Problem 63 - Fundamentals of Physics

Question

A wave has a speed of \(240 \mathrm{~m} / \mathrm{s}\) and a wavelength of \(3.2 \mathrm{~m}\). What are the (a) frequency and (b) period of the wave?

Step-by-Step Solution

Answer

The frequency is 75 Hz and the period is approximately 0.0133 s.

Step 1: Calculate Frequency

We know that the speed of a wave, its frequency, and its wavelength are related by the formula: \[ v = f \times \lambda \]where:- \( v \) is the speed of the wave, given as 240 m/s,- \( f \) is the frequency we need to find,- \( \lambda \) is the wavelength, given as 3.2 m.Rearrange the formula to solve for the frequency:\[ f = \frac{v}{\lambda} = \frac{240}{3.2} \]Calculate the frequency:\[ f = 75 \text{ Hz} \]

Step 2: Calculate Period

The period of a wave, denoted by \( T \), is the inverse of the frequency. The relationship is given by the formula:\[ T = \frac{1}{f} \]Substitute the frequency we calculated:\[ T = \frac{1}{75} \approx 0.0133 \text{ s} \]

Key Concepts

Wave Speed

Wave speed is the rate at which a wave propagates through a medium, such as air, water, or along a string. In physics, the speed of any type of wave can be represented by the symbol \( v \). It is measured in meters per second (m/s). The wave speed depends on the type of wave and the medium through which it is traveling.
To calculate wave speed, you can use the formula \( v = f \times \lambda \), where \( v \) is the wave speed, \( f \) is the frequency of the wave, and \( \lambda \) (lambda) is the wavelength. This means that if you know the frequency and the wavelength of a wave, you can easily determine its speed. In our exercise, the wave speed is already given as 240 m/s, a piece of information that will be useful in further calculations.

Frequency Calculation

Frequency refers to the number of occurrences of a repeating event per unit of time. In the context of waves, it’s the number of wave crests passing a given point per second. It is measured in hertz (Hz), where one hertz is equal to one cycle per second.
To find the frequency of a wave, you rearrange the wave speed equation to solve for \( f \), resulting in the formula \( f = \frac{v}{\lambda} \). In our case, with a wave speed \( v \) of 240 m/s and a wavelength \( \lambda \) of 3.2 m, the frequency comes out to be \( f = \frac{240}{3.2} = 75 \text{ Hz} \). This tells us that 75 wave crests pass a given point every second.

Wavelength

The wavelength of a wave refers to the physical length of one cycle of the wave. It is typically denoted by the Greek letter \( \lambda \) and measured in meters (m). The wavelength is the distance from one crest of the wave to the next.
Wavelength can affect many properties of a wave, such as its frequency and energy. For a constant wave speed, the frequency increases as the wavelength decreases, and vice versa. In simpler terms, shorter wavelengths lead to higher frequencies if the speed remains unchanged. In our exercise, the given wavelength is 3.2 m, which we used along with the wave speed to find the wave's frequency.

Period of Wave

The period of a wave is the time it takes for one complete cycle of the wave to pass a given point. It is represented by the symbol \( T \) and is usually measured in seconds.
The period is directly related to the frequency of the wave. The formula \( T = \frac{1}{f} \) shows that the period is the reciprocal of the frequency. As frequency increases, the period decreases and vice versa. In the exercise given, we calculated the frequency as 75 Hz, which was then used to find the period: \( T = \frac{1}{75} \approx 0.0133 \text{ s} \). This means that each wave takes approximately 0.0133 seconds to complete a cycle.