82 lines
No EOL
3.3 KiB
Markdown
82 lines
No EOL
3.3 KiB
Markdown
# Waves
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> *Definition*: a wave is a propagating disturbance transporting energy and momentum. A $1 + 1$ dimensional wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ travelling can be defined by a linear combination of a right and left travelling function $f,g: \mathbb{R} \to \mathbb{R}$ obtaining
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>
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> $$
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> \Psi(x,t) = f(x - vt) + g(x + vt),
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> $$
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> for all $(x,t) \in \mathbb{R}^2$ and $v \in \mathbb{R}$ the speed of the wave. Satisfies the $1 + 1$ dimensional wave equation
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> $$
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> \partial_x^2 \Psi(x,t) = \frac{1}{v^2} \partial_t^2 \Psi(x,t).
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> $$
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The derivation of the wave equation can be obtained in section...
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> *Theorem*: a right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ with $\lambda, T, A, \varphi \in \mathbb{R}$ the wavelength, period, amplitude and phase of the wave is given by
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> $$
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> \begin{align*}
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> \Psi(x,t) &= A \sin \big(k(x-vt) + \varphi\big), \\
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> &= A \sin(kx-\omega t + \varphi), \\
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> &= A \sin \Big(2\pi \Big(\frac{x}{\lambda} - \frac{t}{T} \Big) + \varphi \Big),
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> \end{align*}
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> $$
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>
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> for all $(x,t) \in \mathbb{R}^2$. With $k = \frac{2\pi}{\lambda}$ the wavenumber, $\omega = \frac{2\pi}{T}$ the angular frequency and $v = \frac{\lambda}{T}$ the wave speed.
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??? note "*Proof*:"
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Will be added later.
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A right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ can also be represented in the complex plane given by
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$$
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\Psi(x,t) = \text{Im} \big(A \exp i(kx - \omega t + \varphi )\big),
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$$
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for all $(x,t) \in \mathbb{R}^2$.
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> *Theorem*: let $\Psi: \mathbb{R}^4 \to \mathbb{R}$ be a $3 + 1$ dimensional wave then it satisfies the $3 + 1$ dimensional wave equation given by
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>
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> $$
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> \nabla^2 \Psi(\mathbf{x},t) = \frac{1}{v^2} \partial_t^2 \Psi(\mathbf{x},t),
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> $$
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>
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> for all $(\mathbf{x},t) \in \mathbb{R}^4$.
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??? note "*Proof*:"
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Will be added later.
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We may formulate various solutions $\Psi: \mathbb{R}^4 \to \mathbb{R}$ for this wave equation.
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The first solution may be the plane wave that follows cartesian symmetry and can therefore best be described in a cartesian coordinate system $\mathbf{v}(x,y,z)$. The solution is given by
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$$
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\Psi(\mathbf{v}, t) = \text{Im}\big(A \exp i(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t + \varphi) \big),
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$$
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for all $(\mathbf{v}, t) \in \mathbb{R}^4$ with $\mathbf{k} \in \mathbb{R}^3$ the wavevector.
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The second solution may be the cylindrical wave that follows cylindrical symmetry and can therefore best be described in a cylindrical coordinate system $\mathbf{v}(r,\theta,z)$. The solution is given by
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$$
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\Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\sqrt{\|\mathbf{v}\|}} \exp i(k \|\mathbf{v} \| - \omega t + \varphi) \Bigg),
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$$
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for all $(\mathbf{v}, t) \in \mathbb{R}^4$.
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The third solution may be the spherical wave that follows spherical symmetry and can therefore best be described in a spherical coordinate system $\mathbf{v}(r, \theta, \varphi)$. The solution is given by
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$$
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\Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\|\mathbf{v}\|} \exp i(k\|\mathbf{v}\| - \omega t + \varphi) \Bigg)
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$$
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for all $(\mathbf{v}, t) \in \mathbb{R}^4$.
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> *Principle*: the principle of superposition is valid for waves, since the solution space of the wave equation is linear.
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From this principle we obtain the property of constructive and destructive interference of waves. |