Wave Mechanics: A Comprehensive Guide

Wave mechanics studies oscillatory disturbances that transfer energy through a medium or vacuum without moving matter. Waves, such as sound, light, water, or seismic waves, are fundamental to physics, described by properties like wavelength, frequency, and amplitude. This MathMultiverse guide explores wave properties, equations (including Doppler effect and standing waves), visualizations, and applications, providing detailed examples and formulas to illustrate their role in science and technology.

Wave Properties

Waves are characterized by key properties: wavelength (\( \lambda \)), the distance between consecutive crests; frequency (\( f \)), cycles per second (Hz); amplitude (\( A \)), maximum displacement; and speed (\( v \)), how fast the wave travels. Additional properties include period, intensity, and energy, crucial for understanding wave behavior.

Key Formulas

  • Wave Speed: Relates frequency and wavelength:
    • \[ v = f \lambda \]
  • Period: Time for one cycle:
    • \[ T = \frac{1}{f} \]
  • Intensity: Power per unit area (for mechanical waves):
    • \[ I = \frac{P}{A} = \frac{1}{2} \rho v \omega^2 A^2 \]
    \[ \text{where } \omega = 2\pi f, \ \rho = \text{medium density} \]
  • Wave Energy: For a harmonic oscillator (e.g., mass on a spring):
    • \[ E = \frac{1}{2} m \omega^2 A^2 \]
    \[ \text{where } m = \text{mass} \]

Example: Wave Speed

A wave with \( \lambda = 2 \, \text{m} \), \( f = 5 \, \text{Hz} \):

\[ v = f \lambda = 5 \times 2 = 10 \, \text{m/s} \]

Example: Period

For \( f = 4 \, \text{Hz} \):

\[ T = \frac{1}{f} = \frac{1}{4} = 0.25 \, \text{s} \]

Example: Intensity

Sound wave with \( f = 100 \, \text{Hz} \), \( \rho = 1.2 \, \text{kg/m}^3 \), \( v = 340 \, \text{m/s} \), \( A = 0.01 \, \text{m} \):

\[ \omega = 2\pi f = 2\pi \times 100 = 200\pi \]
\[ I = \frac{1}{2} \rho v \omega^2 A^2 = \frac{1}{2} \cdot 1.2 \cdot 340 \cdot (200\pi)^2 \cdot (0.01)^2 \]
\[ \approx 0.6 \cdot 340 \cdot 394784 \cdot 0.0001 \approx 80.7 \, \text{W/m}^2 \]

Example: Wave Energy

Mass of 0.5 kg, \( f = 2 \, \text{Hz} \), \( A = 0.1 \, \text{m} \):

\[ \omega = 2\pi f = 2\pi \times 2 = 4\pi \]
\[ E = \frac{1}{2} m \omega^2 A^2 = \frac{1}{2} \cdot 0.5 \cdot (4\pi)^2 \cdot (0.1)^2 \]
\[ \approx 0.25 \cdot 157.91 \cdot 0.01 \approx 0.395 \, \text{J} \]

Wave Equation

The wave equation describes wave propagation, with phenomena like the Doppler effect (frequency shift due to motion) and standing waves (from interference). These are modeled mathematically for various applications.

Key Formulas

  • Wave Speed:
    • \[ v = f \lambda \]
  • Doppler Effect: Frequency shift when source or observer moves:
    • \[ f' = f \frac{v \pm v_o}{v \mp v_s} \]
    \[ \text{(+ for observer approaching, - for receding; - for source approaching, + for receding)} \]
  • Standing Wave Frequency: For a string fixed at both ends:
    • \[ f_n = \frac{n v}{2L}, \quad n = 1, 2, 3, \ldots \]

Example: Wave Speed

Wave with \( \lambda = 0.5 \, \text{m} \), \( f = 10 \, \text{Hz} \):

\[ v = f \lambda = 10 \times 0.5 = 5 \, \text{m/s} \]

Example: Doppler Effect (Approaching)

Source \( f = 440 \, \text{Hz} \), observer moves toward source at \( v_o = 10 \, \text{m/s} \), source at \( v_s = 20 \, \text{m/s} \), \( v = 340 \, \text{m/s} \):

\[ f' = f \frac{v + v_o}{v - v_s} = 440 \frac{340 + 10}{340 - 20} = 440 \frac{350}{320} \approx 481.25 \, \text{Hz} \]

Example: Standing Wave

Fundamental frequency, \( L = 0.8 \, \text{m} \), \( v = 320 \, \text{m/s} \), \( n = 1 \):

\[ f_n = \frac{n v}{2L} = \frac{1 \cdot 320}{2 \cdot 0.8} = \frac{320}{1.6} = 200 \, \text{Hz} \]

Example: Doppler Effect (Receding)

Source \( f = 500 \, \text{Hz} \), observer moves away at \( v_o = 5 \, \text{m/s} \), source at \( v_s = 15 \, \text{m/s} \), \( v = 340 \, \text{m/s} \):

\[ f' = f \frac{v - v_o}{v + v_s} = 500 \frac{340 - 5}{340 + 15} = 500 \frac{335}{355} \approx 471.83 \, \text{Hz} \]

Example: Second Harmonic

Second harmonic, \( n = 2 \), \( L = 1 \, \text{m} \), \( v = 400 \, \text{m/s} \):

\[ f_n = \frac{n v}{2L} = \frac{2 \cdot 400}{2 \cdot 1} = 400 \, \text{Hz} \]

Traveling and Standing Waves

Graphs of a traveling wave \( y = \sin(x - t) \) and a standing wave \( y = 2\sin(x)\cos(t) \) at \( t = 0 \).

Applications

Wave mechanics underpins numerous fields. Below are examples with calculations:

Sound Engineering

Speed of sound \( v = 340 \, \text{m/s} \), \( \lambda = 0.85 \, \text{m} \):

\[ f = \frac{v}{\lambda} = \frac{340}{0.85} = 400 \, \text{Hz} \]

Seismology

Seismic wave, \( v = 5000 \, \text{m/s} \), \( T = 0.02 \, \text{s} \):

\[ f = \frac{1}{T} = \frac{1}{0.02} = 50 \, \text{Hz} \]
\[ \lambda = \frac{v}{f} = \frac{5000}{50} = 100 \, \text{m} \]

Radio Waves

Radio wave, \( f = 1 \times 10^6 \, \text{Hz} \), \( v = 3 \times 10^8 \, \text{m/s} \):

\[ \lambda = \frac{v}{f} = \frac{3 \times 10^8}{1 \times 10^6} = 300 \, \text{m} \]

Doppler Radar

Car approaching radar, \( f = 10 \, \text{GHz} \), \( v_o = 20 \, \text{m/s} \), \( v_s = 0 \), \( v = 3 \times 10^8 \, \text{m/s} \):

\[ f' = f \frac{v + v_o}{v - v_s} = 10 \times 10^9 \frac{3 \times 10^8 + 20}{3 \times 10^8} \approx 10.000067 \times 10^9 \, \text{Hz} \]

Musical Instrument

Guitar string, \( L = 0.6 \, \text{m} \), \( v = 120 \, \text{m/s} \), fundamental frequency:

\[ f_1 = \frac{1 \cdot v}{2L} = \frac{120}{2 \cdot 0.6} = 100 \, \text{Hz} \]

Wave Energy in Water

Wave with \( m = 0.1 \, \text{kg} \), \( f = 2 \, \text{Hz} \), \( A = 0.05 \, \text{m} \):

\[ \omega = 2\pi f = 2\pi \cdot 2 = 4\pi \]
\[ E = \frac{1}{2} m \omega^2 A^2 = \frac{1}{2} \cdot 0.1 \cdot (4\pi)^2 \cdot (0.05)^2 \]
\[ \approx 0.05 \cdot 157.91 \cdot 0.0025 \approx 0.0197 \, \text{J} \]