Question

By rocking a boat, a child produces surface water waves on a previously quiet lake. It is observed that the boat performs 12 oscillations in \(30 \mathrm{~s}\) and also that a given wave crest reaches shore \(15 \mathrm{~m}\) away in \(5.0 \mathrm{~s}\). Find \((a)\) the frequency, \((b)\) the speed, and (c) the wavelength of the waves.

Short answer

The frequency of the waves is 0.4 Hz, the speed is 3 m/s, and the wavelength is 7.5 meters.

Step-by-step solution

Calculate the Frequency

Frequency of any periodic motion is given by the total number of oscillations divided by the total time taken. From the given data, the boat makes 12 oscillations in 30 seconds. So, the frequency (\(f\)) is \(f = \frac{Oscillations}{Time} = \frac{12}{30} Hz\)

Calculate the Speed of the waves

The speed of the waves can be calculated by dividing the total distance the wave travels by the time it takes to travel that distance. Given that a wave crest reaches shore 15m away in 5 seconds, we can calculate the speed (\(v\)) as follows: \(v = \frac{Distance}{Time} = \frac{15m}{5s} m/s\)

Calculate the Wavelength

Lastly, we can use the wave equation \(v = fλ\) (where \(v\) is the speed, \(f\) is the frequency and \(λ\) is the wavelength) to solve for the wavelength (\(λ\)). By rearranging the equation, we get \(λ = \frac{v}{f}\). Substituting our previously obtained values for \(v\) and \(f\), we get \(λ = \frac{3 m/s}{0.4 Hz}\)

Key concepts

Frequency

The frequency of a wave is simply the number of times something happens in a certain period. Here, the boat on the lake performs a set number of oscillations. In physics, frequency ( f ) is defined as the number of complete oscillations or cycles per unit of time. This is often measured in Hertz (Hz), where 1 Hz equals 1 cycle per second.

Using the example, we have 12 oscillations over 30 seconds. Therefore, the frequency can be expressed as:\[ f = \frac{12 \text{ oscillations}}{30 \text{ seconds}} = 0.4 \text{ Hz} \]In practical terms, this means every second, there are 0.4 waves or cycles passing through a point. More simply, the boat creates a wave every 2.5 seconds. Knowing frequency helps in understanding how quickly the waves are produced, influencing everything from sound pitch to how often a swinging pendulum goes back and forth.

Wave Speed

Wave speed represents how quickly a wave travels through a medium, such as water in this exercise. It's similar to how you'd measure speed by considering the distance a car travels over a given time. To find the speed of a wave, divide the distance it travels by the time it takes to cover that distance.

Based on the problem, a wave crest travels 15 meters in 5 seconds. Thus, the wave's speed ( v ) is given by:\[ v = \frac{15 \text{ m}}{5 \text{ s}} = 3 \text{ m/s} \]This tells us that each wave travels 3 meters in one second. Understanding wave speed is crucial because it determines how much time a wave takes to reach a certain point. For example, faster waves can have more impact or carry energy farther than slower waves. These principles are not just limited to water waves, but apply to any type of wave, including sound or light.

Wavelength

Wavelength refers to the physical length of one complete cycle of a wave, measured as the distance between two consecutive wave crests or troughs. It is typically denoted by the symbol \( \lambda \). Knowing the frequency and speed of a wave allows us to calculate its wavelength using the formula:\[ \lambda = \frac{v}{f} \]Substituting the known values from the problem, where the speed \( v \) is 3 m/s and the frequency \( f \) is 0.4 Hz, we find:\[ \lambda = \frac{3 \text{ m/s}}{0.4 \text{ Hz}} = 7.5 \text{ meters} \]This means each wave is 7.5 meters long from crest to crest. Wavelength plays a significant role in physics, impacting how waves interact with one another and with objects. For example, longer wavelengths might bend around objects more easily than shorter wavelengths. This concept is applied across many fields, from explaining colors in light waves to tuning musical instruments based on sound waves.