Difference between revisions of "Wave mechanics"
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so 1/k<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> = 1/ω<sup>2</sup> ∂<sup>2</sup>y/∂t<sup>2</sup> | so 1/k<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> = 1/ω<sup>2</sup> ∂<sup>2</sup>y/∂t<sup>2</sup> | ||
| − | + | or 1/k<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> - 1/ω<sup>2</sup> ∂<sup>2</sup>y/∂t<sup>2</sup> = 0 | |
==Wave Equation in One Dimension== | ==Wave Equation in One Dimension== | ||
Revision as of 13:10, 5 May 2020
Contents
Introduction
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 = -k2Asin(ωt-kx)
∂2y/∂t2 = -ω2Asin(ωt-kx)
so 1/k2 ∂2y/∂x2 = 1/ω2 ∂2y/∂t2
or 1/k2 ∂2y/∂x2 - 1/ω2 ∂2y/∂t2 = 0
Wave Equation in One Dimension
since the speed of a wave (v) is equal to ω/k, the above equation can be expressed in terms of wave velocity by multiplying by ω2:
v2 ∂2y/∂x2 - ∂2y/∂t2 = 0
or:
v2 ∂2y/∂x2 = ∂2y/∂t2
