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In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces.
The wave speed is given by: v 2 = gλ 2π + 2πγ λρ. where g is the gravitational field strength, γ is the surface tension, ρ is the density of the water, and λ the wavelength. As this equation makes clear (wave speed depends on wavelength), water is a dispersive medium.
Water Waves Exact (nonlinear) governing equations for surface gravity waves assuming potential theory y = h(x,z,t) or F(x,y,z,t) = 0 x y z B(x,y,z,t) = 0 Free surface definition: Unknown variables: Velocity fleld: *v (x;y;z;t) = r`(x;y;z;t) Position of free surface: ·(x;z;t) or F (x;y;z;t) Pressure fleld: p(x;y;z;t) Field equation:
Chapter 6 - Water Waves 6.1 Exact (Nonlinear) Governing Equations for Surface Gravity Waves, Assuming Potential Flow Free surface definition B(x, y = η ( x,z,t) or F(x, y,z,t) = 0 x y y z Unknown variables Velocity field: Position of free surface: Pressure field: Governing equations Continuity: Bernoulli for 1P-Flow: Far way, no disturbance ...
As an example, for water waves, v w is the speed of a surface wave; for sound, v w is the speed of sound; and for visible light, v w is the speed of light. The amplitude X is completely independent of the speed of propagation v w and depends only on the amount of energy in the wave.
11 lis 2024 · In deep water, the phase speed depends on wave length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.
All wavelengths travel at the same speed (i.e. non-dispersive), so one can only surf in shallow water. Deep water (kh ≫ 1) ⇒ c = J g/k, so longer waves travel faster, e.g. drop large stone into a pond. B. Capillary Waves: Bo 2 ≪ 1, c = σk tanh kh. Deep water kh ≫ 1 ⇒ c = √ σkρ so short waves travel fastest, e.g. raindrop in a puddle.
