Physics Formulae/Waves Formulae

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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.

Quantity (Common Name/s) (Common) Symbol/s SI Units Dimension
Number of Wave Cycles   dimensionless dimensionless
(Transverse) Displacement   m [L]
(Transverse) Displacement Amplitude   m [L]
(Transverse) Velocity Amplitude   m s-1 [L][T]-1
(Transverse) Acceleration Amplitude   m s-2 [L][T]-2
(Longnitudinal) Displacement   m [L]
Period   s [T]
Wavelength   m [L]
Phase Angle   rad dimensionless

General Derived Quantites


The most general definition of (instantaneous) frequency is:


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


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


Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
(Transverse) Velocity     m s-1 [L][T]-1
(Transverse) Acceleration     m s-2 [L][T]-2
Path Length Difference     m [L]
(Longnitudinal) Velocity     m s-1 [L][T]-1
Frequency     Hz = s-1 [T]-1
Angular Frequency/ Pulsatance     Hz = s-1 [T]-1
Time Delay, Time Lag/Lead     s [T]
Scalar Wavenumber   Two definitions are used:



In the formalism which follows, only the first

definition is used.

m-1 [L]-1
Vector Wavenumber   Again two definitions are possible:



In the formalism which follows, only the first

definition is used.

m-1 [L]-1
Phase Differance     rad dimensionless
Phase   (No standard symbol,   is used

only here for clarity of equivalances )



rad dimensionless
Wave Energy E J [M] [L]2 [T]-2
Wave Power P   W = J s-1 [M] [L]2 [T]-3
Wave Intensity I   W m-2 [M] [T]-3
Wave Intensity (per unit Solid Angle) I  

Often reduces to


W m-2 sr-1 [M] [T]-3


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

wave popagation in +x direction


wave popagation in -x direction


Phase angle can lag if:  

or lead if:  

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


Standing Waves

Harmonic Number  
Harmonic Series  

Propagating Waves


Wave Equation

Any wavefunction of the form


satisfies the hyperbolic PDE:


Principle of Superposition for Waves


General Mechanical Wave Results

Average Wave Power  

Sound Waves


Sound Intensity and Level

Quantity (Common Name/s) (Common) Symbol/s
Sound Level  

Sound Beats and Standing Waves

pipe, two open ends  
Pipe, one open end   for n odd
Acoustic Beat Frequency  

Sonic Doppler Effect

Sonic Doppler Effect  


Mach Cone Angle

(Supersonic Shockwave, Sonic boom)


Sound Wavefunctions

Acoustic Pressure Amplitude  
Sound Displacement Function  
Sound pressure-variation function  

Superposition, Interferance/Diffraction

Phase and Interference  

Constructive Interference


Destructive Interference


n is any integer;


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.

Taut String  
Solid Rods  

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:


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.

  = Spring constant (not wavenumber).

Linear DHO  
Angular SHO  
Low Amplitude Simple Pendulum  
Low Amplitude Physical Pendulum  

Sinusiodal Waves


Equation of a Sinusiodal Wave is


Recall that wave propagation is in   direction for  .

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.

Wave Energy

Quantity (Common Name/s) (Common) Symbol/s
potential harmonic energy  
kinetic harmonic energy  
total harmonic energy  
damped mechanical energy  

General Wavefunctions


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.

Wavefunction Nomenclature Superposition Resultant
Standing Wave  








Coherant Interferance  



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


Non-Solutions to the Wave Equation

Exponentially Damped Waveform  
Solitary Wave

Common Waveforms


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