HL IB Physics Standing Waves & Resonance Notes

Standing Waves

  • Formation of Standing Waves

    • Produced by two waves traveling in opposite directions.

    • Achieved via superposition of a traveling wave with its reflection.

    • Resultant wave pattern has crests and troughs moving vertically.

  • Necessary Conditions for Standing Waves

    • Two waves must travel along the same line with:

    • Same Wavelength

    • Similar Amplitude

    • Resulting wave is stationary with energy stored instead of transfer.

  • Comparison of Progressive and Standing Waves

    • Progressive Waves: Transfer energy.

    • Standing Waves: Store energy; do not transfer it.

Nodes & Antinodes

  • Definition

    • Nodes: Points of zero amplitude, separated by $ rac{1}{2} ext{λ}$.

    • Antinodes: Points of maximum amplitude, oscillating vertically.

  • Oscillation Explanation

    • Nodes:

    • In anti-phase (destructive interference).

    • Antinodes:

    • In phase (constructive interference).

  • Phase Relationship

    • Points with odd nodes are in anti-phase.

    • Points with even nodes are in phase.

    • All points in a loop are in phase.

Boundary Conditions

  • Definition: Conditions under which standing waves form on strings or in pipes.

  • Strings:

    • Can be fixed at both ends, free at both ends, or one end fixed and one free.

    • Frequency depends on string tension and mass per unit length.

  • Pipes:

    • Can be closed at both ends, open at both ends, or one open and one closed.

    • Nodes occur at closed ends; antinodes at open ends.

Harmonics

  • Definition: Specific wave patterns forming at distinct frequencies dependent on boundary conditions.

  • For Strings:

    • First harmonic (fundamental): 1 loop, 2 nodes.

    • Wavelength: $ ext{λ} _1 = 2L$

    • Frequency: $f_1 = rac{v}{2L}$

    • Subsequent harmonics increase complexity, each having distinct numbers of nodes and antinodes.

    • General formula for wavelength: $ ext{λ}_n = rac{2L}{n}$.

  • For Pipes:

    • Open at both ends: first harmonic has 2 antinodes.

    • Open at one end: only odd harmonics appear with unique loop patterns.

    • General wavelength for odd harmonics: $ ext{λ}_n = rac{4L}{n}$, where $n$ is an odd integer.

Resonance

  • Definitions: Occurs when the driving frequency equals the natural frequency of a system.

  • Types of Oscillations:

    • Free Oscillations: Occur without external forces, oscillating at natural frequency.

    • Forced Oscillations: Energy input from an external force sustains oscillations.

  • Resonance: Maximal energy transfer occurs at equal driving and natural frequency, producing maximum amplitude.

    • Example: Swinging to the right frequency yields maximum height.

Damping

  • Definition: The reduction in energy and amplitude of oscillations due to resistive forces.

  • Types of Damping:

    • Light Damping: Exponentially decreasing amplitude over time with constant frequency.

    • Critical Damping: Returns to equilibrium fastest without oscillation.

    • Heavy Damping: Returns to equilibrium slowly.

  • Oscillation Graphs:

    • Lightly damped oscillations show gradual amplitude decay.

    • Critically damped systems return to equilibrium quickly, shown by steep curves.

    • Heavily damped systems show very gradual return with zero oscillations.

Effects of Damping on Resonance

  • As damping increases:

    • Amplitude of resonance decreases and resonance peak broadens.

    • Natural frequency remains unchanged by damping.

  • Key Features:

    • Resonance peaks are lower and shift left on the graph with increased damping.

    • Provides a broader range of frequencies at resonance but reduces amplitude.