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Chapter 15: Problem 23

A sinusoidal wave moving to the left has a wavelength of \(5.0 \mathrm{cm}\) and a frequency of \(50 \mathrm{Hz}\) At \(t=0 \mathrm{s},\) the wave has a crest at \(x=0 \mathrm{cm} .\) What is the earliest time after \(t=0 \mathrm{s}\) at which there is a crest at the position \(x=3.0 \mathrm{cm} ?\)

Short Answer

Expert verified
The earliest time after \(t=0 \mathrm{s}\) at which there is a crest at the position \(x=3.0 \mathrm{cm}\) is \(t = 0.012 \mathrm{s}\).

Step by step solution

01

Compute the velocity of the wave

The velocity of a wave is the product of its wavelength and its frequency. Using the given wavelength \(\lambda = 5.0 \mathrm{cm} = 0.050 \mathrm{m}\) and frequency \(f = 50 \mathrm{Hz}\), the wave velocity \(v\) can be calculated as follows: \(v = \lambda \cdot f = 0.050 \mathrm{m} \cdot 50 \mathrm{Hz} = 2.5 \mathrm{m/s}\).
02

Determine the direction of wave travel

The wave is moving to the left. In standard wave notation, positive velocities correspond to movement to the right, so the actual velocity of the wave is \(v = -2.5 \mathrm{m/s}\). The negative sign indicates movement to the left.
03

Calculate the time at the desired position

Now, knowing the wave's velocity, we can calculate the time it takes for a wave crest to reach the position \(x = 3.0 \mathrm{cm} = 0.03 \mathrm{m}\). The wave travels at a speed of 2.5 m/s to the left, so the time can be found by dividing the position by the absolute velocity: \(t = \frac{x}{|v|} = \frac{0.03 \mathrm{m}}{2.5 \mathrm{m/s}} = 0.012 \mathrm{s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sinusoidal Waves
Sinusoidal waves are a fundamental concept in understanding wave motion. These are waves that have a shape similar to the sine or cosine functions. Because of this, their repeating wave patterns are smooth and regular.

Sinusoidal waves are prevalent in many areas of physics and engineering. They describe various forms of energy propagation, such as sound waves, light waves, and other electromagnetic waves. Their predictable pattern makes them ideal for modeling periodic phenomena.
  • They have crests (highest points) and troughs (lowest points).
  • These waves are characterized by their smooth oscillations.
  • The waves can travel in different directions, including left, right, up, or down.
Understanding sinusoidal behavior is key to analyzing more complex wave interactions and phenomena.
Wave Velocity
Wave velocity refers to the speed at which the wave travels through a medium. It's a crucial aspect of understanding waves, as it tells us how quickly the wave is moving. The velocity of a wave can be determined using the simple formula:

\[ v = \lambda \cdot f \]

where \( v \) is the wave velocity, \( \lambda \) (lambda) is the wavelength, and \( f \) is the frequency of the wave.
  • For example, with a wavelength of 5 cm and a frequency of 50 Hz, the wave velocity is calculated as: \( v = 0.050 \mathrm{m} \times 50 \mathrm{Hz} = 2.5 \mathrm{m/s} \).
  • If a wave's velocity has a positive sign, it is moving to the right; a negative velocity, like in our example, implies movement to the left.
Understanding wave velocity helps predict how fast and in what direction a wave will travel.
Wave Frequency
Wave frequency is the number of oscillations or cycles that occur in one second, measured in Hertz (Hz). It reflects how often the particles in the medium are vibrating as the wave passes. Higher frequency means more cycles per second, leading to greater energy transport.

Frequency is an intrinsic property of waves and is linked to other wave characteristics.
  • In our specific example, the frequency of the wave is 50 Hz, meaning it oscillates 50 times every second.
  • Since frequency is calculated based on cycles per second, it remains constant regardless of the medium.
Recognizing the frequency gives insight into the wave's behavior and its impact on the environment it travels through.
Wavelength
Wavelength is the physical length of one complete oscillation or wave cycle. It is the distance between two consecutive crests or troughs. Wavelength is generally represented by the symbol \( \lambda \) and is measured in meters (m).

Wavelength plays a crucial role in determining the characteristics and effects of waves.
  • In the provided scenario, the wavelength is 5 cm, which converts to 0.05 m for calculations.
  • Shorter wavelengths generally result in higher frequencies, while longer wavelengths correspond to lower frequencies, under identical conditions.
Understanding wavelength helps in visualizing how the wave propagates through its medium over time, affecting how it interacts with the environment.

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