Calculating Wave Frequency from Velocity and Wavelength

This problem asks us to find the frequency of the rocking of a boat caused by waves. The frequency of the boat's rocking will be the same as the frequency of the waves that are causing it to rock. We are given information about the waves: the distance between consecutive crests and the speed of the wave crests.

Understanding Wave Properties: Wavelength, Velocity, and Frequency

Before we solve the problem, let's quickly define the key terms:

• Wavelength ($\lambda$): The distance between two consecutive points on a wave that are in phase, such as two consecutive crests or troughs. It is typically measured in meters (m).
• Wave Velocity ($v$): The speed at which a wave propagates through a medium. It is typically measured in meters per second (ms⁻¹).
• Frequency ($f$): The number of complete wave cycles that pass a given point per unit of time. It is typically measured in Hertz (Hz), which is equivalent to cycles per second (s⁻¹).

These three properties are related by a fundamental equation in wave mechanics:

This equation tells us that the wave velocity is equal to the product of its frequency and wavelength.

Applying the Wave Equation to the Problem

We are given the following information from the question:

• Distance between consecutive crests (Wavelength, $\lambda$) = 125 m
• Velocity of the wave crests (Wave Velocity, $v$) = 25 ms⁻¹

We need to find the frequency ($f$) of the waves (and thus the frequency of the boat's rocking).

Using the wave equation, $v = f\lambda$, we can rearrange it to solve for the frequency ($f$):

Now, we can substitute the given values into this equation:

Simplifying the fraction:

Result

The frequency of the rocking of the boat, which is equal to the frequency of the waves, is 0.20 Hz.

Comparing with Options

Let's compare our calculated frequency with the given options:

• Option 1: 0.20 (Hz)
• Option 2: 100 (Hz)
• Option 3: 625 (Hz)
• Option 4: 250 (Hz)

Our calculated value of 0.20 Hz matches Option 1.

Revision Table: Wave Properties

Note that frequency ($f$) and period ($T$) are inversely related: $f = 1/T$ or $T = 1/f$. The wave equation can also be written as $v = \lambda/T$.

Additional Information: Types of Waves

Waves can be broadly classified based on their nature and how they propagate:

• Mechanical Waves: These waves require a medium (like water, air, or solids) to travel. They involve the oscillation of particles in the medium. Examples include water waves (like the ones rocking the boat), sound waves, and seismic waves.
• Electromagnetic Waves: These waves do not require a medium to travel and can propagate through a vacuum. They consist of oscillating electric and magnetic fields. Examples include light waves, radio waves, microwaves, X-rays, and gamma rays.

Water waves, as described in the problem, are mechanical waves. They cause particles of water to oscillate, which in turn causes the boat to rock with the same frequency as the waves.

The concept of wavelength, velocity, and frequency is fundamental to understanding all types of waves, although the specific way they are produced and propagate differs.