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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

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

At a large fish hatchery the fish are reared in open, water-filled tanks. Each tank is approximately square in shape with curved corners, and the walls are smooth. To create motion in the tanks, water is supplied through a pipe at the edge of the tank. The water is drained from the tank through an opening at the center. (See Video \(\vee 7.9 .)\) A model with a length scale of 1: 13 is to be used to determine the velocity, \(V\), at various locations within the tank. Assume that \(V=f\left(\ell, \ell_{i}, \rho, \mu, g, Q\right)\) where \(\ell\) is some characteristic length such as the tank width, \(\ell\), represents a series of other pertinent lengths, such as inlet pipe diameter, fluid depth, etc.. \(\rho\) is the fluid density, \(\mu\) is the fluid viscosity, \(g\) is the acceleration of gravity, and \(Q\) is the discharge through the tank. (a) Determine a suitable set of dimensionless parameters for this problem and the prediction equation for the velocity. If water is to be used for the model, can all of the similarity requirements be satisfied? Explain and support your answer with the necessary calculations. (b) If the flowrate into the full-sized tank is 250 gpm, determine the required value for the model discharge assuming Froude number similarity. What model depth will correspond to a depth of 32 in. in the full sized tank?

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A student drops two spherical balls of different diameters and different densities. She has a stroboscopic photograph showing the positions of each ball as a function of time. However, she wants to express the velocity of each as a function of time in dimensionless form. Develop the dimensionless group. The equation of motion for each ball is $$m g-\frac{C_{D}}{2} \rho A V^{2}=m \frac{d V}{d t}$$ where \(m\) is ball mass, \(g\) is acceleration of gravity, \(C_{D}\) is a dimensionless and constant drag coefficient, \(\rho\) is air mass density, \(A\) is ball cross-sectional area \(\left(=\pi \mathrm{D}^{2} / 4\right)\) with \(D\) ball diameter, \(V\) is ball velocity, and \(t\) is time.

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