Glossary

Waves are mathematical marvels, described by elegant equations that capture their oscillating nature. These functions reveal how waves move through space and time, showing the interplay between amplitude, , and .

Diving deeper, we see the difference between and . While waves propagate through a medium, particles oscillate in place. This distinction is crucial for understanding wave behavior in various physical phenomena.

Wave Mathematics

Mathematical function for constant velocity waves

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  • A describes a wave moving at constant velocity
    • General form: y(x,t)=Asin⁡(kx−ωt+ϕ)y(x,t) = A \sin(kx - \omega t + \phi) or y(x,t)=Acos⁡(kx−ωt+ϕ)y(x,t) = A \cos(kx - \omega t + \phi)
      • AA represents the
      • kk represents the , calculated as 2π/λ2\pi/\lambda, where λ\lambda is the
      • ω\omega represents the , calculated as 2πf2\pi f, where ff is the frequency
      • ϕ\phi represents the
    • The is the argument of the sine or cosine function, expressed as (kx−ωt+ϕ)(kx - \omega t + \phi)
  • The depends on both position xx and time tt
    • At a fixed position, the wave oscillates in time with a of T=1/fT = 1/f (e.g., a buoy bobbing up and down in the ocean)
    • At a fixed time, the wave oscillates in space with a wavelength of λ\lambda (e.g., a snapshot of a wave on a string)
  • The , a second-order partial differential equation, describes the propagation of waves in a medium

Particle velocity and acceleration in waves

  • Particles in a wave-carrying medium oscillate about their equilibrium positions
    • The y(x,t)y(x,t) gives the displacement of a particle from its
  • The particle velocity is the time derivative of its displacement
    • For a : v(x,t)=∂y∂t=−Aωcos⁡(kx−ωt+ϕ)v(x,t) = \frac{\partial y}{\partial t} = -A\omega \cos(kx - \omega t + \phi)
    • For a : v(x,t)=−Aωsin⁡(kx−ωt+ϕ)v(x,t) = -A\omega \sin(kx - \omega t + \phi)
  • The is the time derivative of its velocity or the second time derivative of its displacement
    • For a sine wave: a(x,t)=∂v∂t=∂2y∂t2=−Aω2sin⁡(kx−ωt+ϕ)a(x,t) = \frac{\partial v}{\partial t} = \frac{\partial^2 y}{\partial t^2} = -A\omega^2 \sin(kx - \omega t + \phi)
    • For a cosine wave: a(x,t)=−Aω2cos⁡(kx−ωt+ϕ)a(x,t) = -A\omega^2 \cos(kx - \omega t + \phi)
  • Example: In a , air particles oscillate back and forth, with their velocity and acceleration determined by the wave function

Wave velocity vs particle velocity

  • The wave velocity, or , is the speed at which the wave propagates through the medium
    • Calculated as vp=ωk=λT=λfv_p = \frac{\omega}{k} = \frac{\lambda}{T} = \lambda f
    • Depends on the medium properties and wave type (e.g., sound waves in air vs. water)
  • The particle velocity is the speed at which the particles oscillate about their equilibrium positions
    • Given by the time derivative of the wave function, v(x,t)=∂y∂tv(x,t) = \frac{\partial y}{\partial t}
    • Varies with position and time, with a maximum value of AωA\omega
  • The wave velocity and particle velocity are generally not the same
    • In most cases, the wave velocity is much greater than the maximum particle velocity (e.g., sound waves in air)
    • Particles do not move with the wave; they oscillate as the wave passes through the medium (e.g., water molecules in ocean waves)
  • In dispersive media, the , which represents the velocity of a wave packet, may differ from the phase velocity

Advanced Wave Concepts

  • The states that when two or more waves overlap, the resulting displacement is the sum of the individual wave displacements
  • occurs when different frequency components of a wave travel at different velocities in a medium
  • allows complex waveforms to be decomposed into simpler sinusoidal components, enabling the study of wave spectra and harmonic content

Key Terms to Review (32)

Angular frequency: Angular frequency, denoted by $\omega$, is the rate of change of angular displacement with time. It is commonly measured in radians per second (rad/s).
Angular Frequency: Angular frequency, often represented by the Greek letter $\omega$ (omega), is a fundamental concept that describes the rate of change of the angular position of an object undergoing rotational or oscillatory motion. It is a crucial parameter in understanding various physical phenomena, including simple harmonic motion, wave propagation, and the behavior of oscillating systems.
Cosine Wave: A cosine wave is a periodic waveform that oscillates between positive and negative values, with the wave shape following the mathematical cosine function. It is a fundamental waveform in the study of wave phenomena and is widely used in various fields, including physics, engineering, and signal processing.
Dispersion: Dispersion is the phenomenon where different components of a wave, such as different frequencies or wavelengths, travel at different velocities through a medium. This causes the wave to spread out and separate into its constituent parts as it propagates.
Equilibrium position: The equilibrium position is the point at which the net force acting on an oscillating system is zero. At this position, the system experiences no acceleration and remains at rest if undisturbed.
Equilibrium Position: The equilibrium position is the point at which a system is in a state of balance, where the net force or torque acting on the system is zero. This concept is fundamental in understanding the behavior of various physical systems, including those related to simple harmonic motion, circular motion, damped oscillations, and wave propagation.
Fourier Analysis: Fourier analysis is a mathematical technique that decomposes a periodic function into an infinite sum of sine and cosine functions. It is a powerful tool for understanding the frequency content of a signal or wave and has numerous applications in physics, engineering, and other scientific fields.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event or phenomenon per unit of time. It is a crucial parameter in various areas of physics, including wave behavior, oscillations, and sound propagation.
Group Velocity: Group velocity is the velocity at which the overall shape or envelope of the wave's amplitudes propagates through space. It describes the speed at which the wave packet, or modulation of the wave, travels rather than the speed of the individual wave crests or troughs.
Linear wave equation: The linear wave equation is a second-order partial differential equation that describes the propagation of linear waves, such as sound or light waves, in a medium. It is typically written as $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$, where $u$ represents the wave function and $c$ is the speed of the wave.
Orbital period: The orbital period is the time taken for a satellite or celestial body to complete one full orbit around another object. It is typically measured in seconds, minutes, hours, or years.
Particle Acceleration: Particle acceleration is the process by which the velocity of a particle is increased over time. This is a fundamental concept in the study of waves, as the acceleration of particles is directly related to the propagation and characteristics of wave motion.
Particle Velocity: Particle velocity refers to the speed and direction of movement of a particle or small object within a wave or oscillating system. It is a fundamental concept in the study of wave mechanics and is essential for understanding the propagation and behavior of various types of waves.
Period: The period of a periodic phenomenon is the time taken for one complete cycle or repetition of the event. This concept is fundamental in understanding various physics topics, including uniform circular motion, simple harmonic motion, and wave phenomena.
Phase: Phase refers to the state or stage of a wave or oscillation at a particular point in time or space. It describes the displacement of a wave relative to a reference point or another wave, and is a fundamental concept in the study of wave phenomena.
Phase Constant: The phase constant, denoted by the Greek letter '$\phi$', is a parameter that determines the initial position or starting point of a wave or oscillating system relative to a reference point. It represents the phase angle or offset of the wave at time '$t=0$'.
Phase of the wave: The phase of a wave describes the position of a point in time on a waveform cycle. It is usually measured in degrees or radians and can indicate the state of oscillation at any given moment.
Phase Velocity: Phase velocity is the rate at which the phase of a wave propagates in space. It represents the speed at which the wave pattern itself moves, which may be different from the speed at which the individual particles in the medium are oscillating.
Pulse: A pulse is a single, non-repeating disturbance that moves through a medium. It can be characterized by its amplitude, duration, and speed.
Sine Wave: A sine wave is a continuous, periodic function that represents a smooth, undulating curve. It is the most fundamental waveform in mathematics and physics, describing the shape of many natural and artificial phenomena, including sound waves, light waves, and alternating current (AC) electrical signals.
Sinusoidal Function: A sinusoidal function is a mathematical function that describes a periodic wave-like oscillation. It is characterized by a sine or cosine curve, which repeats itself at regular intervals, and is commonly used to model various periodic phenomena in the physical world, such as the motion of a pendulum or the vibration of a guitar string.
Sound Wave: A sound wave is a longitudinal wave that propagates through a medium, such as air or water, as a disturbance of particles in the medium. It is a mechanical wave that carries energy through the vibration of particles, causing changes in pressure and density that are perceived as sound.
Superposition Principle: The superposition principle states that for linear systems, the net response caused by two or more stimuli is the sum of the individual responses that each stimulus would cause separately. This principle applies to various physical phenomena, including the behavior of waves, gravitational fields, and normal modes of vibration.
Wave Amplitude: Wave amplitude refers to the maximum displacement of a wave from its resting or equilibrium position. It is a measure of the magnitude or strength of a wave and represents the maximum distance the wave travels from its central axis or midpoint.
Wave Equation: The wave equation is a fundamental mathematical equation that describes the propagation of waves, such as sound waves, light waves, and waves on a string. It governs the relationship between the displacement of a wave and the variables that determine its behavior, including time, position, and the properties of the medium through which the wave is traveling.
Wave function: A wave function is a mathematical representation that describes the amplitude of a wave at each point in space and time. It encapsulates the properties of waves such as frequency, wavelength, and phase.
Wave Function: The wave function is a mathematical representation of the quantum state of an object or a system. It provides a complete description of the quantum mechanical behavior of an entity and allows for the calculation of the probability of its measurable properties.
Wave number: Wave number is a measure of the number of wavelengths per unit distance, typically expressed in reciprocal meters (m^-1). It is mathematically defined as the spatial frequency of a wave, represented by the symbol $k$ and calculated as $k = \frac{2\pi}{\lambda}$, where $\lambda$ is the wavelength.
Wave Number: The wave number is a fundamental property of a wave that represents the number of waves per unit distance. It is a measure of the spatial frequency of a wave and is inversely proportional to the wavelength of the wave.
Wave Velocity: Wave velocity is the speed at which a wave propagates through a medium. It is a fundamental property of waves that describes how quickly the wave disturbance travels through the medium, such as air, water, or a solid material.
Wavelength: Wavelength is the distance between successive crests or troughs of a wave. It is typically represented by the Greek letter lambda ($\lambda$).
Wavelength: Wavelength is a fundamental characteristic of waves, representing the distance between consecutive peaks or troughs of a wave. It is a crucial parameter that describes the spatial properties of various wave phenomena, including light, sound, and other types of oscillations.
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