A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (Fig. below). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width w of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 859 seats around a stadium in 40.0 s, with spectators requiring about 1.80 s to respond to the wave's passage by standing and then sitting. What are (a) the wave speed v (in seats per second) and (b) width w (in number of seats)?

To find the wave speed, we can use the formula:

wave speed (v) = distance traveled / time taken

In this case, the distance traveled is given as 859 seats and the time taken is given as 40.0 seconds. Plugging these values into the formula, we can calculate:

v = 859 seats / 40.0 s
v ≈ 21.475 seats/s (rounded to three significant figures)

So, the wave speed is approximately 21.475 seats per second.

Next, we can find the width of the wave. The width is defined as the distance from the leading edge to the trailing edge of the wave.

The time required for a spectator to respond to the wave's passage and stand up and then sit down is given as 1.80 seconds.

Considering this, we can calculate the number of waves passing through in 1.8 seconds:

Number of waves = 40.0 s / 1.80 s = 22.222 waves (rounded to three decimal places)

Since each wave travels a distance of 859 seats, the total distance covered in 1.8 seconds is:

Total distance covered = 859 seats × 22.222 waves = 19,097.318 seats (rounded to three decimal places)

Therefore, the width of the wave is approximately 19,097 seats.

In conclusion, the answers are:
(a) The wave speed is approximately 21.475 seats per second.
(b) The width of the wave is approximately 19,097 seats.