Question

Project: Graph the function that is the sum of the following four functions: $$\begin{aligned} &y=\sin x\\\ &y=(1 / 3) \sin 3 x\\\ &y=(1 / 5) \sin 5 x\\\ &y=(1 / 7) \sin 7 x \end{aligned}$$ Use a graphics calculator or computer or manual addition of ordinates on graph paper. What can you say about the shape of the composite curve? What do you think would happen if you added in more sine waves \([y=(1 / 9) \sin 9 x \text { ) and so on }] ?\)

Short answer

The composite curve of the sum of the given sine wave functions will have a complex shape with unique peaks and troughs that become more intricate with the addition of more sine waves with higher frequencies and smaller amplitudes.

Step-by-step solution

Understand the Components

The exercise involves adding the ordinates (y-values) of four different sine wave functions at each x-value. Each function has a different amplitude and frequency: 1. \(y = \text{sin}(x) \) with amplitude 1 and fundamental frequency, 2. \(y = \frac{1}{3}\text{sin}(3x) \) with amplitude \( \frac{1}{3} \) and three times the fundamental frequency, 3. \(y = \frac{1}{5}\text{sin}(5x) \) with amplitude \( \frac{1}{5} \) and five times the fundamental frequency, and 4. \(y = \frac{1}{7}\text{sin}(7x) \) with amplitude \( \frac{1}{7} \) and seven times the fundamental frequency.

Graph each Function Individually

Use a graphing calculator or manually plot each function on graph paper. Draw one period of each sine wave, ensuring that the x-axis has the same scale for all graphs. This will allow you to accurately add the ordinates later.

Sum the Functions

To graph the sum of the functions, at every x-value, add the y-values of all four functions together. This composite y-value is the ordinate for the sum of the functions at that specific x-value. Plot these points to get the composite curve.

Analyze the Composite Curve

The resulting graph will show peaks and troughs at different locations than the individual sine waves due to the constructive and destructive interference. The sum makes a more complex pattern known as a Fourier Series expansion.

Predict the Addition of More Sine Waves

If more sine waves such as \(y = \frac{1}{9}\text{sin}(9x)\) are added in, the composite curve will become increasingly complex. More peaks and valleys would appear, but a pattern related to the frequencies and amplitudes of the added waves will emerge.

Key concepts

Fourier Series Expansion

When we think about sound, light, or any periodic signal, it often involves not just one simple wave, but rather a complex pattern that can be broken down into simpler components. This decomposition is made possible through the concept of Fourier series expansion. A Fourier series is a way to represent a function as the sum of simple sine and cosine waves.

In mathematical terms, the Fourier series of a periodic function is defined as: \[ f(x) = a_0 + \textcolor{gray}{\sum_{n=1}^{\infty}} (a_n \cos(nx) + b_n \sin(nx))\, \] where \( a_0, a_n, \) and \( b_n \) are the Fourier coefficients that can be determined through integration.

The Fourier series captures the essence of a complex wave by adding up these simpler waves with different frequencies and amplitudes. When you graph functions like the one from the exercise, each sine function you add makes the graph look more intricate. If you were to continue adding sine waves with odd multiples of the original frequency (like the 3x, 5x, and 7x), the pattern would become even more complex. This principle is the foundation for many areas of signal processing, physics, and engineering.

Superposition of Waves

The concept of superposition is a critical principle in physics, particularly in wave theory. It states that when two or more waves cross paths, their displacements at any point and time are simply summed. This phenomenon doesn't change the waves themselves; they continue as if they were alone, but we see the effect as a combined wave.

This principle is what's being applied when we graph the sum of multiple sine waves in our exercise. Each wave is graphed according to its amplitude and frequency, and the values (ordinates) at each point on the x-axis are added together to produce a new wave form. Whether it's sound waves, light waves, or the movement of a string, the superposition principle holds true.

In essence, the graph of the combined function represents what you would see or hear if multiple waves were interacting in real space. For example, if you have two tuning forks vibrating nearby, you'd hear a sound that's the sum of the two individual sounds because of wave superposition.

Sine Wave Interference

When we superimpose sine waves, like in our exercise, we observe what is known as interference. Interference can be constructive or destructive, depending on the relative phases of the waves. When waves are in phase, meaning their peaks and troughs align, they create constructive interference, resulting in a wave with a higher amplitude. Contrarily, destructive interference happens when waves are out of phase, causing the peaks of one wave to fill in the troughs of another, leading to cancellation or a wave with reduced amplitude.

In the graphing of our exercise, where multiple sine waves with different frequencies are added, both types of interference occur. The result is a composite sine wave that has a more intricate pattern due to the alignment and misalignment of the component sine waves' peaks and troughs. As you add more sine waves with higher frequencies and smaller amplitudes, this phenomenon creates an even more complex wave pattern that encapsulates the behavior of real-world phenomena like musical tones or electronic signals.