sinusoidal steady state
Indicates every voltage and current in a system is sinusoidal with the same angular frequency ω.
Examples of sinusoidal steady state in the following topics:

Phasors
 Phasors are used to analyze electrical systems in sinusoidal steady state and with a uniform angular frequency.
 This can be particularly useful because the frequency factor (which includes the timedependence of the sinusoid) is often common to all the components of a linear combination of sinusoids.
 Sinusoids can be represented mathematically as the sum of two complexvalued functions:
 Sinusoidal Steady State and the Series RLC CircuitPhasors may be used to analyze the behavior of electrical and mechanical systems that have reached a kind of equilibrium called sinusoidal steady state.
 In the sinusoidal steady state, every voltage and current (or force and velocity) in a system is sinusoidal with angular frequency ω.

Resistors in AC Circuits
 If the source varies periodically, particularly sinusoidally, the circuit is known as an alternatingcurrent circuit.
 The voltage and current are sinusoidal and are in phase for a simple resistance circuit.

Driven Oscillations and Resonance
 This equation can be solved exactly for any driving force, using the solutions z(t) which satisfy the unforced equation: $\frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0$, and which can be expressed as damped sinusoidal oscillations $z(t) = A \mathrm{e}^{\zeta \omega_0 t} \ \sin \left( \sqrt{1\zeta^2} \ \omega_0 t + \varphi \right)$in the case where ζ ≤ 1.
 In the case of a sinusoidal driving force: $\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t)$, where $\!
 \omega$ is the driving frequency for a sinusoidal driving mechanism.
 The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude $\!
 Steady state variation of amplitude with frequency and damping of a driven simple harmonic oscillator.

Position, Velocity, and Acceleration as a Function of Time
 This equation simply states that the acceleration of the waveform (Left: second derivative with respect to time) is proportional to the Laplacian (Right: second spatial derivative) of the same waveform.
 A general form of a sinusoidal wave is $y(x,t) = A sin(kx\omega t + \phi)$, where A is the amplitude of the wave, $\omega$ is the wave's angular frequency, k is the wavenumber, and $\phi$ is the phase of the sine wave given in radians.
 Since a wave with an arbitrary shape can be represented by a sum of many sinusoidal waves (this is called Fourier analysis), we can generate a great variety of solutions of the wave equation by translating and summing sine waves that we just looked closely into.

Spherical Accretion
 We continue looking at steady flows with two specific applications: matter flowing onto an object (accretion) and matter flowing away from an object (winds).
 We will assume that the accretion is steady at a rate ${\dot M}$ and that the pressure $P \propto \rho^\gamma$ with $1 < \gamma < 5/3$.
 Let's use the equation of state to eliminate $p$ from the Euler equation and use the continuity equation to eliminate $\rho$,

Different Types of Currents
 This solution gives the circuit voltages and currents when the circuit is in DC steady state.
 The solution to these equations usually contain a time varying or transient part as well as constant or steady state part.
 It is this steady state part that is the DC solution.

Time
 The second is now operationally defined as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom."
 It follows that the hyperfine splitting in the ground state of the cesium 133 atom is exactly 9,192,631,770 hertz .
 In other words, cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted.

Maxwell's Equations
 Gauss's law for magnetism states that there are no "magnetic charges (or monopoles)" analogous to electric charges, and that magnetic fields are instead generated by magnetic dipoles.
 Ampere's law originally stated that magnetic field could be created by electrical current.
 Electric (red) and magnetic (blue) waves propagate in phase sinusoidally, and perpendicularly to one another.

B.2 Chapter 2
 For the linearly polarized wave, the particle moves up and down sinusoidally.
 The velocity varies sinusoidally, the power magnetic force vary as $\sin^2 \omega t$.
 This states that the divergence of the current must vanish, which means that either charge is not conserved or that the charge density is constant (neither is good).

Detonation Waves
 Because the flux of the flow is conserved through the transition, the state of the gas must remain on the chord .
 There is a minimum flux that can pass through the detonation front, , and this flux also corresponds to the minimum velocity jump through the front where the final state is or the Jouguet point.
 Furthermore, for final states above $O$ along $a'$ the gas leaves the front subsonically.
 If the final state lies at $O$, the gas leaves the detonation front right at the speed of sound in the downstream flow.
 At this point the postshock gas leaves the front at the sound speed so the rarefaction wave no longer overtakes the shock and the combined detonation front and rarefaction wave achieves a steady state.