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. But the wave profile (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 |
General derived quantities
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:
Modulation indices
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 |
Acoustics
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 |
Equations
In what follows n, m are any integers (Z = set of integers); .
Standing waves
Physical situation | Nomenclature | Equations |
---|---|---|
Harmonic frequencies | fn = nth mode of vibration, nth harmonic, (n-1)th overtone |
Propagating waves
Sound waves
Physical situation | Nomenclature | Equations |
---|---|---|
Average wave power | P0 = Sound power due to source | |
Sound intensity | Ω = Solid angle |
|
Acoustic beat frequency |
|
|
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 waves
Gravitational radiation for two orbiting bodies in the low-speed limit.[1]
Physical situation | Nomenclature | Equations |
---|---|---|
Radiated power |
|
|
Orbital radius decay | ||
Orbital lifetime |
|
Superposition, interference, and diffraction
Physical situation | Nomenclature | Equations |
---|---|---|
Principle of superposition |
|
|
Resonance |
|
|
Phase and interference |
|
Constructive interference Destructive interference |
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 |
|
|
Dispersion relation | Implicit form
Explicit form
| |
Amplitude modulation, AM | ||
Frequency modulation, FM |
General wave functions
Wave equations
Physical situation | Nomenclature | Wave equation | General solution/s |
---|---|---|---|
Non-dispersive Wave Equation in 3d |
|
||
Exponentially damped waveform |
|
||
Korteweg–de Vries equation[2] |
|
Sinusoidal solutions to the 3d wave equation
- N different sinusoidal waves
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 |
See also
- Defining equation (physical chemistry)
- List of equations in classical mechanics
- List of equations in fluid mechanics
- List of equations in gravitation
- List of equations in nuclear and particle physics
- List of equations in quantum mechanics
- List of photonics equations
- List of relativistic equations
- SI electromagnetism units
Footnotes
- ^ "Gravitational Radiation" (PDF). Retrieved 2012-09-15.
- ^ 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
Sources
- P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
- G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
- A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4.
- R.G. Lerner, G.L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4.
- C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3.
- P.A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W.H. Freeman and Co. ISBN 9-781429-202657.
- L.N. Hand, J.D. Finch (2008). Analytical Mechanics. Cambridge University Press,. ISBN 978-0-521-57572-0.
{{cite book}}
: CS1 maint: extra punctuation (link) - T.B. Arkill, C.J. Millar (1974). Mechanics, Vibrations and Waves. John Murray,. ISBN 0-7195-2882-8.
{{cite book}}
: CS1 maint: extra punctuation (link) - H.J. Pain (1983). The Physics of Vibrations and Waves (3rd ed.). John Wiley & Sons,. ISBN 0-471-90182-2.
{{cite book}}
: CS1 maint: extra punctuation (link) - J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley,. ISBN 978-0-470-01460-8.
{{cite book}}
: CS1 maint: extra punctuation (link) - G.A.G. Bennet (1974). Electricity and Modern Physics (2nd ed.). Edward Arnold (UK). ISBN 0-7131-2459-8.
- I.S. Grant, W.R. Phillips, Manchester Physics (2008). Electromagnetism (2nd Edition). John Wiley & Sons. ISBN 978-0-471-92712-9.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley,. ISBN 81-7758-293-3.
{{cite book}}
: CS1 maint: extra punctuation (link)
Further reading
- L.H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0.
- J.B. Marion, W.F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2.
- A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
- H.D. Young, R.A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 0-321-50130-6.