# Waves

Corresponding Wikipedia article: Wave

## Definition

The waves are the oscillations, which propagate along a compressible or deformable continuum with a finite velocity. The compression (elastic deformation) of the continuum should cause in it a pressure gradient or a stress.

Two types of waves
Type Oscillation direction Conditions
Longitudinal (I) Along the propagation. Compressible continuum.
Transverse Perpendicular to the propagation. Shear stresses (for example, SAW) or a deformation in the force field (for example, the waves on a liquid surface).

## Longitudinal waves

Next, only the longitudinal waves in a homogeneous continuum without any force fields are considered, as the most general form of the waves, and also the aetheric waves. The density fluctuations $$\delta\rho$$ are considered to be negligible in comparison with the average density $$\rho_0$$. The average pressure $$P_0$$ is a constant. $\rho=\rho_0+\delta\rho\;\;\;\;\;\delta\rho\ll\rho_0=const$ $P=P_0+\delta P\;\;\;\;\;\;\;\;\;\;\;\;P_0=const$ The Euler’s and continuity equations ("Continuum", 1 и 2), disregarding $$\delta\rho$$ in sum with $$\rho_0$$, are: $\frac{\partial\rho}{\partial t}+\rho_0div\mathbf{\overrightarrow{v}}=0\tag{1}$ $\rho_0\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}+grad\;P=0\tag{2}$ The differentiation of (1) with respect to time, and substitution $$\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}$$ from (2) gives the wave equation: $\frac{\partial^2\rho}{\partial t^2}+\rho_0div\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}=0\tag{3}$ $\boxed{\partial^2\rho/\partial t^2-\Delta P=0}\tag{4}$ $$\Delta=div\;grad$$ (Laplace operator, Laplacian). The general solution of the wave equation is: $\delta\rho=\sum\rho_A\psi(\mathbf{\overrightarrow{x}},t)$ $\delta P=\sum P_A\psi(\mathbf{\overrightarrow{x}},t)$ The integration constants are included in $$\rho_0$$ and $$P_0$$. The amplitudes $$\rho_A$$ and $$P_A$$ are determined by the boundary conditions.

The waves propagate on the Huygens-Fresnel principle, that is, they are sums of the secondary waves, which propagate from the primary wave front. The function of a wave, which propagates from the coordinate $$\mathbf{\overrightarrow{x_0}}$$, is: $\psi(\mathbf{\overrightarrow{x}},t)=\sin(\omega t-k|\mathbf{\overrightarrow{x}}-\mathbf{\overrightarrow{x_0}}|+\varphi_0)$

Kinematic parameters of a wave
Formula Name Reference frame dependence
$\omega$ Angular (circular, radial etc.) frequency at a point. Dependent.
$\nu=\frac{\omega}{2\pi}$ Frequency at a point.
$k$ Wavenumber (spatial frequency). Independent.
$\lambda=\frac{2\pi}{k}$ Wavelength.
$c=\frac{\omega}{k}=\lambda\nu$ Phase velocity is the wave front velocity (of the points in the same phase). Dependent.

The substitution of the general solution to the wave equation gives: $\rho_A\omega^2=P_Ak^2$ The phase velocity of a wave is: $c=\frac{\omega}{k}=\sqrt{\frac{P_A}{\rho_A}}\tag{5}$ According to the Huygens-Fresnel principle, the interface of mediums (continuums) with different phase velocities is the reflective and refractive surface.

## Transverse waves

The transverse waves differ from the longitudinal waves by its direction and by various origins of the forces, which cause a pressure (stress). The transverse waves also can have a polarization:

• Linear, oscillations in a single plane.
• Circular (elliptical), oscillations in two orthogonal planes with a phase shift at the right angle. The oscillating field vector rotates at each point (in time) and along the wave (in space).

## Сoncentrated oscillations

The oscillation is a degenerate wave, which does not propagate in space. These oscillations occur in the heterogeneous mediums with the lumped (non-distributed) parameters, for example in the pendulum and in the electric oscillating circuit. The elastic deformation is not required for these oscillations. The general oscillation equation of parameter $$x$$ is: $\frac{\mathrm{d}^2x}{\mathrm{d}t^2}+\omega^2x=f(x,t)$ The circular (elliptical) motion is the mechanical oscillations in two orthogonal planes with a phase shift at the right angle. The angular frequency of both oscillations is equal to the angular velocity of motion.

## Standing waves

The standing (stationary) wave is two opposite waves of the same wavelength on the same axis. Here’s an example of the in-phase waves: $\psi(x,t)+\psi(-x,t)=\sin(\omega t-kx)+\sin(\omega t+kx)=2\cos kx\sin\omega t$ The standing waves are the oscillatory processes, and they should not be confused with the stationary waves of another kind, which arise in a helical vortical flow and called the solitons.

The number of standing waves in a continuum is considered as a number of the half-waves or of the nodes/anti-nodes. The number of one-dimensional standing waves on a line segment of length $$L$$ is: $n=L\frac{2}{\lambda}=L\frac{\omega}{\pi c}=L\frac{2\nu}{c}$ As the waves propagate by the circles (spheres), then in case of the two-dimensional standing waves on a square area $$S$$, the number of waves should be corrected by the area ratio of disk to square (the disk is inscribed into the square) $$\pi/4$$: $n=\frac{\pi}{4}S\frac{4}{\lambda^2}=S\frac{\pi}{\lambda^2}=S\frac{\omega^2}{4\pi c^2}=S\frac{\pi\nu^2}{c^2}\tag{6}$ Similarly, for the three-dimensional (in the cubic volume $$V$$) standing waves, the number should be corrected by the volume ratio of ball to cube (the ball is inscribed into the cube) $$\pi/6$$: $n=\frac{\pi}{6}V\frac{8}{\lambda^3}=V\frac{4\pi}{3\lambda^3}=V\frac{\omega^3}{6\pi^2c^3}=V\frac{4\pi\nu^3}{3c^3}\tag{7}$ These corrections for (6) and (7) would not be required when using the hyperspherical units of area and volume.

## Resonance

The resonance is an occurrence of the standing waves with a frequency called the natural frequency. The resonance of a non-wave oscillator is a free oscillation with a natural frequency. The resonance is not an energy source, but only a maximal energy manifestation.