Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary 'return force' for water piled up in the wave crests is due to the gravitational attraction of the Earth. Thus the speed v (m/s) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that υ might also depend on water depth h and the wave's wavelength λ. Assume the wave speed is given by the functional form v = Cgᵅ hᵝ λᵞ, where α , β , c and C are numbers without dimension.
(a) In deep water, the water deep below the surface does not affect the motion of waves at the surface. Thus υ should be independent of depth h (i.e., β = 0). Using only dimensional analysis (Section 1–7 and Appendix D), determine the formula for the speed of surface ocean waves in deep water.

Verified step by step guidance

1. Identify the dimensions of each variable in the equation. The speed of the wave, v, has dimensions of LT^-1 (length per time). The acceleration due to gravity, g, has dimensions of LT^-2. The wavelength, λ, has dimensions of L (length).
2. Since in deep water the speed of the wave is independent of the water depth h, set β = 0 in the equation v = Cgᵅ hᵝ λᵞ.
3. Apply dimensional analysis to find the values of α and γ. The dimensions on both sides of the equation must match. The left side (v) has dimensions of LT^-1. The right side, with β set to 0, simplifies to Cgᵅλᵞ.
4. Set up the dimensional equation based on the simplified equation: [LT^-1] = [LT^-2]^α[L]^γ. Solve for α and γ by equating the powers of L and T on both sides of the equation.
5. Substitute the values of α and γ back into the equation to get the final formula for the speed of surface ocean waves in deep water.

Key Concepts

Dimensional Analysis

Dimensional analysis is a mathematical technique used to convert one set of units to another and to derive relationships between physical quantities based on their dimensions. It involves checking the consistency of equations by ensuring that both sides have the same dimensions, which helps in identifying the fundamental relationships between variables. This method is particularly useful in physics for simplifying complex problems and deriving formulas without needing detailed knowledge of the underlying phenomena.

Wave Speed in Deep Water

In deep water, the speed of surface waves is primarily influenced by gravity and is independent of water depth. This is because the restoring force acting on the wave is due to gravity, which acts uniformly regardless of how deep the water is. As a result, the wave speed can be expressed as a function of gravitational acceleration alone, leading to the conclusion that the depth of the water does not play a significant role in determining wave speed in deep water conditions.

Functional Form of Wave Speed

The functional form of wave speed, expressed as v = Cgᵅ hᵝ λᵞ, indicates that wave speed is a product of various factors, including gravitational acceleration (g), water depth (h), and wavelength (λ). In this equation, the exponents (α, β, γ) represent how each factor influences the wave speed. By applying dimensional analysis, one can determine the values of these exponents, particularly noting that in deep water, the depth does not affect wave speed, leading to β = 0.