The Physics
Hypertextbook
Opus in profectus

# Interference & Superposition

## Practice

Write something.

### practice problem 2

Write something else.

### practice problem 3

Graph the following Fourier series with enough detail that you can determine basic shape of each wave.
1.  y = sin x + 1 sin 3x + 1 sin 5x + 1 sin 7x +… 3 5 7

 y = ∑ 1 sin(2n − 1)x 2n − 1
2.  y = sin x − 1 sin 2x + 1 sin 3x − 1 sin 4x +… 2 3 4

 y = ∑ (− 1)n +1 sin nx n
3.  y = cos x + 1 cos 3x + 1 cos 5x + 1 cos 7x +… 9 25 49

 y = ∑ 1 cos(2n − 1)x (2n − 1)2

#### solution

I summed the first ten terms for all three functions. You don't need to take the series out that far to get an idea of the basic shape, however. Each resulting "wave" is named for its shape.

1. Here we have a square wave (although this one is more rectangular than square).

2. This is a sawtooth wave, also known as a ramp wave.

3. And we end with a triangle wave. This one converges to its final shape very quickly.

### practice problem 4

Write something completely different.