16.6 Exercise 42

A trough with dimensions 10.00 meters by 0.10 meters by 0.10 meters is partially filled with water. Small-amplitude surface water waves are produced from both ends of the trough by paddles oscillating in simple harmonic motion. The height of the water waves are modeled with two sinusoidal wave equations, $y_{1} \left(x , t\right) = 0.3 \text{m} \text{sin} \left(4 \text{m}^{−1} x - 3 \text{s}^{−1} t\right)$ and $y_{2} \left(x , t\right) = 0.3 \text{m} \text{cos} \left(4 \text{m}^{−1} x + 3 \text{s}^{−1} t - \frac{\pi}{2}\right) .$ What is the wave function of the resulting wave after the waves reach one another and before they reach the end of the trough (i.e., assume that there are only two waves in the trough and ignore reflections)? Use a spreadsheet to check your results. (Hint: Use the trig identities $\text{sin} \left(u \pm v\right) = \text{sin} u \text{cos} v \pm \text{cos} u \text{sin} v$ and $\text{cos} \left(u \pm v\right) = \text{cos} u \text{cos} v +- \text{sin} u \text{sin} v \left.\right)$