List of equations in wave theory

This article summarizes equations in the theory of waves.

Definitions

=General fundamental quantities=

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.

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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. $\mathbf{A} = A \mathbf{\hat{e}}_{\parallel} \,\!$ for longitudinal waves, $\mathbf{A} = A \mathbf{\hat{e}}_{\bot} \,\!$ 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, v_m. Here v₀ is used. \| m s⁻¹ \| [L][T]⁻¹
(Oscillatory) acceleration amplitude \| A, a₀, a_m. 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 $\mathbf{r} \equiv r \mathbf{\hat{e}}_{\parallel} \equiv \mathbf{d} - \mathbf{r}_0 \,\!$ \| m \| [L]
Phase angle \| δ, ε, φ \| rad \| dimensionless

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Number of wave cycles

| N

| 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.

$\mathbf{A} = A \mathbf{\hat{e}}_{\parallel} \,\!$ for longitudinal waves,

$\mathbf{A} = A \mathbf{\hat{e}}_{\bot} \,\!$ 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, v_m. Here v₀ is used.

| m s⁻¹

| [L][T]⁻¹

(Oscillatory) acceleration amplitude

| A, a₀, a_m. 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

$\mathbf{r} \equiv r \mathbf{\hat{e}}_{\parallel} \equiv \mathbf{d} - \mathbf{r}_0 \,\!$

| m

| [L]

Phase angle

| δ, ε, φ

| rad

| dimensionless

=General derived quantities=

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Wavelength \| λ \|General definition (allows for FM): $\lambda = \mathrm{d} r/\mathrm{d} N \,\!$ For non-FM waves this reduces to: $\lambda = \Delta r/\Delta N \,\!$ \| m \| [L]
Wavenumber, k-vector, Wave vector \| k, σ \|Two definitions are in use: $\mathbf{k} = \left ( 2\pi/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!$ $\mathbf{k} = \left ( 1/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!$ \| m⁻¹ \| [L]⁻¹
Frequency \| f, ν \|General definition (allows for FM): $f = \mathrm{d} N/\mathrm{d} t \,\!$ For non-FM waves this reduces to: $f = \Delta N/\Delta t \,\!$ In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation: $f = 1/T \,\!$ \| Hz = s⁻¹ \| [T]⁻¹
Angular frequency/ pulsatance \| ω \| $\omega = 2\pi f = 2\pi / T \,\!$ \| Hz = s⁻¹ \| [T]⁻¹
Oscillatory velocity \|v, v_t, v \| Longitudinal waves: $\mathbf{v} = \mathbf{\hat{e}}_{\parallel} \left ( \partial A/\partial t \right ) \,\!$ Transverse waves: $\mathbf{v} = \mathbf{\hat{e}}_{\bot} \left ( \partial A/\partial t \right ) \,\!$	m s⁻¹	[L][T]⁻¹
Oscillatory acceleration	a, a_t	Longitudinal waves: $\mathbf{a} = \mathbf{\hat{e}}_{\parallel} \left ( \partial^2 A/\partial t^2 \right ) \,\!$ Transverse waves: $\mathbf{a} = \mathbf{\hat{e}}_{\bot} \left ( \partial^2 A/\partial t^2 \right ) \,\!$ \| m s⁻² \| [L][T]⁻²
Path length difference between two waves \| L, ΔL, Δx, Δr \| $\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1 \,\!$ \| m \| [L]
Phase velocity \| v_p \|General definition: $\mathbf{v}_\mathrm{p} = \mathbf{\hat{e}}_{\parallel} \left ( \Delta r /\Delta t \right ) \,\!$ In practice reduces to the useful form: $\mathbf{v}_\mathrm{p} = \lambda f \mathbf{\hat{e}}_{\parallel} = \left ( \omega/k \right ) \mathbf{\hat{e}}_{\parallel} \,\!$ \| m s⁻¹ \| [L][T]⁻¹
(Longitudinal) group velocity \| v_g \| $\mathbf{v}_\mathrm{g} = \mathbf{\hat{e}}_{\parallel} \left ( \partial \omega /\partial k \right ) \,\!$ \| m s⁻¹ \| [L][T]⁻¹
Time delay, time lag/lead \| Δt \| $\Delta t = t_2 - t_1 \,\!$ \| s \| [T]
Phase difference \| δ, Δε, Δϕ \| $\Delta \phi = \phi_2 - \phi_1 \,\!$ \| rad \| dimensionless
Phase \| No standard symbol \| $\mathbf{k} \cdot \mathbf{r} \mp \omega t + \phi= 2\pi N \,\!$ 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

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Wavelength

| λ

|General definition (allows for FM):

$\lambda = \mathrm{d} r/\mathrm{d} N \,\!$

For non-FM waves this reduces to:

$\lambda = \Delta r/\Delta N \,\!$

| m

| [L]

Wavenumber, k-vector, Wave vector

| k, σ

|Two definitions are in use:

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

$\mathbf{k} = \left ( 1/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!$

| m⁻¹

| [L]⁻¹

Frequency

| f, ν

|General definition (allows for FM):

$f = \mathrm{d} N/\mathrm{d} t \,\!$

For non-FM waves this reduces to:

$f = \Delta N/\Delta t \,\!$

In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation:

$f = 1/T \,\!$

| Hz = s⁻¹

| [T]⁻¹

Angular frequency/ pulsatance

| ω

| $\omega = 2\pi f = 2\pi / T \,\!$

| Hz = s⁻¹

| [T]⁻¹

Oscillatory velocity

|v, v_t, v

| Longitudinal waves:

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

Transverse waves:

$\mathbf{v} = \mathbf{\hat{e}}_{\bot} \left ( \partial A/\partial t \right ) \,\!$

m s⁻¹

[L][T]⁻¹

Oscillatory acceleration

a, a_t

Longitudinal waves:

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

Transverse waves:

$\mathbf{a} = \mathbf{\hat{e}}_{\bot} \left ( \partial^2 A/\partial t^2 \right ) \,\!$

| m s⁻²

| [L][T]⁻²

Path length difference between two waves

| L, ΔL, Δx, Δr

| $\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1 \,\!$

| m

| [L]

Phase velocity

| v_p

|General definition:

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

In practice reduces to the useful form:

$\mathbf{v}_\mathrm{p} = \lambda f \mathbf{\hat{e}}_{\parallel} = \left ( \omega/k \right ) \mathbf{\hat{e}}_{\parallel} \,\!$

| m s⁻¹

| [L][T]⁻¹

(Longitudinal) group velocity

| v_g

| $\mathbf{v}_\mathrm{g} = \mathbf{\hat{e}}_{\parallel} \left ( \partial \omega /\partial k \right ) \,\!$

| m s⁻¹

| [L][T]⁻¹

Time delay, time lag/lead

| Δt

| $\Delta t = t_2 - t_1 \,\!$

| s

| [T]

Phase difference

| δ, Δε, Δϕ

| $\Delta \phi = \phi_2 - \phi_1 \,\!$

| rad

| dimensionless

Phase

| No standard symbol

| $\mathbf{k} \cdot \mathbf{r} \mp \omega t + \phi= 2\pi N \,\!$

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:

$\frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!$

=Modulation indices=

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AM index: \| h, h_AM \| $h_{AM} = A/A_m \,\!$ A = carrier amplitude A_m = peak amplitude of a component in the modulating signal \| dimensionless \| dimensionless
FM index: \|h_FM \| $h_{FM} = \Delta f/f_m \,\!$ Δf = max. deviation of the instantaneous frequency from the carrier frequency f_m = peak frequency of a component in the modulating signal \| dimensionless \| dimensionless
PM index: \| h_PM \| $h_{PM} = \Delta \phi \,\!$ Δϕ = peak phase deviation \| dimensionless \| dimensionless

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AM index:

| h, h_AM

| $h_{AM} = A/A_m \,\!$

A = carrier amplitude

A_m = peak amplitude of a component in the modulating signal

| dimensionless

FM index:

|h_FM

| $h_{FM} = \Delta f/f_m \,\!$

Δf = max. deviation of the instantaneous frequency from the carrier frequency

f_m = peak frequency of a component in the modulating signal

| dimensionless

PM index:

| h_PM

| $h_{PM} = \Delta \phi \,\!$

Δϕ = peak phase deviation

| dimensionless

=Acoustics=

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Acoustic impedance \| Z \| $Z = \rho v\,\!$ v = speed of sound, ρ = volume density of medium \| kg m⁻² s⁻¹ \| [M] [L]⁻² [T]⁻¹
Specific acoustic impedance \| z \| $z = ZS\,\!$ S = surface area \| kg s⁻¹ \| [M] [T]⁻¹
Sound Level	β \| $\beta = \left ( \mathrm{dB} \right ) 10 \log \left \| \frac{I}{I_0} \right \| \,\!$ \| dimensionless \| dimensionless

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Acoustic impedance

| Z

| $Z = \rho v\,\!$

v = speed of sound,

ρ = volume density of medium

| kg m⁻² s⁻¹

| [M] [L]⁻² [T]⁻¹

Specific acoustic impedance

| z

| $z = ZS\,\!$

S = surface area

| kg s⁻¹

| [M] [T]⁻¹

Sound Level

| $\beta = \left ( \mathrm{dB} \right ) 10 \log \left | \frac{I}{I_0} \right | \,\!$

| dimensionless

Equations

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

=Standing waves=

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Harmonic frequencies \|f_n = nth mode of vibration, nth harmonic, (n-1)th overtone \| $f_n = \frac{v}{\lambda_n} = \frac{nv}{2L} = n f_1\,\!$

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Harmonic frequencies

|f_n = nth mode of vibration, nth harmonic, (n-1)th overtone

| $f_n = \frac{v}{\lambda_n} = \frac{nv}{2L} = n f_1\,\!$

=Propagating waves=

==Sound waves==

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Average wave power \| P₀ = Sound power due to source \| $\langle P \rangle = \mu v \omega^2 x_m^2/2\,\!$
Sound intensity \|Ω = Solid angle \| $I = P_0/(\Omega r^2)\,\!$ $I = P/A = \rho v \omega^2 s^2_m/2\,\!$
Acoustic beat frequency \|f₁, f₂ = frequencies of two waves (nearly equal amplitudes) \| $f_\mathrm{beat} = \left \| f_2 - f_1 \right \| \,\!$
Doppler effect for mechanical waves \| {{plainlist}} V = speed of sound wave in medium f₀ = Source frequency f_r = Receiver frequency v₀ = Source velocity v_r = Receiver velocity {{endplainlist}} \| $f_r = f_0 \frac{V \pm v_r}{V \mp v_0}\,\!$ upper signs indicate relative approach, lower signs indicate relative recession.
Mach cone angle (Supersonic shockwave, sonic boom) \| {{plainlist}} v = speed of body v_s = local speed of sound θ = angle between direction of travel and conic envelope of superimposed wavefronts {{endplainlist}} \| $\sin \theta = \frac{v}{v_s}\,\!$
Acoustic pressure and displacement amplitudes \| {{plainlist}} p₀ = pressure amplitude s₀ = displacement amplitude v = speed of sound ρ = local density of medium {{endplainlist}} \| $p_0 = \left ( v \rho \omega \right ) s_0\,\!$
Wave functions for sound \| \|Acoustic beats $s = \left [ 2 s_0 \cos \left ( \omega' t \right ) \right ] \cos \left ( \omega t \right )\,\!$ Sound displacement function $s = s_0\cos(k r - \omega t)\,\!$ Sound pressure-variation $p = p_0 \sin(k r - \omega t)\,\!$

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Average wave power

| P₀ = Sound power due to source

| $\langle P \rangle = \mu v \omega^2 x_m^2/2\,\!$

Sound intensity

|Ω = Solid angle

| $I = P_0/(\Omega r^2)\,\!$

$I = P/A = \rho v \omega^2 s^2_m/2\,\!$

Acoustic beat frequency

|f₁, f₂ = frequencies of two waves (nearly equal amplitudes)

| $f_\mathrm{beat} = \left | f_2 - f_1 \right | \,\!$

Doppler effect for mechanical waves

| {{plainlist}}

V = speed of sound wave in medium
f₀ = Source frequency
f_r = Receiver frequency
v₀ = Source velocity
v_r = Receiver velocity

| $f_r = f_0 \frac{V \pm v_r}{V \mp v_0}\,\!$

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

Mach cone angle (Supersonic shockwave, sonic boom)

| {{plainlist}}

v = speed of body
v_s = local speed of sound
θ = angle between direction of travel and conic envelope of superimposed wavefronts

| $\sin \theta = \frac{v}{v_s}\,\!$

Acoustic pressure and displacement amplitudes

| {{plainlist}}

p₀ = pressure amplitude
s₀ = displacement amplitude
v = speed of sound
ρ = local density of medium

| $p_0 = \left ( v \rho \omega \right ) s_0\,\!$

Wave functions for sound

|Acoustic beats

$s = \left [ 2 s_0 \cos \left ( \omega' t \right ) \right ] \cos \left ( \omega t \right )\,\!$

Sound displacement function

$s = s_0\cos(k r - \omega t)\,\!$

Sound pressure-variation

$p = p_0 \sin(k r - \omega t)\,\!$

==Gravitational waves==

Gravitational radiation for two orbiting bodies in the low-speed limit.{{cite web |url=http://www.eftaylor.com/exploringblackholes/GravWaves100707V2.pdf |title=Gravitational Radiation |access-date=2012-09-15 |archive-url=https://web.archive.org/web/20120402135140/http://www.eftaylor.com/exploringblackholes/GravWaves100707V2.pdf |archive-date=2012-04-02 |url-status=dead }}

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Radiated power \| {{plainlist}} P = Radiated power from system, t = time, r = separation between centres-of-mass m₁, m₂ = masses of the orbiting bodies {{endplainlist}} \| $P = \frac{\mathrm{d}E}{\mathrm{d}t} = - \frac{32}{5}\, \frac{G^4}{c^5}\, \frac{(m_1m_2)^2 (m_1+m_2)}{r^5}$
Orbital radius decay \| \| $\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}\$
Orbital lifetime \|r₀ = initial distance between the orbiting bodies \| $t = \frac{5}{256} \frac{c^5}{G^3} \frac{r_0^4}{(m_1m_2)(m_1+m_2)}\$

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Radiated power

| {{plainlist}}

P = Radiated power from system,
t = time,
r = separation between centres-of-mass
m₁, m₂ = masses of the orbiting bodies

| $P = \frac{\mathrm{d}E}{\mathrm{d}t} = - \frac{32}{5}\, \frac{G^4}{c^5}\, \frac{(m_1m_2)^2 (m_1+m_2)}{r^5}$

Orbital radius decay

| $\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}\$

Orbital lifetime

|r₀ = initial distance between the orbiting bodies

| $t = \frac{5}{256} \frac{c^5}{G^3} \frac{r_0^4}{(m_1m_2)(m_1+m_2)}\$

=Superposition, interference, and diffraction=

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Principle of superposition \|N = number of waves \| $y_\mathrm{net} = \sum_{i=1}^N y_i \,\!$
Resonance \| {{plainlist}} ω_d = driving angular frequency (external agent) ω_nat = natural angular frequency (oscillator) {{endplainlist}} \| $\omega_d = \omega_\mathrm{nat} \,\!$
Phase and interference \| {{plainlist}} Δr = path length difference φ = phase difference between any two successive wave cycles {{endplainlist}} \| $\frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!$ Constructive interference $n = \frac{\lambda}{\Delta x}\,\!$ Destructive interference $n+\frac{1}{2} = \frac{\lambda}{\Delta x}\,\!$

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Principle of superposition

|N = number of waves

| $y_\mathrm{net} = \sum_{i=1}^N y_i \,\!$

Resonance

| {{plainlist}}

ω_d = driving angular frequency (external agent)
ω_nat = natural angular frequency (oscillator)

| $\omega_d = \omega_\mathrm{nat} \,\!$

Phase and interference

| {{plainlist}}

Δr = path length difference
φ = phase difference between any two successive wave cycles

| $\frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!$

Constructive interference

$n = \frac{\lambda}{\Delta x}\,\!$

Destructive interference

$n+\frac{1}{2} = \frac{\lambda}{\Delta x}\,\!$

=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.

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Idealized non-dispersive media \| {{plainlist}} p = (any type of) Stress or Pressure, ρ = Volume Mass Density, F = Tension Force, μ = Linear Mass Density of medium {{endplainlist}} \| $v = \sqrt{\frac{p}{\rho}} = \sqrt{\frac{F}{\mu}} \,\!$
Dispersion relation \| \| Implicit form $D \left ( \omega, k \right ) = 0$ Explicit form $\omega = \omega \left ( k \right )$
Amplitude modulation, AM \| \| $A = A \left ( t \right )$
Frequency modulation, FM \| \| $f = f \left ( t \right )$

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Idealized non-dispersive media

| {{plainlist}}

p = (any type of) Stress or Pressure,
ρ = Volume Mass Density,
F = Tension Force,
μ = Linear Mass Density of medium

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

Dispersion relation

| Implicit form

$D \left ( \omega, k \right ) = 0$

Explicit form

$\omega = \omega \left ( k \right )$

Amplitude modulation, AM

| $A = A \left ( t \right )$

Frequency modulation, FM

| $f = f \left ( t \right )$

=General wave functions=

==Wave equations==

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

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! scope="col" style="width:10px;"| Wave equation

! scope="col" style="width:10px;"| General solution/s

Non-dispersive Wave Equation in 3d

|A = amplitude as function of position and time

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

| $A \left ( \mathbf{r}, t \right ) = A \left ( x - v_{\parallel} t \right ) \,\!$

Exponentially damped waveform

| {{plainlist}}

A₀ = Initial amplitude at time t = 0
b = damping parameter

| $A = A_0 e^{-bt} \sin \left ( k x - \omega t + \phi \right ) \,\!$

Korteweg–de Vries equationEncyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3

|α = constant

| $\frac{\partial y}{\partial t} + \alpha y \frac{\partial y}{\partial x} + \frac{\partial^3 y}{\partial x^3} = 0 \,\!$

| $A(x,t) = \frac{3v_{\parallel}}{\alpha} \mathrm{sech}^2 \left [ \frac{\sqrt{v_{\parallel}}}{2} \left ( x-v_{\parallel} t \right ) \right ] \,\!$

==Sinusoidal solutions to the 3d wave equation==

;N different sinusoidal waves

Complex amplitude of wave n

$A_n = \left | A_n \right | e^{i \left ( \mathbf{k}_\mathrm{n}\cdot\mathbf{r} - \omega_n t + \phi_n \right )} \,\!$

Resultant complex amplitude of all N waves

$A = \sum_{n=1}^{N} A_n \,\!$

Modulus of amplitude

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

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.

class="wikitable"
scope="col" style="width:100px;"\| Wavefunction ! scope="col" style="width:250px;"\| Nomenclature ! scope="col" style="width:10px;"\| Superposition ! scope="col" style="width:10px;"\| Resultant
Standing wave \| \| $\begin{align} y_1+y_2 & = A \sin \left ( k x - \omega t \right ) \\ & + A \sin \left ( k x + \omega t \right ) \end{align}\,\!$ \| $y = 2A \sin \left ( k x \right ) \cos \left ( \omega t \right ) \,\!$
Beats \|{{plainlist}} $\langle \omega \rangle = \frac{\omega_1 + \omega_2}{2} \,\!$ $\langle k \rangle = \frac{k_1 + k_2}{2} \,\!$ $\Delta \omega = \omega_1 - \omega_2 \,\!$ $\Delta k = k_1 - k_2 \,\!$ {{endplainlist}} \| $\begin{align} y_1 + y_2 & = A \sin \left ( k_1 x - \omega_1 t \right ) \\ & + A \sin \left ( k_2 x + \omega_2 t \right ) \end{align}\,\!$ \| $y = 2 A \sin \left ( \langle k \rangle x - \langle \omega \rangle t \right ) \cos \left ( \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t \right ) \,\!$
Coherent interference \| \| $\begin{align} y_1+y_2 & = 2A \sin \left ( k x - \omega t \right ) \\ & + A \sin \left ( k x + \omega t + \phi \right ) \end{align}\,\!$ \| $y = 2 A \cos \left ( \frac{\phi}{2} \right ) \sin \left ( k x - \omega t + \frac{\phi}{2} \right ) \,\!$

class="wikitable"

scope="col" style="width:100px;"| Wavefunction

! scope="col" style="width:250px;"| Nomenclature

! scope="col" style="width:10px;"| Superposition

! scope="col" style="width:10px;"| Resultant

Standing wave

| $\begin{align} y_1+y_2 & = A \sin \left ( k x - \omega t \right ) \\
& + A \sin \left ( k x + \omega t \right )
\end{align}\,\!$

| $y = 2A \sin \left ( k x \right ) \cos \left ( \omega t \right ) \,\!$

Beats

|{{plainlist}}

$\langle \omega \rangle = \frac{\omega_1 + \omega_2}{2} \,\!$
$\langle k \rangle = \frac{k_1 + k_2}{2} \,\!$
$\Delta \omega = \omega_1 - \omega_2 \,\!$
$\Delta k = k_1 - k_2 \,\!$

| $\begin{align} y_1 + y_2 & = A \sin \left ( k_1 x - \omega_1 t \right ) \\
& + A \sin \left ( k_2 x + \omega_2 t \right )
\end{align}\,\!$

| $y = 2 A \sin \left ( \langle k \rangle x - \langle \omega \rangle t \right ) \cos \left ( \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t \right ) \,\!$

Coherent interference

| $\begin{align} y_1+y_2 & = 2A \sin \left ( k x - \omega t \right ) \\
& + A \sin \left ( k x + \omega t + \phi \right )
\end{align}\,\!$

| $y = 2 A \cos \left ( \frac{\phi}{2} \right ) \sin \left ( k x - \omega t + \frac{\phi}{2} \right ) \,\!$

Footnotes

Sources

{{cite book|author1=P.M. Whelan |author2=M.J. Hodgeson | title=Essential Principles of Physics| publisher=John Murray|edition=2nd| year=1978 | isbn=0-7195-3382-1}}
{{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| url=https://archive.org/details/cambridgehandboo0000woan| url-access=registration| publisher=Cambridge University Press| year=2010| isbn=978-0-521-57507-2}}
{{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=Mc Graw Hill| year=1988| isbn=978-0-07-025734-4}}
{{cite book|pages=12–13|author1=R.G. Lerner |author2=G.L. Trigg | title=Encyclopaedia of Physics| publisher=VHC Publishers, Hans Warlimont, Springer|edition=2nd| year=2005| isbn=978-0-07-025734-4}}
{{cite book| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park}}
{{cite book|author1=P.A. Tipler |author2=G. Mosca | title=Physics for Scientists and Engineers: With Modern Physics| publisher=W.H. Freeman and Co|edition=6th| year=2008| isbn=978-1-4292-0265-7}}
{{cite book|title=Analytical Mechanics|author1=L.N. Hand |author2=J.D. Finch |publisher=Cambridge University Press |year=2008|isbn=978-0-521-57572-0}}
{{cite book|title=Mechanics, Vibrations and Waves|author1=T.B. Arkill |author2=C.J. Millar |publisher=John Murray |year=1974|isbn=0-7195-2882-8}}
{{cite book|title=The Physics of Vibrations and Waves|edition=3rd|author=H.J. Pain|publisher=John Wiley & Sons |year=1983|isbn=0-471-90182-2}}
{{cite book|title=Dynamics and Relativity|author1=J.R. Forshaw |author2=A.G. Smith |publisher=Wiley |year=2009|isbn=978-0-470-01460-8}}
{{cite book|title=Electricity and Modern Physics |edition=2nd|author=G.A.G. Bennet|publisher=Edward Arnold (UK)|year=1974|isbn=0-7131-2459-8}}
{{cite book|title=Electromagnetism|author1=I.S. Grant |author2=W.R. Phillips |author3=Manchester Physics |publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9|edition=2nd }}
{{cite book|title=Introduction to Electrodynamics|edition=3rd |author=D.J. Griffiths|publisher=Pearson Education, Dorling Kindersley |year=2007|isbn=978-81-7758-293-2}}

List of equations in wave theory

Definitions

=General fundamental quantities=

=General derived quantities=

=Modulation indices=

=Acoustics=

Equations

=Standing waves=

=Propagating waves=

==Sound waves==

==Gravitational waves==

=Superposition, interference, and diffraction=

=Wave propagation=

=General wave functions=

==Wave equations==

==Sinusoidal solutions to the 3d wave equation==

See also

Footnotes

Sources

Further reading