Wave and Oscillation Vocabulary

Wave Terminology and Simulation Setup

• Learning Objective: Identify and describe wave terms (crest, trough, wavelength, amplitude, frequency, period).
• Simulation: PhET: Wave on a String
• Setup:
  • Mode: Oscillate
  • Damping: None
  • End: Fixed
  • Enable rulers and stopwatch
• Measurements:
  • Wavelength: Distance between two consecutive crests.
  • Amplitude: Distance from equilibrium to crest.

Wave Pulse Interference and Superposition

• Learning Goal: Explore superposition of wave pulses and interference.
• Simulation: Wave Pulse Interference - oPhysics
• Observations:
  • When identical pulses overlap, constructive interference occurs.
  • When opposite pulses overlap, destructive interference occurs.
• Principle of Superposition: Waves pass through each other, and their effects add together.

Wave Reflection at Boundaries

• Learning Outcome: Investigate wave reflection at fixed and free ends.
• Simulation: PhET: Wave on a String (Pulse mode)
• Fixed End: Reflected pulse is inverted and has the same speed.
• Free End: Reflected wave is not inverted and has the same shape.

Standing Waves: Nature and Formation

• Definition: Stationary wave pattern formed by two identical waves moving in opposite directions.
• Formation:
  • Two waves with the same frequency, wavelength, and amplitude travel in opposite directions.
  • Waves superpose, and displacements add algebraically.
  • Constructive interference: Antinodes (maximum amplitude).
  • Destructive interference: Nodes (zero displacement).
• Conditions for Formation:
  • Travel in opposite directions.
  • Superpose on the same medium.
  • Same frequency.
  • Same amplitude.
• Visual Features:
  • Nodes: Points of zero displacement.
  • Antinodes: Points of maximum displacement.
• Wavelength: \lambda is twice the distance between adjacent nodes or antinodes.

Progressive vs Standing Waves

• Progressive Waves:
  • Move forward through a medium.
  • Energy is transferred in the direction of propagation.
  • No distinct nodes or antinodes.
• Standing Waves:
  • Appear stationary.
  • No energy transfer; energy oscillates between nodes and antinodes.
  • Distinct nodes and antinodes.

Nodes and Antinodes in Standing Waves

• Amplitude:
  • Nodes: Zero amplitude (A = 0).
  • Antinodes: Maximum amplitude (A = Max).
• Phase:
  • Between two adjacent nodes: Same phase.
  • Between adjacent nodal regions: 180° phase difference.
• Distance: Distance between adjacent nodes or antinodes is half the wavelength (\frac{\lambda}{2}).

Resonance and Oscillations

• Free Oscillation:
  • Occurs with only internal forces (no external force or energy input).
  • System vibrates at its natural frequency.
• Forced Oscillation:
  • Occurs with a periodic external force.
  • System oscillates at the frequency of the external force.
• Resonance:
  • Driving frequency equals the system's natural frequency.
  • Amplitude of oscillation reaches its maximum.

Standing Waves on a String

• Formation:
  1. Wave is sent along the string.
  2. Wave reflects at fixed ends.
  3. Incident and reflected waves interfere.
  4. Constructive interference occurs at natural frequencies.
  5. Standing wave forms with nodes at fixed ends and antinodes in between.
• Wavelengths: Only certain wavelengths "fit" the string length (L = \frac{n\lambda}{2}, where n = 1, 2, 3,…).
• Harmonics: These patterns are called harmonics or modes of vibration.

Standing Waves: Wavelength and Frequency

• Formulas
  • Wavelength: \lambda = \frac{2L}{n}
  • Frequency: f = \frac{nv}{2L} where:
    • L = string length
    • n = harmonic number
    • v = wave speed

Standing Waves in Closed Pipes

• Formation:
  1. Sound wave is sent into the pipe.
  2. Wave reflects at the closed end.
  3. Incident and reflected waves interfere.
  4. Constructive interference occurs at natural frequencies.
  5. Standing wave forms with a node at the closed end and an antinode at the open end.
• Wavelengths: Only certain wavelengths "fit" the pipe's length.
  • Pipe length equals \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, … of the wavelength.
• Harmonics: Only odd harmonics form (n = 1, 3, 5,…).
• Formulas
  • Wavelength: \lambda_n = \frac{4L}{n}
  • Frequency: f_n = \frac{nv}{4L}

Standing Waves in Open Pipes

• Formation:
  1. Sound wave is introduced into the pipe.
  2. Wave reflects at both open ends.
  3. Incident and reflected waves interfere.
  4. Constructive interference occurs at natural frequencies.
  5. Standing wave forms with antinodes at both open ends and nodes in between.
• Wavelengths: Only certain wavelengths "fit" the pipe's length.
  • Pipe length equals \frac{1}{2}, 1, \frac{3}{2}, … of the wavelength.
• Harmonics: All harmonics can form (n = 1, 2, 3,…).
• Formulas
  • Wavelength: \lambda = \frac{2L}{n}
  • Frequency: f = \frac{nv}{2L}

Modes of Vibration of Standing Waves

• Harmonic Identification by Boundary Conditions:
  • String (fixed ends): N-N, count each loop (\frac{1}{2} \lambda), n = 1, 2, 3, …, \lambda = \frac{2L}{n}, f = \frac{nv}{2L}
  • Closed Pipe (N - AN): N - AN, count each quarter loop (\frac{1}{4} \lambda), n = 1, 3, 5, …, \lambda = \frac{4L}{n}, f = \frac{nv}{4L}
  • Open Pipe (AN - AN): AN - AN, count each loop (\frac{1}{2} \lambda), n = 1, 2, 3, …, \lambda = \frac{2L}{n}, f = \frac{nv}{2L}

Effect of Damping on Amplitude & Resonant Frequency

• Effect on Amplitude:
  • Damping reduces the amplitude of oscillations over time.
  • Increasing damping lowers the peak amplitude in a resonance curve.
  • Strong damping makes the system less responsive to the driving frequency.
• Effect on Resonant Frequency:
  • Damping causes a slight decrease in the resonant frequency.
  • The system resonates at a frequency slightly lower than the natural frequency.