Spectrogram. 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.

2.0 sin(t) 1.5 sin(3t) A sin(t)+sin(3t) 1.0 A 0.5 0.0 -0.5 -1.0 -1.5 -2.0 0 2 W 4 6 7 time(t)