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.
for longitudinal waves,
For non-FM waves this reduces to:
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:
In practice reduces to the useful form:
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.
Relation between space, time, angle analogues used to describe the phase:
A = carrier amplitude Am = peak amplitude of a component in the modulating signal Δf = max.
deviation of the instantaneous frequency from the carrier frequency fm = peak frequency of a component in the modulating signal Δϕ = peak phase deviation v = speed of sound, ρ = volume density of medium S = surface area In what follows n, m are any integers (Z = set of integers);
upper signs indicate relative approach, lower signs indicate relative recession.
Sound displacement function
Gravitational radiation for two orbiting bodies in the low-speed limit.
[1] Constructive 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.
Complex amplitude of wave n
Resultant complex amplitude of all N waves
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.