List of equations in wave theory

From KYNNpedia

This article summarizes equations in the theory of waves.

Definitions

General fundamental quantities

Main articles: Transverse wave and Longitudinal wave

A wave can be longitudinal where the oscillations are parallel (or antiparallel) to the propagation direction, or transverse where the oscillations are perpendicular to the propagation direction. These oscillations are characterized by a periodically time-varying displacement in the parallel or perpendicular direction, and so the instantaneous velocity and acceleration are also periodic and time varying in these directions. (the apparent motion of the wave due to the successive oscillations of particles or fields about their equilibrium positions) propagates at the phase and group velocities parallel or antiparallel to the propagation direction, which is common to longitudinal and transverse waves. Below oscillatory displacement, velocity and acceleration refer to the kinematics in the oscillating directions of the wave - transverse or longitudinal (mathematical description is identical), the group and phase velocities are separate.

Quantity (common name/s) (Common) symbol/s SI units Dimension
Number of wave cycles N dimensionless dimensionless
(Oscillatory) displacement Symbol of any quantity which varies periodically, such as h, x, y (mechanical waves), x, s, η (longitudinal waves) I, V, E, B, H, D (electromagnetism), u, U (luminal waves), ψ, Ψ, Φ (quantum mechanics). Most general purposes use y, ψ, Ψ. For generality here, A is used and can be replaced by any other symbol, since others have specific, common uses.

<math>\mathbf{A} = A \mathbf{\hat{e}}_{\parallel} \,\!</math> for longitudinal waves,
<math>\mathbf{A} = A \mathbf{\hat{e}}_{\bot} \,\!</math> for transverse waves.

m [L]
(Oscillatory) displacement amplitude Any quantity symbol typically subscripted with 0, m or max, or the capitalized letter (if displacement was in lower case). Here for generality A0 is used and can be replaced. m [L]
(Oscillatory) velocity amplitude V, v0, vm. Here v0 is used. m s−1 [L][T]−1
(Oscillatory) acceleration amplitude A, a0, am. Here a0 is used. m s−2 [L][T]−2
Spatial position
Position of a point in space, not necessarily a point on the wave profile or any line of propagation 		d, r m [L]
Wave profile displacement
Along propagation direction, distance travelled (path length) by one wave from the source point r0 to any point in space d (for longitudinal or transverse waves) 		L, d, r

<math> \mathbf{r} \equiv r \mathbf{\hat{e}}_{\parallel} \equiv \mathbf{d} - \mathbf{r}_0 \,\!</math>

 m [L]
Phase angle δ, ε, φ rad dimensionless

General derived quantities

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Wavelength λ General definition (allows for FM):

<math>\lambda = \mathrm{d} r/\mathrm{d} N \,\!</math>

For non-FM waves this reduces to:
<math>\lambda = \Delta r/\Delta N \,\!</math>

m [L]
Wavenumber, k-vector, Wave vector k, σ Two definitions are in use:

<math>\mathbf{k} = \left ( 2\pi/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!</math>
<math>\mathbf{k} = \left ( 1/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!</math>

m−1 [L]−1
Frequency f, ν General definition (allows for FM):

<math>f = \mathrm{d} N/\mathrm{d} t \,\!</math>

For non-FM waves this reduces to:
<math>f = \Delta N/\Delta t \,\!</math>

In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation:
<math>f = 1/T \,\!</math>

Hz = s−1 [T]−1
Angular frequency/ pulsatance ω <math>\omega = 2\pi f = 2\pi / T \,\!</math> Hz = s−1 [T]−1
Oscillatory velocity v, vt, v Longitudinal waves:

<math> \mathbf{v} = \mathbf{\hat{e}}_{\parallel} \left ( \partial A/\partial t \right ) \,\!</math>

Transverse waves:
<math> \mathbf{v} = \mathbf{\hat{e}}_{\bot} \left ( \partial A/\partial t \right ) \,\!</math>

m s−1 [L][T]−1
Oscillatory acceleration a, at Longitudinal waves:

<math> \mathbf{a} = \mathbf{\hat{e}}_{\parallel} \left ( \partial^2 A/\partial t^2 \right ) \,\!</math>

Transverse waves:
<math> \mathbf{a} = \mathbf{\hat{e}}_{\bot} \left ( \partial^2 A/\partial t^2 \right ) \,\!</math>

m s−2 [L][T]−2
Path length difference between two waves L, ΔL, Δx, Δr <math> \mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1 \,\!</math> m [L]
Phase velocity vp General definition:

<math> \mathbf{v}_\mathrm{p} = \mathbf{\hat{e}}_{\parallel} \left ( \Delta r /\Delta t \right ) \,\!</math>

In practice reduces to the useful form:
<math> \mathbf{v}_\mathrm{p} = \lambda f \mathbf{\hat{e}}_{\parallel} = \left ( \omega/k \right ) \mathbf{\hat{e}}_{\parallel} \,\!</math>

m s−1 [L][T]−1
(Longitudinal) group velocity vg <math> \mathbf{v}_\mathrm{g} = \mathbf{\hat{e}}_{\parallel} \left ( \partial \omega /\partial k \right ) \,\!</math> m s−1 [L][T]−1
Time delay, time lag/lead Δt <math> \Delta t = t_2 - t_1 \,\!</math> s [T]
Phase difference δ, Δε, Δϕ <math> \Delta \phi = \phi_2 - \phi_1 \,\!</math> rad dimensionless
Phase No standard symbol <math> \mathbf{k} \cdot \mathbf{r} \mp \omega t + \phi= 2\pi N \,\!</math>

Physically;
upper sign: wave propagation in +r direction
lower sign: wave propagation in −r direction

Phase angle can lag if: ϕ > 0
or lead if: ϕ < 0.

rad dimensionless

Relation between space, time, angle analogues used to describe the phase:

<math> \frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!</math>

Modulation indices

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
AM index:
 h, hAM <math> h_{AM} = A/A_m \,\!</math>

A = carrier amplitude
Am = peak amplitude of a component in the modulating signal

dimensionless dimensionless
FM index:
 hFM <math> h_{FM} = \Delta f/f_m \,\!</math>

Δf = max. deviation of the instantaneous frequency from the carrier frequency
fm = peak frequency of a component in the modulating signal

dimensionless dimensionless
PM index:
 hPM <math> h_{PM} = \Delta \phi \,\!</math>

Δϕ = peak phase deviation

dimensionless dimensionless

Acoustics

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Acoustic impedance Z <math>Z = \rho v\,\!</math>

v = speed of sound, ρ = volume density of medium

kg m−2 s−1 [M] [L]−2 [T]−1
Specific acoustic impedance z <math>z = ZS\,\!</math>

S = surface area

kg s−1 [M] [T]−1
Sound Level β \frac{I}{I_0} \right | \,\!</math> dimensionless dimensionless

Equations

In what follows n, m are any integers (Z = set of integers); <math>n, m \in \mathbf{Z} \,\!</math>.

Standing waves

Physical situation Nomenclature Equations
Harmonic frequencies fn = nth mode of vibration, nth harmonic, (n-1)th overtone <math>f_n = \frac{v}{\lambda_n} = \frac{nv}{2L} = n f_1\,\!</math>

Propagating waves

Sound waves

Physical situation Nomenclature Equations
Average wave power P0 = Sound power due to source <math>\langle P \rangle = \mu v \omega^2 x_m^2/2\,\!</math>
Sound intensity Ω = Solid angle <math> I = P_0/(\Omega r^2)\,\!</math>

<math> I = P/A = \rho v \omega^2 s^2_m/2\,\!</math>

Acoustic beat frequency f1, f2 = frequencies of two waves (nearly equal amplitudes) f_2 - f_1 \right | \,\!</math>
Doppler effect for mechanical waves
  • V = speed of sound wave in medium
  • f0 = Source frequency
  • fr = Receiver frequency
  • v0 = Source velocity
  • vr = Receiver velocity
 <math>f_r = f_0 \frac{V \pm v_r}{V \mp v_0}\,\!</math>

upper signs indicate relative approach, lower signs indicate relative recession.

Mach cone angle (Supersonic shockwave, sonic boom)
  • v = speed of body
  • vs = local speed of sound
  • θ = angle between direction of travel and conic envelope of superimposed wavefronts
 <math> \sin \theta = \frac{v}{v_s}\,\!</math>
Acoustic pressure and displacement amplitudes
  • p0 = pressure amplitude
  • s0 = displacement amplitude
  • v = speed of sound
  • ρ = local density of medium
 <math> p_0 = \left ( v \rho \omega \right ) s_0\,\!</math>
Wave functions for sound Acoustic beats

<math> s = \left [ 2 s_0 \cos \left ( \omega' t \right ) \right ] \cos \left ( \omega t \right )\,\!</math>

Sound displacement function <math>s = s_0\cos(k r - \omega t)\,\!</math>

Sound pressure-variation <math> p = p_0 \sin(k r - \omega t)\,\!</math>

Gravitational waves

Main article: Gravitational wave § Sources of gravitational waves

Gravitational radiation for two orbiting bodies in the low-speed limit.<ref name="Gravitational Radiation">"Gravitational Radiation" (PDF). Archived from the original (PDF) on 2012-04-02. Retrieved 2012-09-15.</ref>

Physical situation Nomenclature Equations
Radiated power
  • P = Radiated power from system,
  • t = time,
  • r = separation between centres-of-mass
  • m1, m2 = masses of the orbiting bodies
 <math>P = \frac{\mathrm{d}E}{\mathrm{d}t} = - \frac{32}{5}\, \frac{G^4}{c^5}\, \frac{(m_1m_2)^2 (m_1+m_2)}{r^5}</math>
Orbital radius decay <math>\frac{\mathrm{d}r}{\mathrm{d}t} = - \frac{64}{5} \frac{G^3}{c^5} \frac{(m_1m_2)(m_1+m_2)}{r^3}\ </math>
Orbital lifetime r0 = initial distance between the orbiting bodies <math> t = \frac{5}{256} \frac{c^5}{G^3} \frac{r_0^4}{(m_1m_2)(m_1+m_2)}\ </math>

Superposition, interference, and diffraction

Physical situation Nomenclature Equations
Principle of superposition N = number of waves <math> y_\mathrm{net} = \sum_{i=1}^N y_i \,\!</math>
Resonance
  • ωd = driving angular frequency (external agent)
  • ωnat = natural angular frequency (oscillator)
 <math>\omega_d = \omega_\mathrm{nat} \,\!</math>
Phase and interference
  • Δr = path length difference
  • φ = phase difference between any two successive wave cycles
 <math> \frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!</math>

Constructive interference <math>n = \frac{\lambda}{\Delta x}\,\!</math>

Destructive interference <math>n+\frac{1}{2} = \frac{\lambda}{\Delta x}\,\!</math>

Wave propagation

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

Physical situation Nomenclature Equations
Idealized non-dispersive media
  • p = (any type of) Stress or Pressure,
  • ρ = Volume Mass Density,
  • F = Tension Force,
  • μ = Linear Mass Density of medium
 <math>v = \sqrt{\frac{p}{\rho}} = \sqrt{\frac{F}{\mu}} \,\!</math>
Dispersion relation Implicit form

<math> D \left ( \omega, k \right ) = 0 </math>

Explicit form <math> \omega = \omega \left ( k \right ) </math>

Amplitude modulation, AM <math> A = A \left ( t \right ) </math>
Frequency modulation, FM <math> f = f \left ( t \right ) </math>

General wave functions

Wave equations

Physical situation Nomenclature Wave equation General solution/s
Non-dispersive Wave Equation in 3d A = amplitude as function of position and time <math> \nabla^2 A = \frac{1}{v_{\parallel}^2} \frac{\partial ^2 A}{\partial t^2}\,\!</math> <math> A \left ( \mathbf{r}, t \right ) = A \left ( x - v_{\parallel} t \right ) \,\!</math>
Exponentially damped waveform
  • A0 = Initial amplitude at time t = 0
  • b = damping parameter
 <math> A = A_0 e^{-bt} \sin \left ( k x - \omega t + \phi \right ) \,\!</math>
Korteweg–de Vries equation<ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3</ref> α = constant <math> \frac{\partial y}{\partial t} + \alpha y \frac{\partial y}{\partial x} + \frac{\partial^3 y}{\partial x^3} = 0 \,\!</math> <math> A(x,t) = \frac{3v_{\parallel}}{\alpha} \mathrm{sech}^2 \left [ \frac{\sqrt{v_{\parallel}}}{2} \left ( x-v_{\parallel} t \right ) \right ] \,\!</math>

Sinusoidal solutions to the 3d wave equation

N different sinusoidal waves

Complex amplitude of wave n
<math> A_n = \left | A_n \right | e^{i \left ( \mathbf{k}_\mathrm{n}\cdot\mathbf{r} - \omega_n t + \phi_n \right )} \,\!</math>

Resultant complex amplitude of all N waves
<math> A = \sum_{n=1}^{N} A_n \,\!</math>

Modulus of amplitude
<math> A = \sqrt{AA^{*}} = \sqrt{\sum_{n=1}^N \sum_{m=1}^N \left | A_n \right | \left | A_m \right | \cos \left [ \left ( \mathbf{k}_n - \mathbf{k}_m \right ) \cdot \mathbf{r} + \left ( \omega_n - \omega_m \right ) t + \left ( \phi_n - \phi_m \right ) \right ]} \,\!</math>

The transverse displacements are simply the real parts of the complex amplitudes.

1-dimensional corollaries for two sinusoidal waves

The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.

Wavefunction Nomenclature Superposition Resultant
Standing wave <math> \begin{align} y_1+y_2 & = A \sin \left ( k x - \omega t \right ) \\

& + A \sin \left ( k x + \omega t \right ) \end{align}\,\!</math>

<math> y = 2A \sin \left ( k x \right ) \cos \left ( \omega t \right ) \,\!</math>
Beats
  • <math> \langle \omega \rangle = \frac{\omega_1 + \omega_2}{2} \,\!</math>
  • <math> \langle k \rangle = \frac{k_1 + k_2}{2} \,\!</math>
  • <math> \Delta \omega = \omega_1 - \omega_2 \,\!</math>
  • <math> \Delta k = k_1 - k_2 \,\!</math>
 <math> \begin{align} y_1 + y_2 & = A \sin \left ( k_1 x - \omega_1 t \right ) \\

& + A \sin \left ( k_2 x + \omega_2 t \right ) \end{align}\,\!</math>

<math> y = 2 A \sin \left ( \langle k \rangle x - \langle \omega \rangle t \right ) \cos \left ( \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t \right ) \,\!</math>
Coherent interference <math> \begin{align} y_1+y_2 & = 2A \sin \left ( k x - \omega t \right ) \\

& + A \sin \left ( k x + \omega t + \phi \right ) \end{align}\,\!</math>

 <math> y = 2 A \cos \left ( \frac{\phi}{2} \right ) \sin \left ( k x - \omega t + \frac{\phi}{2} \right ) \,\!</math>

See also

Footnotes

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Sources

Further reading

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