Question

A snapshot (frozen in time) of a water wave is described by the function \(z=1+\sin (x-y),\) where \(z\) gives the height of the wave relative to a reference point and \((x, y)\) are coordinates in a horizontal plane. a. Use a graphing utility to graph \(z=1+\sin (x-y)\) b. The crests and the troughs of the waves are aligned in the direction in which the height function has zero change. Find the direction in which the crests and troughs are aligned. c. If you were surfing on this wave and wanted the steepest descent from a crest to a trough, in which direction would you point your surfboard (given in terms of a unit vector in the \(x y\) -plane)? d. Check that your answers to parts (b) and (c) are consistent with the graph of part (a).

Short answer

a. Parallel to the line \(x = y - \frac{\pi}{2}\) b. Parallel to the line \(x = y\) c. Parallel to the line \(x = y + \frac{\pi}{2}\) d. Parallel to the line \(x = y + \pi\) Answer: c. Parallel to the line \(x = y + \frac{\pi}{2}\)

Step-by-step solution

Understand the problem

We are given the function of the wave \(z = 1 + \sin(x - y)\). This function describes the wave height relative to a reference point. Our first task is to find the direction in which the crests and troughs are aligned. To do this, we will find the partial derivatives of the wave function with respect to \(x\) and \(y\), and set them to zero.

Find partial derivatives

Find the partial derivatives of the function \(z\) with respect to \(x\) and \(y\), which are denoted as \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) respectively. \[\frac{\partial z}{\partial x} = \cos(x - y)\] \[\frac{\partial z}{\partial y} = -\cos(x - y)\]

Determine the direction of crests and troughs

As mentioned earlier, the crests and troughs are aligned in the direction in which the height function has zero change. Therefore, we have to find the direction vector for which \(\frac{\partial z}{\partial x}=0\) and \(\frac{\partial z}{\partial y}=0\). We can notice that: \[\cos(x - y) = 0\] This implies that: \[x - y = \frac{\pi}{2} \Rightarrow x = y + \frac{\pi}{2}\] Thus, the direction of the crests and troughs is parallel to the line \(x = y + \frac{\pi}{2}\).

Find the steepest descent direction

To find the steepest descent direction, we need to compute the gradient of the function. This is done by finding the gradient vector, which is given by the partial derivatives of the function. The gradient vector is: \[\nabla z = \left\langle \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y} \right\rangle = \left\langle \cos(x - y), -\cos(x - y) \right\rangle\] Now, we need to find the unit vector in the direction of the gradient vector: \[\hat{u} = \frac{\nabla z}{\|\nabla z\|} = \frac{\left\langle \cos(x - y), -\cos(x - y) \right\rangle}{\sqrt{\cos^2(x - y) + (-\cos(x - y))^2}} = \left\langle \frac{\cos(x - y)}{\cos(x - y)}, -\frac{\cos(x - y)}{\cos(x - y)} \right\rangle = \left\langle 1, -1 \right\rangle\] The unit vector \(\hat{u} = \left\langle 1, -1 \right\rangle\) represents the direction in which the surfboard should be pointed for the steepest descent from a crest to a trough.

Check consistency with the graph

In a graphing utility, when the function \(z = 1 + \sin(x - y)\) is plotted, we can see that the crests and troughs are indeed parallel to the line \(x = y + \frac{\pi}{2}\). Also, the unit vector \(\hat{u} = \left\langle 1, -1 \right\rangle\) represents the steepest descent direction from a crest to a trough, which is consistent with the graph.

Key concepts

Wave Height Function

The wave height function is a mathematical way to describe how high a wave is at any given point. In this case, it's given by the equation, \(z = 1 + \sin(x - y)\). Here, \(z\) represents the height of the wave at a point \((x, y)\), where \(x\) and \(y\) are coordinates in a flat, horizontal plane. Understanding how this function works can help us analyze the features of the wave, such as its peaks (crests) and lows (troughs).
The sine function, \(\sin(x - y)\), creates a periodic wave pattern. This means that the height repeats its values in a regular pattern, causing repeated peaks and valleys. The equation includes a constant term "1", which sets the baseline height of the wave, ensuring the minimum height is always at least 1. So, no part of the wave could be below a height of 1 unit.

Gradient Vector

The gradient vector is a key tool in multivariable calculus that helps identify the direction of the steepest ascent for a function. For our wave height function \(z = 1 + \sin(x - y)\), the gradient vector is found using the partial derivatives of \(z\) with respect to \(x\) and \(y\).
The partial derivatives are:

• \(\frac{\partial z}{\partial x} = \cos(x - y)\)
• \(\frac{\partial z}{\partial y} = -\cos(x - y)\)

These form the gradient vector: \(abla z = \langle \cos(x - y), -\cos(x - y) \rangle\).
This vector points in the direction where the height of the wave increases the fastest. In simpler terms, it shows you the steepest climb up the wave's surface.

Steepest Descent Direction

When surfing a wave, understanding the steepest descent direction can be crucial, as it allows you to navigate down the wave efficiently. Using the gradient vector, we find the steepest ascent, but to move downhill (or downwave in this context), we go in the opposite direction.
The gradient vector obtained for our wave function indicates the steepest ascent direction. To find the steepest descent, you would travel in the opposite direction. For this wave, the gradient vector is \(abla z = \langle \cos(x-y), -\cos(x-y) \rangle\). By flipping this direction, the steepest descent direction is \(\langle -\cos(x-y), \cos(x-y) \rangle\).
Normalizing this, which makes sure our direction vector has a length of one unit, results in the unit vector \(\langle 1, -1 \rangle\). This means, pointing your surfboard along \(\langle 1, -1 \rangle\) will let you ride the wave from its crest to its trough as steeply as possible.

Direction of Crests and Troughs

The alignment of the crests and troughs in a wave tells us where there is no change in height. For our wave function, this direction is crucial to understand the wave's overall form.
To find this alignment, we look for the direction where the partial derivatives are zero, meaning the height doesn't change. From \(\frac{\partial z}{\partial x} = \cos(x-y)\) and \(\frac{\partial z}{\partial y} = -\cos(x-y)\), we see these derivatives are zero when \(\cos(x-y) = 0\).
This happens when \(x-y\) equals \(\frac{\pi}{2}\), implying a line \(x = y + \frac{\pi}{2}\). Along this line, the height remains constant, meaning the crests and troughs are aligned. Imagine skimming along these lines, and you'd notice no vertical change, just horizontal movement across the wave.