1.5 Wave Equation and Wave Speed

Waves are fundamental to physics, describing how energy moves through space and time. The wave equation and wave speed are key concepts, helping us understand how waves behave in different media and situations.

This section dives into the math behind waves, showing how they're described and how fast they move. We'll see how these ideas apply to real-world examples, from guitar strings to light in diamonds.

Derivation of the Wave Equation

Fundamental Principles and Concepts

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• Wave equation describes propagation of waves in a medium using second-order partial differential equation
• Derivation utilizes Newton's Second Law of Motion and Hooke's Law as foundational principles
• Process involves analyzing forces acting on small element of the medium
• Equation relates spatial and temporal derivatives of wave function
• One-dimensional wave equation takes form $\frac{\partial^2y}{\partial t^2} = v^2\frac{\partial^2y}{\partial x^2}$
  • y represents displacement
  • t denotes time
  • x indicates position
  • v signifies wave speed
• Extension to higher dimensions allows for more complex wave propagation scenarios
  • Two-dimensional equation used for waves on surfaces (membranes)
  • Three-dimensional equation applied to sound waves in air or seismic waves in Earth's crust

Mathematical Derivation Steps

• Begin with small element of medium (string segment)
• Apply Newton's Second Law $F = ma$ to element
• Express forces in terms of tension and displacement using Hooke's Law
• Derive equation of motion for element
• Take limit as element size approaches zero
• Simplify resulting equation to obtain wave equation
• Identify wave speed in terms of medium properties (tension and mass density for string)

Wave Speed in Different Media

Mechanical Waves

• Wave speed depends on medium properties through which wave propagates
• General formula for mechanical waves $v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}}$
• Specific formulas for different media:
  • Strings or wires: $v = \sqrt{\frac{T}{\mu}}$ (T tension, μ linear mass density)
  • Sound waves in fluids: $v = \sqrt{\frac{B}{\rho}}$ (B bulk modulus, ρ density)
  • Longitudinal waves in solids: $v = \sqrt{\frac{Y}{\rho}}$ (Y Young's modulus)
• Examples of wave speeds:
  • Guitar string (nylon, medium tension) approximately 500 m/s
  • Sound in air at room temperature roughly 343 m/s
  • Seismic P-waves in granite about 5000 m/s

Electromagnetic Waves

• Speed of electromagnetic waves in medium given by $v = \frac{c}{n}$
  • c represents speed of light in vacuum (299,792,458 m/s)
  • n denotes refractive index of medium
• Examples of electromagnetic wave speeds:
  • Light in water (n ≈ 1.33) approximately 225,000,000 m/s
  • Light in diamond (n ≈ 2.42) roughly 124,000,000 m/s

Factors Affecting Wave Speed

• Temperature influences wave speed
  • Sound waves travel faster in warmer air
  • Affects speed of waves in solids (thermal expansion)
• Pressure impacts wave propagation
  • Higher pressure increases speed of sound in gases
  • Affects wave speed in compressed solids
• Other physical properties affect wave speed
  • Salinity changes speed of sound in water
  • Humidity alters speed of sound in air

Solving the Wave Equation

Boundary Conditions and Their Effects

• Boundary conditions specify wave behavior at medium edges or interfaces
• Common boundary conditions:
  • Fixed ends (string tied at both ends)
  • Free ends (string with loose ends)
  • Periodic boundaries (circular membrane)
• Fixed ends lead to standing waves with nodes at boundaries
  • Example: guitar string produces harmonics
• Free ends allow antinodes at boundaries
  • Example: open-ended organ pipe
• Periodic boundaries result in waves repeating in space
  • Example: vibrations on surface of drum

Solution Methods and Applications

• Separation of variables technique often employed to solve wave equation
  • Assumes solution can be written as product of functions of space and time
  • Leads to two separate ordinary differential equations
• Solutions with boundary conditions introduce concepts of:
  • Normal modes (standing wave patterns)
  • Natural frequencies (resonant frequencies of system)
• Examples of applications:
  • Calculating resonant frequencies of musical instruments
  • Designing acoustic spaces for optimal sound quality
  • Analyzing vibrations in engineering structures (bridges, buildings)

Dispersion Relation for Waves

Concept and Mathematical Representation

• Dispersion relation describes relationship between wave frequency and wavenumber
• General form $\omega = \omega(k)$
  • ω represents angular frequency
  • k denotes wavenumber
• Non-dispersive media have linear dispersion relation $\omega = vk$
  • v signifies constant wave speed
• Dispersive media exhibit frequency-dependent phase velocity
  • Phase velocity $v_p = \frac{\omega}{k}$ varies with frequency
  • Group velocity $v_g = \frac{d\omega}{dk}$ differs from phase velocity

Examples and Phenomena

• Water waves in deep water show dispersion
  • Dispersion relation $\omega = \sqrt{gk}$ (g gravitational acceleration)
  • Longer wavelengths travel faster than shorter ones
• Electromagnetic waves in certain materials display dispersion
  • Leads to phenomena like chromatic aberration in lenses
• Analysis of dispersion relation reveals:
  • Wave packet spreading in dispersive media
  • Anomalous dispersion in certain frequency ranges
    • Example: light passing through glass near absorption frequencies