Sound consists of the superposition of many waves of different frequencies and amplitudes. The human ear can detect sound waves with frequencies from approximately 20 Hz to 20 kHz.

A. Use Desmos (or Excel or Capstone) to produce a graph of several periods of a sine wave. Add to this a second sine wave of equal amplitude and three times the frequency. Finally, produce the curve that is the sum of these two waves. Label the vertical-axis as "delta p" and the horizontal-axis as "time". The vertical axis represents the small change in pressure created by a sound wave at a person's ear. The horizontal axis is a small amount of time in the life of these waves. If you continued the graph for larger values of time, you would simply see these patterns repeated forever. If the frequency of the first wave were 440 Hz and the time axis were in seconds, what would be the value of t (time, in seconds) at the end of the sketch of one period? If these waves were continued in exactly this form for a long time (many, many oscillations), the resulting graph would be a representation of a steady sound. What one hears is the superposition of the two original waves -- the third wave you produced. The wave formed by the superposition has all the information of the two original waves. Some people report hearing two different pitches while others describe a single, more complex sound.

B. There is more to the story. Produce a new graph showing one hundred periods of a sine wave. Add a second wave to this graph, with three times the amplitude but with a frequency two hundred times lower. This second wave thus starts from zero amplitude and ends with zero amplitude. Instead of the superposition of the two waves, produce a graph of the product of these two waves. If the resulting waveform were the entire "picture" of the pressure fluctuations associated with some sound (for all time), how might a person describe what she heard? You may assume the frequency of the first wave is 440 Hz.

C. It is possible to produce this pattern (of two waves multiplied) with a superposition (sum) of two different waves? Figure out what the frequencies, amplitudes, and functions of those two waves should be. Write down the resulting expression.

D. A spectrogram is a way to depict superpositions of many waves, each of whose frequencies, amplitudes, and durations vary in time. Go to the spectrum analyzer at https://academo.org/demos/spectrum-analyzer/. Play one of the sound samples (with your computer's sound on, so that you can hear it) and study the spectrogram that is produced. What are the x- and y- axes? What do the colors represent? Which sound sample have you selected? Describe the relationship between what you see in the spectrogram and what you hear. Now record your own voice, preferably producing a constant pitch (try your best), upload the file into the spectrum analyzer, and analyze it. Attach the resulting spectrogram with your answer. What is the fundamental frequency (first harmonic) of the sound you produced? Explain how you determined that. HINT: You may find switching to a logarithmic frequency scale more useful. However, you will need to practice how to read a log scale. https://www.wikihow.com/Read-a-Logarithmic-Scale Pick the first three harmonics that your voice produces. Try your best to estimate the relative amplitudes of those harmonic and using Desmos (or similar) plot the resulting wave form. Clearly report your values for the frequency and the amplitudes of your harmonics. Attach the graph with your answers.