Wave Equation
Wave Equation - Related Videos
01:02
Equations of Wave Motion
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2023
Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered. From the ratio, using the value of θ, and...
01:24
Electromagnetic Wave Equation
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2025
Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete. However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations: What...
03:10
Chemical Equations
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2020
Chemical equations represent the identities and relative quantities of substances involved in a chemical reaction. The substances undergoing reaction are called reactants, and their formulas are placed on the left side of the equation. The substances generated by the reaction are called products, and their formulas are placed on the right side of the equation. Plus signs (+) separate individual reactant and product formulas, and an arrow (→) separates the reactant and product (left and right)...
02:59
The Nernst Equation
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2020
Nonstandard Reaction Conditions The interconnection between standard cell potentials and various thermodynamic parameters such as the standard free energy change ΔG° and equilibrium constant K has been previously explored. For example, a redox reaction involving zinc(II) and tin(II) ions at 1 M concentration with Eºcell = +0.291 V and ΔG° = −56.2 kJ is spontaneous. Discharge of this cell, however, results in a change in the reactant concentration and a steady decrease of the cell potential. At...
02:55
Thermochemical Equations
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2020
For a chemical reaction (the system) carried out at constant pressure – with the only work done caused by expansion or contraction – the enthalpy of reaction (also called the heat of reaction, ΔHrxn) is equal to the heat exchanged with the surroundings (qp). The change in enthalpy is an extensive property, and it depends on the amounts of the reactants participating in the reaction (or the number of moles of reactants). The change in enthalpy is specific to the reaction, and the physical...
FAQs
The differential equation alone permits many mathematically possible motions. Initial conditions describe the system at the starting point of the analysis, while boundary conditions constrain behavior at the limits of the modeled region. Together, they select the particular solution that represents the physical setup, allowing predictions of wave behavior to be matched with experimental observations.
Wave speed appears as its square multiplying the second spatial derivative. This coefficient links the spatial variation of a displacement or field to its change over time, setting the rate at which a disturbance propagates through the modeled system. Altering the wave speed changes predicted propagation behavior while preserving the equation’s basic mathematical structure.
Solutions describe the resulting motion throughout space and time, making characteristic wave behaviors visible in the predicted pattern. Researchers can examine those solutions for overlapping disturbances, returning waves, or repeated responses associated with resonance. This makes the equation useful not only for calculating propagation, but also for connecting mathematical results with recognizable physical effects.
A typical analysis identifies the displacement or field being studied, writes the appropriate wave-equation form, and specifies the relevant initial and boundary conditions. The resulting solution then predicts how the disturbance evolves. Researchers can compare those predictions with observations to assess wave speed, spatial behavior, and effects such as reflection or resonance.
The same mathematical framework can be applied to sound, vibrating strings, water-surface disturbances, and electromagnetic waves. Each case assigns the modeled displacement or field to a different physical phenomenon, while the equation supports analysis of propagation and wave behavior. This broad applicability makes it a useful bridge between distinct areas of physics.
Its solutions generate predictions about measurable wave behavior, including propagation speed, interference, reflection, and resonance. Those predictions can be compared with experimental observations from systems such as strings, sound, water surfaces, or electromagnetic fields. Agreement links the mathematical description to physical behavior, while differing results can indicate that the chosen conditions or model require reconsideration.