Waves are fascinating phenomena that transfer energy through matter and space. From ripples in a pond to light from distant stars, waves shape our world in countless ways. They come in various forms, each with unique properties and behaviors.

Understanding wave motion is crucial for grasping how energy moves through different media. This topic explores the fundamental concepts, characteristics, and types of waves, setting the stage for deeper insights into oscillations and wave phenomena.

Wave motion and its characteristics

Fundamental concepts of wave motion

• Wave motion transfers energy through a medium without transferring matter
• Particles in the medium oscillate or vibrate as the wave passes through
• Energy propagates in the direction of wave motion
• Examples include water waves, sound waves, and light waves

Key characteristics of waves

• Amplitude measures the maximum displacement from equilibrium position
  • Directly related to the energy carried by the wave
  • Larger amplitude waves carry more energy
• Wavelength (λ) represents the distance between consecutive wave peaks or troughs
  • Measured in units of length (meters)
  • Inversely related to frequency
• Frequency (f) counts the number of complete wave cycles per unit time
  • Measured in Hertz (Hz)
  • Higher frequency waves oscillate more rapidly
• Period (T) measures the time for one complete wave cycle
  • Calculated as the reciprocal of frequency (T = 1/f)
  • Measured in seconds
• Wave speed (v) calculates the rate of wave propagation through a medium
  • Determined by the product of wavelength and frequency (v = λf)
  • Depends on the properties of the medium

Mathematical relationships and wave equations

• Wave equation relates spatial and temporal derivatives of wave displacement
  • Describes wave propagation in various media
  • Takes the form $\frac{\partial^2y}{\partial t^2} = v^2 \frac{\partial^2y}{\partial x^2}$
• Frequency-wavelength relationship expressed as $f = \frac{v}{\lambda}$
• Period-frequency relationship given by $T = \frac{1}{f}$
• Wave speed calculation using $v = \frac{\lambda}{T} = \lambda f$

Types of waves: Transverse vs Longitudinal

Transverse waves

• Particles oscillate perpendicular to the direction of wave propagation
• Create peaks (crests) and valleys (troughs) as the wave moves
• Examples include waves on a string, electromagnetic waves, and water surface waves
• Characterized by their ability to be polarized
• Transverse waves on a string described by the equation $y(x,t) = A \sin(kx - \omega t)$
  • A represents amplitude
  • k represents wave number
  • ω represents angular frequency

Longitudinal waves

• Particles oscillate parallel to the direction of wave propagation
• Create regions of compression and rarefaction as the wave moves
• Examples include sound waves in air, compression waves in springs
• Cannot be polarized due to parallel oscillation
• Described by the displacement equation $s(x,t) = s_0 \sin(kx - \omega t)$
  • s₀ represents the maximum displacement
  • Other variables similar to transverse wave equation

Comparison and unique properties

• Transverse waves visible as oscillations perpendicular to motion
• Longitudinal waves visible as alternating regions of compression and expansion
• Both types can exhibit phenomena like reflection, refraction, and diffraction
• Transverse waves support polarization, longitudinal waves do not
• Some waves (surface waves) combine both transverse and longitudinal motion
• Examples of surface waves include ocean waves, seismic waves (Rayleigh waves)

Properties of waves: Mechanical vs Electromagnetic

Mechanical waves

• Require a physical medium for propagation
• Transfer energy through oscillations of matter particles
• Speed depends on properties of the medium (density, elasticity)
• Examples include sound waves, water waves, seismic waves
• Cannot travel through a vacuum
• Obey Newton's laws of motion
• Experience energy loss due to friction in the medium

Electromagnetic waves

• Propagate through vacuum and matter
• Consist of oscillating electric and magnetic fields
• Travel at the speed of light in vacuum (c ≈ 3 × 10⁸ m/s)
• Examples include light, radio waves, X-rays, gamma rays
• Described by Maxwell's equations
• Do not require a medium for propagation
• Carry both energy and momentum

Common wave properties

• Both mechanical and electromagnetic waves exhibit:
  • Reflection when encountering boundaries
  • Refraction when entering a new medium
  • Diffraction around obstacles
  • Interference when multiple waves overlap
• Obey the principle of superposition
• Experience Doppler effect when source or observer moves
• Can form standing waves under specific conditions
• Transmit energy from one point to another

Wave propagation in different media

Factors affecting wave propagation

• Medium properties influence wave behavior:
  • Density affects wave speed (generally slower in denser media)
  • Elasticity impacts wave speed (faster in more elastic media)
  • Temperature can change medium properties, altering wave speed
• Boundary conditions at interfaces determine wave behavior:
  • Reflection occurs when waves encounter a boundary
  • Transmission happens when waves pass through an interface
  • Refraction bends waves when entering a new medium
• Snell's law describes refraction: $\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$
  • θ₁, θ₂ represent angles of incidence and refraction
  • v₁, v₂ represent wave speeds in respective media
  • n₁, n₂ represent refractive indices of media

Wave interactions and phenomena

• Superposition principle governs wave interactions:
  • Waves can overlap and combine algebraically
  • Results in constructive or destructive interference
• Standing waves form when waves reflect and interfere:
  • Occur in musical instruments, resonant cavities
  • Characterized by nodes (zero amplitude) and antinodes (maximum amplitude)
• Beats result from interference of waves with slightly different frequencies:
  • Beat frequency equals the difference between wave frequencies
• Dispersion causes wave components to travel at different speeds:
  • Results in separation of wave components (prism separating light)
  • Group velocity (vg) differs from phase velocity (vp) in dispersive media

Wave attenuation and energy transfer

• Attenuation decreases wave amplitude as it propagates:
  • Caused by energy dissipation in the medium
  • Follows exponential decay: $A(x) = A_0e^{-\alpha x}$
    • A₀ represents initial amplitude
    • α represents attenuation coefficient
    • x represents distance traveled
• Energy transfer in waves:
  • Energy density proportional to square of amplitude
  • Power transmitted proportional to square of amplitude and frequency
• Intensity of waves decreases with distance:
  • Follows inverse square law in 3D propagation
  • $I \propto \frac{1}{r^2}$ where r represents distance from source