Question

Traveling waves in general Generalize Exercise 79 by considering a set of waves described by the function \(z=A+\sin (a x-b y),\) where \(a, b,\) and \(A\) are real numbers. a. Find the direction in which the crests and troughs of the waves are aligned. Express your answer as a unit vector in terms of \(a\) and \(b\). b. Find the surfer's direction- that is, the direction of steepest descent from a crest to a trough. Express your answer as a unit vector in terms of \(a\) and \(b\).

Short answer

(a) The unit vector for the direction of wave crests and troughs is \(\frac{1}{\sqrt{a^2 + b^2}}\cdot\langle -b, a \rangle\). (b) The unit vector for the surfer's direction is \(\frac{1}{\sqrt{a^2 + b^2}}\cdot\langle a, -b \rangle\).

Step-by-step solution

Find the gradient vector of the wave function

First, we need to find the gradient vector of the given wave function. The gradient vector is the vector of partial derivatives with respect to \(x\) and \(y\): $$\nabla z = \langle \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y} \rangle$$ Compute the partial derivatives: $$\frac{\partial z}{\partial x} = a \cos(ax - by)$$ and $$\frac{\partial z}{\partial y} = -b \cos(ax - by)$$ So, the gradient vector is: $$\nabla z = \langle a \cos(ax - by), -b \cos(ax - by) \rangle$$

Find the direction of wave crests and troughs

The direction in which the crests and troughs of the waves are aligned is perpendicular to the gradient vector. To find this direction, we can take the gradient vector and swap its components while negating one of them: $$\text{Crests and troughs direction} = \langle -b \cos(ax - by), a \cos(ax - by) \rangle$$ To express this direction as a unit vector, we need to normalize it. First, find the magnitude of the vector: $$\text{Magnitude} = \sqrt{(-b \cos(ax - by))^2 + (a \cos(ax - by))^2} = |a|\cdot|\cos(ax - by)| \cdot \sqrt{a^2 + b^2}$$ Finally, to make it a unit vector, divide each component of the direction vector by the magnitude: $$\text{Unit vector} = \frac{1}{|a|\cdot|\cos(ax - by)| \cdot \sqrt{a^2 + b^2}} \cdot \langle -b \cos(ax - by), a \cos(ax - by) \rangle = \frac{1}{\sqrt{a^2 + b^2}}\cdot\langle -b, a \rangle$$

Find the surfer's direction

The surfer's direction is the direction in which the waves descend most steeply from a crest to a trough. This direction is the same as the gradient vector's direction. We have already found the gradient vector in Step 1: $$\nabla z = \langle a \cos(ax - by), -b \cos(ax - by) \rangle$$ To express the surfer's direction as a unit vector, we need to normalize the gradient vector. First, find the magnitude of the gradient vector: $$\text{Magnitude} = \sqrt{(a \cos(ax - by))^2 + (-b \cos(ax - by))^2} = |a|\cdot|\cos(ax - by)| \cdot \sqrt{a^2 + b^2}$$ Finally, to make it a unit vector, divide each component of the gradient vector by the magnitude: $$\text{Unit vector} = \frac{1}{|a|\cdot|\cos(ax - by)| \cdot \sqrt{a^2 + b^2}} \cdot \langle a \cos(ax - by), -b \cos(ax - by) \rangle = \frac{1}{\sqrt{a^2 + b^2}}\cdot\langle a, -b \rangle$$ In conclusion, the unit vector for the direction of the wave crests and troughs is \(\frac{1}{\sqrt{a^2 + b^2}}\cdot\langle -b, a \rangle\) and the unit vector for the surfer's direction is \(\frac{1}{\sqrt{a^2 + b^2}}\cdot\langle a, -b \rangle\).

Key concepts

Direction of Wave Crests and Troughs

Understanding the direction of wave crests and troughs is vital for interpreting wave patterns, whether in oceanography or when analyzing mathematical wave functions. Imagine waves in the ocean; their crests and troughs form lines that reveal the wave direction. Similarly, in the exercise, we consider a wave function z defined by an equation involving sine. The crests and troughs of this mathematical wave are lines where the wave achieves its maximum and minimum values, respectively.

When we discuss the alignment of these crests and troughs, we are seeking a vector that runs along these lines. A crest or trough line is perpendicular to the direction in which the wave's height changes most rapidly – this rapid change is quantified by the gradient vector. Therefore, to find a unit vector aligned with the wave's crests and troughs, we find a vector perpendicular to the gradient. This is done by swapping the components of the gradient and negating one, which results in a vector that points in the desired direction. Normalizing this vector gives us the direction as a unit vector, making it independent of the wave's amplitude.

Steepest Descent Direction

The steepest descent direction is essential in fields like gradient descent algorithms in machine learning and understanding physical phenomena like how a surfer would descend from a wave's crest to a trough. The direction of steepest descent at a point on the wave is the same as the direction of the gradient vector at that point. This is because the gradient vector points in the direction of the greatest rate of increase of the function, and thus, by going in the opposite direction, one would descend most steeply.

In our wave example, a surfer would ride in the direction opposite to the gradient to find the quickest path down a wave. Normalizing this vector provides a unit vector that indicates the surfer's direction, conceptually representing the surfer's path in pure directional terms without regard to the wave's size or intensity. This provides a clear and concise direction for descent, invaluable for navigating complex wave patterns or multidimensional data in other applications.

Partial Derivatives

Partial derivatives are fundamental to multivariable calculus, and they represent the rate of change of a function with respect to one variable while holding other variables constant. In the context of our wave example, the wave function z depends on two variables, x and y. The partial derivative with respect to x tells us how the wave height changes as we move horizontally, and the partial derivative with respect to y shows the change as we move vertically.

The computation of these partial derivatives lays the foundation for finding the gradient, which aggregates both partial derivatives into a vector. This gradient then informs us about both the steepest ascent direction and its perpendicular direction, which corresponds to the wave crests and troughs alignment. Understanding partial derivatives is crucial for students to grasp how changes in different directions affect a function's output, a concept that applies to numerous scientific and engineering fields.

Normalizing Vectors

Normalizing vectors is an important process in vector calculus that involves converting a vector into a unit vector—a vector with a magnitude of 1 in the same direction. This operation simplifies many problems by focusing only on the direction, not the length, of the vector. In our wave problem, normalizing helps express both the wave's direction of crests and troughs and the surfer's descent direction in standardized forms.

To normalize a vector, we divide each component by the vector's magnitude. For the gradient vector of our wave function, this process involved dividing by the square root of the sum of the squares of its components, resulting in a unit vector that indicates the same direction with a fixed length of 1. By normalizing vectors, students can compare directions more easily and apply vector operations without the complexity introduced by varying magnitudes, making the analysis more intuitive.