Real wave fields consist of many components with varying amplitudes, frequencies, and directions that follow statistical distributions. Common measures used to describe wave heights include significant wave height (Hs), which corresponds to the average height of the highest one-third of waves. Wave periods are also measured, including significant wave period (Ts) and peak period (Tp).
Wave heights and periods can be analyzed statistically. Deep water wave heights often follow a Rayleigh distribution defined by the root-mean-square wave height (Hrms). Wave energy is represented by wave spectra such as the Bretschneider and JONSWAP spectra, which define the distribution of energy across frequencies. Spectral data can be used to determine key wave parameters like significant
This document discusses statistics and irregular waves. It provides information on:
1. Measures used to describe wave height and period such as significant wave height and peak period.
2. Probability distributions that describe wave heights, particularly the Rayleigh distribution for narrow-banded seas.
3. Wave energy spectra including typical models like the Bretschneider and JONSWAP spectra, and how these relate to significant wave height.
Linear wave theory assumes wave amplitudes are small, allowing second-order effects to be ignored. It accurately describes real wave behavior including refraction, diffraction, shoaling and breaking. Waves are described by their amplitude, wavelength, frequency, period, wavenumber and phase/group velocities. Phase velocity is the speed at which the wave profile propagates, while group velocity (always lower) is the speed at which wave energy is transmitted. Wave energy is proportional to the square of the amplitude and is divided equally between kinetic and potential components on average.
This document provides an overview of sonar technology. It discusses the history of sonar from its early developments in the late 19th/early 20th century to its modern applications. Key developments include Fessenden's early experiments in 1914, the creation of ASDIC by the British during WWI, and the transfer of technology between Britain and the US during WWII. The document also examines how sonar works using sound waves, and describes different sonar systems like active/passive sonar and side-scan/multi-beam sonar used for tasks like submarine detection, fishing, and seafloor mapping.
The document discusses various types of tsunamis including those caused by landslides, meteorological conditions, and human activities. It provides examples of destructive meteotsunamis and discusses attempts to artificially trigger tsunamis through explosions. The characteristics of tsunamis are explained, noting that while waves have short wavelengths in deep ocean, they have much longer wavelengths and travel very quickly. The document also discusses drawback effects, forecasting tsunami probability, anatomy of tsunamis, facts about tsunamis, and preparation and safety during and after tsunamis.
Floods are a common natural disaster caused by heavy rainfall or snowmelt that leads to overflowing rivers, streams, lakes or oceans overtaking dry land. They can cause widespread damage, injury and loss of life. Some key points about floods are:
- They occur globally in every country and region near bodies of water or with heavy precipitation. Common types include river floods, flash floods, coastal floods, and urban floods.
- Factors like intensity and duration of rainfall, soil conditions, terrain, development in floodplains, and blocked drainage systems influence flood risks and impacts.
- Floods destroy property and infrastructure, damage crops and land, and disrupt lives. They are among the most costly
This document discusses tsunamis, which are giant waves caused by earthquakes or volcanic eruptions under the sea. Tsunamis can cause massive damage and loss of life by destroying homes and infrastructure. While impossible to prevent, their effects can be minimized through preparation measures like building in safe areas, establishing evacuation routes, and early warning systems. The document outlines dos and don'ts for before, during, and after a tsunami and emphasizes the importance of quickly evacuating coastal areas if a tsunami is detected or warned.
Flood Warning Systems - A practical approachChris Goding
A practical overview of the components typically used in a flood warning system.
Contact Greenspan for more information:
cgoding at greenspan dot com dot sg
+65 98214182
Surface ocean currents are driven by wind and develop circular gyre patterns. They transfer heat between latitudes and influence climate. Upwelling brings nutrients to surface waters. Deep currents are driven by density differences from temperature and salinity changes. The global conveyor belt model depicts a circulation pattern from the Atlantic to the Pacific and back. Waves are characterized by height, wavelength, period, and fetch. Breaking waves increase in height and decrease in wavelength near shore. Tides have spring and neap variations due to lunar phases. Shoreline processes include abrasion, refraction, and longshore transport, which shape features like barrier islands.
This document discusses statistics and irregular waves. It provides information on:
1. Measures used to describe wave height and period such as significant wave height and peak period.
2. Probability distributions that describe wave heights, particularly the Rayleigh distribution for narrow-banded seas.
3. Wave energy spectra including typical models like the Bretschneider and JONSWAP spectra, and how these relate to significant wave height.
Linear wave theory assumes wave amplitudes are small, allowing second-order effects to be ignored. It accurately describes real wave behavior including refraction, diffraction, shoaling and breaking. Waves are described by their amplitude, wavelength, frequency, period, wavenumber and phase/group velocities. Phase velocity is the speed at which the wave profile propagates, while group velocity (always lower) is the speed at which wave energy is transmitted. Wave energy is proportional to the square of the amplitude and is divided equally between kinetic and potential components on average.
This document provides an overview of sonar technology. It discusses the history of sonar from its early developments in the late 19th/early 20th century to its modern applications. Key developments include Fessenden's early experiments in 1914, the creation of ASDIC by the British during WWI, and the transfer of technology between Britain and the US during WWII. The document also examines how sonar works using sound waves, and describes different sonar systems like active/passive sonar and side-scan/multi-beam sonar used for tasks like submarine detection, fishing, and seafloor mapping.
The document discusses various types of tsunamis including those caused by landslides, meteorological conditions, and human activities. It provides examples of destructive meteotsunamis and discusses attempts to artificially trigger tsunamis through explosions. The characteristics of tsunamis are explained, noting that while waves have short wavelengths in deep ocean, they have much longer wavelengths and travel very quickly. The document also discusses drawback effects, forecasting tsunami probability, anatomy of tsunamis, facts about tsunamis, and preparation and safety during and after tsunamis.
Floods are a common natural disaster caused by heavy rainfall or snowmelt that leads to overflowing rivers, streams, lakes or oceans overtaking dry land. They can cause widespread damage, injury and loss of life. Some key points about floods are:
- They occur globally in every country and region near bodies of water or with heavy precipitation. Common types include river floods, flash floods, coastal floods, and urban floods.
- Factors like intensity and duration of rainfall, soil conditions, terrain, development in floodplains, and blocked drainage systems influence flood risks and impacts.
- Floods destroy property and infrastructure, damage crops and land, and disrupt lives. They are among the most costly
This document discusses tsunamis, which are giant waves caused by earthquakes or volcanic eruptions under the sea. Tsunamis can cause massive damage and loss of life by destroying homes and infrastructure. While impossible to prevent, their effects can be minimized through preparation measures like building in safe areas, establishing evacuation routes, and early warning systems. The document outlines dos and don'ts for before, during, and after a tsunami and emphasizes the importance of quickly evacuating coastal areas if a tsunami is detected or warned.
Flood Warning Systems - A practical approachChris Goding
A practical overview of the components typically used in a flood warning system.
Contact Greenspan for more information:
cgoding at greenspan dot com dot sg
+65 98214182
Surface ocean currents are driven by wind and develop circular gyre patterns. They transfer heat between latitudes and influence climate. Upwelling brings nutrients to surface waters. Deep currents are driven by density differences from temperature and salinity changes. The global conveyor belt model depicts a circulation pattern from the Atlantic to the Pacific and back. Waves are characterized by height, wavelength, period, and fetch. Breaking waves increase in height and decrease in wavelength near shore. Tides have spring and neap variations due to lunar phases. Shoreline processes include abrasion, refraction, and longshore transport, which shape features like barrier islands.
Waves are caused by wind dragging across the surface of water, causing the water to oscillate rather than move as a solid block. Individual water particles move in circles while the wave energy travels forward. Key characteristics of waves include their height from trough to crest, wavelength between two crests, and period of time for one full wavelength to pass. Wave size depends on wind speed, duration, and the distance of open water over which the wind blows known as fetch. As waves approach shore, their speed decreases causing their height to increase and wavelength to decrease, sometimes causing waves to break.
DSD-INT 2014 - SWAN Advanced Course - 01 - General introduction to waves and ...Deltares
This document provides an overview of waves and the SWAN wave model. It discusses wave generation by wind, propagation processes like shoaling and refraction, and dissipation mechanisms like whitecapping and bottom friction. The SWAN model is introduced as solving the action density balance equation. Source terms like wind input and dissipation parameterizations are explained. The document concludes with details on the SWAN North Sea implementation, including grid specifications, boundary conditions, and validation results.
Ocean waves are characterized by their amplitude, wavelength, frequency, and period. Amplitude refers to the height of the wave from still water level to the crest. Wavelength is the distance between two identical points on successive waves. Frequency is the number of waves passing a fixed point per second, while period is the time for one full wave cycle. Wave height depends on factors like fetch (distance over water the wind blows) and breaking occurs when the wave steepness exceeds about 1/7. Breaking waves can be spilling, plunging, or surging.
A PowerPoint about storm surges and how it affects the weather. A brief case study about storm surges in the North Sea and Bangladesh are also included.
This document contains formulas and concepts related to engineering economics including:
1) Formulas for calculating the future and present value of amounts given an interest rate and number of periods.
2) Formulas for calculating the future and present value of a series of payments or annuities.
3) Formulas for cash flow analysis of revenue-dominated and cost-dominated projects.
4) Formulas and methods for calculating depreciation including straight-line, declining balance, sum-of-years digits, and sinking fund methods.
5) Formulas for calculating rate of return, benefit-cost ratios, economic order quantities, total costs for make-or-buy decisions, and break-even points
6161103 3.4 three dimensional force systemsetcenterrbru
1) Three-dimensional force systems involve resolving forces into x, y, and z components and using the equations of equilibrium to solve for unknown forces.
2) Examples are provided of using free body diagrams and the equations of equilibrium to solve for tensions in cables, magnitudes of applied forces, and stretches of springs in static systems with multiple forces.
3) Unknown forces and stretches are determined by setting the vector sum of the forces in x, y, and z directions equal to zero and solving the resulting simultaneous equations.
East Coast MARE Ocean Lecture May 16, 2012 - Surf's Up! All About Waves at th...coseenow
The document discusses waves and coastal processes. It describes how waves form, grow, and change as they approach the shoreline. This includes wave shoaling, refraction, and breaking. It also discusses surf zone currents and sediment transport. Coastal changes occur over various timescales from storms to sea level rise. Long-term trends include shoreline erosion and accretion. Rising sea levels are projected to increase coastal flooding risks in the future.
Tsunami are powerful waves created by earthquakes, volcanic eruptions, or landslides that displace large volumes of water. They can reach heights over 30 meters and speeds over 700 km/hr, destroying everything in their path. The devastating 2004 Indian Ocean tsunami killed over 118,000 people across several countries, displaced millions, and caused widespread damage to infrastructure and the environment. Relief efforts provided temporary housing, food, water and medical aid to victims, but recovery is a long process and many remain in need of support years later.
This document discusses tsunamis and their management. It defines tsunamis as long ocean waves caused by underwater seismic events like earthquakes. Tsunamis can cause significant damage through flooding and high-energy waves. Management involves identifying tsunami-prone areas, protecting coastlines, establishing early warning systems, educating communities on evacuation procedures, and conducting drills. The post-disaster phase focuses on search and rescue, relief efforts, and long-term recovery and rehabilitation programs. Proper management can help reduce risks from future tsunami disasters.
A tsunami is caused by earthquakes, volcanic eruptions, or meteorite impacts under water which generates a wave that grows much larger as it reaches the coast. This wall of water goes back and forth, devastating everything with strong currents. Tsunamis can cause severe flooding, destruction of buildings, and loss of life. The most important safety measures after a tsunami warning are to evacuate beaches and seek higher ground in buildings or mountains.
1) The document discusses remote sensing and provides definitions and explanations of key concepts such as the electromagnetic spectrum, atmospheric interaction with electromagnetic waves, and atmospheric windows.
2) It describes the seven elements of remote sensing including the energy source, interaction with the atmosphere and target, sensor recording, processing, interpretation, and application.
3) The electromagnetic spectrum is divided into regions including radio waves, microwaves, infrared, visible light, ultraviolet, and others. Certain regions have high atmospheric transmittance and are considered atmospheric windows for remote sensing.
This document discusses the application of remote sensing and geographical information systems in civil engineering. It begins by defining remote sensing as the acquisition of information about an object without physical contact, typically by measuring electromagnetic radiation. It then defines geographical information systems as a system for capturing, storing, analyzing and presenting spatially referenced data. The document provides examples of how remote sensing data from sources like Google Earth can be spatially analyzed using a GIS. It proceeds to discuss key concepts in remote sensing including the electromagnetic spectrum, atmospheric interactions with radiation, and radiation measurement principles.
The document discusses operational remote sensing applications in India. It outlines the institutional mechanisms for natural resource management using remote sensing data from Indian Space Research Organization (ISRO) satellites. Some key applications discussed are natural resource inventories, land use/land cover mapping, agriculture monitoring, drought and flood assessment, and the FASAL program for multi-crop forecasting using remote sensing, agrometeorology and ground data. National level inventories are done at 180m, 60m and 24m resolutions with finer scales at state and district levels.
Differential GPS (DGPS) improves the accuracy of standard GPS by using a stationary reference unit to calculate errors in the GPS signal caused by things like atmospheric conditions. This error data is transmitted to a mobile GPS unit in real-time to correct its position reading. DGPS can achieve sub-meter level accuracy compared to the 15 meter accuracy of standard GPS. It works by having a base station calculate positioning errors compared to its known location and sending corrections to a roving unit. This allows the roving unit to correct its reported position and improve accuracy.
The document summarizes key concepts about ocean waves. It describes how wind generates sea and swell, with sea being wind-driven waves and swell being uniform waves that travel outward from storm areas. Wave energy is affected by wind speed, duration, and fetch (distance over which wind blows). Wave height increases as waves enter shallower water and become steeper, eventually breaking in different types - spilling, plunging, or surging - depending on sea floor slope. The document also discusses wave refraction, reflection, tsunamis, and historical tsunami events.
Storm surges are increases in seawater levels caused by intense winds and low pressure from tropical cyclones. They are the primary cause of casualties and property damage in coastal areas during tropical cyclones. The Philippines is highly prone to storm surges due to its location and extensive coastline. Storm surge height depends on factors like storm intensity, size, forward speed, angle of approach, and local geography. Preparing for surges involves securing property, creating emergency plans and kits, and evacuating if instructed.
DGPS improves upon standard GPS accuracy by using a fixed reference station to calculate and broadcast differential corrections for errors caused by atmospheric delays of GPS signals. Receivers equipped with DGPS can then apply these corrections to achieve sub-meter accuracy, as low as 10 cm in some cases. It works by having a stationary receiver at a known location calculate differential errors compared to GPS satellites and broadcasting correction signals to enable mobile DGPS receivers to determine their position with much greater precision.
Wave breaking is a complex phenomenon characterized by energy dissipation and turbulence. The study analyzed wave breaking through laboratory tests using wave gauges and an acoustic Doppler velocimeter. Fifteen wave conditions were tested in a wave channel with a variable slope bottom profile designed to induce breaking. Timeseries and spectral analysis of free surface elevation data provided insights into wave propagation and breaking behavior under different conditions. Empirical formulations were also evaluated based on the experimental results.
Quick refresher on the physics of coaxial cable(draftone)foxtrot jp R
This document provides an overview of modeling a twin-lead wire transmission line as a two-port network with resistance, inductance, conductance and capacitance per unit length. Performing circuit analysis yields differential equations that can be written as wave equations in phasor form, containing a propagation constant γ. γ can be resolved into attenuation and phase constants, allowing the total wave voltage to be expressed as the sum of incident and reflected wave components propagating in opposite directions with different attenuation profiles.
Waves are caused by wind dragging across the surface of water, causing the water to oscillate rather than move as a solid block. Individual water particles move in circles while the wave energy travels forward. Key characteristics of waves include their height from trough to crest, wavelength between two crests, and period of time for one full wavelength to pass. Wave size depends on wind speed, duration, and the distance of open water over which the wind blows known as fetch. As waves approach shore, their speed decreases causing their height to increase and wavelength to decrease, sometimes causing waves to break.
DSD-INT 2014 - SWAN Advanced Course - 01 - General introduction to waves and ...Deltares
This document provides an overview of waves and the SWAN wave model. It discusses wave generation by wind, propagation processes like shoaling and refraction, and dissipation mechanisms like whitecapping and bottom friction. The SWAN model is introduced as solving the action density balance equation. Source terms like wind input and dissipation parameterizations are explained. The document concludes with details on the SWAN North Sea implementation, including grid specifications, boundary conditions, and validation results.
Ocean waves are characterized by their amplitude, wavelength, frequency, and period. Amplitude refers to the height of the wave from still water level to the crest. Wavelength is the distance between two identical points on successive waves. Frequency is the number of waves passing a fixed point per second, while period is the time for one full wave cycle. Wave height depends on factors like fetch (distance over water the wind blows) and breaking occurs when the wave steepness exceeds about 1/7. Breaking waves can be spilling, plunging, or surging.
A PowerPoint about storm surges and how it affects the weather. A brief case study about storm surges in the North Sea and Bangladesh are also included.
This document contains formulas and concepts related to engineering economics including:
1) Formulas for calculating the future and present value of amounts given an interest rate and number of periods.
2) Formulas for calculating the future and present value of a series of payments or annuities.
3) Formulas for cash flow analysis of revenue-dominated and cost-dominated projects.
4) Formulas and methods for calculating depreciation including straight-line, declining balance, sum-of-years digits, and sinking fund methods.
5) Formulas for calculating rate of return, benefit-cost ratios, economic order quantities, total costs for make-or-buy decisions, and break-even points
6161103 3.4 three dimensional force systemsetcenterrbru
1) Three-dimensional force systems involve resolving forces into x, y, and z components and using the equations of equilibrium to solve for unknown forces.
2) Examples are provided of using free body diagrams and the equations of equilibrium to solve for tensions in cables, magnitudes of applied forces, and stretches of springs in static systems with multiple forces.
3) Unknown forces and stretches are determined by setting the vector sum of the forces in x, y, and z directions equal to zero and solving the resulting simultaneous equations.
East Coast MARE Ocean Lecture May 16, 2012 - Surf's Up! All About Waves at th...coseenow
The document discusses waves and coastal processes. It describes how waves form, grow, and change as they approach the shoreline. This includes wave shoaling, refraction, and breaking. It also discusses surf zone currents and sediment transport. Coastal changes occur over various timescales from storms to sea level rise. Long-term trends include shoreline erosion and accretion. Rising sea levels are projected to increase coastal flooding risks in the future.
Tsunami are powerful waves created by earthquakes, volcanic eruptions, or landslides that displace large volumes of water. They can reach heights over 30 meters and speeds over 700 km/hr, destroying everything in their path. The devastating 2004 Indian Ocean tsunami killed over 118,000 people across several countries, displaced millions, and caused widespread damage to infrastructure and the environment. Relief efforts provided temporary housing, food, water and medical aid to victims, but recovery is a long process and many remain in need of support years later.
This document discusses tsunamis and their management. It defines tsunamis as long ocean waves caused by underwater seismic events like earthquakes. Tsunamis can cause significant damage through flooding and high-energy waves. Management involves identifying tsunami-prone areas, protecting coastlines, establishing early warning systems, educating communities on evacuation procedures, and conducting drills. The post-disaster phase focuses on search and rescue, relief efforts, and long-term recovery and rehabilitation programs. Proper management can help reduce risks from future tsunami disasters.
A tsunami is caused by earthquakes, volcanic eruptions, or meteorite impacts under water which generates a wave that grows much larger as it reaches the coast. This wall of water goes back and forth, devastating everything with strong currents. Tsunamis can cause severe flooding, destruction of buildings, and loss of life. The most important safety measures after a tsunami warning are to evacuate beaches and seek higher ground in buildings or mountains.
1) The document discusses remote sensing and provides definitions and explanations of key concepts such as the electromagnetic spectrum, atmospheric interaction with electromagnetic waves, and atmospheric windows.
2) It describes the seven elements of remote sensing including the energy source, interaction with the atmosphere and target, sensor recording, processing, interpretation, and application.
3) The electromagnetic spectrum is divided into regions including radio waves, microwaves, infrared, visible light, ultraviolet, and others. Certain regions have high atmospheric transmittance and are considered atmospheric windows for remote sensing.
This document discusses the application of remote sensing and geographical information systems in civil engineering. It begins by defining remote sensing as the acquisition of information about an object without physical contact, typically by measuring electromagnetic radiation. It then defines geographical information systems as a system for capturing, storing, analyzing and presenting spatially referenced data. The document provides examples of how remote sensing data from sources like Google Earth can be spatially analyzed using a GIS. It proceeds to discuss key concepts in remote sensing including the electromagnetic spectrum, atmospheric interactions with radiation, and radiation measurement principles.
The document discusses operational remote sensing applications in India. It outlines the institutional mechanisms for natural resource management using remote sensing data from Indian Space Research Organization (ISRO) satellites. Some key applications discussed are natural resource inventories, land use/land cover mapping, agriculture monitoring, drought and flood assessment, and the FASAL program for multi-crop forecasting using remote sensing, agrometeorology and ground data. National level inventories are done at 180m, 60m and 24m resolutions with finer scales at state and district levels.
Differential GPS (DGPS) improves the accuracy of standard GPS by using a stationary reference unit to calculate errors in the GPS signal caused by things like atmospheric conditions. This error data is transmitted to a mobile GPS unit in real-time to correct its position reading. DGPS can achieve sub-meter level accuracy compared to the 15 meter accuracy of standard GPS. It works by having a base station calculate positioning errors compared to its known location and sending corrections to a roving unit. This allows the roving unit to correct its reported position and improve accuracy.
The document summarizes key concepts about ocean waves. It describes how wind generates sea and swell, with sea being wind-driven waves and swell being uniform waves that travel outward from storm areas. Wave energy is affected by wind speed, duration, and fetch (distance over which wind blows). Wave height increases as waves enter shallower water and become steeper, eventually breaking in different types - spilling, plunging, or surging - depending on sea floor slope. The document also discusses wave refraction, reflection, tsunamis, and historical tsunami events.
Storm surges are increases in seawater levels caused by intense winds and low pressure from tropical cyclones. They are the primary cause of casualties and property damage in coastal areas during tropical cyclones. The Philippines is highly prone to storm surges due to its location and extensive coastline. Storm surge height depends on factors like storm intensity, size, forward speed, angle of approach, and local geography. Preparing for surges involves securing property, creating emergency plans and kits, and evacuating if instructed.
DGPS improves upon standard GPS accuracy by using a fixed reference station to calculate and broadcast differential corrections for errors caused by atmospheric delays of GPS signals. Receivers equipped with DGPS can then apply these corrections to achieve sub-meter accuracy, as low as 10 cm in some cases. It works by having a stationary receiver at a known location calculate differential errors compared to GPS satellites and broadcasting correction signals to enable mobile DGPS receivers to determine their position with much greater precision.
Wave breaking is a complex phenomenon characterized by energy dissipation and turbulence. The study analyzed wave breaking through laboratory tests using wave gauges and an acoustic Doppler velocimeter. Fifteen wave conditions were tested in a wave channel with a variable slope bottom profile designed to induce breaking. Timeseries and spectral analysis of free surface elevation data provided insights into wave propagation and breaking behavior under different conditions. Empirical formulations were also evaluated based on the experimental results.
Quick refresher on the physics of coaxial cable(draftone)foxtrot jp R
This document provides an overview of modeling a twin-lead wire transmission line as a two-port network with resistance, inductance, conductance and capacitance per unit length. Performing circuit analysis yields differential equations that can be written as wave equations in phasor form, containing a propagation constant γ. γ can be resolved into attenuation and phase constants, allowing the total wave voltage to be expressed as the sum of incident and reflected wave components propagating in opposite directions with different attenuation profiles.
1) A sound wave is a mechanical disturbance that travels outward as a longitudinal wave, causing increases and decreases in pressure relative to the atmospheric pressure.
2) Ultrasound waves reflect at tissue interfaces and scatter within tissue, sending echoes back to the ultrasound probe. Diagnostic information comes mainly from scattered echoes.
3) The speed of sound is determined by the density and compressibility of the medium and does not depend on frequency. It is higher in tissues that are less compressible like bone.
1. Waves are disturbances that transfer energy through a medium, such as water. They can be regular (single frequency/height) or irregular/random (variable frequency/height).
2. Important wave parameters include wavelength, period, frequency, speed, height, amplitude, and water elevation.
3. Ocean waves are classified based on their period/frequency and include capillary, gravity, and infragravity waves.
4. Wind generates waves by transferring energy and momentum to water. Wave characteristics depend on wind speed, fetch (distance over which wind blows), and duration. Fully developed seas occur when energy input balances dissipation.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
Quick refresher on the physics of coaxial cable asdeqfoxtrot jp R
This document provides an overview of modeling a twin-lead wire transmission line as a two-port network and derives the governing differential equations. It defines the key parameters of the transmission line model, including the characteristic impedance Z0, propagation constant γ, attenuation constant α, and phase constant β. The analysis yields expressions for these parameters in terms of the per-unit-length resistance R', inductance L', conductance G', and capacitance C' of the twin-lead wire.
Basic transmission line refresher notes twin lead wirefoxtrot jp R
This document provides an overview of modeling a twin-lead wire transmission line as a two-port network with resistance, inductance, conductance and capacitance per unit length. Performing circuit analysis yields differential equations that can be written as wave equations, from which the propagation constant γ is derived. γ is resolved into the attenuation constant α and phase constant β. The characteristic impedance zq, relating voltage and current waves, is defined in terms of these parameters and the per-unit-length circuit elements. Specific solutions are given for incident and reflected voltage and current waves propagating along the twin-lead line.
This document discusses wave interference patterns produced by two point sources of waves. It explains that when two waves are perfectly in phase, constructive interference occurs, shown as circular wave fronts. The points of constructive and destructive interference can be determined mathematically based on the path difference between the sources being equal to integer multiples of the wavelength. As an example, it analyzes the interference pattern between two boats emitting waves 80m apart, finding the point of constructive interference occurs at x = -8.90995 due to the symmetry of the setup.
1. The document provides a syllabus for RMS and average values, steady state analysis of RLC circuits with sinusoidal excitation, self and mutual inductances, and resonance in series and parallel circuits.
2. Key concepts covered include RMS and average values, form factors, steady state analysis using phasors, self and mutual inductances, dot convention, bandwidth and Q factor.
3. Example calculations are provided for average value, RMS value, form factor, and peak factor of different waveforms.
This document summarizes research using the Whole Atmosphere Community Climate Model (WACCM) to characterize high- to medium-frequency gravity waves in the mesosphere lower thermosphere region. The model was used to analyze 8 days of data from February 4-11, 2015 between 80-110km in altitude and 39-41°N, 100-120°W in latitude and longitude. A promising gravity wave was identified on February 8th passing through 40.6641°N, 113.4375°W with a period of 1.02 hours. Preliminary results matching the wave's temperature, period, and vertical wavelength to 1.02 hours, 13.63km respectively were promising for validating the model. Further
Electro magnetic resonance & its relation with frequency,wave length and wave...SohailPattan
This document discusses electromagnetic radiation and its relationship to frequency, wavelength, and wave number. It defines electromagnetic radiation as a type of energy transmitted through space at enormous velocities. Electromagnetic radiation has both wave and particle properties. The key relationships discussed are:
- Frequency (ν) is the number of wavelengths passing a point per unit time and is measured in hertz.
- Wavelength (λ) is the distance between wave peaks and is typically expressed in nanometers.
- Wave number (V) is the number of waves per centimeter and is related to wavelength as V = 1/λ.
- Velocity (c) of electromagnetic radiation depends on the medium but is 3x10^8 m/s in
1. The document discusses various topics related to wave motion including the characteristics of waves, types of waves, and the formation of stationary waves.
2. It provides definitions for key wave concepts like amplitude, wavelength, frequency, longitudinal and transverse waves. Equations are given for plane progressive waves traveling in different directions.
3. Reflection of waves at fixed and free ends is explained. The principle of superposition is described and used to show how two identical waves traveling in opposite directions can form a stationary standing wave with nodes and antinodes.
IOSR Journal of Applied Physics (IOSR-JAP) is an open access international journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This PowerPoint describes briefly about the ultrasonic absorption technique. I briefly discussed the various techniques and theoretical concepts involved in the absorption technique.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
FOR HUMANITY: (V4) A BREAKTHROUGH IN TOKAMAK APPLIED PHYSICS GRAVITATIONAL WA...GLOBAL HEAVYLIFT HOLDINGS
The document discusses using a Tokamak fusion reactor to generate detectable gravitational waves. It finds that utilizing plasma drift currents could generate strain amplitudes of around 10-25 five meters above the center of the Tokamak, allowing possible detection in the coming decade. This is an order of magnitude greater than estimates using simple Ohm's law. The document outlines the theory and provides equations, figures, and a table of strain estimates for different Tokamaks to support these conclusions.
Modalities such as X-ray, CT, ultrasound, nuclear medicine, and MRI provide different types of anatomical and functional medical imaging. X-ray and CT measure attenuation coefficients, ultrasound measures acoustic reflectivity, nuclear medicine provides functional imaging using radioactive tracers, and MRI uses magnetic fields to produce multiparametric images related to proton density and relaxation times. Each modality has strengths and limitations for different clinical applications depending on safety, image quality, and other factors. The economics of medical imaging require significant capital investments and operational costs to support clinical services.
1) Offset refers to the horizontal distance between a seismic source and receiver. It causes a delay in the arrival time of reflections that can be corrected before stacking seismic traces.
2) Acoustic impedance is the product of density and seismic velocity, which varies between rock layers and affects the reflection coefficient at layer boundaries.
3) A seismogram contains traces recorded from a single shot point, and multiple seismograms make up a seismic section.
This document provides an introduction to the physical principles of medical diagnostic ultrasound. It begins with a preface describing the intended audience and chapters of varying difficulty. Chapter 1 introduces ultrasound as a diagnostic imaging modality. Chapter 2 discusses the basics of ultrasound, including wave propagation and the 1D wave equation. Chapter 3 describes different types of ultrasound waves, including plane and spherical waves. Chapter 4 discusses the generation of ultrasound using piezoelectric transducers and the acoustic field produced by a disk transducer.
This document provides an overview of acoustic emission testing and its applications presented by Arvind Vishavkarma under the guidance of Dr. Sudhir Mishra at the Indian Institute of Technology Kanpur. It discusses acoustic emission sources in materials like metals, composites, and concrete due to cracking, fracture, phase transformations, and corrosion. Parameters of acoustic emission signals like amplitude, duration, rise time, and counts are described. Techniques for acoustic emission data analysis including source location and load monitoring are summarized. Finally, applications of acoustic emission testing for corrosion monitoring in reinforced concrete structures are briefly outlined.
This document discusses linear wave theory and the governing equations for water wave mechanics. It introduces key wave parameters like amplitude, height, wavelength, frequency, period, and phase speed. It then covers the linearized equations of motion, including continuity, irrotationality, and the time-dependent Bernoulli equation. Boundary conditions at the bed and free-surface are also presented, including the kinematic and dynamic free-surface boundary conditions. The linearized equations and boundary conditions form the basis for solving for the velocity potential using separation of variables.
This document contains solutions to examples related to wave motion. It begins by finding the period and phase speed of a wave given its wavelength or depth, using the dispersion relationship. It then calculates wave properties like height, velocity, energy, and power from pressure sensor readings. Further sections determine wave characteristics in deep water, shallow water, and when a current is present. The document solves for wavelength, period, phase speed and direction in examples involving deep water, shallow water and coastal refraction.
The document discusses wave loading on coastal structures. It provides equations to calculate the maximum wave pressure and force on both surface-piercing and fully-submerged structures. For surface-piercing structures, the force is proportional to wave height and depends on water depth. In shallow water it is approximately hydrostatic, and in deep water it is independent of depth. For fully-submerged structures the force is always less than for surface-piercing ones. Methods are given to calculate loads on vertical breakwaters by dividing them into pressure distributions and calculating individual forces and moments.
Waves undergo several transformations as they propagate towards shore:
- Refraction causes waves to change direction as their speed changes in varying water depths, bending towards parallel to depth contours. This is governed by Snell's law.
- Shoaling causes waves to increase in height as their speed decreases in shallower water, to conserve shoreward energy flux. Wave height is related to the refraction and shoaling coefficients.
- Breaking occurs once waves steepen enough, dissipating energy. Types of breakers depend on the relative beach slope and wave steepness via the Iribarren number. Common breaking criteria include the Miche steepness limit and breaker height/depth indices.
The document provides mathematical derivations of key concepts in fluid dynamics, including:
1) Definitions of hyperbolic functions like sinh, cosh, and tanh and their basic properties.
2) The fundamental fluid flow equations - continuity, irrotationality/use of a velocity potential, and the time-dependent Bernoulli equation - that are used to model wave behavior.
3) The derivation of the wave field and dispersion relationship by applying Laplace's equation, kinematic and dynamic boundary conditions, and making linear approximations to obtain solutions for a sinusoidal wave.
1. The document provides answers to example problems involving wave propagation and hydraulics. It analyzes wave characteristics such as wavelength, phase speed, and acceleration for different water depths.
2. Methods like iteration of the dispersion relationship are used to determine wave numbers and properties for scenarios with and without current.
3. Key wave parameters like height and wavelength are calculated from pressure readings using linear wave theory and shoaling equations. Different cases consider deep, intermediate, and shallow water conditions.
The document discusses various processes of wave transformation as waves propagate into shallower water, including refraction, shoaling, breaking, diffraction, and reflection. It provides definitions and equations for each process. As examples, it works through calculations of wave properties for a given scenario involving wave refraction and shoaling as depth decreases.
This document discusses wave loading on structures. It describes the pressure distribution on surface-piercing and fully-submerged structures. For surface-piercing structures, the maximum pressure is at the water surface and decreases with depth. For fully-submerged structures, the maximum pressure is always less. It also provides an example calculation of wave forces and overturning moment on a caisson breakwater, determining the required caisson height, maximum horizontal force, and maximum overturning moment.
The document contains 23 multi-part questions related to wave properties and behavior. The questions cover topics such as calculating wave properties like wavelength, phase speed and particle motion from given parameters; estimating wave properties at different depths and under the influence of currents; applying wave theories to problems involving wave propagation over varying bathymetry; and analyzing wave loads on coastal structures. Sample questions provided seek solutions for wave characteristics at offshore measurement locations, during propagation to shore, and at breaking.
This document outlines the contents of a course on hydraulic waves, including linear wave theory, wave transformation processes like refraction and shoaling, random wave statistics, and wave loading on coastal structures. The topics are organized into sections covering main wave parameters, dispersion relationships, velocity and pressure, energy transfer, particle motion, shallow and deep water behavior, waves on currents, refraction, shoaling, breaking, diffraction, reflection, statistical measures of waves, wave spectra, reconstruction of wave fields, wave climate prediction, pressure distributions, and loads on surface-piercing, submerged, and vertical breakwater structures. Mathematical derivations are included in an appendix. Recommended textbooks on coastal engineering and water wave mechanics are provided.
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1. Hydraulics 3 Waves: Random Waves and Statistics – 1 Dr David Apsley
3. RANDOM WAVES AND STATISTICS AUTUMN 2022
Real wave fields are not regular, but a combination of many components of different amplitude,
frequency and direction, which may be assumed to follow some statistical distribution. For
design, models must reflect the appropriate probability distributions of heights and spectral
distribution of frequencies (and, ideally, direction).
3.1 Measures of Wave Height and Period
At a fixed location, a depth-monitoring device (such as a wave buoy, pressure sensor or
acoustic sounder) measures sea-surface elevation η(𝑡). From this, successive wave heights may
be measured. Suppose that a set of 𝑁 wave heights 𝐻𝑖 are measured and put in descending
order. The following are common measures of the sea state.
𝐻max Largest wave height in the sample 𝐻max = max
𝑖
(𝐻𝑖)
𝐻av Mean wave height 𝐻av =
1
𝑁
∑ 𝐻𝑖
𝐻rms Root-mean-square wave height 𝐻rms = √
1
𝑁
∑ 𝐻𝑖
2
𝐻1 3
⁄ Average of the highest 𝑁/3 waves 𝐻1 3
⁄ =
1
𝑁/3
∑ 𝐻𝑖
𝑁/3
1
𝐻𝑚0 Estimate based on the rms surface elevation 𝐻𝑚0 = 4( 𝜂2
̅̅̅ )
1 2
⁄
𝐻𝑠 Significant wave height Either 𝐻1 3
⁄ or 𝐻𝑚0
For a typical probability distribution of wave heights (Rayleigh distribution)
𝐻1 3
⁄ = 1.416 𝐻rms, whilst for a typical spectral distribution of frequencies (Bretschneider
spectrum – see later) 𝐻𝑚0 = √2 𝐻rms. Thus, 𝐻1/3 and 𝐻𝑚0 are almost equal and either may be
taken to define the significant wave height 𝐻𝑠, depending on what is used to measure sea state.
The “significant wave height” (allegedly) corresponds to “what an experienced mariner would
judge the height of waves in a storm”.
There are also common measures of wave period:
𝑇𝑠 Significant wave period (average over highest 𝑁/3 waves)
𝑇𝑝 Peak period (corresponds to the peak frequency of the energy spectrum)
𝑇𝑒
Energy period (period of a regular wave with same significant wave height and
power density; used in wave-energy prediction; derived from the energy spectrum)
𝑇𝑧 Mean zero up-crossing period
2. Hydraulics 3 Waves: Random Waves and Statistics – 2 Dr David Apsley
3.2 Probability Distribution of Wave Heights
For a narrow-banded sea state (i.e. a small range of frequencies) deep-water waves have been
observed to follow a Rayleigh Distribution, for which the probability that wave heights exceed
𝐻 is given by
𝑃(height > 𝐻) = exp[−(𝐻/𝐻rms)2]
Thus,
cumulative distribution function: 𝐹(𝐻) = 𝑃(height < 𝐻) = 1 − e−(𝐻/𝐻rms)2
probability density function: 𝑓(𝐻) =
d𝐹
𝑑𝐻
= 2
𝐻
𝐻rms
2
e−(𝐻/𝐻rms)2
By definition of a probability density function, the probability that an individual wave height
lies between 𝐻1 and 𝐻2 is
𝑃(𝐻1 < 𝐻 < 𝐻2) = ∫ 𝑓(𝐻) d𝐻
𝐻2
𝐻1
= 𝐹(𝐻2) − 𝐹(𝐻1)
= e−(𝐻1/𝐻rms)2
− e−(𝐻2/𝐻rms)2
The only parameter of this distribution is the root-mean-square height, 𝐻rms. Exercise: verify
from probability theory, i.e.
𝐸(𝐻2) ≡ ∫ 𝐻2
𝑓(𝐻) d𝐻
∞
0
that E(𝐻2) = 𝐻rms
2
.
Other key wave statistics may be determined by the usual rules for probability distributions:
𝐻av ≡ 𝐸(𝐻) = ∫ 𝐻 𝑓(𝐻) d𝐻
∞
0
𝐻𝑝 = average of highest fraction 𝑝 of waves =
1
𝑝
∫ 𝐻 𝑓(𝐻) d𝐻
∞
𝐹−1(1−𝑝)
3. Hydraulics 3 Waves: Random Waves and Statistics – 3 Dr David Apsley
With this we find1
:
𝐻av =
√π
2
𝐻rms = 0.886 𝐻rms
𝐻1/3 = 1.416 𝐻rms
𝐻1 10
⁄ = 1.800 𝐻rms
𝐻1 100
⁄ = 2.359 𝐻rms
Example:
Near a pier, 400 consecutive wave heights are measured. Assume that the sea state is narrow-
banded.
(a) How many waves are expected to exceed 2𝐻rms?
(b) If the significant wave height is 2.5 m, what is 𝐻rms?
(c) Estimate the wave height exceeded by 80 waves.
(d) Estimate the number of waves with a height between 1.0 m and 3.0 m.
1
A certain amount of fiddly mathematics gives for the average of the highest fraction 𝑝:
𝐻𝑝
𝐻rms
= √ln(1/𝑝) +
√π
2𝑝
erfc(√ln(1/𝑝))
Here erfc() is the complementary error function, which most modern computer languages will
provide as a library function. Here it is in Python:
from math import sqrt, log, erfc, pi
def prob( p ):
x = sqrt( log( 1 / p ) )
return x + sqrt( pi ) / ( 2 * p ) * erfc( x )
p = float( input( "Enter p: " ) )
print( "H/Hrms = ", prob( p ) )
4. Hydraulics 3 Waves: Random Waves and Statistics – 4 Dr David Apsley
3.3 Wave Spectra
For regular waves of a single frequency, the wave energy (per unit surface area) is given by
𝐸 =
1
2
𝜌𝑔𝐴2
= 𝜌𝑔𝜂2(𝑡) =
1
8
𝜌𝑔𝐻2
i.e. the wave energy is proportional to the square of the amplitude 𝐴 (or height 𝐻) of the
harmonically-varying surface displacement 𝜂(𝑡).
Real wave fields contain many frequencies. The energy spectrum or power-spectral density
𝑆(𝑓) is such that the amount of energy (divided by ρ𝑔) in the small frequency interval d𝑓 is
𝑆(𝑓) d𝑓
or, equivalently, that the energy from waves between frequencies 𝑓1 and 𝑓2 is (𝜌𝑔 times)
∫ 𝑆(𝑓) d𝑓
𝑓2
𝑓1
(This is rather like a continuous probability distribution). Note that we can just as well work in
wave angular frequency 𝜔, where 𝜔 = 2π𝑓. In that case,
𝑆(𝜔) d𝜔
is the energy (divided by 𝜌𝑔) between wave angular frequencies 𝜔1 and 𝜔2.
3.3.1 Bretschneider Spectrum
Various model spectra are used in design. The Bretschneider spectrum is recommended for use
in open-ocean conditions:
𝑆(𝑓) =
5
16
𝐻𝑚0
2
𝑓
𝑝
4
𝑓5
exp (−
5
4
𝑓𝑝
4
𝑓4
)
where 𝐻𝑚0 is a measure of significant wave height 𝐻𝑠 based on the total energy (see below)
and 𝑓𝑝 is the peak frequency (= 1/𝑇𝑝, where 𝑇𝑝 is the peak period).
5. Hydraulics 3 Waves: Random Waves and Statistics – 5 Dr David Apsley
If we integrate over all frequencies we obtain total energy (strictly, energy divided by 𝜌𝑔):
𝐸
𝜌𝑔
≡ ∫ 𝑆(𝑓) d𝑓
∞
0
=
1
16
𝐻𝑚0
2
Hence
𝐻𝑚0 = 4√𝐸/𝜌𝑔 = 4√𝜂2(𝑡)
For a regular wave with height 𝐻rms (there is only one wave height, so it is the rms value):
𝐸
𝜌𝑔
=
1
8
𝐻rms
2
Thus, the complete spectrum with height parameter 𝐻𝑚0 will have the same energy density as
a regular wave with parameter 𝐻rms provided
𝐻𝑚0 = √2𝐻𝑟𝑚𝑠 = 1.414𝐻rms
But, for a Rayleigh distribution of wave heights, we have already seen that
𝐻𝑠 = 𝐻1/3 = 1.416𝐻rms
Hence, in practice, either 𝐻𝑚0 and 𝐻1/3 can be used synonymously for 𝐻𝑠.
3.3.2 Use of Spectral Data to Determine height and Period Parameters
From, e.g, wave-buoy data for 𝜂(𝑡), the power spectrum 𝑆(𝑓) can be determined by a discrete
Fourier transform of 𝜂2
. From this, as seen above, we can deduce immediately the peak period
𝑇𝑝 from the peak frequency in 𝑆(𝑓):
𝑇𝑝 =
1
𝑓𝑝
and the significant wave height
𝐻𝑠 = 𝐻𝑚0 = 4√𝑚0
where 𝑚0 is the zeroth moment: the area under the 𝑆(𝑓) curve. Other moments can be defined:
𝑚𝑛 = ∫ 𝑓𝑛
𝑆(𝑓) d𝑓
∞
0
and found from an experimentally-derived spectrum by numerical integration2
. A particularly
important one is 𝑚−1, since this can be used to determine the energy period 𝑇𝑒 (the period of
a regular wave with the same significant wave height and power density, which is widely used
in wavepower prediction):
𝑇𝑒 =
𝑚−1
𝑚0
and the zero up-crossing period 𝑇𝑧, which can be estimated from the wave spectrum by:
2
For the Bretschneider spectrum, some moderate mathematics produces 𝑚𝑛 =
1
16
𝐻𝑚0
2
𝑓𝑝
𝑛
(
5
4
)
𝑛
4
Γ(1 −
𝑛
4
) , where
Γ() is the gamma function (a generalisation of a factorial function).
6. Hydraulics 3 Waves: Random Waves and Statistics – 6 Dr David Apsley
𝑇𝑧 = √
𝑚0
𝑚2
For the Bretschneider spectrum these give
𝑇𝑒 = 0.857𝑇𝑝 (energy period)
𝑇𝑧 = 0.710𝑇𝑝 (zero up-crossing period)
3.3.3 The JONSWAP Spectrum
Another widely-used spectrum recommended for fetch-limited conditions (based on extensive
wave data from the North Sea) is the JONSWAP spectrum
𝑆(𝑓) = 𝐶𝐻𝑚0
2
𝑓𝑝
4
𝑓5
exp (−
5
4
𝑓
𝑝
4
𝑓4
) 𝛾𝑏
Here, the peak of the spectrum is enhanced (i.e. a greater proportion of the total energy is
clustered around the peak frequency) by the factor 𝛾𝑏
, where 𝛾 may be fitted to real
measurements, but is typically 3.3 and
𝑏 = exp {−
1
2
(
𝑓/𝑓
𝑝 − 1
𝜎
)
2
} , 𝜎 = {
0.07 𝑓 < 𝑓
𝑝
0.09 𝑓 > 𝑓
𝑝
C is the constant required to get the correct total energy (e.g.by numerical integration). A
JONSWAP spectrum is recommended for seas with more limited fetch.
For a JONSWAP spectrum (with 𝛾 = 3.3) numerical integration gives other design periods:
𝑇𝑒 = 0.903𝑇𝑝 (energy period)
𝑇𝑧 = 0.778𝑇𝑝 (zero up-crossing period)
3.3.4 Multi-Modal Spectra
Real sea states may contain waves from multiple sources – often waves of lower frequency
from a far-off storm (“swell”) and higher-frequency waves from a local storm (“wind”).
Complex statistical techniques can be used to extract the separate contributions from the
combined spectrum.
frequency f (Hz)
spectral
density
S
(m^2
s)
swell
wind
7. Hydraulics 3 Waves: Random Waves and Statistics – 7 Dr David Apsley
3.4 Constructing a Representative Wave Field From a Spectrum
For most spectra there is negligible energy associated with frequencies less than 0.5 𝑓
𝑝 or
greater than 3𝑓
𝑝, where 𝑓𝑝 is the peak frequency. If we break this or a larger frequency range
up into discrete intervals of length Δ𝑓, we can simulate a realistic spectrum (either in a
numerical simulation, or in a wave tank with programmable wave paddle) as a sum of
individual harmonic components:
𝜂(𝑡) = ∑ 𝑎𝑖cos (𝑘𝑖𝑥 − 𝜔𝑖𝑡 − 𝜙𝑖)
where 𝜔𝑖 and 𝑘𝑖 are the wavenumbers associated with frequency 𝑓𝑖, the 𝜙𝑖 are random phases,
and the correct amount of energy (𝐸𝑖) at this frequency occurs if we take
𝑆(𝑓𝑖)Δ𝑓 = 𝐸𝑖 =
1
2
𝑎𝑖
2
or
𝑎𝑖 = √2𝑆(𝑓𝑖) Δ𝑓
The wave tanks at the University of Manchester are equipped with programmable wave paddles
that can create such realistic random wave fields for a given spectrum.
Note that, as different wavenumbers travel with different speeds, this wave form is not
propagated unchanged (as it would be for a regular wave), but evolves with time.
If, instead of choosing random phases 𝜙𝑖 we choose them deliberately such that waves
travelling at different speeds arrive at the same point at the same time it is possible to generate
focused wave groups, with the focusing producing a very large-amplitude disturbance at a
single instant. These worst-case scenarios are used to simulate extreme-wave events.
8. Hydraulics 3 Waves: Random Waves and Statistics – 8 Dr David Apsley
Example.
An irregular wavefield at a deep-water location is characterised by peak period of 8.7 s and
significant wave height of 1.5 m.
(a) Provide a sketch of a Bretschneider spectrum, labelling both axes with variables and
units and indicating the frequencies corresponding to both the peak period and the
energy period.
Note: Calculations are not needed for this part.
(b) Determine the power density (in kW m–1
) of a regular wave component with frequency
0.125 Hz that represents the frequency range 0.12 to 0.13 Hz of the irregular wave field.
Example.
Wave measurements are obtained from a stationary sensor located in deep water. The measured surface
elevation of an irregular wave can be modelled as the sum of four regular wave components:
(a) In the context of modelling an irregular wave, explain the meaning of the following
terms:
(i) significant wave height;
(ii) significant wave period;
(iii) duration-limited.
(b) Obtain the total power conveyed by these deep-water wave components per metre width
of wave crest if conditions were measured:
(i) with zero current;
(ii) with an opposing current of 1.0 m s–1
.
Period (s) 6 7 8 9
Amplitude (m) 0.8 1.2 0.8 0.4
9. Hydraulics 3 Waves: Random Waves and Statistics – 9 Dr David Apsley
3.5 Prediction of Wave Climate
Models for wave spectra generally require one to specify a representative wave height (e.g. 𝐻𝑠)
and period (𝑇𝑠 or 𝑇𝑝).
Waves are generated by wind stress on the water surface, whilst gravity provides the restoring
force. Thus, wave height and period are expected to be functions of:
• wind speed 𝑈 (conventionally the wind speed at 10 m above the surface);
• fetch 𝐹 (the distance over which the wind blows);
• duration 𝑡 of the storm;
• gravity, 𝑔.
By dimensional analysis,
𝐻𝑠~𝑈, 𝐹, 𝑡, 𝑔
This gives 5 variables, 2 independent dimensions (length and time), and hence 3 dimensionless
Π groups:
𝑔𝐻𝑠
𝑈2
= function(
𝑔𝐹
𝑈2
,
𝑔𝑡
𝑈
)
Similarly
𝑔𝑇𝑝
𝑈
= function(
𝑔𝐹
𝑈2
,
𝑔𝑡
𝑈
)
Extensive wave data has led to empirical forms for these. Two of the commonest are SMB
(Sverdrup, Monk and Bretschneider) and JONSWAP (JOint North Sea WAve Project).
These correlations can be used to predict wave climate (usually to construct a wave spectrum)
from a weather forecast (forecasting), ongoing weather (nowcasting) and reconstructing wave
climate from measured wind records (hindcasting).
The following are standard correlations for deep-water waves.
3.5.1 JONSWAP (Hasselman et al., 1973)
If the wind has blown long enough for wave energy to propagate right across the fetch then the
wave parameters become functions only of the fetch 𝐹 and cease to be dependent on the storm
duration 𝑡. These are called fetch-limited waves and an empirical correlation is
𝑔𝐻𝑠
𝑈2
= 0.0016 (
𝑔𝐹
𝑈2
)
1 2
⁄
(up to maximum 0.2433)
𝑔𝑇𝑝
𝑈
= 0.286 (
𝑔𝐹
𝑈2
)
1 3
⁄
(up to maximum 8.134)
The (fairly rare) “maximum” conditions correspond to a “fully-developed sea” – one for which
energy dissipation equals energy input and wave conditions become independent of fetch.
The minimum duration for fetch-limited waves, 𝑡min is given, in non-dimensional form by
10. Hydraulics 3 Waves: Random Waves and Statistics – 10 Dr David Apsley
(
𝑔𝑡
𝑈
)
min
= 68.8 (
𝑔𝐹
𝑈2
)
2 3
⁄
(fully − developed sea: 7.15 × 104
)
If the storm duration 𝑡 < 𝑡min then the wave conditions are said to be duration-limited and the
non-dimensional fetch 𝑔𝐹/𝑈2
in the equations for 𝐻𝑠 and 𝑇𝑃 has to be replaced by an effective
fetch 𝐹eff determined by rearranging the last equation in terms of the actual duration t:
(
𝑔𝐹
𝑈2
)
eff
= (
1
68.8
𝑔𝑡
𝑈
)
3 2
⁄
Note:
(1) The period parameter predicted here is the peak period 𝑇𝑝, since this is what is required in
the Jonswap spectrum. If required, the significant wave period can be estimated by
𝑇𝑠 ≈ 0.945𝑇𝑝
(2) It is a considerable nuisance to have to keep writing the Π groups out in full. The JONSWAP
equations are conveniently written (ignoring the fully-developed limit) as
𝐻
̂𝑠 = 0.0016𝐹
̂1 2
⁄
𝑇
̂𝑝 = 0.286𝐹
̂1 3
⁄
𝑡̂min = 68.8𝐹
̂2 3
⁄
where
𝐹
̂ ≡
𝑔𝐹
𝑈2
, 𝑡̂ ≡
𝑔𝑡
𝑈
, 𝐻
̂𝑠 ≡
𝑔𝐻𝑠
𝑈2
, 𝑇
̂𝑝 ≡
𝑔𝑇𝑝
𝑈
distance travelled by wave energy greater than fetch F
F
distance travelled by wave energy less than fetch F
F
Feff
FETCH-LIMITED
DURATION-LIMITED
11. Hydraulics 3 Waves: Random Waves and Statistics – 11 Dr David Apsley
3.5.2 SMB (Bretschneider, 1970)
This alternative correlation has slightly(!) more complex formulae. Note that the representative
period here is 𝑇𝑠, the significant wave period, rather than 𝑇𝑝, the peak period. In non-
dimensional form:
𝐻
̂𝑠 = 0.283 tanh{0.0125𝐹
̂0.42
}
𝑇
̂𝑠 = 7.54 tanh{0.077𝐹
̂0.25
}
The minimum storm duration for fetch-limited waves is given by
𝑡̂min = 𝐾 exp {√[𝐴(ln 𝐹
̂)
2
− 𝐵 ln 𝐹
̂ + 𝐶] + 𝐷 ln 𝐹
̂}
where 𝐾 = 6.5882, 𝐴 = 0.0161, 𝐵 = 0.3692, 𝐶 = 2.2024, 𝐷 = 0.8798.
Prediction curves; solid: JONSWAP; dashed: SMB
Example.
(a) Wind has blown at a consistent 𝑈 = 20 m s−1
over a fetch 𝐹 = 100 km for 𝑡 = 6 hrs.
Determine 𝐻𝑠 and 𝑇𝑝 using the JONSWAP curves.
(b) If the wind blows steadily for another 4 hours what are 𝐻𝑠 and 𝑇𝑝?