Problem 6 - Physics

Question

A person lying on an air mattress in the ocean rises and falls through one complete cycle every five seconds. The crests of the wave causing the motion are \(20.0 \mathrm{~m}\) apart. Determine (a) the frequency and (b) the speed of the wave.

Step-by-Step Solution

Answer

(a) Frequency is 0.2 Hz; (b) Speed is 4.0 m/s.

Step 1: Identify the Known Variables

We know that the period (T) is the time it takes for one complete cycle, which is given as 5 seconds. The distance between the crests of the wave, known as the wavelength (\(\lambda\)), is 20.0 meters.

Step 2: Calculate the Frequency

Frequency (\(f\)) is the reciprocal of the period (\(T\)). Use the formula \(f = \frac{1}{T}\). Substituting the given period, we have \(f = \frac{1}{5 \text{ seconds}} = 0.2 \text{ Hz}\).

Step 3: Calculate the Wave Speed

The speed of the wave (\(v\)) can be determined using the formula \(v = f \cdot \lambda\). We have already found \(f = 0.2 \text{ Hz}\) and the wavelength \(\lambda = 20.0 \text{ m}\). So, \(v = 0.2 \text{ Hz} \times 20.0 \text{ m} = 4.0 \text{ m/s}\).

Key Concepts

Frequency

Frequency refers to how often a repeating event occurs per unit of time. In the context of wave motion, it is the number of times a wave crest passes a fixed point every second.
The unit of frequency is the Hertz (Hz), which is equivalent to one cycle per second.To find frequency, we use the period, which is the time for one complete cycle. The formula is:

• \( f = \frac{1}{T} \)

where \( T \) is the period.
In our example, the period \( T \) was 5 seconds per cycle, and dividing gives us a frequency of 0.2 Hz. This means every 5 seconds, 0.2 cycles of the wave occur.

Wavelength

Wavelength is the distance between successive crests of a wave. This is an important property of waves, as it determines some aspects of their behavior.
Waves in different mediums, like in our ocean scenario, travel with distinct wavelengths.The wavelength is often denoted by the Greek letter lambda (\(\lambda\)). In this problem, we know the crests are 20 meters apart:

• \( \lambda = 20.0 \text{ m} \)

Understanding wavelength helps in calculating wave speed and anticipating how waves will interact with obstacles or other waves.

Wave Speed

Wave speed indicates how fast the wave travels through a medium. It shows the distance a wave travels over a period of time.
The formula to find wave speed \( v \) is:

• \( v = f \cdot \lambda \)

Here, \( f \) is the frequency and \( \lambda \) is the wavelength. In our exercise, the wave speed was calculated using the frequency 0.2 Hz and wavelength 20 meters:

• \( v = 0.2 \text{ Hz} \times 20.0 \text{ m} = 4.0 \text{ m/s} \)

This tells us that every second, the wave travels 4 meters in the ocean.