Wave motion

• for the wave traveling in the postive \(x\)-direction: \(\psi(x, t) = f(x - vt)\)
• for the wave traveling in the negative \(x\)-direction (\(v > 0\)): \(\psi(x, t) = f(x + vt)\)

\[\psi(x, t)\Big |_{t = 0} = f(x) = A\sin kx\] where \(A\) is Amplitude 振幅, \(k\) is Propagation number 波矢

\[\psi(x, t) = f(x - vt) = A\sin k(x - vt)\]

• Spatial period is known as the wavelength \(\lambda\). \[\psi(x, t) = \psi(x\pm \lambda, t)\Rightarrow k = \frac{2\pi}{\lambda}\]
• Temporal period \(\tau\) \[\psi(x, t) = \psi(x, t\pm \tau) \Rightarrow kv\tau = 2\pi \Rightarrow \tau = \frac{\lambda}{v}\]
• Temporal frequency \(\nu\) \[\nu \equiv \frac{1}{\tau}\]
• Angular temporal frequency \(\omega\) \[\omega \equiv \frac{2\pi}{\tau} = 2\pi\nu\]
• Wave number or spatial frequency \(\kappa\) \[\kappa\equiv \frac{1}{\lambda}\]

The traveling harmonic wave in alternative expressions

\begin{aligned} \psi &= A\sin k(x\mp vt) \\ &= A\sin 2\pi \left( \frac{x}{\lambda}\mp \frac{t}{\tau} \right) \\ &= A\sin 2\pi (\kappa x \mp \nu t) \\ &= A\sin (kx \mp \omega t)\\ &= A\sin 2\pi\nu \left( \frac{x}{v} \mp t \right) \end{aligned}

Harmonic wave is monochromatic(单色的) wave.

A real wave is never purely monochromatic.

\[\psi(x, t) = A\sin k(x - vt)\]

phase and initial phase

define phase \(\varphi\) \[\varphi = kx - \omega t\] Generally, \[\psi(x, t) = A\sin k(x - vt + \varepsilon) \] hence \(\varphi = (kx -\omega t + \varepsilon)\), where \(\varepsilon\) is the initial phase.

The phase describes the status of a disturbance \(\psi(x, t)\)

• Rate-of-change of phase with time: \[\left| \left( \frac{\partial\varphi}{\partial t} \right)_x \right| = \omega\]
• Rate-of -change of phase with distance \[\left| \left( \frac{\partial\varphi}{\partial x} \right)_t \right| = k\]

Suppose \(\psi_1\), \(\psi_2\) are each separate solutions fo the wave equation; it follows that \((\psi_1 + \psi_2)\) is also a solution. \[\frac{\partial^2\psi_1}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2\psi_1}{\partial t^2},\; \frac{\partial^2\psi_2}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2\psi_2}{\partial t^2} \] \[\Rightarrow \frac{\partial^2 (\psi_1 + \psi_2)}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2(\psi_1 + \psi_2)}{\partial t^2}\]

3D wave equation

\[\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} = \frac{1}{v^2}\frac{\partial^2\psi}{\partial t^2}\] This is also a linear equation: suppose \(\psi_1(\vec{r}, t), \psi_2(\vec{r}, t), \ldots, \psi_n(\vec{r}, t)\) are each solutions, then \[\psi(\vec{r}, t) = \sum_{i=1}^n C_i\psi_i(\vec{r}, t)\] is also a solution.

The Euler formula: \[\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta\] \[\Rightarrow\psi(x, t) = \Re \left[ A\mathrm{e}^{\mathrm{i}(\omega t-kx+\varepsilon)} \right]\]