Glossary

14.3 Waves in Continuous Media

2 min read•july 22, 2024

Waves in elastic media are governed by the , linking displacement to stress and density. This fundamental equation describes various phenomena, from sound waves to seismic activity, and forms the basis for understanding wave propagation in different materials.

Dispersion relations connect frequency and , revealing how waves behave in different media. have constant , while show frequency-dependent speeds. Understanding these relationships is crucial for analyzing wave behavior in various physical systems.

Wave Equation and Dispersion Relations

Wave equation in elastic media

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  • Describes propagation of waves in continuous elastic media such as solids, liquids, and gases
  • Derived from equation of motion relating density ρ\rho, displacement uu, and stress σ\sigma
  • Incorporates Hooke's law connecting stress σ\sigma and strain ∂u∂x\frac{\partial u}{\partial x} via Young's modulus EE
  • Resulting wave equation: ∂2u∂t2=Eρ∂2u∂x2\frac{\partial^2 u}{\partial t^2} = \frac{E}{\rho} \frac{\partial^2 u}{\partial x^2}, where Eρ\frac{E}{\rho} represents square of wave speed c2c^2
  • Governs various wave phenomena (sound waves in air, seismic waves in Earth's crust)

Dispersion relations for various media

  • Connect ω\omega and wavenumber kk, characterizing wave propagation in different media
  • Non-dispersive media exhibit constant wave speed for all frequencies (ω=ck\omega = ck)
    • Examples: sound waves in air, light in vacuum
  • Dispersive media have frequency-dependent wave speed (ω=ω(k)\omega = \omega(k))
    • Examples: water waves, light in optical fibers
  • Longitudinal waves involve displacement parallel to propagation direction (sound waves)
  • Transverse waves have displacement perpendicular to propagation direction (electromagnetic waves)
  • Surface waves propagate along interface between two media (water waves, Rayleigh waves)

Wave Velocities and Boundary Conditions

Phase vs group velocities

  • vp=ωkv_p = \frac{\omega}{k} represents speed of individual frequency component
  • vg=dωdkv_g = \frac{d\omega}{dk} describes speed of wave packet envelope
  • Non-dispersive media: phase and group velocities are equal and constant (vp=vg=cv_p = v_g = c)
  • Dispersive media: phase and group velocities differ (vp≠vgv_p \neq v_g), causing dispersion
    • Examples: light in dispersive materials (prisms), water waves of different wavelengths
  • Dispersion leads to separation of frequency components and broadening of wave packets over time

Boundary conditions for wave reflection

  • Reflection occurs when wave encounters boundary and returns to original medium
    • Reflected wave maintains frequency, may have altered amplitude and phase
  • Transmission happens when wave crosses boundary into another medium
    • Transmitted wave can have different speed, wavelength, and amplitude
  • Boundary conditions ensure continuity of displacement uu and stress σ\sigma at interface
    • Fixed boundary: u=0u = 0 (nodes of standing waves on strings)
    • Free boundary: σ=0\sigma = 0 (open end of organ pipe)
  • Reflection and transmission coefficients quantify amplitude ratios of reflected and transmitted waves relative to incident wave
    • Determined by applying boundary conditions at interface
    • Examples: light reflecting off mirror (high reflection coefficient), sound transmitting through wall (low transmission coefficient)

Key Terms to Review (20)

Acoustics: Acoustics is the branch of physics that deals with the study of sound, including its production, transmission, and effects. It plays a crucial role in understanding how sound waves propagate through different media, which is essential for applications such as audio engineering, architecture, and environmental noise control. By analyzing the behavior of sound waves in various contexts, acoustics helps to optimize sound quality and manage noise levels effectively.
Angular Frequency: Angular frequency is a measure of how quickly an object rotates or oscillates, expressed in radians per second. It connects the frequency of oscillation with the circular motion, representing the rate of change of the phase of a sinusoidal waveform. This concept is crucial for understanding systems involving periodic motion, such as those found in mechanics, electrical circuits, quantum mechanics, and wave phenomena.
Augustin-Jean Fresnel: Augustin-Jean Fresnel was a French engineer and physicist known for his groundbreaking work on the wave theory of light, particularly the development of the Fresnel equations and the design of the Fresnel lens. His contributions laid the foundation for understanding light propagation and diffraction, significantly influencing optics and wave behavior in continuous media.
Christiaan Huygens: Christiaan Huygens was a Dutch mathematician and physicist best known for his work on the wave theory of light and the development of the pendulum clock. His ideas laid the groundwork for understanding wave phenomena, particularly in continuous media, where he introduced concepts that explain how waves propagate through different materials.
Diffraction: Diffraction is the bending and spreading of waves when they encounter an obstacle or pass through an aperture that is comparable in size to their wavelength. This phenomenon illustrates how waves can interact with their environment, leading to patterns that can be observed in various contexts, such as sound, light, and water waves. Understanding diffraction is crucial for analyzing wave behavior in continuous media, revealing insights about interference, wavelength, and the nature of wave propagation.
Dispersion relation: A dispersion relation describes the relationship between the frequency of a wave and its wavenumber, revealing how wave speed varies with wavelength in a given medium. This concept is crucial for understanding how different waves propagate in continuous media, as it helps predict behaviors such as phase and group velocity. The dispersion relation often depends on the physical properties of the medium, such as density and elasticity.
Dispersive media: Dispersive media are materials in which the phase velocity of a wave depends on its frequency, resulting in the spreading of wave packets over time. This phenomenon is significant because it affects how signals and information propagate through the medium, leading to various applications in technology and science. In contexts like signal processing and quantum mechanics, dispersive effects are crucial for understanding the behavior of waves, while in continuous media, they influence wave interactions and energy transfer.
Group Velocity: Group velocity is the speed at which the overall envelope of a wave packet or group of waves travels through space. This concept is essential when considering how energy and information propagate through continuous media, as it often differs from the phase velocity, which is the speed of individual wave crests.
Interference: Interference is a phenomenon that occurs when two or more waves overlap and combine, resulting in a new wave pattern. This can lead to regions of constructive interference, where the waves amplify each other, and destructive interference, where they cancel each other out. This concept is crucial for understanding various physical behaviors of waves in continuous media, including sound and light waves, and their interactions.
Longitudinal wave: A longitudinal wave is a type of mechanical wave in which the particle displacement is parallel to the direction of wave propagation. This means that as the wave travels through a medium, particles of that medium move back and forth along the same direction as the wave itself, creating areas of compression and rarefaction. Longitudinal waves are essential in understanding how sound travels through various materials.
Non-dispersive media: Non-dispersive media are materials where waves propagate without changing their shape or speed over time. In such media, all frequency components of a wave travel at the same velocity, allowing for the preservation of waveforms and enabling clearer communication and signal transmission. This characteristic is vital in various physical applications, ensuring that signals maintain their integrity as they travel through the medium.
Phase Velocity: Phase velocity is the speed at which a particular phase of a wave propagates through space. It is defined mathematically as the ratio of the wavelength to the wave's period, represented as $$v_p = \frac{\lambda}{T}$$, where $$\lambda$$ is the wavelength and $$T$$ is the period. Understanding phase velocity is crucial for analyzing wave behaviors in continuous media, as it helps describe how energy and information are transmitted through different materials.
Potential Energy: Potential energy is the stored energy in an object due to its position, condition, or configuration. This energy can be converted into kinetic energy when the object is set in motion. It plays a crucial role in various physical systems, including mechanical systems and waves, influencing how objects move and interact with one another.
Seismology: Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth. It involves understanding how seismic waves, generated by natural events or human activities, travel through different geological materials. Seismology plays a crucial role in predicting seismic hazards and assessing the Earth's internal structure, which is essential for understanding wave behavior in continuous media.
Superposition Principle: The superposition principle states that when two or more waves overlap in a medium, the resultant displacement at any point is equal to the sum of the individual displacements caused by each wave. This principle is fundamental in understanding wave behavior and allows for the analysis of complex waveforms by breaking them down into simpler components, making it particularly relevant when examining wave equations and their solutions or waves propagating through continuous media.
Surface Wave: A surface wave is a type of wave that travels along the interface between two different media, such as water and air, or between solid and fluid materials. These waves are characterized by particle motion that is typically elliptical or circular, with energy concentrated near the surface. This unique behavior makes surface waves significant in various physical phenomena, including ocean waves and seismic activities.
Transverse Wave: A transverse wave is a type of wave where the oscillations of the medium are perpendicular to the direction of the wave's propagation. This characteristic leads to the formation of crests and troughs, allowing transverse waves to exhibit unique behaviors in various contexts, such as when analyzing wave equations or studying waves in continuous media.
Wave equation: The wave equation is a second-order partial differential equation that describes the propagation of waves, such as sound waves, light waves, and water waves, through various media. It captures how the displacement of a wave changes over time and space, highlighting essential properties like speed, frequency, and wavelength. The wave equation plays a critical role in both mathematical modeling and the analysis of dynamic systems.
Wave speed: Wave speed is the speed at which a wave travels through a medium, defined as the distance a wave travels per unit of time. This speed is influenced by various properties of the medium, such as its density and elasticity, as well as the type of wave being transmitted. Understanding wave speed is crucial for analyzing how different types of waves propagate through continuous media and for solving problems related to wave behavior.
Wavenumber: Wavenumber is a measure of spatial frequency defined as the number of wavelengths per unit distance, typically expressed in reciprocal meters (m\^{-1}). It relates to the wavelength and frequency of a wave and is crucial in understanding wave behavior in continuous media, linking various wave properties like energy and momentum.
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