• 1 Wave propagation
  • 1.1 Phase velocity
  • 1.2 Group velocity
  • 1.3 Incoming and outgoing wave
• 2 Oscillation
  • 2.1 Simple harmonic oscillator
  • 2.2 Coupled oscillator
  • 2.3 Continuous systems - waves

1 Wave propagation

1.1 Phase velocity

The phase velocity \(v_{p}\) is the speed at which a point of constant phase of the wave will travel for a discrete frequency and is given as

    \begin{align}
    v_{p} &= \frac{x}{t} \\
    &= \frac{\lambda}{T} \\
    &= \frac{\omega}{k} \\
    \end{align}

\(\lambda\) is the wave length and T is the period. \(\omega\) is the angular frequency, \(k\) is the wavenumber. Formally, the phase is defined as follows

    \begin{equation}
    \Phi = k x - \omega t
    \end{equation}

Then the phase velocity is given as follows

    \begin{align}
    v_{p} &= \frac{dx}{dt} \\
    &= \frac{d \Phi}{dt} / \frac{d \Phi}{dx} \\
    &= \omega / k
    \end{align}

And the angular frequency is related to the wavenumber through the dispersion relationship.

    \begin{equation}
    \omega = \Omega(k)
    \end{equation}

• A special case: \(\Omega(k) = c k\), with \(c\) a constant. This kind of wave is called non-dispersive, since all frequencies travel at the same phase speed \(c\).
• In other cases, we have dispersive waves.
• The dispersion relationship depends on the medium and the type of waves, like sound waves or water waves.

1.2 Group velocity

The group velocity is the speed at which a wave packet from a narrow range of frequencies will travel and is determined from the gradient of the dispersion relation

    \begin{equation}
    v_{g} = \frac{\partial \omega}{\partial k}
    \end{equation}

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave.

1.3 Incoming and outgoing wave

In quantum theory, there is a concept of incoming and outgoing wave. They distinguish with each by \(e^{ikr}\) and \(e^{-ikr}\). It seems that if \(r\) increases then \(e^{ikr}\) increases while \(e^{-ikr}\) decreases with respect to positive \(k\). However, in flow instability theory, things seem to be opposite.

2 Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value, often a point of equilibrium, or between two or more different states.

2.1 Simple harmonic oscillator

It’s the system where the restoring force on a body is directly proportional to its displacement. And in this system, the regular motion is referred as simple harmonic motion.

2.2 Coupled oscillator

The simple harmonic oscillator has a single degree of freedom. Coupled oscillator has more degrees of freedom and deeper interpretation is given by resolving the motion into normal modes.

2.3 Continuous systems - waves

Waves can be regarded as systems with an infinite number of degrees of freedom and normal modes.