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.