Physics Formulae/Waves Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Waves.

General Fundamental Quantites
For transverse directions, the remaining cartesian unit vectors i and j can be used.

General Derived Quantites
The most general definition of (instantaneous) frequency is:

$$f = \frac{\partial N}{\partial t} \,\!$$

For a monochromatic (one frequency) waveform the change reduces to the linear gradient:

$$f = \frac{\Delta N}{\Delta t} \,\!$$

but common pratice is to set N = 1 cycle, then setting t = T = time period for 1 cycle gives the more useful definition:

 Phase 

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point. Physically;

Relation between quantities of space, time, and angle analogues used to describe the phase $$ \Phi \,\!$$ is summarized simply:

Propagating Waves
 Wave Equation 

Any wavefunction of the form

$$ y = y \left ( x - v_{\parallel} t \right )  \,\!$$

satisfies the hyperbolic PDE:


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 * $$ \nabla^2 y = \frac{1}{v_{\parallel}^2} \frac{\partial ^2 y}{\partial t^2}\,\!$$
 * }

 Principle of Superposition for Waves 


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 * $$ y_\mathrm{net} = \sum_i \left ( y_i \right ) \,\!$$
 * }

Sound Waves
Sound Intensity and Level

Sound Beats and Standing Waves

Sonic Doppler Effect

Sound Wavefunctions

Phase Velocities in Various Media
The general equation for the phase velocity of any wave is (equivalent to the simple "speed-distance-time" relation, using wave quantities):

The general equation for the group velocity of any wave is:

A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media.

In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency.

The phase velocity is the rate at which the phase of the wave propagates in space.

The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.

Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function below called the  Dispersion Relation , given in explicit form and implicit form respectively.

The use of ω(k) for explicit form is standard, since the phase velocity ω/k and the group velocity dω/dk usually have convenient representations by this function.

For more specific media through which waves propagate, phase velocities are tabulated below. All cases are idealized, and the media are non-dispersive, so the group and phase velocity are equal.

The generalization for these formulae is for any type of stress or pressure p, volume mass density ρ, tension force F, linear mass density μ for a given medium:

$$v = \sqrt{\frac{p}{\rho}} = \sqrt{\frac{F}{\mu}} \,\!$$

Pulsatances of Common Osscilators
Pulatances (angular frequencies) for simple osscilating systems, the linear and angular Simple Harmonic Oscillator (SHO) and Damped Harmonic Oscillator (DHO) are summarized in the table below. They are often useful shortcuts in calculations.

$$ k_H \,\!$$ = Spring constant (not wavenumber).

Sinusiodal Waves
Equation of a Sinusiodal Wave is


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 * $$ y = A\sin\left ( kx - \omega t + \phi \right ) \,\!$$
 * }

Recall that wave propagation is in $$ \pm x \,\!$$ direction for $$ \mp \omega \,\!$$.

Sinusiodal waves are important since any waveform can be created by applying the principle of superposition to sinusoidal waves of varying frequencies, amplitudes and phases. The physical concept is easily manipulated by application of Fourier Transforms.

Sinusiodal Solutions to the Wave Equation
The following may be duduced by applying the principle of superposition to two sinusiodal waves, using trigonometric identities. Most often the angle addition and sum-to-product formulae are useful; in more advanced work complex numbers and Fourier series and transforms are often used.

Note: When adding two wavefunctions togther the following trigonometric identity proves very useful:

$$ \sin A \pm \sin B = 2A \sin \left ( \frac{A \pm B}{2} \right ) \cos \left ( \frac{A \mp B}{2} \right )\,\!$$

Non-Solutions to the Wave Equation
Common Waveforms