There is no definition of a wave that is simple and general enough to be useful, but in a rough way we can think of a wave as
A moving signal, typically moving at a rate distinct from the motion of the medium
In a fluid whose signal could be, for example, an acoustic pressure pulse, the surface elevation of the ocean in a gravity wave, the rippling of the 500 mb surface in the troposphere due to a cyclone wave,
or the distortion of deep isopycnals in the thermocline due to internal gravity waves, the wave moves faster and further than the individual fluid elements.
Thus, usually, if
- \( u \) = characteristic velocity of the fluid element in the wave
- \( c \) = signal speed of the wave
then
\[
\frac{u}{c} \ll 1
\]
For simple systems and for small amplitude waves, we often find solutions to the equations of motion in the form of a plane wave.
This usually requires the medium to be, at least locally on the scale of the wave, homogeneous.
If \( \phi(\vec{x}, t) \) is a field variable such as pressure
\[
\phi(\vec{x}, t) = \phi(x_i, t) = \text{Re} \left( A e^{i(\vec{K} \cdot \vec{x} - \omega t)} \right)
\]
- \( A \) = the wave amplitude (complex, including a constant phase factor)
- \( \vec{K} \) = the wave vector
- \( \omega \) = the wave frequency
- Re = implies taking the real part of the expression