This article summarizes equations in the theory of waves.
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.
for longitudinal 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
|
m | [L] |
Phase angle | δ, ε, φ | rad | dimensionless |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Wavelength | λ | General definition (allows for
FM):
For non-FM waves this reduces to: |
m | [L] |
Wavenumber, k-vector, Wave vector | k, σ | Two definitions are in use:
|
m−1 | [L]−1 |
Frequency | f, ν | General definition (allows for
FM):
For non-FM waves this reduces to: In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation: |
Hz = s−1 | [T]−1 |
Angular frequency/ pulsatance | ω | Hz = s−1 | [T]−1 | |
Oscillatory velocity | v, vt, v | Longitudinal waves:
Transverse waves: |
m s−1 | [L][T]−1 |
Oscillatory acceleration | a, at | Longitudinal waves:
Transverse waves: |
m s−2 | [L][T]−2 |
Path length difference between two waves | L, ΔL, Δx, Δr | m | [L] | |
Phase velocity | vp | General definition:
In practice reduces to the useful form: |
m s−1 | [L][T]−1 |
(Longitudinal) group velocity | vg | m s−1 | [L][T]−1 | |
Time delay, time lag/lead | Δt | s | [T] | |
Phase difference | δ, Δε, Δϕ | rad | dimensionless | |
Phase | No standard symbol |
Physically; Phase angle can lag if: ϕ > 0 |
rad | dimensionless |
Relation between space, time, angle analogues used to describe the phase:
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
AM index: |
h, hAM |
A = carrier amplitude |
dimensionless | dimensionless |
FM index: |
hFM |
Δf = max. deviation of the instantaneous frequency from the carrier frequency |
dimensionless | dimensionless |
PM index: |
hPM |
Δϕ = peak phase deviation |
dimensionless | dimensionless |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Acoustic impedance | Z |
v = speed of sound, ρ = volume density of medium |
kg m−2 s−1 | [M] [L]−2 [T]−1 |
Specific acoustic impedance | z |
S = surface area |
kg s−1 | [M] [T]−1 |
Sound Level | β | dimensionless | dimensionless |
In what follows n, m are any integers (Z = set of integers); .
Physical situation | Nomenclature | Equations |
---|---|---|
Harmonic frequencies | fn = nth mode of vibration, nth harmonic, (n-1)th overtone |
Physical situation | Nomenclature | Equations |
---|---|---|
Average wave power | P0 = Sound power due to source | |
Sound intensity | Ω = Solid angle |
|
Acoustic beat frequency | f1, f2 = frequencies of two waves (nearly equal amplitudes) | |
Doppler effect for mechanical waves |
|
upper signs indicate relative approach, lower signs indicate relative recession. |
Mach cone angle (Supersonic shockwave, sonic boom) |
|
|
Acoustic pressure and displacement amplitudes |
|
|
Wave functions for sound | Acoustic beats
Sound displacement function Sound pressure-variation |
Gravitational radiation for two orbiting bodies in the low-speed limit. [1]
Physical situation | Nomenclature | Equations |
---|---|---|
Radiated power |
|
|
Orbital radius decay | ||
Orbital lifetime | r0 = initial distance between the orbiting bodies |
Physical situation | Nomenclature | Equations |
---|---|---|
Principle of superposition | N = number of waves | |
Resonance |
|
|
Phase and interference |
|
Constructive interference Destructive interference |
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.
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 |
|
|
Dispersion relation | Implicit form
Explicit form
| |
Amplitude modulation, AM | ||
Frequency modulation, FM |
Physical situation | Nomenclature | Wave equation | General solution/s |
---|---|---|---|
Non-dispersive Wave Equation in 3d | A = amplitude as function of position and time | ||
Exponentially damped waveform |
|
||
Korteweg–de Vries equation [2] | α = constant |
Complex amplitude of wave n
Resultant complex amplitude of all N waves
Modulus of amplitude
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 | |||
Beats | |||
Coherent interference |
This article summarizes equations in the theory of waves.
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.
for longitudinal 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
|
m | [L] |
Phase angle | δ, ε, φ | rad | dimensionless |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Wavelength | λ | General definition (allows for
FM):
For non-FM waves this reduces to: |
m | [L] |
Wavenumber, k-vector, Wave vector | k, σ | Two definitions are in use:
|
m−1 | [L]−1 |
Frequency | f, ν | General definition (allows for
FM):
For non-FM waves this reduces to: In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation: |
Hz = s−1 | [T]−1 |
Angular frequency/ pulsatance | ω | Hz = s−1 | [T]−1 | |
Oscillatory velocity | v, vt, v | Longitudinal waves:
Transverse waves: |
m s−1 | [L][T]−1 |
Oscillatory acceleration | a, at | Longitudinal waves:
Transverse waves: |
m s−2 | [L][T]−2 |
Path length difference between two waves | L, ΔL, Δx, Δr | m | [L] | |
Phase velocity | vp | General definition:
In practice reduces to the useful form: |
m s−1 | [L][T]−1 |
(Longitudinal) group velocity | vg | m s−1 | [L][T]−1 | |
Time delay, time lag/lead | Δt | s | [T] | |
Phase difference | δ, Δε, Δϕ | rad | dimensionless | |
Phase | No standard symbol |
Physically; Phase angle can lag if: ϕ > 0 |
rad | dimensionless |
Relation between space, time, angle analogues used to describe the phase:
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
AM index: |
h, hAM |
A = carrier amplitude |
dimensionless | dimensionless |
FM index: |
hFM |
Δf = max. deviation of the instantaneous frequency from the carrier frequency |
dimensionless | dimensionless |
PM index: |
hPM |
Δϕ = peak phase deviation |
dimensionless | dimensionless |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Acoustic impedance | Z |
v = speed of sound, ρ = volume density of medium |
kg m−2 s−1 | [M] [L]−2 [T]−1 |
Specific acoustic impedance | z |
S = surface area |
kg s−1 | [M] [T]−1 |
Sound Level | β | dimensionless | dimensionless |
In what follows n, m are any integers (Z = set of integers); .
Physical situation | Nomenclature | Equations |
---|---|---|
Harmonic frequencies | fn = nth mode of vibration, nth harmonic, (n-1)th overtone |
Physical situation | Nomenclature | Equations |
---|---|---|
Average wave power | P0 = Sound power due to source | |
Sound intensity | Ω = Solid angle |
|
Acoustic beat frequency | f1, f2 = frequencies of two waves (nearly equal amplitudes) | |
Doppler effect for mechanical waves |
|
upper signs indicate relative approach, lower signs indicate relative recession. |
Mach cone angle (Supersonic shockwave, sonic boom) |
|
|
Acoustic pressure and displacement amplitudes |
|
|
Wave functions for sound | Acoustic beats
Sound displacement function Sound pressure-variation |
Gravitational radiation for two orbiting bodies in the low-speed limit. [1]
Physical situation | Nomenclature | Equations |
---|---|---|
Radiated power |
|
|
Orbital radius decay | ||
Orbital lifetime | r0 = initial distance between the orbiting bodies |
Physical situation | Nomenclature | Equations |
---|---|---|
Principle of superposition | N = number of waves | |
Resonance |
|
|
Phase and interference |
|
Constructive interference Destructive interference |
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.
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 |
|
|
Dispersion relation | Implicit form
Explicit form
| |
Amplitude modulation, AM | ||
Frequency modulation, FM |
Physical situation | Nomenclature | Wave equation | General solution/s |
---|---|---|---|
Non-dispersive Wave Equation in 3d | A = amplitude as function of position and time | ||
Exponentially damped waveform |
|
||
Korteweg–de Vries equation [2] | α = constant |
Complex amplitude of wave n
Resultant complex amplitude of all N waves
Modulus of amplitude
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 | |||
Beats | |||
Coherent interference |