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the higher the tension, the higher the frequency of the fundamental; the lighter the string, the higher the frequency of the fundamental; Moreover, if we take the nth harmonic as having a wavelength given by = /, then we easily get an expression for the frequency of the nth harmonic:
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).
The equation says the matter wave frequency in vacuum varies with wavenumber (= /) in the non-relativistic approximation. The variation has two parts: a constant part due to the de Broglie frequency of the rest mass ( ℏ ω 0 = m 0 c 2 {\displaystyle \hbar \omega _{0}=m_{0}c^{2}} ) and a quadratic part due to kinetic energy.
The wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests (on top), or troughs (on bottom), or corresponding zero crossings as shown. In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats.
By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x component for the x axis) of a vector wave without sources of waves ...
where ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and v p is the phase velocity of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a dispersion relation .
The relationship between velocity and frequency (or wavelength) is inherent in the characteristic equations. In the case of the plate, these equations are not simple and their solution requires numerical methods. This was an intractable problem until the advent of the digital computer forty years after Lamb's original work.
If the group velocity (see below) is wavelength-independent, this equation can be simplified as: [14] (,) = (+), showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation .