14.6 Wave Interference and Standing Waves

Wave Interference

Wave interference is a fundamental wave phenomenon that occurs when multiple waves overlap in the same region of space. Unlike solid objects that bounce off each other, waves can pass through one another, creating fascinating patterns of reinforcement and cancellation.

Interaction of Wave Pulses

When two or more waves meet, they combine according to specific physical principles. This interaction can lead to temporary increases or decreases in wave amplitude as the waves pass through each other. 🌊

• The waves maintain their individual characteristics (frequency, wavelength, speed) before and after the interaction
• The interaction is temporary, lasting only during the period when the waves overlap
• After passing through each other, each wave continues on its original path unchanged

Travel Through Each Other

Unlike material objects, waves don't bounce off each other when they meet. Instead, they pass through one another in a process called superposition.

• During superposition, each wave maintains its individual properties
• The waves appear to "pass through" each other without permanent alteration
• This behavior is fundamentally different from particle collisions, where momentum and energy transfer occur

Superposition of Waves

The principle of superposition is the mathematical foundation for understanding wave interference. It states that when multiple waves overlap, the resulting displacement at any point equals the sum of the individual wave displacements at that point.

• For water waves, this means the height of the combined wave equals the sum of heights of individual waves
• For sound waves, the resulting pressure variation equals the sum of individual pressure variations
• For electromagnetic waves, the net electric and magnetic fields equal the vector sum of individual fields

Constructive vs Destructive Interference

The way waves combine depends on their relative phase—whether their peaks and troughs align or oppose each other.

• Constructive interference occurs when waves are in phase (peaks align with peaks, troughs with troughs)
  • The amplitudes add together, creating a larger combined amplitude
  • Maximum constructive interference happens when waves are exactly in phase
  • Example: Two identical sound waves in phase create a louder sound
• Destructive interference occurs when waves are out of phase (peaks align with troughs)
  • The amplitudes subtract, creating a smaller combined amplitude
  • Complete cancellation occurs when identical waves are exactly 180° out of phase
  • Example: Noise-cancelling headphones use destructive interference to eliminate unwanted sound

Visual Representations

Graphical representations help visualize the principles of wave interference and superposition.

• Wave diagrams show how individual wave profiles combine at each point
• These visualizations make it easier to predict interference patterns
• Amplitude vs. position graphs clearly demonstrate nodes (zero amplitude) and antinodes (maximum amplitude)

Beat Frequency

When two waves with slightly different frequencies interfere, they create a phenomenon called beats—regular fluctuations in amplitude that are perceived as periodic variations in loudness. 🥁

• The waves alternately interfere constructively and destructively as they move in and out of phase
• This creates a pulsating effect where the combined amplitude varies with time
• The beat frequency equals the absolute difference between the two original frequencies:

$|f_{\text{beat}}| = |f_1 - f_2|$

• Musicians use beats to tune instruments by listening for these amplitude fluctuations
• Example: A 440 Hz tuning fork and a 442 Hz guitar string will produce beats at 2 Hz (2 pulses per second)

Standing Waves

Standing waves represent a special case of wave interference where waves confined to a region create stationary patterns of vibration. These patterns are crucial in understanding musical instruments, resonance, and many wave applications.

Confined Waves

Standing waves form when waves are confined within a region and reflect back and forth, creating interference patterns that appear to "stand still." 🎻

• They result from the superposition of two identical waves traveling in opposite directions
• Unlike traveling waves, standing waves don't appear to move in any direction
• The energy in a standing wave oscillates between kinetic and potential forms
• Standing waves only form at specific frequencies determined by the confining medium's properties

Nodes and Antinodes

Standing waves create distinctive patterns with fixed points of zero amplitude (nodes) and maximum amplitude (antinodes).

• Nodes are points where the medium never moves from equilibrium
• Antinodes are points where the medium oscillates with maximum amplitude
• The distance between adjacent nodes or adjacent antinodes equals half a wavelength
• The pattern of nodes and antinodes remains fixed in position

The fundamental mode (first harmonic) has the simplest pattern with the fewest nodes:

• For a string fixed at both ends: one antinode in the middle, nodes at the ends
• For a pipe open at both ends: antinodes at the ends, one node in the middle
• For a pipe closed at one end: node at closed end, antinode at open end

Wavelength Determination

The possible wavelengths for standing waves depend on the boundary conditions and dimensions of the confining region.

For a string fixed at both ends (or a pipe open at both ends):

• The length L must equal a whole number of half-wavelengths
• $L = n \cdot \frac{\lambda}{2}$ where n = 1, 2, 3, ...
• Therefore, $\lambda_n = \frac{2L}{n}$ for the nth harmonic

For a pipe closed at one end and open at the other:

• The length L must equal an odd number of quarter-wavelengths
• $L = (2n-1) \cdot \frac{\lambda}{4}$ where n = 1, 2, 3, ...
• Therefore, $\lambda_n = \frac{4L}{2n-1}$ for the nth harmonic

The frequency of each standing wave mode relates to wavelength through the wave speed:

• $f_n = \frac{v}{\lambda_n}$ where v is the wave speed

Practice Problem 1: Wave Interference

Solution

To find the beat frequency, we need to calculate the absolute difference between the two frequencies:

$|f_{\text{beat}}| = |f_1 - f_2|$

Substituting the given values: $|f_{\text{beat}}| = |256 \text{ Hz} - 260 \text{ Hz}| = |{-4 \text{ Hz}}| = 4 \text{ Hz}$

Therefore, a listener would hear 4 beats per second as these sound waves interfere.

Practice Problem 2: Standing Waves

Solution

For a string fixed at both ends, the wavelengths of the harmonics are given by: $\lambda_n = \frac{2L}{n}$ where n = 1, 2, 3, ...

The frequency is related to wavelength by: $f_n = \frac{v}{\lambda_n}$

Substituting the wavelength formula: $f_n = \frac{v}{\frac{2L}{n}} = \frac{nv}{2L}$

For the first harmonic (n = 1): $f_1 = \frac{1 \times 320 \text{ m/s}}{2 \times 0.65 \text{ m}} = \frac{320 \text{ m/s}}{1.3 \text{ m}} = 246.2 \text{ Hz}$

For the second harmonic (n = 2): $f_2 = \frac{2 \times 320 \text{ m/s}}{2 \times 0.65 \text{ m}} = \frac{640 \text{ m/s}}{1.3 \text{ m}} = 492.3 \text{ Hz}$

For the third harmonic (n = 3): $f_3 = \frac{3 \times 320 \text{ m/s}}{2 \times 0.65 \text{ m}} = \frac{960 \text{ m/s}}{1.3 \text{ m}} = 738.5 \text{ Hz}$

Therefore, the frequencies of the first three harmonics are 246.2 Hz, 492.3 Hz, and 738.5 Hz.