Ripples in a shallow puddle propagate at \(34 \mathrm{cm} / \mathrm{s}\). If the wave frequency is \(5.2 \mathrm{Hz}\), find (a) the period and (b) the wavelength.

When studying waves, one fundamental concept is the wave period. The wave period, denoted by the symbol T, refers to the time it takes for one complete wave cycle to occur. In simpler terms, it's the time between two consecutive crests or troughs passing a fixed point.

Mathematically, the wave period is the reciprocal of the frequency, which is the number of cycles per second. The frequency is denoted by the symbol f and measured in hertz (Hz). To find the period, you can use the formula: \( T = \frac{1}{f} \). Therefore, if a wave's frequency is known, its period can be easily calculated. For example, a wave with a frequency of 5.2 Hz has a period of \( T = \frac{1}{5.2} \) seconds, which is approximately 0.192 seconds. This value represents how long it takes one wave cycle to pass, which is essential in many aspects of physics and engineering applications.

The concept of wavelength is equally vital in understanding wave properties. Wavelength, represented by the Greek letter lambda \( \lambda \), is the physical distance between two identical points on successive waves, such as from crest to crest. To visualize this, imagine measuring the length of one complete wave cycle along the medium of propagation.

To calculate the wavelength of a wave, you can use the relationship involving velocity (V) and frequency (f), expressed as \( V = f \cdot \lambda \). If you rearrange this equation to solve for wavelength, you get \( \lambda = \frac{V}{f} \). For a wave traveling at 34 cm/s with a frequency of 5.2 Hz, the wavelength calculation would be \( 6.54 \text{cm} \), showing how far the wave travels within one period. Accurate knowledge of wavelength is critical for understanding various wave phenomena such as diffraction, interference, and resonance.

Understanding the relationship between velocity, frequency, and wavelength is pivotal in wave mechanics. This interconnection is captured by the basic wave equation, which ties these three properties together: \( V = f \cdot \lambda \). Here, V represents the velocity of the wave, f stands for frequency, and \( \lambda \) is the wavelength.

The formula suggests that the wave's velocity is directly proportional to both its frequency and its wavelength. In terms of practical application, if you know two of the values, you can determine the third. For instance, knowing that the frequency of ripples in a puddle is 5.2 Hz and they travel at 34 cm/s, you can deduce that each ripple must be 6.54 cm long. This velocity-frequency relationship is a core principle in various fields, from telecommunications to acoustics, and it is crucial for problems involving wave motion.