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Chapter 4: Problem 1

A pulse of the form \(y=a e^{-b x^{2}}\) is formed in a rope, where \(a\) and \(b\) are constants and \(x\) is in centimeters. Sketch this pulse. Then write an equation that represents the pulse moving in the negative direction at \(10 \mathrm{cm} / \mathrm{s}\).

Short Answer

Expert verified
The moving pulse is given by \( y = a e^{-b(x + 10t)^2} \).

Step by step solution

01

Understanding the Initial Function

The given function for the pulse is exponential: \( y = a e^{-bx^2} \). It represents a Gaussian function, characterized by its bell-shaped curve, where \( a \) affects the amplitude of the wave and \( b \) affects the spread.
02

Sketching the Pulse

To sketch \( y = a e^{-bx^2} \), note that when \( x = 0 \), \( y = a \), which is the peak. As \( x \) moves away from 0, \( y \) decreases rapidly towards zero due to the \( e^{-bx^2} \) term. The curve is symmetric about the origin.
03

Deriving the Moving Pulse Equation

For a pulse moving in the negative x-direction, substitute \( x + vt \) for \( x \) in the original equation, where \( v \) is the velocity. Here, \( v = 10 \) cm/s and since it's moving in the negative direction, substitute \( x + 10t \). Thus, the equation becomes: \[ y = a e^{-b(x + 10t)^2} \]
04

Final Expression for the Moving Pulse

In the new equation \( y = a e^{-b(x + 10t)^2} \), the \( 10t \) term signifies a shift in the pulse position leftwards over time, at a speed of 10 cm/s. This gives us the equation of the pulse moving in negative x-direction.

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Key Concepts

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

Gaussian Pulse
A Gaussian pulse is a specific type of wave that has a shape described by a Gaussian function. This type of wave is most well-known for its smooth, symmetrical, bell-shaped curve. The equation that typically represents a Gaussian pulse is \(y = a e^{-b x^2}\), where \(a\) and \(b\) are constants:
  • \(a\) determines the height of the pulse, also known as the amplitude.
  • \(b\) affects how "spread out" the bell curve is.
The Gaussian pulse has significant applications in areas such as optics and signal processing because it can model how light or waves propagate in a medium. This is due to its ability to remain localized in space while moving along its path.
Wave Motion
Wave motion refers to the transfer of energy through a medium without the permanent displacement of the particles in the medium. In the context of a Gaussian pulse moving through a medium, this means the pulse or wave travels without carrying material along with it.
  • In wave motion, particles of the medium oscillate around a fixed point, passing energy to neighboring particles.
  • The wave itself is the disturbance that travels, carrying energy from one place to another.
This is essential in understanding how Gaussian pulses can travel along ropes or other mediums, maintaining their shape while transferring energy through the system.
Exponential Function
The exponential function is a mathematical function of the form \(e^x\). It's a crucial part of the equation \(y = a e^{-b x^2}\) used to describe a Gaussian pulse.
  • In our context, \(e^{-b x^2}\) gives the pulse its distinct bell-shaped appearance.
  • As \(x\) moves away from zero, \(e^{-b x^2}\) decreases rapidly, ensuring that the pulse decays to zero swiftly.
Exponential functions are particularly powerful due to their capability to describe growth and decay processes, making them applicable in various scientific fields, including optics, where understanding how wave amplitudes diminish over distance is vital.
Amplitude
Amplitude, denoted by \(a\) in the Gaussian pulse equation, is a measure of the wave's height, or the maximum displacement from its position of equilibrium.
  • In our equation \(y = a e^{-b x^2}\), \(a\) sets the peak value of the wave.
  • The larger the \(a\), the taller the wave's peak, and thus, the more energy it carries.
Amplitude is a fundamental property in wave physics and optics because it is directly related to the intensity of the wave. In optics, greater amplitude can correlate with brighter light intensity or louder sound in acoustics.
Velocity
Velocity refers to the speed and direction in which a wave pulse travels through a medium. In our scenario, we have a Gaussian pulse moving in the negative x-direction with a velocity of \(10\) cm/s.
  • To express this mathematically, we modify the equation to account for motion: \(y = a e^{-b(x + 10t)^2}\).
  • The term \(10t\) indicates that the wave moves leftwards (negative x-direction) as time \(t\) increases.
Understanding velocity is crucial in contexts like wave propagation in optics, where controlling the speed and direction of light pulses can have significant impacts on technologies, such as communications and imaging systems.

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Most popular questions from this chapter

How fast does one have to approach a red traffic light to see a green signal? So that we all get the same answer, say that a good red is \(640 \mathrm{nm}\) and a good green is \(540 \mathrm{nm}\).

a. Show that if the maximum positive displacement of a sinusoidal wave occurs at distance \(x_{0} \mathrm{cm}\) from the origin when \(t=0\), its initial phase angle \(\varphi_{0}\) is given by $$\varphi_{0}=\frac{\pi}{2}-\left(\frac{2 \pi}{\lambda}\right) x_{0}$$ where the wavelength \(\lambda\) is in centimeters. b. Determine the initial phase and sketch the wave when \(\lambda=10 \mathrm{cm}\) and \(x_{0}=0, \frac{5}{6}, \frac{5}{2}, 5,\) and \(-\frac{1}{2} \mathrm{cm}\) c. What are the appropriate initial phase angles for (b) when a cosine function is used instead?

Two waves of the same amplitude, speed, and frequency travel together in the same region of space. The resultant wave may be written as a sum of the individual waves. $$\psi(y, t)=A \sin (k y+\omega t)+A \sin (k y-\omega t+\pi)$$ With the help of complex exponentials, show that $$\psi(y, t)=2 A \cos (k y) \sin (\omega t)$$

Use the constant phase condition to determine the velocity of each of the following waves in terms of the constants \(A, B, C,\) and \(D .\) Distances are in meters and time in seconds. Verify your results dimensionally. a. \(f(y, t)=A(y-B t)\) b. \(f(x, t)=A(B x+C t+D)^{2}\) c. \(f(z, t)=A \exp \left(B z^{2}+B C^{2} t^{2}-2 B C z t\right)\)

a. The light from a 220 -W lamp spreads uniformly in all directions. Find the irradiance of these optical electromagnetic waves and the amplitude of their \(\overrightarrow{\mathbf{E}}\) -field at a distance of \(10 \mathrm{m}\) from the lamp. Assume that \(5 \%\) of the lamp energy is converted to light. b. Suppose a 2000 -W laser beam is concentrated by a lens into a cross- sectional area of about \(1 \times 10^{-6} \mathrm{cm}^{2} .\) Find the corresponding irradiance and amplitudes of the \(\overrightarrow{\mathbf{E}}\) and B-fields there.
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