This document discusses the basic principles of seismic waves. It introduces longitudinal (P) waves and shear (S) waves, and derives the one-dimensional wave equation. It discusses wave phenomena like reflection, transmission, and refraction based on Snell's law at boundaries between layers. It also discusses the different arrivals of direct, reflected, and refracted/head waves that can be measured at the surface for seismic exploration purposes.
This document provides a question bank for Physics I with questions related to three topics:
1) Waves and Oscillations: Questions cover types of waves, wave equations, phase and group velocity, standing waves, forced vibrations and resonance.
2) Fields: Questions cover vector and scalar fields, gradient, divergence, curl, Gauss's theorem, Stokes' theorem and their applications.
3) Electromagnetic Theory: Questions cover Gauss's law, electric potential, dielectrics, Ampere's law, Faraday's law, inductance, Maxwell's equations and electromagnetic waves. The document provides 25 questions for the first topic, 9 questions for the second topic and 22 questions for the third topic, for
This document provides 3 key points about angular impulse and momentum:
1) It defines angular momentum as the moment of linear momentum about a point, and derives equations relating angular momentum, moment of forces, and rate of change of angular momentum.
2) It discusses examples of applying the principle of conservation of angular momentum, including a ball on a cylinder and a ballistic pendulum.
3) It introduces the principle of angular impulse, which states that the angular impulse on a particle equals its change in angular momentum, and can be used to analyze impulsive forces.
- The document discusses gravitational waves and binary systems, including perturbative computations of gravitational wave flux from binary systems up to order v7/c7.
- It covers the effective one body (EOB) method for modeling binary coalescence, including resummations of post-Newtonian results and the addition of ringdown effects. This provides the first complete waveforms for binary black hole coalescences.
- Developments are discussed such as extending EOB to include spinning bodies, comparisons to numerical relativity results, and using gravitational self-force calculations to improve EOB modeling.
Alessandra Buonanno gave a lecture on the analytical and numerical relativity approaches used to model gravitational waveforms from inspiraling binary systems. She discussed how post-Newtonian theory, effective one body theory, and numerical relativity are used to approximately and exactly solve Einstein's field equations. She emphasized the crucial synergy between analytical and numerical relativity approaches to develop accurate gravitational waveform models like EOBNR and Phenom that have been used to infer astrophysics from LIGO/Virgo detections.
1) Gravitational waves are predicted by Einstein's theory of general relativity and are generated by accelerating masses like binary star systems.
2) Modeling the motion and gravitational wave emission of compact binary systems like neutron stars and black holes requires using techniques like post-Newtonian theory, effective field theory approaches, and numerical relativity simulations.
3) Understanding strong gravitational fields like those near black holes requires tools from general relativity like multipolar expansions, matched asymptotic expansions, and analytic continuation techniques.
1) The document presents three theoretical physics problems involving spinning balls, charged particles in loops, and laser cooling of atoms.
2) Problem 1 considers a spinning ball falling and rebounding, calculating the rebound angle, horizontal distance traveled, and minimum spin rate. Problem 2 analyzes relativistic effects on charged particles in a moving loop in an electric field.
3) Problem 3 describes using lasers to cool atoms by resonant absorption. It calculates the laser frequency needed, velocity range absorbed, direction change upon emission, maximum velocity decrease, number of absorption events to slow to zero velocity, and distance traveled during cooling.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
The document discusses gravitational waves and binary systems. It provides context on the history of gravitational wave detection, from Einstein's early work developing the theory of gravitational waves to Joseph Weber's pioneering efforts to detect them in the 1960s. It also summarizes the development of laser interferometer gravitational wave detectors by researchers in the US, Germany, Italy, and the UK beginning in the 1970s and 1980s. Key detections by LIGO and Virgo are noted, including GW150914 in 2015. Theoretical work on modeling gravitational waveforms from coalescing compact binaries is summarized, from early perturbative approaches to more recent analytical methods like the effective one-body formalism.
This document provides a question bank for Physics I with questions related to three topics:
1) Waves and Oscillations: Questions cover types of waves, wave equations, phase and group velocity, standing waves, forced vibrations and resonance.
2) Fields: Questions cover vector and scalar fields, gradient, divergence, curl, Gauss's theorem, Stokes' theorem and their applications.
3) Electromagnetic Theory: Questions cover Gauss's law, electric potential, dielectrics, Ampere's law, Faraday's law, inductance, Maxwell's equations and electromagnetic waves. The document provides 25 questions for the first topic, 9 questions for the second topic and 22 questions for the third topic, for
This document provides 3 key points about angular impulse and momentum:
1) It defines angular momentum as the moment of linear momentum about a point, and derives equations relating angular momentum, moment of forces, and rate of change of angular momentum.
2) It discusses examples of applying the principle of conservation of angular momentum, including a ball on a cylinder and a ballistic pendulum.
3) It introduces the principle of angular impulse, which states that the angular impulse on a particle equals its change in angular momentum, and can be used to analyze impulsive forces.
- The document discusses gravitational waves and binary systems, including perturbative computations of gravitational wave flux from binary systems up to order v7/c7.
- It covers the effective one body (EOB) method for modeling binary coalescence, including resummations of post-Newtonian results and the addition of ringdown effects. This provides the first complete waveforms for binary black hole coalescences.
- Developments are discussed such as extending EOB to include spinning bodies, comparisons to numerical relativity results, and using gravitational self-force calculations to improve EOB modeling.
Alessandra Buonanno gave a lecture on the analytical and numerical relativity approaches used to model gravitational waveforms from inspiraling binary systems. She discussed how post-Newtonian theory, effective one body theory, and numerical relativity are used to approximately and exactly solve Einstein's field equations. She emphasized the crucial synergy between analytical and numerical relativity approaches to develop accurate gravitational waveform models like EOBNR and Phenom that have been used to infer astrophysics from LIGO/Virgo detections.
1) Gravitational waves are predicted by Einstein's theory of general relativity and are generated by accelerating masses like binary star systems.
2) Modeling the motion and gravitational wave emission of compact binary systems like neutron stars and black holes requires using techniques like post-Newtonian theory, effective field theory approaches, and numerical relativity simulations.
3) Understanding strong gravitational fields like those near black holes requires tools from general relativity like multipolar expansions, matched asymptotic expansions, and analytic continuation techniques.
1) The document presents three theoretical physics problems involving spinning balls, charged particles in loops, and laser cooling of atoms.
2) Problem 1 considers a spinning ball falling and rebounding, calculating the rebound angle, horizontal distance traveled, and minimum spin rate. Problem 2 analyzes relativistic effects on charged particles in a moving loop in an electric field.
3) Problem 3 describes using lasers to cool atoms by resonant absorption. It calculates the laser frequency needed, velocity range absorbed, direction change upon emission, maximum velocity decrease, number of absorption events to slow to zero velocity, and distance traveled during cooling.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
The document discusses gravitational waves and binary systems. It provides context on the history of gravitational wave detection, from Einstein's early work developing the theory of gravitational waves to Joseph Weber's pioneering efforts to detect them in the 1960s. It also summarizes the development of laser interferometer gravitational wave detectors by researchers in the US, Germany, Italy, and the UK beginning in the 1970s and 1980s. Key detections by LIGO and Virgo are noted, including GW150914 in 2015. Theoretical work on modeling gravitational waveforms from coalescing compact binaries is summarized, from early perturbative approaches to more recent analytical methods like the effective one-body formalism.
The document discusses using gravitational wave waveform models to infer astrophysical properties from observations of gravitational wave events. It describes how waveform models encode information about binary black hole parameters like mass and spin, and how Bayesian inference can be used to estimate these parameters from the detected gravitational wave signal. It also addresses assessing confidence in detections and evaluating potential modeling systematics by comparing waveform models to numerical relativity simulations.
This document contains physics formulae related to mechanics, thermodynamics, electromagnetism, optics, modern physics and more. Some key formulae include:
Density = mass / volume, Force = rate of change of momentum, Kinetic energy = (1/2)mv^2, Ohm's law: V=IR, Index of refraction n=c/v, Half life of radioactive element t1/2=ln(2)/λ, Bohr's model: L=nh/2π.
The document discusses harmonic motion and traveling waves. It defines periodic and harmonic motion, and notes that harmonic motion can be described by a sinusoidal function. Hooke's law relating force and displacement in springs is introduced. Equations of motion for simple harmonic oscillators like masses on springs and pendulums are derived. The relationship between wavelength, frequency and propagation velocity is defined for traveling waves. Solutions to the wave equation for strings and their properties are also summarized.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
This document summarizes a lecture on using gravitational wave waveform models to test general relativity and probe the nature of compact objects through gravitational wave observations. It discusses how waveform models can be used to bound post-Newtonian coefficients, constrain phenomenological merger-ringdown parameters, and probe the quasi-normal modes of black hole ringdowns. Measuring multiple modes could verify the no-hair theorem and black hole uniqueness properties. Future observations from LIGO and Virgo at design sensitivity may allow high-precision black hole spectroscopy and tests of general relativity in the strong, dynamical gravity regime.
1) The document discusses using the effective one body (EOB) formalism to model gravitational wave templates for LIGO and LISA. It summarizes the number and type of templates used in LIGO's first two observing runs.
2) It also discusses using EOB, post-Newtonian theory, numerical relativity simulations, and quantum field theory to model gravitational wave emission from binary black hole and binary neutron star mergers across different mass ratio and velocity regimes.
3) The document focuses on recent work extending EOB models to higher post-Minkowskian orders and including the effects of spin and tidal interactions, with the goal of more accurate gravitational wave template modeling.
This document discusses hoisting and dynamics of rotation. It provides examples and explanations of:
1) The forces, torques, and equations of motion involved when a hoist drum raises or lowers a load while accelerating or decelerating. This includes the inertia couple of the drum opposing changes in rotation and friction torque opposing rotation.
2) Specific examples that calculate the torque required to raise a load or bring it to a stop, given information like the drum's moment of inertia, load mass, acceleration, and friction torque.
3) Diagrams illustrate the forces and torques acting on the hoist drum and load in different scenarios like raising or lowering while accelerating versus coming to a stop
1) A mass attached to a spring can move up and down from its equilibrium position. Its movement is modeled by a differential equation that depends on factors like the mass, spring constant, damping, and any external forcing.
2) There are three types of behavior for damped systems without external forcing: underdamped systems oscillate with a decaying amplitude, critically damped systems do not oscillate and cross the equilibrium at most once, and overdamped systems do not oscillate and may or may not cross the equilibrium.
3) For an undamped system with external periodic forcing, the behavior depends on whether the forcing frequency matches the natural frequency. If they are different, the displacement takes the form of a
This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include:
- Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades.
- Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments.
- Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales.
- The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.
Large scale coherent structures and turbulence in quasi-2D hydrodynamic modelsColm Connaughton
This document discusses turbulence in two-dimensional systems and the inverse energy cascade phenomenon. It begins with an overview of turbulence in 3D and 2D, describing the inverse energy cascade in 2D systems whereby energy is transferred to larger scales rather than smaller scales. It then discusses how finite size effects can generate large-scale coherent structures by blocking the inverse cascade. The document concludes by noting that extracting coherent flow from turbulent fluctuations is challenging and that diagnostics like the third-order structure function may not be reliable indicators of the energy cascade direction due to the presence of coherent structures.
Nonequilibrium statistical mechanics of cluster-cluster aggregation, School o...Colm Connaughton
Colm Connaughton presented on nonequilibrium statistical mechanics models of cluster-cluster aggregation. He discussed simple models where particles move randomly and merge upon contact. More sophisticated models track the size distribution of clusters as they aggregate. The Smoluchowski equation describes this process. For certain collision kernels, clusters of arbitrarily large size can form in finite time, known as gelation. While some kernels mathematically describe instantaneous gelation, physical models avoid this with a cluster size cutoff. Stationary states can be reached with a particle source.
This document contains physics formulas and concepts related to waves, electricity, electromagnetism, electronics, and radioactivity. It includes equations for oscillation, wavelength, interference, charge, current, potential difference, resistance, electromotive force, transformers, alpha decay, beta decay, gamma emission, and nuclear energy. The document provides definitions and explanations for key terms as well as examples of applying the formulas.
A Pedagogical Discussion on Neutrino Wave Packet EvolutionCheng-Hsien Li
This document discusses the time evolution of a neutrino wave packet. It presents a method to calculate higher-order corrections to the wave packet solution by expanding the momentum distribution and energy terms. The results show that including higher-order terms causes the wave packet to evolve into a spherical shape as expected by relativity. While the solution is limited to early times, it demonstrates that higher-order terms are needed to accurately model the wave packet's evolution into a spherical wavefront.
The document provides an overview of general dynamics concepts including:
1) Linear and angular velocity, acceleration, and their relationships. Equations for uniformly accelerated linear and angular motion are presented.
2) The concepts of work, power, kinetic energy, and potential energy are introduced. Work is defined as force multiplied by distance. Kinetic energy and potential energy equations are provided.
3) The principle of conservation of energy is described as energy cannot be created or destroyed, only transformed between different forms.
4) Objectives of the unit are to understand general dynamics concepts and be able to solve problems involving equations of motion, different types of acceleration and forces, and conservation of energy and momentum.
"Squeezed States in Bose-Einstein Condensate"Chad Orzel
1. The document discusses the formation of squeezed quantum states in Bose-Einstein condensates trapped in optical lattices. By slowly ramping up the depth of the optical lattice, the atoms can be prepared in a number-squeezed state.
2. Releasing the atoms from the lattice allows their wavefunctions to overlap and interfere, providing a way to probe the quantum phase state of the atoms. Number-squeezed states are observed to produce interference patterns with higher contrast than coherent states.
3. Variational calculations of the quantum state dynamics during lattice ramping and dephasing agree qualitatively with experimental observations of the transition between coherent and squeezed states.
1. The document discusses potential low frequency gravitational wave sources that could be detected by LISA, including galactic white dwarf binaries, massive black hole binaries, and extreme mass ratio inspirals.
2. LISA could detect thousands of massive black hole binaries and provide precise measurements of their parameters like mass and spin, enabling tests of general relativity and learning about black hole formation mechanisms.
3. Extreme mass ratio inspirals where a compact object spirals into a massive black hole could occur at a rate of 10-7 per year in our galaxy, allowing precision cosmology and tests of the no-hair theorem.
The document discusses key concepts from Einstein's special theory of relativity, including:
1) Frames of reference and the distinction between inertial and non-inertial frames. The laws of motion only hold in inertial frames.
2) The postulates of special relativity - that the laws of physics are the same in all inertial frames, and that the speed of light is constant.
3) Consequences of these postulates, including Lorentz transformations, length contraction, time dilation, and the relativity of simultaneity.
4) Experiments that motivated relativity, like the Michelson-Morley experiment, and equations like Lorentz transformations that were developed to be consistent with the postulates
1. The document provides conceptual problems and solutions related to superposition and standing waves. It discusses topics like wave pulses traveling in opposite directions, fundamental frequencies of open and closed organ pipes, and using resonance frequencies to estimate air temperature.
2. It also covers problems involving interference of two waves with different phases and frequencies, and deriving an expression for the envelope of a superposed wave.
3. For one problem, it plots the total displacement of a superposed wave at t=0, and the envelope function at t=0, 5, and 10 seconds. From these plots, it estimates the speed of the envelope and compares it to the theoretical value obtained from the problem parameters.
This is a talk I gave at the end of my first visiting professorship at Stanford in 2004. It gives a preview of Rocky Mountain Institute's Winning the Oil Endgame study, which was released in September 2004. http://www.oilendgame.com
A man watched as a butterfly struggled to emerge from its cocoon. He helped by cutting open the cocoon, but the butterfly emerged with a shriveled body and wings that could not support flight. Struggling to exit the cocoon forces fluid into the wings, allowing them to expand fully. While kindness can seem helpful, nature prepares organisms through difficulties and obstacles.
Harnessing Reflective Thinking in GovernmentGovLoop
An analysis of how people spend their time found that 28% is spent on interruptions from things that are not urgent or important, such as unnecessary emails. 25% is spent on productive content creation like writing emails. 20% is spent in meetings and 15% is spent searching through content. Only 5% of time is spent thinking and reflecting. The Petraeus Big Idea Framework suggests getting the "big ideas" right first, then communicating those ideas within an organization, overseeing and implementing the big ideas, and capturing and sharing refinements to the big ideas.
The document discusses using gravitational wave waveform models to infer astrophysical properties from observations of gravitational wave events. It describes how waveform models encode information about binary black hole parameters like mass and spin, and how Bayesian inference can be used to estimate these parameters from the detected gravitational wave signal. It also addresses assessing confidence in detections and evaluating potential modeling systematics by comparing waveform models to numerical relativity simulations.
This document contains physics formulae related to mechanics, thermodynamics, electromagnetism, optics, modern physics and more. Some key formulae include:
Density = mass / volume, Force = rate of change of momentum, Kinetic energy = (1/2)mv^2, Ohm's law: V=IR, Index of refraction n=c/v, Half life of radioactive element t1/2=ln(2)/λ, Bohr's model: L=nh/2π.
The document discusses harmonic motion and traveling waves. It defines periodic and harmonic motion, and notes that harmonic motion can be described by a sinusoidal function. Hooke's law relating force and displacement in springs is introduced. Equations of motion for simple harmonic oscillators like masses on springs and pendulums are derived. The relationship between wavelength, frequency and propagation velocity is defined for traveling waves. Solutions to the wave equation for strings and their properties are also summarized.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
This document summarizes a lecture on using gravitational wave waveform models to test general relativity and probe the nature of compact objects through gravitational wave observations. It discusses how waveform models can be used to bound post-Newtonian coefficients, constrain phenomenological merger-ringdown parameters, and probe the quasi-normal modes of black hole ringdowns. Measuring multiple modes could verify the no-hair theorem and black hole uniqueness properties. Future observations from LIGO and Virgo at design sensitivity may allow high-precision black hole spectroscopy and tests of general relativity in the strong, dynamical gravity regime.
1) The document discusses using the effective one body (EOB) formalism to model gravitational wave templates for LIGO and LISA. It summarizes the number and type of templates used in LIGO's first two observing runs.
2) It also discusses using EOB, post-Newtonian theory, numerical relativity simulations, and quantum field theory to model gravitational wave emission from binary black hole and binary neutron star mergers across different mass ratio and velocity regimes.
3) The document focuses on recent work extending EOB models to higher post-Minkowskian orders and including the effects of spin and tidal interactions, with the goal of more accurate gravitational wave template modeling.
This document discusses hoisting and dynamics of rotation. It provides examples and explanations of:
1) The forces, torques, and equations of motion involved when a hoist drum raises or lowers a load while accelerating or decelerating. This includes the inertia couple of the drum opposing changes in rotation and friction torque opposing rotation.
2) Specific examples that calculate the torque required to raise a load or bring it to a stop, given information like the drum's moment of inertia, load mass, acceleration, and friction torque.
3) Diagrams illustrate the forces and torques acting on the hoist drum and load in different scenarios like raising or lowering while accelerating versus coming to a stop
1) A mass attached to a spring can move up and down from its equilibrium position. Its movement is modeled by a differential equation that depends on factors like the mass, spring constant, damping, and any external forcing.
2) There are three types of behavior for damped systems without external forcing: underdamped systems oscillate with a decaying amplitude, critically damped systems do not oscillate and cross the equilibrium at most once, and overdamped systems do not oscillate and may or may not cross the equilibrium.
3) For an undamped system with external periodic forcing, the behavior depends on whether the forcing frequency matches the natural frequency. If they are different, the displacement takes the form of a
This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include:
- Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades.
- Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments.
- Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales.
- The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.
Large scale coherent structures and turbulence in quasi-2D hydrodynamic modelsColm Connaughton
This document discusses turbulence in two-dimensional systems and the inverse energy cascade phenomenon. It begins with an overview of turbulence in 3D and 2D, describing the inverse energy cascade in 2D systems whereby energy is transferred to larger scales rather than smaller scales. It then discusses how finite size effects can generate large-scale coherent structures by blocking the inverse cascade. The document concludes by noting that extracting coherent flow from turbulent fluctuations is challenging and that diagnostics like the third-order structure function may not be reliable indicators of the energy cascade direction due to the presence of coherent structures.
Nonequilibrium statistical mechanics of cluster-cluster aggregation, School o...Colm Connaughton
Colm Connaughton presented on nonequilibrium statistical mechanics models of cluster-cluster aggregation. He discussed simple models where particles move randomly and merge upon contact. More sophisticated models track the size distribution of clusters as they aggregate. The Smoluchowski equation describes this process. For certain collision kernels, clusters of arbitrarily large size can form in finite time, known as gelation. While some kernels mathematically describe instantaneous gelation, physical models avoid this with a cluster size cutoff. Stationary states can be reached with a particle source.
This document contains physics formulas and concepts related to waves, electricity, electromagnetism, electronics, and radioactivity. It includes equations for oscillation, wavelength, interference, charge, current, potential difference, resistance, electromotive force, transformers, alpha decay, beta decay, gamma emission, and nuclear energy. The document provides definitions and explanations for key terms as well as examples of applying the formulas.
A Pedagogical Discussion on Neutrino Wave Packet EvolutionCheng-Hsien Li
This document discusses the time evolution of a neutrino wave packet. It presents a method to calculate higher-order corrections to the wave packet solution by expanding the momentum distribution and energy terms. The results show that including higher-order terms causes the wave packet to evolve into a spherical shape as expected by relativity. While the solution is limited to early times, it demonstrates that higher-order terms are needed to accurately model the wave packet's evolution into a spherical wavefront.
The document provides an overview of general dynamics concepts including:
1) Linear and angular velocity, acceleration, and their relationships. Equations for uniformly accelerated linear and angular motion are presented.
2) The concepts of work, power, kinetic energy, and potential energy are introduced. Work is defined as force multiplied by distance. Kinetic energy and potential energy equations are provided.
3) The principle of conservation of energy is described as energy cannot be created or destroyed, only transformed between different forms.
4) Objectives of the unit are to understand general dynamics concepts and be able to solve problems involving equations of motion, different types of acceleration and forces, and conservation of energy and momentum.
"Squeezed States in Bose-Einstein Condensate"Chad Orzel
1. The document discusses the formation of squeezed quantum states in Bose-Einstein condensates trapped in optical lattices. By slowly ramping up the depth of the optical lattice, the atoms can be prepared in a number-squeezed state.
2. Releasing the atoms from the lattice allows their wavefunctions to overlap and interfere, providing a way to probe the quantum phase state of the atoms. Number-squeezed states are observed to produce interference patterns with higher contrast than coherent states.
3. Variational calculations of the quantum state dynamics during lattice ramping and dephasing agree qualitatively with experimental observations of the transition between coherent and squeezed states.
1. The document discusses potential low frequency gravitational wave sources that could be detected by LISA, including galactic white dwarf binaries, massive black hole binaries, and extreme mass ratio inspirals.
2. LISA could detect thousands of massive black hole binaries and provide precise measurements of their parameters like mass and spin, enabling tests of general relativity and learning about black hole formation mechanisms.
3. Extreme mass ratio inspirals where a compact object spirals into a massive black hole could occur at a rate of 10-7 per year in our galaxy, allowing precision cosmology and tests of the no-hair theorem.
The document discusses key concepts from Einstein's special theory of relativity, including:
1) Frames of reference and the distinction between inertial and non-inertial frames. The laws of motion only hold in inertial frames.
2) The postulates of special relativity - that the laws of physics are the same in all inertial frames, and that the speed of light is constant.
3) Consequences of these postulates, including Lorentz transformations, length contraction, time dilation, and the relativity of simultaneity.
4) Experiments that motivated relativity, like the Michelson-Morley experiment, and equations like Lorentz transformations that were developed to be consistent with the postulates
1. The document provides conceptual problems and solutions related to superposition and standing waves. It discusses topics like wave pulses traveling in opposite directions, fundamental frequencies of open and closed organ pipes, and using resonance frequencies to estimate air temperature.
2. It also covers problems involving interference of two waves with different phases and frequencies, and deriving an expression for the envelope of a superposed wave.
3. For one problem, it plots the total displacement of a superposed wave at t=0, and the envelope function at t=0, 5, and 10 seconds. From these plots, it estimates the speed of the envelope and compares it to the theoretical value obtained from the problem parameters.
This is a talk I gave at the end of my first visiting professorship at Stanford in 2004. It gives a preview of Rocky Mountain Institute's Winning the Oil Endgame study, which was released in September 2004. http://www.oilendgame.com
A man watched as a butterfly struggled to emerge from its cocoon. He helped by cutting open the cocoon, but the butterfly emerged with a shriveled body and wings that could not support flight. Struggling to exit the cocoon forces fluid into the wings, allowing them to expand fully. While kindness can seem helpful, nature prepares organisms through difficulties and obstacles.
Harnessing Reflective Thinking in GovernmentGovLoop
An analysis of how people spend their time found that 28% is spent on interruptions from things that are not urgent or important, such as unnecessary emails. 25% is spent on productive content creation like writing emails. 20% is spent in meetings and 15% is spent searching through content. Only 5% of time is spent thinking and reflecting. The Petraeus Big Idea Framework suggests getting the "big ideas" right first, then communicating those ideas within an organization, overseeing and implementing the big ideas, and capturing and sharing refinements to the big ideas.
The social-media-report-2012-by-NielsenYour Social
1) Social media usage continues to grow rapidly and is now integral to daily life for many globally. Mobile access is driving much of this growth, now accounting for over 60% of time spent on social media.
2) Key trends include the rise of mobile/tablet usage, proliferation of new social platforms like Pinterest, and transformation of TV viewing into a shared social experience.
3) Marketers now have opportunities to engage consumers who are using social media to make informed purchase decisions and provide customer service, though many still find ads annoying.
Two visitors from Japan, Yuki and Makoto, are introducing themselves and explaining that they want to create an app to help people learn more about what is happening locally from those who live there. The app would allow locals to post about events, which others could upvote or downvote, building a database of local information and history that helps visitors and residents understand what to expect from different neighborhoods and locations.
Industry updates on key mobile trends 8 02 13Performics
A new Forrester report found that while consumers' use of mobile technologies is increasing rapidly, marketers are not adapting their strategies to keep up. The gap between how consumers use mobile and how marketers plan for it will lead to missed opportunities and an outdated approach. The report also found that marketers tend to overfocus on building apps and adopting trendy technologies without fully utilizing more mainstream mobile options. Additionally, Google's Android platform surpassed Apple's iOS in total app downloads for the first time, though iOS still generates twice as much revenue for developers.
The document provides information about a 2-day disaster preparedness and risk reduction training hosted by NDRN CAMANAVA from November 16-17, 2012. The training will cover topics such as disaster response, risk reduction, and management. It will take place at SM Valenzuela from 10am to 4pm with free admission. NDRN CAMANAVA works to create a community-based organization that advocates for disaster response programs and uses holistic integrated approaches to address climate change issues in CAMANAVA.
The document discusses wave optics and electromagnetic waves. It defines key concepts like wavefronts, which connect points of equal phase, and rays, which describe the direction of wave propagation perpendicular to wavefronts. It explains Huygens' principle, which states that each point on a wavefront acts as a secondary source of spherical wavelets to determine the new wavefront position. The principle of superposition states that multiple waves add linearly at each point in space to determine the resulting disturbance. Interference occurs when waves are out of phase and their amplitudes diminish or vanish.
Fourier series are used to represent periodic functions as the sum of simple oscillating functions like sines and cosines. This allows periodic functions, including discontinuous ones, to be broken down into their constituent frequencies or harmonics. Applications include representing sound waves, light waves, radio signals, and other physical phenomena involving wave motion or vibration. The Fourier coefficients determine the relative importance of each harmonic in the overall signal.
Classical mechanics failed to explain certain phenomena observed at the microscopic level like black body radiation and the photoelectric effect. This led to the development of quantum mechanics, with key aspects being the wave function Ψ, Schrodinger's time-independent and time-dependent wave equations, and operators like differentiation that act on wave functions to produce other wave functions. The wave function Ψ relates to the probability of finding a particle, with |Ψ|2 representing the probability.
This document discusses different types of mechanical waves and their properties. It defines mechanical waves as oscillations that transfer energy through a medium. Key points include:
- Mechanical waves can be transverse (perpendicular to direction of travel) or longitudinal (parallel to direction of travel).
- They transport energy through the medium and require a medium, like air or water, to propagate.
- Examples of mechanical waves include water waves, sound waves, and waves on a string or rope.
- Harmonic waves have a sinusoidal shape described by a mathematical function involving amplitude, wavelength, frequency, and phase.
Standing waves occur when two waves of equal amplitude, wavelength, and frequency travel in opposite directions and superimpose. The result is a wave with a position-dependent amplitude described by A(x)=2Asin(kx), where k is the wavenumber. Nodes occur at integer multiples of half wavelengths, where the amplitude is zero. Anti-nodes occur at odd integer multiples of quarter wavelengths, where the amplitude is maximum (2A). The full equation for a standing wave is D(x,t)=2Asin(kx)cos(ωt), relating amplitude to position and time using sines and cosines respectively.
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1. Chapter 2
Basic principles of the seismic
method
In this chapter we introduce the basic notion of seismic waves. In the earth, seismic
waves can propagate as longitudinal (P) or as shear (S) waves. For free space, the one-
dimensional wave equation is derived. The wave phenomena occurring at a boundary
between two layers are discussed, such as Snell’s Law, reflection and transmission. For
seismic-exploration purposes, where measurements are taking place at the surface, the
different arrivals of direct waves, reflected waves and refracted/head waves are discussed.
2.1 Introduction
The seismic method makes use of the properties of the velocity of sound. This velocity
is different for different rocks and it is this difference which is exploited in the seismic
method. When we create sound at or near the surface of the earth, some energy will be
reflected back (bounced back). They can be characterized as echoes. From these echoes
we can determine the velocities of the rocks, as well as the depths where the echoes came
from. In this chapter we will discuss the basic principles behind the behaviour of sound in
solid materials. When we use the seismic method, we usually discuss two types of seismic
methods, depending on whether the distance from the sound source to the detector (the
”ear”) is large or small with respect to the depth of interest: the first is known as refraction
seismics, the other as reflection seismics. Of course, there is some overlap between those
two types and that will be discussed in this chapter. When features really differ, then that
will be discussed in next chapter for refraction and the chapters on reflection seismics.
The overlap lies in the physics behind it, so we will deal with these in this chapter. In the
following chapters will deal with instrumentation, field techniques, corrections (which are
not necessary for refraction data) and interpretation.
18
2. 2.2 Basic physical notions of waves
Everybody knows what waves are when we are talking about waves at sea. Sound in
materials has the same kind of behaviour as these waves, only they travel much faster
than the waves we see at sea. Waves can occur in several ways. We will discuss two of
them, namely the longitudinal and the shear waves. Longitudinal waves behave like waves
in a large spring. When we push from one side of a spring, we will observe a wave going
through the spring which characterizes itself by a thickening of the wires running through
the spring in time (see also figure (2.1)). A property of this type of wave is that the
motion of a piece of the wire is in the same direction as the wave moves. These waves are
also called Push-waves, abbreviated to P-waves, or compressional waves. Another type
of wave is the shear wave. A shear wave can be compared with a chord. When we push
a chord upward from one side, a wave will run along the chord to the other side. The
movement of the chord itself is only up- and downward: characteristic of this wave is that
a piece of the chord is moving perpendicular to the direction of that of the wave (see
also figure (2.1)). These types of waves are referred to as S-waves, also called dilatational
waves. Characteristic of this wave is that a piece of the chord is pulling its ”neighbour”
upward, and this can only occur when the material can support shear strain. In fluids,
one can imagine that a ”neighbour” cannot be pulled upward simply because it is a fluid.
Therefore, in fluids only P-waves exist, while in a solid both P- and S-waves exist.
Waves are physical phenomena and thus have a relation to basic physical laws. The
two laws which are applicable are the conservation of mass and Newton’s second law.
These two have been used in appendix A to derive the two equations governing the wave
motion due to a P-wave. There are some simplifying assumptions in the derivation, one
of them being that we consider a 1-dimensional wave. When we denote p as the pressure
and vx as the particle velocity, the conservation of mass leads to:
1 ∂p ∂vx
=− (2.1)
K ∂t ∂x
in which K is called the bulk modulus. The other relation follows from application of
Newton’s law:
∂p ∂vx
− =ρ (2.2)
∂x ∂t
where ρ denotes the mass density. This equation is called the equation of motion. The
combination of these two equations leads, for constant density ρ, to the equation which
describes the behaviour of waves, namely the wave equation:
∂2p 1 ∂2p
− 2 2 =0 (2.3)
∂x2 c ∂t
in which c can be seen as the velocity of sound, for which we have: c = K/ρ. The
19
3. Figure 2.1: Particle motion of P and S waves
solution to this equation is:
p(x, t) = s(t ± x/c) (2.4)
where s(t) is some function. Note the dependency on space and time via (t ± x/c), which
denotes that it is a travelling wave. The sign in the argument is depending on which
direction the wave travelling in.
Often, seismic responses are analyzed in terms of frequencies, i.e., the Fourier spectra
as given in Chapter 1. The definition we use here is:
+∞
G(ω) = g(t) exp(−iωt) dt (2.5)
−∞
which is the same as equation (1.1) from Chapter 1 but we used the radial frequency ω
instead: ω = 2πf. Using this convention, it is easy to show that a differentiation with
respect to time is equivalent to multiplication with iω in the Fourier domain. When we
transform the solution of the wave equation (equation(2.4)) to the Fourier domain, we
obtain:
P (x, ω) = S(ω) exp(±iωx/c). (2.6)
20
4. Note here that the time delay x/c (in the time domain) becomes a linear phase in the
Fourier domain, i.e., (ωx/c).
In the above, we gave an expression for the pressure, but one can also derive the
equivalent expression for the particle velocity v x . To that purpose, we can use the equation
of motion as expressed in equation (2.2), but then in its Fourier-transformed version, which
is:
1 ∂P (x, ω)
Vx (x, ω) = − . (2.7)
iωρ ∂x
When we subsitute the solution for the pressure from above (equation (2.6)), we get for
the negative sign:
1 −iω
Vx (x, ω) = − S(ω) exp(−iωx/c)
iωρ c
1
= S(ω) exp(−iωx/c). (2.8)
ρc
Notice that the particle velocity is a scaled version of the pressure:
P (x, ω)
Vx (x, ω) = . (2.9)
ρc
The scaling factor is (ρc), being called the seismic impedance.
In the previous analysis, we considered one-dimensional waves. Normally in the real
world, we deal with three dimensions, so a wave will spread in three directions. In a
homogeneous medium (so the properties of the material are everywhere constant and the
same) the wave will spread out like a sphere. The outer shell of this sphere is called the
wave front. Another way of describing this wave front is in terms of the normal to the
wavefront: the ray. We are used to rays in optics and we can use the same notion in
the seismic method. When we were explaining the behaviour of P- and S-waves, we are
already using the term ”neighbour”. This was an important feature otherwise the wave
would not move forward. A fundamental notion included in this, is Huygens’ principle.
When a wave front arrives at a certain point, that point will behave also as a source for
the wave, and so will all its neighbours. The new wavefront is then the envelope of all the
waves which were generated by these points. This is illustrated in figure (2.2). Again, the
ray can then be defined as the normal to that envelope which is also given in the figure.
So far, we only discussed the way in which the wave moves forward, but there is also
another property of the wave we haven’t discussed yet, namely the amplitude : how does
the amplitude behave as the wave moves forward? We have already mentioned spherical
spreading when the material is everywhere the same. The total energy will be spreaded
out over the area over the sphere. This type of energy loss is called spherical divergence.
It simply means that if we put our ”ear” at a larger distance, the sound will be less loud.
21
5. secondary sources
t at t = t0
wav efron
c∆
t
= t0 + ∆t
fr on t at t
wave
Figure 2.2: Using Huygens’ principle to locate new wavefronts.
Material velocity Material velocity
(m/s) (m/s)
Air 330 Sandstone 2000-4500
Water 1500 Shales 3900-5500
Soil 20-300 Limestone 3400-7000
Sands 600-1850 Granite 4800-6000
Clays 1100-2500 Ultra-mafic rocks 7500-8500
Table 2.1: Seismic wave velocities for common materials and rocks.
There is also another type of energy loss, and that is due to losses within the material,
which mainly consists of internal friction losses. This means that the amplitude of a wave
will be extra damped because of this property. S-waves usually show higher friction losses
than P-waves. Finally, we give a table of common rocks and their seismic wave velocities
in table (2.1).
2.3 The interface : Snell’s law, refraction and reflection
So far, we discussed waves in a material which had everywhere the same constant wave
velocity. When we include a boundary between two different materials, some energy is
bounced back, or reflected, and some energy is going through to the other medium. It
is nice to perform Huygens’ principle graphically on such a configuration to see how the
wavefront moves forward (propagates), especially into the second medium. From this
22
6. A"
0
e
im
tt
t
ta
e
im
n
ro
tt
ef
ta
av
on
w
θ1
r
ef
A’
av
w
velocity c1
A B’ velocity c2
θ2 B
Figure 2.3: Snell’s law
picture, we could also derive the ray concept. In this discussion, we will only consider
the notion of rays. A basic notion in the ray concept, is Snell’s law. Snell’s law is a
fundamental relation in the seismic method. It tells us the relation between angle of
incidence of a wave and velocity in two adjacent layers (see Figure (2.3)).
AA A is part of a plane wave incident at angle θ 1 to a plane interface between medium
1 of velocity c1 , and medium 2 of velocity c2 . The velocities c1 and c2 are constant. In a
time t the wave front moves to the position AB and are normals to the wave front. So the
time t is given by
AB AB
t= = (2.10)
c1 c2
Considering the two triangles and this may be written as:
AB sin θ1 AB sin θ2
t= = (2.11)
c1 c2
23
7. Hence,
sin θ1 sin θ2
= (2.12)
c1 c2
which is Snell’s law for transmission. So far, we have taken general velocities c 1 and c2 .
However, in a solid, two velocities exist, namely P- and S-wave velocities. Generally, when
a P-wave is incident on a boundary, it can transmit as a P-wave into the second medium,
but also as a S-wave. So in the case of the latter, Snell’s law reads:
sin θP sin θS
= (2.13)
cP cS
where cP is the P-wave velocity, and cS the S-wave velocity. The same holds for reflection:
a P-wave incident on the boundary generates a reflected P-wave and a reflected S-wave.
Finally, the same holds for an incident S-wave: it generates a reflected P-wave, a reflected
S-wave, a transmitted P-wave and a transmitted S-wave.
A special case of Snell’s law is of interest in refraction prospecting. If the ray is
refracted along the interface (that is, if θ 2 = 90 deg), we have
sin θc 1
= (2.14)
c1 c2
where θc is known as the critical angle.
So far, we have looked at basic notions of refraction and reflection at an interface.
When we measure in the field, and there would be one boundary below it, we could
observe several arrivals: a direct ray, a reflected ray and a refracted ray. We will derive
the arrival time of each ray as depicted in figure 2.4.
The direct ray is very simple: it is the horizontal distance divided by the velocity of
the wave, i.e.,:
x
t= (2.15)
c1
When we look at the reflected ray, we have that the angle of incidence is the same as the
angle of reflection. This also follows from Snell’s law: when the velocities are the same,
the angles must also be the same. When we use Pythagoras’ theorem, we obtain for the
traveltime:
1/2
4z 2 + x2
t= (2.16)
c1
When we square this equation, we see that it is the equation of a hyperbola:
2 2
2z x
t2 = + (2.17)
c1 c1
24
8. distance x
A D
direct
thickness z
re
fle
ct
io
n
velocity c1
B refraction C velocity c2
Figure 2.4: The direct, reflected and refracted ray.
When we look at the refracted ray the derivation is a bit more complicated. Take each
ray element, so take the paths AB, BC and CD separately. Then, for the first element,
as shown in figure (2.5), we obtain the traveltime:
∆s1 + ∆s2 ∆x1 sin θc z cos θc
∆t1 = = + (2.18)
c1 c1 c1
where θc is the critical angle.
We can do this also for the paths BC and CD, and we obtain the total time as:
t = ∆t1 + ∆t2 + ∆t3 (2.19)
∆x1 sin θc z cos θc ∆x2 ∆x3 sin θc z cos θc
= + + + + (2.20)
c1 c1 c2 c1 c1
where ∆x2 = BC and ∆x3 is the horizontal distance between C and D. When we use
now Snell’s law, i.e., sin θc /c1 = 1/c2 , in the terms with ∆x1 and ∆x3 , then we can add
all the terms with 1/c2 , using x = ∆x1 + ∆x2 + ∆x3 to obtain:
x 2z cos θc
t= + (2.21)
c2 c1
We recognize this equation as the equation of a straight line when t is considered as a
function of distance x, the line along which we do our measurements. We will use this
25
9. A
∆s
1
θc
z
∆s
2
∆x 1 B refracted ray
Figure 2.5: An element of the ray with critical incidence
equation later in the next chapter. We have now derived the equations for the three rays,
and we can plot the times as a function of x. This is done in figure (2.6).
This picture is an important one. When we measure data in the field the characteristics
in this plot can most of the time be observed.
Now we have generated this figure, we can specify better when we are performing
a reflection survey, or a refraction survey. In refraction seismics we are interested in
the refractions and only in the travel times. This means that we can only observe the
traveltimes well if it is not masked by the reflections or the direct ray, which means
that we must measure at a relatively large distance with respect to the depth of interest.
This is different with reflection seismics. There, the reflections will always be masked by
refractions or the direct ray, but there are ways to enhance the reflections. What we are
interested in, is the arrival at relatively small offsets, thus distances of the sound source
to the detector which are small with respect to the depth we are interested in.
Before discussing any more differences between the refraction method and the reflection
method, we would like to discuss amplitude effects at the boundary. Let us first introduce
the acoustic impedance, which is the product of the density ρ with the wave velocity c,
i.e., ρc. When a ray encounters a boundary, some energy will be reflected back, and some
will be transmitted to the next layer. The amount of energy reflected back is characterized
by the reflection coefficient R:
ρ2 c2 − ρ 1 c1
R= (2.22)
ρ2 c2 + ρ 1 c1
26
10. traveltime
n
on
io
ti
rac
ct
ref
fle
re
t
ec
dir
0
distance x
Figure 2.6: Time-distance (t, x) curve for direct, reflected and refracted ray.
That this is the case, will be derived from basic physical principles in the chapter on
processing (Chapter 5). Obviously, the larger the impedance contrast between two layers,
the higher the amplitude of the reflected wave will be. Notice that it is the impedance
contrast which determines whether energy is reflected back or not; it may happen that
the velocities and densities are different between two layers, but that the impedance is
(nearly) the same. In that case we will see no reflection. We can now state another
difference between refraction and reflection seismics. With refraction seismics we are only
interested in traveltimes of the waves, so this means that we are interested in contrasts in
velocities. This is different in reflection seismics. Then we are interested in the amplitude
of the waves, and we will only measure an amplitude if there is a contrast in acoustic
impedance in the subsurface.
Generally speaking the field equipment for refraction and reflection surveys have the
same functionality: we need a source, detectors and recording equipment. Since reflection
seismics gives us a picture of the subsurface, it is much used by the oil industry and
therefore, they put high demands on the quality of the equipment. As said before, a
difference between the two methods is that in reflection seismics we are interested in
amplitudes as well, so this means that high-precision instruments are necessary to pick
27
11. REFRACTION SEISMICS REFLECTION SEISMICS
Based on contrasts in : Based on contrasts in :
seismic wave speed (c) seismic wave impedances (ρc)
Material property determined : Material properties determined:
wave speed only wave speed and wave impedance
Only traveltimes used Traveltimes and amplitudes used
No need to record amplitudes completely : Must record amplitudes correctly :
relatively cheap instruments relatively expensive instruments
Source-receiver distances large compared to Source-receiver distances small compared to
investigation depth investigation depth
Table 2.2: Important differences between refraction and reflection seismics.
those up accurately. Source, detectors and recording equipment will be discussed in the
chapter on seismic instrumentation (chapter 3).
Finally, we tabulate the most important differences between reflection and refraction
seismic in table 2.2.
28