2. A water wave is moving outwards with a frequency of 1 wave every 10 seconds. If the wave has a velocity of 50.0cm/s and the water bug is traveling left at a speed of 20.0 cm/s , what is the wave frequency as the bug: a. Gets closer to you? b. Gets further from you?

Here's how to solve this problem using the Doppler effect for waves:<br /><br />**Understanding the Doppler Effect**<br /><br />The Doppler effect describes the change in frequency of a wave (like sound or water waves) as the source and observer move relative to each other. When they move closer, the frequency increases, and when they move apart, the frequency decreases.<br /><br />**Formula**<br /><br />The formula for the Doppler effect with waves is:<br /><br />f' = f * (v ± v₀) / (v ± vs)<br /><br />Where:<br /><br />* f' is the observed frequency<br />* f is the source frequency<br />* v is the wave velocity<br />* v₀ is the observer velocity (positive if moving towards the source, negative if moving away)<br />* vs is the source velocity (positive if moving away from the observer, negative if moving towards)<br /><br />In this case, the water bug is the observer, and the wave source is stationary (vs = 0).<br /><br />**a. Bug moving towards the source (closer to you)**<br /><br />* f = 1 wave / 10 s = 0.1 Hz<br />* v = 50.0 cm/s<br />* v₀ = +20.0 cm/s (positive because the bug is moving towards the source)<br />* vs = 0 cm/s<br /><br />f' = 0.1 Hz * (50.0 cm/s + 20.0 cm/s) / (50.0 cm/s)<br />f' = 0.1 Hz * 70.0 cm/s / 50.0 cm/s<br />f' = 0.14 Hz<br /><br />**b. Bug moving away from the source (further from you)**<br /><br />* f = 0.1 Hz<br />* v = 50.0 cm/s<br />* v₀ = -20.0 cm/s (negative because the bug is moving away from the source)<br />* vs = 0 cm/s<br /><br />f' = 0.1 Hz * (50.0 cm/s - 20.0 cm/s) / (50.0 cm/s)<br />f' = 0.1 Hz * 30.0 cm/s / 50.0 cm/s<br />f' = 0.06 Hz<br /><br />**Answer:**<br /><br />a. As the bug gets closer, the observed frequency is **0.14 Hz**.<br />b. As the bug gets further away, the observed frequency is **0.06 Hz**.<br />