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. 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. Step 1: Begin by analyzing the given functional form of the wave speed: v = Cgᵅ hᵝ λᵞ. Here, v is the wave speed (m/s), g is the acceleration due to gravity (m/s²), h is the water depth (m), and λ is the wavelength (m). The constants α, β, and γ are dimensionless exponents, and C is a dimensionless constant.
2. Step 2: Since the problem specifies deep water, the wave speed v is independent of the water depth h. This means β = 0. The functional form simplifies to v = Cgᵅ λᵞ.
3. Step 3: Perform dimensional analysis. The dimensions of v are [L][T]⁻¹ (length per time), the dimensions of g are [L][T]⁻² (acceleration), and the dimensions of λ are [L] (length). Substitute these into the simplified equation: [L][T]⁻¹ = [L][T]⁻²ᵅ × [L]ᵞ.
4. Step 4: Equate the dimensions of length [L] and time [T] on both sides of the equation. For length [L]: 1 = α + γ. For time [T]: -1 = -2α. Solve these equations simultaneously to find the values of α and γ.
5. Step 5: Solve the equations. From the time dimension equation, α = 1/2. Substitute α = 1/2 into the length dimension equation to find γ = 1/2. Thus, the formula for the wave speed in deep water is v = C√(gλ), where C is a dimensionless constant.

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 wave speed is proportional to the square root of the gravitational acceleration, typically represented as v = √(g/λ).

Non-dimensional Parameters

Non-dimensional parameters are quantities that have no units and are used to simplify the analysis of physical systems. In the context of wave motion, parameters like α, β, and γ in the equation v = Cgᵅ hᵝ λᵞ help to express the relationships between different physical quantities without the complications of units. By setting β = 0 for deep water, we eliminate the dependence on depth, allowing for a clearer understanding of how wave speed relates to gravity and wavelength.