16-5 Interference of Waves

16-5 Interference of Waves#

Prompts

  • When two waves overlap on the same string, how do you find the resultant displacement? What principle governs this?

  • Two identical sinusoidal waves (same amplitude, wavelength) travel in the same direction but differ in phase by \(\phi\). Write the amplitude of the resultant wave in terms of \(y_m\) and \(\phi\).

  • When is interference fully constructive? Fully destructive? What phase difference \(\phi\) gives each?

  • Express phase difference in wavelengths: \(\phi = 2\pi\) corresponds to how many wavelengths? For 0.50 wavelength phase difference, what type of interference?

Lecture Notes#

Overview#

  • Interference is the combining of waves that overlap. By superposition (section 16-4), the resultant displacement is the sum of the individual displacements.

  • Two identical sinusoidal waves (same amplitude, wavelength, direction) that differ only in phase produce a resultant wave whose amplitude depends on the phase difference \(\phi\).

  • Fully constructive (\(\phi = 0\)): peaks align → amplitude doubles. Fully destructive (\(\phi = \pi\)): peaks align with valleys → complete cancellation.

Superposition and interference#

Principle of superposition: When two or more waves traverse the same medium, the displacement at any point is the algebraic sum of the displacements each wave would produce alone.

(100)#\[ y'(x,t) = y_1(x,t) + y_2(x,t) \]

  • Overlapping waves add; they do not alter each other’s propagation.

  • Interference refers to this combining—whether the result is larger (constructive), smaller (destructive), or in between (intermediate).

Two identical waves with phase difference#

Consider two waves of the same amplitude \(y_m\), wavelength \(\lambda\), and frequency, traveling in the \(+x\) direction:

(101)#\[ y_1 = y_m \sin(kx - \omega t), \qquad y_2 = y_m \sin(kx - \omega t + \phi) \]

The phase difference \(\phi\) is the constant shift between the two waves. Using \(\sin a + \sin b = 2\sin\frac{a+b}{2}\cos\frac{a-b}{2}\):

(102)#\[ y' = y_1 + y_2 = \left[2y_m \cos\frac{\phi}{2}\right] \sin\left(kx - \omega t + \frac{\phi}{2}\right) \]

The amplitude of the resultant wave is

(103)#\[ y'_m = \left|2y_m \cos\frac{\phi}{2}\right| \]

Constructive, destructive, and intermediate interference#

Phase difference \(\phi\)

Amplitude \(y'_m\)

Type

\(0\) (or \(2\pi\), \(4\pi\), …)

\(2y_m\)

Fully constructive

\(\pi\) rad (180°)

\(0\)

Fully destructive

Other values

\(0 < y'_m < 2y_m\)

Intermediate

  • Fully constructive (\(\phi = 0\)): peaks align with peaks, valleys with valleys → maximum amplitude.

  • Fully destructive (\(\phi = \pi\)): peaks align with valleys → complete cancellation; string is flat.

  • Intermediate: e.g., \(\phi = 2\pi/3\) gives \(y'_m = y_m\).

Phase in wavelengths

A phase difference of \(\phi = 2\pi\) corresponds to a shift of one wavelength. So \(\phi/(2\pi)\) = fraction of a wavelength. Discard integer wavelengths: 2.40 and 0.40 wavelength are equivalent. Quick check: 0 → constructive; 0.50 → destructive; 0.25 or 0.75 → intermediate.

Summary#

  • Superposition: \(y' = y_1 + y_2\); overlapping waves add.

  • Resultant amplitude: \(y'_m = |2y_m \cos(\phi/2)|\) for two identical waves with phase difference \(\phi\).

  • \(\phi = 0\): fully constructive (\(y'_m = 2y_m\)). \(\phi = \pi\): fully destructive (\(y'_m = 0\)).

  • Phase in wavelengths: \(\phi = 2\pi\) ↔ 1\(\lambda\); use fractional part to classify interference.