Complex Waves, Another Example of Wave Interference: Shown below are three wave

Question

Complex Waves, Another Example of Wave Interference: Shown below are three wave patterns. Each pattern is the variation in pressure at different positions in space at one instant of time. The resultant wave at the bottom is simply the sum of the pressure variations caused by the two top waves if present at the same time. You can produce the same pattern with the simulation by adding two waves each of amplitude +30 with the first at frequency f1 = 2.0 kHz and the second at twice the frequency, or f2 = 4.0 kHz. In the music business, the first wave is called the fundamental, and the second is the second harmonic, or the first overtone. The resultant complex wave produced by adding the two is the pattern seen in the simulation and reproduced below. Show that the bottom wave is just the sum of the two waves at each position of space. SUMMARY Beats are produced by adding two waves of slightly different frequencies. The beat frequency is the absolute value of the difference in frequency of the two waves: fbeat = f2 - f1 Complex waves are formed by adding a fundamental-frequency wave with higher-frequency harmonics that are integral multiples of the frequency of the fundamental wave.

Explanation / Answer

P1 +P2 = Asin(2*pi*2t+kx)+Asin(2*pi*4t +kx) = 2Asin(2*pi*3t +kx)*cos(2*pi*t) = 2Acos(2pi*t)*sin(2*pi*3t +kx) hence P1+p2 is a traveling wave whose amplitude 2Acos(2pi*t) ,

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