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Chapter 15: Problem 71

Write the y-equation for a wave traveling in the negative x-direction with wavelength 50 cm, speed 4.0 m/s, and amplitude 5.0 cm

Short Answer

Expert verified
The y-equation for the wave traveling in the negative x-direction is \(y(x,t) = 5.0 \cos\left(4\pi x - 16\pi t\right)\).

Step by step solution

01

Identify the given variables

The amplitude A is 5.0 cm. The wavelength \(\lambda\) is 50 cm, but needs to be in meters, so convert it to 0.5 m. The speed \(v\) is given as 4.0 m/s.
02

Calculate the wave number \(k\)

The wave number \(k\) can be found using the formula: \(k = 2\pi/\lambda\). Substituting the value for \(\lambda\), we get \(k = 2\pi/0.5\) which equals \(4\pi\).
03

Calculate the angular frequency \(w\)

The angular frequency \(w\) can be calculated using the formula: \(w = v \cdot k\). Substituting values for \(v\) and \(k\), we get \(w = 4.0 \cdot 4\pi\) which equals \(16\pi\).
04

Substitute into the wave equation

Substitute \(A\), \(k\), and \(w\) into the wave equation given as: \(y(x,t) = A \cos(kx - wt)\). After substitution, we get the wave equation as: \(y(x,t) = 5.0 \cos\left(4\pi x - 16\pi t\right)\).

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Key Concepts

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

Wavelength
The wavelength of a wave is a fundamental concept in wave dynamics. It refers to the distance between consecutive crests (or troughs) of the wave. In simpler terms, it鈥檚 the length of one complete wave cycle. For example, if you could "freeze" a wave in time, the wavelength would be the distance from one peak to the next.
  • The symbol for wavelength is \( \lambda \).
  • It's usually measured in meters (m), even if given in other units like centimeters (cm). Conversion may be necessary.
  • In our case, starting with 50 cm, we converted this to meters, resulting in \( \lambda = 0.5 \) m.
Understanding wavelength is crucial because it affects how waves behave and interact with objects and other waves they encounter.
Wave Speed
Wave speed is the rate at which a wave travels through a medium. Essentially, it is the speed at which a spot on the wave (like a crest) propagates along the x-direction. Think of wave speed as how fast energy or information carried by a wave moves from one place to another.
  • Denoted by the letter \( v \).
  • Typically measured in meters per second (m/s).
In our exercise, the wave speed is a given value of 4.0 m/s. This means a crest on our wave travels 4 meters every second.
Amplitude
Amplitude is a measure of the strength or intensity of a wave. It's the height from the rest position to the crest of the wave. For sound waves, a larger amplitude means a louder sound, while for light waves, it often means a brighter light. However, in this context, amplitude tells us how much the wave oscillates.
  • Symbolized by \( A \).
  • Measured in units of length, in this case, centimeters (cm).
For the problem at hand, the amplitude is 5.0 cm, which is the maximum vertical displacement of particles the wave influences.
Wave Number
The wave number represents the number of wavelengths per unit distance and is closely related to the concept of wavelength. It is used when describing how many complete wave cycles are packed within a spatial distance.
  • Wave number is denoted by \( k \).
  • It is given by the formula \( k = \frac{2\pi}{\lambda} \), providing units in radians per meter.
By calculating using \( \lambda = 0.5 \) m, the wave number \( k \) becomes \( 4\pi \), illustrating a tightly packed wave with more cycles per unit distance.
Angular Frequency
Angular frequency provides insight into how quickly a wave oscillates over time. Unlike regular frequency, which is measured in cycles per second, angular frequency accounts for the angular aspect, like how a rotating object moves through angles.
  • Angular frequency is denoted \( \omega \).
  • Defined as \( \omega = v \cdot k \) and measured in radians per second.
For our example, multiplying 4.0 m/s by \( 4\pi \) yields an angular frequency of \( 16\pi \). This tells us how the wave oscillates over time at a rapid pace, completing several oscillations within short intervals.

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A whistle you use to call your hunting dog has a frequency of 21 kHz, but your dog is ignoring it. You suspect the whistle may not be working, but you can鈥檛 hear sounds above 20 kHz. To test it, you ask a friend to blow the whistle, then you hop on your bicycle. In which direction should you ride (toward or away from your friend) and at what minimum speed to know if the whistle is working?

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A wave on a string is described by \(y(x, t)=(3.0 \mathrm{cm}) \times\) \(\cos [2 \pi(x /(2.4 \mathrm{m})+t /(0.20 \mathrm{s}))] .\) where \(x\) is in \(\mathrm{m}\) and \(t\) in \(\mathrm{s}\) a. In what direction is this wave traveling? b. What are the wave speed, frequency, and wavelength? c. At \(t=0.50 \mathrm{s},\) what is the displacement of the string at \(x=0.20 \mathrm{m} ?\)

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