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## Products of harmonic functions and projections: Instructor's Guide

### Main Ideas

• Harmonic functions.
• Projections.
• Symmetry.

Estimated Time:

1. The students were presented with variation of the harmonic basis functions and asked to plot of the functions. Then, student were asked to establish graphically whether the integral of the given functions over the time interval $T = 2 \pi / \omega$ are zero or non-zero.
2. To calculate the same integrals analytically then compared the results to the previous task.

### Prerequisite Knowledge

The integral over one complete period (or more) of a sinusoidal function is zero, e.g. $\int\limits_{0}^{T}{\cos \left( n\omega _{0}t \right)dt}=0$

The average value over one complete period of the square of the sine or cosine function is 1/2: $\frac{1}{2\pi }\int\limits_{0}^{T}{\cos ^{2}\left( \omega _{0}t \right)dt}=\frac{1}{2}$

### Activity: Introduction

This is a discovery exercise, so no introduction is given, other than reminding the students about how to conduct work efficiently in groups. During the activity, if some groups finish the graphical exercise, they are encouraged to evaluate the integrals analytically.

### Activity: Student Conversations

Group members are often not sure what it means to multiply two functions together. Often, they try to find an analytical form of the product function, and plot that. Some training is needed to find the useful short cut of identifying the zeros of the product function, and determining the sign of the product function between the zeroes. Students usually discover the symmetry themselves after that: e.g. “If this section of the function looks like this, then this other section must be the same but of opposite sign.”

When asked to decide whether the integral of the product function is zero or non zero, students who have discovered the symmetry are easily able to answer. Some students don't know that the “area under the curve” has a different sign above and below the axis, but there is almost always someone in the group who does.

In most low-integer combinations it is easy to spot the symmetry, but there are some product functions where the symmetry is not so obvious, e.g. $\int\limits_{0}^{T}{\sin \left( \omega _{0}t \right)\sin \left( 3\omega _{0}t \right)dt}$ . Students often try new functions on their own and discover this for themselves.

To foreshadow the idea of projections, it is helpful for the instructor to plant the idea that each function can be thought of as a set of points (a very large vector), and that the point-by-point multiplication is similar to the multiplication of vector components when one computes a dot product. They students might already associate a dot product with a projection, and the remaining connection is the discrete sum in a dot product of a vector translating to an integral in the case of this continuous set of points.

### Activity: Wrap-up

Groups present their results to the class. Choose examples of products of two sines of different frequencies, a sine and a cosine, and the a cosine-squared function to get most of the cases. The students should tabulate the results and be persuaded that $\frac{2}{T}\int\limits_{0}^{T}{\sin \left( n\omega _{0}t \right)\sin \left( m\omega _{0}t \right)dt}=\delta _{mn}$ and similarly for cosines. Also point out that any sine/cosine combinations integrate to zero.

The $\frac{2}{T}$ factor is not obvious to most students, and they often forget, later in the course, where it came from.

The projection language can be introduced now, and students tend to use the language “how much of this function A is contained in this function B”. Mostly the know that “sine and cosine are independent”, and this exercise formalizes that notion a little. Now they learn that $\cos \left( \omega _{0}t \right)$ and $\cos \left( 2\omega _{0}t \right)$ are independent in the same sense.

### Extensions

It is instructive, but often too much information at this stage, to consider what happens if one of the functions has a frequency that is half of what is being called the fundamental frequency. The orthogonality does not apply in general, e.g. $\int\limits_{0}^{T}{\sin \left( \frac{1}{2}\omega _{0}t \right)\cos \left( \omega _{0}t \right)dt}\ne 0$ but it is easy to find particular instances where the product is zero. This can be a homework problem or an item for later discussion.

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