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Chapter 7: Problem 25

The speed of deep ocean waves depends on the wave length and gravitational acceleration. What are the appropriate dimensionless parameters?

Short Answer

Expert verified
The appropriate dimensionless parameter for the problem is v 鈭�(位/g).

Step by step solution

01

Identifying the physical quantities and their dimensions

Identify the physical quantities involved and their respective dimensions in the SI system. These are: speed (v) with dimensions M^0 L^1 T^-1, wavelength (位) with dimensions M^0 L^1 T^0, and gravitational acceleration (g) with dimensions M^0 L^1 T^-2.
02

Deriving dimensionless parameters

The objective is to make a quantity that has M^0 L^0 T^0 by multiplying or dividing these quantities. There is only one way of obtaining a dimensionless quantity from these variables, which is by multiplying the speed by the square root of the ratio of the wavelength to the gravitational acceleration. It can be written as: 螤 = v 鈭�(位/g)
03

Verifying dimensions

Ensure the proposed parameter is indeed dimensionless by confirming that its dimensions are M^0 L^0 T^0. For the parameter found in Step 2, this can be verified via: [螤] = [v 鈭�(位/g)] = M^0 L^1 T^-1 * 鈭歔(M^0 L^1 T^0) / (M^0 L^1 T^-2)] = M^0 L^0 T^0. So, it鈥檚 indeed dimensionless. Therefore, the dimensionless parameter associated with the problem is 螤 = v 鈭�(位/g).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dimensional Analysis
Dimensional analysis is a powerful tool in fluid mechanics and other fields of physics and engineering. It helps identify relationships between different physical quantities by comparing their dimensions. In this exercise, we use dimensional analysis to find a dimensionless parameter that involves wave speed, wavelength, and gravitational acceleration.

The main idea is to combine these quantities in such a way that all dimensions are canceled out, resulting in a dimensionless number. This is usually done by forming products and ratios of the quantities based on their dimensions. It is an essential step towards simplifying complex physical phenomena into understandable relations.
  • Dimensions are usually expressed in terms of mass (M), length (L), and time (T).
  • Dimensionless parameters can reveal underlying scaling laws and relationships.
  • They are used to simplify and normalize equations governing physical systems.
Wave Speed
Wave speed is crucial in analyzing the behavior of waves, including ocean waves. It refers to the distance a wave travels per unit of time. In this context, it forms part of our dimensionless parameter along with wavelength and gravitational acceleration.

The speed of waves can vary significantly based on factors like wavelength and the medium through which the wave is traveling. For deep water waves, the wave speed is particularly influenced by the wavelength and gravitational acceleration.
  • Expressed dimensionally as M^0 L^1 T^-1.
  • Key to understanding how energy and momentum are transferred through waves.
  • In deep water, typically increases with longer wavelengths due to the conservation of energy principles.
Gravitational Acceleration
Gravitational acceleration is a constant at Earth's surface, influencing how objects accelerate when falling. It plays a significant role in various fluid mechanics phenomena, including wave motion.

In the context of wave mechanics, gravitational acceleration helps to determine the wave speed based on the balance of forces involved in wave motion. It's vital when forming the dimensionless parameter, as it impacts the basic wave characteristics.
  • Expressed dimensionally as M^0 L^1 T^-2.
  • Provides the force required for waves in water to propagate.
  • Forms part of the dimensionless parameter by relating to the wavelength and wave speed.
Wavelength
Wavelength is the distance between successive crests (or troughs) of a wave, and it is crucial in determining the wave's characteristics. Alongside gravitational acceleration, it influences the speed of ocean waves.

Longer wavelengths tend to travel faster in deep water, as they expend less energy vertically and more horizontally. This is especially relevant in the expression of the dimensionless parameter where wavelength plays a pivotal role.
  • Expressed dimensionally as M^0 L^1 T^0.
  • Directly impacts how fast the wave can travel in deep waters.
  • A key factor in designing and analyzing coastal structures and ship stability.

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Most popular questions from this chapter

The dimensionless parameters for a ball released and falling from rest in a fluid are $$C_{D}, \quad \frac{g t^{2}}{D}, \quad \frac{\rho}{\rho_{b}}, \quad \text { and } \quad \frac{V_{t}}{D}$$ where \(\left.C_{\mathbf{D}} \text { is a drag coefficient (assumed to be constant at } 0,4\right)\) \(g\) is the acceleration of gravity, \(D\) is the ball diameter, \(t\) is the time after it is released, \(\rho\) is the density of the fluid in which it is dropped, and \(\rho_{b}\) is the density of the ball. Ball 1 , an aluminum ball \(\left(\rho_{b_{1}}=2710 \mathrm{kg} / \mathrm{m}^{3}\right)\) having a diameter \(D_{1}=1.0 \mathrm{cm},\) is dropped in water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\right) .\) The ball velocity \(V_{\mathrm{t}}\) is \(0.733 \mathrm{m} / \mathrm{s}\) at \(t_{1}=\) 0.10 s. Find the corresponding velocity \(V_{2}\) and time \(t_{2}\) for ball 2 having \(D_{2}=2.0 \mathrm{cm}\) and \(\rho_{b_{2}}=\rho_{b_{1}} .\) Next, use the computed value of \(V_{2}\) and the equation of motion,$$\rho_{b} V_{g}-\frac{\mathrm{C}_{\mathrm{D}}}{2} \rho A V^{2}=\rho_{b} V \frac{d V}{d t}$$ where \(Y\) is the volume of the ball and \(A\) is its cross-sectional area. to verify the value of \(t_{2}\). Should the two values of \(t_{2}\) agree?

A very small needle valve is used to control the flow of air in a \(\frac{1}{8}-\) in. air line. The valve has a pressure drop of 4.0 psi at a flow rate of \(0.005 \mathrm{ft}^{3} / \mathrm{s}\) of \(60^{\circ} \mathrm{F}\) air. Tests are performed on a large, geometrically similar valve and the results are used to predict the performance of the smaller valve. How many times larger can the model valve be if \(60^{\circ} \mathrm{F}\) water is used in the test and the water flow rate is limited to 7.0 gal/min?

Develop the Froude number by starting with estimates of the fluid kinetic energy and fluid potential energy.

A mixing basin in a sewage filtration plant is stirred by a mechanical agitator with a power input \(\dot{W} \doteq F \cdot L / T\). Other parameters describing the performance of the mixing process are the fluid absolute viscosity \(\mu \doteq F \cdot T / L^{2},\) the basin volume \(V \doteq L^{3}\) and the velocity gradient \(G \doteq 1 / T\). Determine the form of the dimensionless relationship.

Develop the Weber number by starting with estimates for the inertia and surface tension forces.
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