DOPPLER EFFECT – ADVANCED READING
Waves
Wave function
For theoretical description we must consider a wide range of possible waves. From water waves, through sound waves to electromagnetic waves. DIfferent physical quantities are changing during the wave motion (e.g. temperature, pressure, velocity, density, electric field, magnetic field etc.). For well-arranged description we use one symbol for all quantities which is greek letter Psi: ψ(t, x). We call this function as a wave function and we can use it for everything which is in wave montion. If we put t = const, we get time image of wave motion, like a photo of stormed sea. If we put x = const, we get time behaviour of wave motion in the one point – oscillating (quantity depends only on the time). It's useful to use complex functions for mathematical description of the wave motion. Their real parts have meaning as measureable quantities. Complex functions are decomposable to two parts, amplitude and phase:
| ψ(t, x) = A(t, x) exp [iφ(t, x)] | (1) |
Amplitude A predicates about wave size and phase φ describes the position on the wave where the set point is. Surfaces with the same phase are called wavefronts, ande are described by equation
| φ(t, x) = konst | (2) |
Wavefronts are muving with phase velocity, which is not connected with information or mass transport, so its value can be superluminal.
Figure of longitudal wave motion – points are oscillating in the direction of wave motion. In this case the wave function can represent particle density. It's obvious that wavefronts are moving to the right direction so their phase velocity isn't zero. Otherwise the mass is not moving, every point is oscillating around fixed position.
Figure of transversal wave motion – points are oscillating in the direction perpendicular to the wave motion. Phase velocity is nonzero and the velocity of mass motion is zero as in the previous case. nulová. |
Angular frequency, frequency, period
Angular frequency is the time derivative of the phase:
| ω ≡ ∂φ/∂t. | (3) |
Angular frequency of the wave can change with time or/and with position. If the wave is regular (e.g. described by only one sine or cosine function) than angular frequency don't change. Than we can denote wave period as a time interval T after which the phase will change for 360°, or 2π and we obtain the simple relation for angular frequency
| ω ≡ 2π /T. | (4) |
Angular frequency is measured in units rad·s−1. Also we can measure a number of repeating of the physical process during one second. This quantity is called frequency, it is measured in units Hertz (Hz = s−1) and it's defined by relation
| f ≡ 1 /T. | (5) |
For example the frequency 50 Hz means that the physical process is repeating fifty times per second. There is simple relation between both frequencies.
| ω ≡ 2π f. | (6) |
Wave vector, wave length
Wave vector is described as space derivative of phase:
| kx ≡ ∂φ/∂x, ky ≡ ∂φ/∂y, kz ≡ ∂φ/∂z, neboli zkráceně k ≡ grad φ. | (7) |
Wave vector can change with time or/and with position. Vlnový vektor se může měnit jak s časem, tak místo od místa. If the wave is regular (e.g. described by only one sine or cosine function) than wave vector don't change. Than we can denote wave period as a space interval λ after which the phase will change for 360°, or 2π. If we choose one of the axis as a direction of wave motion than the wave vector has only one nonzero part in this direction and we obtain the simple relation.
| k = 2π/λ. | (8) |
