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Chapter 3: Problem 11

Sketch the graph of the harmonic wave \(\phi(x, t)=\sin 2 \pi(x-t)\) as a function of \(x(-1 \leq x \leq 2)\) for values of time \(t\), (i) \(t=0\), (ii) \(t=1 / 4\), (iii) \(t=1 / 2\).

Short Answer

Expert verified
Plot the sine wave for each given \( t \) value, adjusting position and shape accordingly.

Step by step solution

01

Understand the Wave Function

The given wave function is \( \phi(x, t) = \sin 2\pi(x-t) \). It's important to identify that this function represents a wave traveling along the x-axis, where \( x \) is the position and \( t \) is the time. The argument of the sine function, \( 2\pi(x - t) \), indicates a wave with a period of 1.
02

Plot for t=0

Substitute \( t = 0 \) into the wave function: \( \phi(x, 0) = \sin 2\pi x \). This is a standard sine wave with a wavelength of 1, which means it completes one full cycle over the interval from 0 to 1. Plot this wave for \( x \) from -1 to 2, considering one full cycle over every interval of length 1.
03

Plot for t=1/4

Substitute \( t = 1/4 \) into the wave function: \( \phi(x, 1/4) = \sin 2\pi(x - 1/4) = \sin (2\pi x - \pi/2) \). This represents the original wave shifted to the right by 1/4 of its cycle. Adjust the plot of \( \sin 2\pi x \) to start at \( x = 1/4 \) and end at \( x = 1.25 \) over one complete cycle.
04

Plot for t=1/2

Substitute \( t = 1/2 \) into the wave function: \( \phi(x, 1/2) = \sin 2\pi(x - 1/2) = \sin (2\pi x - \pi) \). This corresponds to the wave being shifted to the right by 1/2 of its cycle or a half-wavelength. Effectively, this is the same as reflecting the wave: \( \phi(x, 1/2) = -\sin 2\pi x \), resulting in an inverted wave with peaks where there were troughs.

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Key Concepts

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

Wave Function
Understanding the concept of a wave function is key to analyzing harmonic waves. A wave function, such as \[ \phi(x, t) = \sin 2\pi(x-t) \]especially in the context of harmonic waves, is a mathematical description that tells us the displacement of the wave at any given point in space and time. Here,
  • \(x\): represents the position along the wave's path.
  • \(t\): represents the time, showing how the wave evolves over time.
  • \(\sin\): indicates the type of wave, which in this case is sinusoidal or a sine wave.
The argument of the sine function, \(2\pi(x-t)\), suggests that the wave is periodic, meaning it repeats its shape at regular intervals. The wave propagates in the positive x-direction, and the term \(x-t\) indicates that as time increases, the wave shifts along its path. The function shows how the wave's amplitude changes over its cycle at given positions and times.
Sine Wave
A sine wave is one of the most fundamental types of waveforms. It is characterized by its smooth, repetitive oscillation. The wave function given \\( \phi(x, t) = \sin 2\pi(x-t) \) is an example of a sine wave, which naturally occurs in various physical phenomena such as sound waves, light waves, and water waves.
  • Properties: Sine waves have a distinct wave pattern, where they begin at zero, rise to a peak, descend through zero to a trough, and then return to zero, completing one full cycle.
  • Cycle: A single cycle refers to the wave moving through one complete oscillation, standardly defined from 0 to 2\(\pi\) for the sine function, before repeating again.
  • Amplitude: The maximum height reached by the wave from its central axis. For the function \( \sin 2\pi(x-t) \), the amplitude is 1, indicating the wave's peaks occur at \(\pm 1\).
Therefore, understanding sine waves is essential because they represent a pure form of motion found throughout wave mechanics and various applications.
Wave Period
The wave period is a critical concept when analyzing periodic waves, like the sine waves described by our wave function \[ \phi(x, t) = \sin 2\pi(x-t) \].The period of a wave is the distance over which the wave's shape repeats, and it is inversely related to frequency.
  • Definition: It can be seen as the horizontal length required to complete one full wave cycle.
  • Calculation: For our function, since the argument is \(2\pi(x-t)\), the wave completes one cycle for every unit interval of \(x\), giving a period of 1.
  • Implication: A period of 1 means that as \(x\) advances from 0 to 1, the wave transitions through its complete pattern from crest to trough and back to crest.
Understanding the wave period helps in predicting how waves behave over different spatial and temporal intervals and how they interact when combined with other waves.
Wave Shift
Wave shift is an essential concept for understanding changes in a wave's position over time. It describes how a wave can move horizontally along its axis. In the context of the wave function \[ \phi(x, t)=\sin 2\pi(x-t) \], various shifts occur as the time \(t\) changes.
  • General Definition: A phase shift occurs when the graph of the sine wave slides left or right by a certain amount, without altering its shape.
  • Example at \(t=1/4\): Substituting \(t=1/4\) results in a wave function \(\phi(x, 1/4) = \sin (2\pi x - \pi/2)\), showing a shift to the right by 1/4 of the cycle, equivalent to \(\pi/2\) in the sine function's phase.
  • Example at \(t=1/2\): For \(t=1/2\), the function becomes \(\phi(x, 1/2) = -\sin 2\pi x\), indicating a half-cycle shift or phase shift, which mirrors the wave across the x-axis.
Wave shifts provide insights into the dynamic nature of waves, showing how they adjust over time and in response to various conditions beyond just spatial dimensions.

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Most popular questions from this chapter

For a circle of radius \(r=4\), find (i) the angle subtended at the centre of the circle by arc of length 6 , (ii) the length of arc that subtends angle \(\pi / 10\) at the centre of the circle, (iii) the length of arc that subtends angle \(\pi / 2\) at the centre of the circle, (iv) the circumference of the circle.

By considering the limit \(\theta \rightarrow 0\) of an internal angle of a right-angled triangle, show that (i) \(\sin 0=0\) (ii) \(\cos 0=1\)

The function \(\psi(x, t)=\sin \pi x \cos 2 \pi t\) represents a standing wave. Find the values of time \(t\) for which \(\psi\) has (i) maximum amplitude, (ii) zero amplitude. (iii) Sketch the wave function between \(x=0\) and \(x=3\) at (a) \(t=0\), (b) \(t=1 / 8\).

Express the following angles in radians: (i) \(5^{\circ}\) (ii) \(87^{\circ}\) (iii) \(120^{\circ}\) (iv) \(260^{\circ}\) (v) \(540^{\circ}\) (vi) \(720^{\circ}\)

Find the polar coordinates of the points whose cartesian coordinates are (i) \((3,2)\), (ii) \((3,-2)\).
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