Wave mechanics

Basic Equation

The basic equation of wave mechanics is a function that describes the position of a particle as a function of time and displacement, where the function is wavelike, such as sine or cosine:

y(x,t) = Ψ = Asin(kx - ωt)

A = amplitude

k = 2π/λ where λ = wavelength

w = 2πf = 2π/T where T = period

So if given the amplitude, wavelength and frequency of a particle that was behaving in a sine fashion, its position could be calculated at a particular time on the t axis using this equation. It would also give the particle position as a function of time at a particular distance on the x axis. Both plots y vs. x and y vs. t would be sine plots.

Partial Derivatives of basic wave equation

∂y/∂x = Akcos(kx-ωt)

∂y/∂t = -Aωcos(kx-ωt)

Second Partial Derivatives of basic wave equation

∂²y/∂x² = Akcos(kx-ωt) = k²Asin(kx-ωt)

∂²y/∂t² = -Aωcos(kx-ωt) = ω²Asin(kx-ωt)