Standing waves form when identical waves traveling in opposite directions interfere. They create fixed nodes and antinodes, appearing stationary. This phenomenon occurs in various media, like guitar strings and pipe organs, producing distinct patterns of vibration.

Resonance amplifies oscillations when a system is driven at its natural frequency. It's crucial in musical instruments for sound production and amplification. However, resonance can also be dangerous in structures, potentially causing damage if not properly managed by engineers.

Standing Waves

Formation of standing waves

• Two identical waves traveling in opposite directions interfere constructively and destructively at specific points
  • Waves have the same frequency, amplitude, and wavelength enabling superposition
  • Constructive interference creates antinodes with maximum displacement (crests and troughs align)
  • Destructive interference creates nodes with no displacement (crests align with troughs)
• Standing waves appear stationary because nodes and antinodes remain fixed in position
  • No net energy transfer occurs along the medium
  • Distance between adjacent nodes or antinodes equals half the wavelength ($\frac{\lambda}{2}$)
• Standing waves form in various media
  • Strings fixed at both ends (guitar strings, violin strings)
  • Air columns open at one end and closed at the other (pipe organs) or closed at both ends (flutes)
  • Membranes fixed along the edges (drumheads, speaker diaphragms)

Modes and nodes on strings

• Each mode of a standing wave on a string corresponds to a specific frequency and wavelength
  • Fundamental mode (1st harmonic) has the lowest frequency with one antinode at the center and nodes at the ends
    • Wavelength $\lambda_1$ equals twice the string length ($2L$)
    • Frequency $f_1$ equals wave speed $v$ divided by twice the string length ($\frac{v}{2L}$)
  • Higher harmonics (2nd, 3rd, etc.) have frequencies that are integer multiples of the fundamental frequency
    • Wavelengths $\lambda_n$ equal twice the string length divided by the harmonic number ($\frac{2L}{n}$)
    • Frequencies $f_n$ equal the harmonic number multiplied by the fundamental frequency ($n \cdot f_1$ or $n \cdot \frac{v}{2L}$)
• Number of nodes $N_n$ equals the harmonic number plus one ($n + 1$)
  • Fundamental mode has 2 nodes, 2nd harmonic has 3 nodes, etc.
• Number of antinodes $A_n$ equals the harmonic number ($n$)
  • Fundamental mode has 1 antinode, 2nd harmonic has 2 antinodes, etc.
• Normal modes represent the specific patterns of vibration that satisfy the boundary conditions of the system

Wave function and boundary conditions

• The wave function describes the displacement of the medium at any point and time
• Boundary conditions determine how the wave behaves at the ends of the medium (e.g., fixed or free ends)
• The combination of the wave function and boundary conditions defines the possible standing wave patterns

Resonance

Resonance in real-world applications

• Resonance amplifies oscillations when a system is driven at its natural frequency
  • Energy efficiently transfers from the driving force to the system causing large amplitude vibrations
• Musical instruments rely on resonance to produce and amplify sound
  • String instruments (guitars, violins) have natural frequencies determined by string length, tension, and mass per unit length
  • Wind instruments (flutes, clarinets) have natural frequencies determined by air column length and end conditions (open or closed)
  • Resonance boxes (guitar bodies, violin bodies) amplify sound by resonating at the same frequencies as the strings
• Bridges and structures have natural frequencies depending on material, size, and shape
  • External forces (wind, earthquakes, marching soldiers) with frequencies matching natural frequencies can induce resonance
    • Large-amplitude vibrations may cause structural damage or collapse (Tacoma Narrows Bridge, 1940)
  • Engineers design structures to avoid resonance by ensuring natural frequencies differ from expected external frequencies
• Forced oscillation occurs when an external periodic force is applied to a system, potentially leading to resonance
• Damping reduces the amplitude of oscillations over time, affecting the resonance behavior of a system