Question

Use Exercise 69 to do the following calculations. a. Find the velocity of a wave where \(\lambda=50 \mathrm{m}\) and \(d=20 \mathrm{m}\). b. Determine the depth of the water if a wave with \(\lambda=15 \mathrm{m}\) is traveling at \(v=4.5 \mathrm{m} / \mathrm{s}\).

Short answer

Answer: The velocity of the wave is approximately 32.558 m/s. Question 2: What is the depth of water for a wave with a wavelength of 15 meters and a velocity of 4.5 m/s? Answer: The depth of the water is approximately 3.492 meters.

Step-by-step solution

Part a: Find the velocity with given wavelength and depth

Given that \(\lambda = 50 \mathrm{m}\) and \(d = 20 \mathrm{m}\). Let's use the formula for the velocity of a wave in shallow water: \(v = \sqrt{gd} * \sqrt{\frac{\lambda}{\lambda + 2d}}\), where \(g = 9.81 \mathrm{m/s^2}\) is the acceleration due to gravity. First, let's find the second part of the equation: \(\sqrt{\frac{\lambda}{\lambda + 2d}} = \sqrt{\frac{50}{50 + 2*20}} = \sqrt{\frac{50}{90}} = \frac{\sqrt{50}}{3}\). Next, calculate the first part of the equation: \(\sqrt{gd} = \sqrt{9.81*20} = \sqrt{196.2} = 14\). Now, we can find the velocity by multiplying the two parts: \(v = 14 * \frac{\sqrt{50}}{3} \approx 32.558 \mathrm{m/s}\). So, the velocity of the wave is approximately \(32.558 \mathrm{m/s}\).

Part b: Find the depth with given wavelength and velocity

Given that \(\lambda = 15 \mathrm{m}\) and \(v = 4.5 \mathrm{m/s}\). Let's use the formula for the velocity of a wave in shallow water again and solve for the depth (d): \(v = \sqrt{gd} * \sqrt{\frac{\lambda}{\lambda + 2d}}\). First, isolate the depth (d) in the equation: \(v^2 = gd * \frac{\lambda}{\lambda + 2d}\). Next, we need to plug in the given values and solve for d: \((4.5)^2 = 9.81d * \frac{15}{15 + 2d}\). Now we have a quadratic equation in terms of d. To solve it, first multiply both sides by \(15 + 2d\): \((4.5)^2 * (15 + 2d) = 9.81d * 15\). Now, expand the equation and collect all d terms on one side: \(20.25 * (15 + 2d) = 147.15d\). Solving for d, we get: \(d \approx 3.492 \mathrm{m}\). So, the depth of the water is approximately \(3.492 \mathrm{m}\).

Key concepts

Shallow Water Wave Equation

The shallow water wave equation is a very useful equation in understanding wave behavior in shallow waters, like those near shorelines. It helps us calculate the wave velocity or the depth of water, given one and other parameters.
The formula is:
\[ v = \sqrt{gd} \times \sqrt{\frac{\lambda}{\lambda + 2d}} \]
In this equation:

• \(v\) is the velocity of the wave.
• \(g\) is the acceleration due to gravity, typically taken as \(9.81 \mathrm{m/s^2}\).
• \(d\) is the depth of water.
• \(\lambda\) is the wavelength of the wave.

This formula can help find the wave's velocity when you know the wavelength and depth. Similarly, you can rearrange it to solve for either the wavelength or the depth, depending on the known variables.

Wavelength

Wavelength is a critical factor in wave dynamics. It refers to the distance between successive crests (or troughs) of a wave. In simpler terms, it tells us how far apart these "peaks" or "valleys" are.
In the context of shallow water waves, knowing the wavelength is essential for calculating either the wave’s velocity or the depth of water if one of these variables is provided.
For example, in the first part of the original exercise, the wavelength \(\lambda\) was given as 50 meters. This was used to determine the wave's velocity along with the known depth.
Wavelength greatly impacts the energy and speed of the wave. Longer wavelengths often signify waves that travel faster and carry more energy. Understanding wavelength helps us predict the behavior of waves, especially as they approach the shore.

Velocity

Wave velocity is the speed at which a wave travels through a medium, like water. In shallow water, wave velocity can change based on both the wavelength and water depth.
In shallow water conditions, you can often find the velocity using the shallow water wave equation:
\[ v = \sqrt{gd} \times \sqrt{\frac{\lambda}{\lambda + 2d}} \]
Velocity gives insight into how quickly the wave will reach different points. In the exercise, the known values of wavelength and depth helped calculate a wave velocity of approximately 32.558 m/s.
Understanding velocity is not only crucial for predicting wave arrival times but also for assessing their potential impacts and energy. The faster the wave, the more energy it potentially carries, affecting marine structures and coastal lines.

Acceleration Due to Gravity

Acceleration due to gravity, symbolized as \(g\), is a constant that describes the rate at which objects accelerate towards the Earth’s surface. In wave calculations, \(g\) is a fundamental constant necessary for finding wave velocity.
It is often approximated as \(9.81 \mathrm{m/s^2}\) on Earth. This measurement is vital in the shallow water wave equation, contributing to the calculation of wave velocity and the interpretation of how gravity influences wave movements.

• Gravitational acceleration ensures waves are pulled back towards the water's surface and helps control the speed at which they propagate.
• When solving for depth or velocity, this constant remains unchanged, providing a reliable factor in wave dynamics across different scenarios.

Understanding gravity’s role clarifies why waves do not endlessly accelerate and how they stabilize at certain speeds.