Wave Mechanics

Wave mechanics is the study of waves, which are oscillatory disturbances that transfer energy through a medium or vacuum without transporting matter. These phenomena are central to physics, appearing in sound waves, light waves, water waves, and seismic waves. Wave mechanics builds on principles of oscillation and interference, described by mathematical relationships involving wavelength, frequency, amplitude, and speed. This guide provides a comprehensive overview with all basic formulas (wave speed, period, intensity, energy, Doppler effect, standing waves), detailed examples, and practical applications to illustrate their significance in science and technology.

Wave Properties (with Period, Intensity, Energy)

Waves are defined by several key properties. Wavelength (\(\lambda\)) is the distance between consecutive crests, frequency (\(f\)) is the number of cycles per second, and amplitude is the maximum displacement. The speed (\(v\)) relates to frequency and wavelength.

Basic Formulas:

Wave Speed: \( v = f \lambda \)
Period: \( T = \frac{1}{f} \)
Intensity: \( I = \frac{P}{A} = \frac{1}{2} \rho v \omega^2 A^2 \) (where \( P \) is power, \( A \) is area, \( \rho \) is medium density, \( \omega = 2\pi f \) is angular frequency, \( A \) is amplitude)
Wave Energy: \( E = \frac{1}{2} m \omega^2 A^2 \) (for a mass on a spring, where \( m \) is mass)

Example 1: Wave Speed

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

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

Example 2: Period

Find the period for a wave with \(f = 4 \, \text{Hz}\):

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

Example 3: Intensity

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

\[ \omega = 2\pi f \] \[ = 2\pi \times 100 \] \[ = 200\pi \] \[ I = \frac{1}{2} \rho v \omega^2 A^2 \] \[ = \frac{1}{2} \cdot 1.2 \cdot 340 \cdot (200\pi)^2 \cdot (0.01)^2 \] \[ = 0.6 \cdot 340 \cdot 40000\pi^2 \cdot 0.0001 \] \[ \approx 0.6 \cdot 340 \cdot 3958400 \cdot 0.0001 \] \[ \approx 80.7 \, \text{W/m}^2 \]

Example 4: Wave Energy

Energy of a 0.5 kg mass on a spring with \(f = 2 \, \text{Hz}\), \(A = 0.1 \, \text{m}\):

\[ \omega = 2\pi f \] \[ = 2\pi \times 2 \] \[ = 4\pi \] \[ E = \frac{1}{2} m \omega^2 A^2 \] \[ = \frac{1}{2} \cdot 0.5 \cdot (4\pi)^2 \cdot (0.1)^2 \] \[ = 0.25 \cdot 16\pi^2 \cdot 0.01 \] \[ \approx 0.25 \cdot 157.91 \cdot 0.01 \] \[ \approx 0.395 \, \text{J} \]

Wave Equation (with Doppler Effect, Standing Waves)

The wave equation relates speed, frequency, and wavelength. Additional phenomena include the Doppler effect (frequency shift due to relative motion) and standing waves (resulting from interference).

Basic Formulas:

Wave Speed: \( v = f \lambda \)
Doppler Effect: \( f' = f \frac{v \pm v_o}{v \mp v_s} \) (where \( v_o \) is observer speed, \( v_s \) is source speed, \( v \) is wave speed, signs depend on approach/recede)
Standing Wave Frequency: \( f_n = \frac{n v}{2L} \) (where \( n \) is harmonic number, \( L \) is length)

Example 1: Wave Speed

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

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

Example 2: Doppler Effect (Approaching)

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

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

Example 3: Standing Wave

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

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

Example 4: Doppler Effect (Receding)

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

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

Example 5: Standing Wave (Second Harmonic)

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

\[ f_n = \frac{n v}{2L} \] \[ = \frac{2 \cdot 400}{2 \cdot 1} \] \[ = \frac{800}{2} \] \[ = 400 \, \text{Hz} \]

Applications

Waves are essential in various fields. Here are detailed examples with calculations:

Example 1: Sound Engineering

Speed of sound in air (\( v = 340 \, \text{m/s} \)), \(\lambda = 0.85 \, \text{m}\). Find frequency:

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

Example 2: Seismology

A seismic wave with \( v = 5000 \, \text{m/s} \), \( T = 0.02 \, \text{s} \). Find \(\lambda\):

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

Example 3: Radio Waves

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

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

Example 4: Doppler Radar

A car moving toward a radar at \( v_o = 20 \, \text{m/s} \), source \( f = 10 \, \text{GHz} \), \( v = 3 \times 10^8 \, \text{m/s} \), \( v_s = 0 \):

\[ f' = f \frac{v + v_o}{v - v_s} \] \[ = 10 \times 10^9 \frac{3 \times 10^8 + 20}{3 \times 10^8 - 0} \] \[ \approx 10 \times 10^9 \frac{3 \times 10^8}{3 \times 10^8} \] \[ \approx 10.000067 \times 10^9 \, \text{Hz} \]

Example 5: Musical Instrument

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

\[ f_n = \frac{n v}{2L} \] \[ = \frac{1 \cdot 120}{2 \cdot 0.6} \] \[ = \frac{120}{1.2} \] \[ = 100 \, \text{Hz} \]

Example 6: Wave Energy in Water

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

\[ \omega = 2\pi f \] \[ = 2\pi \cdot 2 \] \[ = 4\pi \] \[ E = \frac{1}{2} m \omega^2 A^2 \] \[ = \frac{1}{2} \cdot 0.1 \cdot (4\pi)^2 \cdot (0.05)^2 \] \[ = 0.05 \cdot 16\pi^2 \cdot 0.0025 \] \[ \approx 0.05 \cdot 157.91 \cdot 0.0025 \] \[ \approx 0.0197 \, \text{J} \]