_1| \]

The phase difference at P is

(125)#\[ \phi = \frac{\Delta L}{\lambda} \cdot 2\pi \quad \Leftrightarrow \quad \frac{\phi}{2\pi} = \frac{\Delta L}{\lambda} \]

  • Each extra wavelength of path adds \(2\pi\) rad of phase.

  • \(\Delta L/\lambda\) = number of wavelengths by which one wave leads or lags the other.

Fully constructive and fully destructive interference#

Same logic as transverse waves (section 16-5): two identical waves with phase difference \(\phi\) give resultant amplitude \(s'_m = |2s_m \cos(\phi/2)|\).

Type

Phase \(\phi\)

Path length \(\Delta L/\lambda\)

Fully constructive

\(m(2\pi)\), \(m = 0, 1, 2, \ldots\)

\(0, 1, 2, \ldots\)

Fully destructive

\((2m+1)\pi\), \(m = 0, 1, 2, \ldots\)

\(0.5, 1.5, 2.5, \ldots\)

  • Constructive: waves arrive in phase; peaks align with peaks.

  • Destructive: waves arrive out of phase; peaks align with valleys.

  • Intermediate: e.g., \(\Delta L/\lambda = 0.25\) or \(0.75\) → neither maximum nor zero.

Poll: Possible wavelength for destructive interference

Two speakers are 4.0 m apart. You stand at a point 3.0 m from one speaker and farther from the other. You hear destructive interference. Which of the following is a possible wavelength of the sound?

[FIGURE: Two speakers 4.0 m apart; observer 3.0 m from one speaker; indicate distances to both speakers so \(\Delta r = 2\) m]

  • (A) 1.5 m

  • (B) 2.5 m

  • (C) 4.0 m

Poll: Phase shift to change destructive → constructive

Same setup: you hear destructive interference. Someone adjusts the phase of one speaker so you now hear constructive interference. What phase shift was applied?

  • (A) 90° ahead (leads the other)

  • (B) 90° behind (lags the other)

  • (C) 180°, either way

Phase in wavelengths

\(\phi = 2\pi\) ↔ 1\(\lambda\). Use the fractional part of \(\Delta L/\lambda\): 2.40\(\lambda\) and 0.40\(\lambda\) are equivalent. Quick checks: 0 → constructive; 0.50 → destructive.

Converting phase units#

Conversion

Formula

Radians ↔ wavelengths

\(\phi/(2\pi) = \Delta L/\lambda\)

Radians ↔ degrees

\(\phi_{\text{deg}} = \phi_{\text{rad}} \times (180°/\pi)\)

Degrees ↔ wavelengths

\(\phi_{\text{deg}}/360° = \Delta L/\lambda\)

Example: Two in-phase sources#

Two point sources S\(_1\) and S\(_2\), separated by \(D = 1.5\lambda\), emit spherical sound waves in phase.

  • On the perpendicular bisector: \(\Delta L = 0\) → fully constructive.

  • Along the line through the sources (far side): \(\Delta L = D = 1.5\lambda\) → fully destructive.

  • Around a circle through both sources: \(\Delta L\) varies with angle; constructive and destructive points alternate.

Poll: Path difference

Two in-phase speakers are 2.0 m apart. You stand on the perpendicular bisector, 4.0 m from the midpoint. For sound with \(\lambda = 0.50\) m, what is \(\Delta L\) at your position? What type of interference do you hear?

  • (A) \(\Delta L = 0\), constructive

  • (B) \(\Delta L = 0.5\lambda\), destructive

  • (C) \(\Delta L = 2\lambda\), constructive

Summary#

  • \(\phi = (\Delta L/\lambda) \cdot 2\pi\)—path length difference sets phase difference for in-phase sources.

  • Constructive: \(\Delta L/\lambda = 0, 1, 2, \ldots\). Destructive: \(\Delta L/\lambda = 0.5, 1.5, 2.5, \ldots\).

  • Use fractional part of \(\Delta L/\lambda\) to classify; convert between rad, deg, and wavelengths as needed.