Chapter 8.5, Problem 64AYU

Challenge Problem Beats When two sinusoidal waves travel through the same medium, a third wave is formed that is the sum of the two original waves. If the two waves have slightly different frequencies, the sum of the waves results in an interference pattern known as a beat. Musicians use this idea when tuning an instrument with the aid of a tuning fork. It the instrument and the tuning fork. If the instrument and the tuning fork play the same frequency, no beat is heard. Suppose two waves given by the functions, $y_1=3\cos \left({\omega }_{2}t\right)$ and $y_2=3\cos \left({\omega }_{2}t\right)$ where ${\omega }_1>{\omega }_2$ pass through the same medium, and each has a maximum at $t=0$ sec.

How long does it take the sum function $y_3=y_1+y_2$ to equal $0$ for the first time?

If the periods of the two functions $y_1$ and $y_2$ are $T_1=19$ sec and $T_2=20$ sec, respectively, find the first time the sum $y_3=y_1+y_2=0$.

Use the values from part (b) to graph $y_3$ over the interval $0\le x\le 600$. Do the waves appear to be in tune?