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Chapter 7: Problem 20

Find the amplitude, period, frequency, wave velocity, and wavelength of the given wave. By computer, plot on the same axes, \(y\) as a function of \(x\) for the given values of \(t,\) and label each graph with its value of \(t .\) Similarly, plot on the same axes, \(y\) as a function of \(t\) for the given values of \(x,\) and label each curve with its value of \(x.\) $$y=3 \sin \pi\left(x-\frac{1}{2} t\right) ; \quad t=0,1,2,3 ; \quad x=0, \frac{1}{2}, 1, \frac{3}{2}, 2$$

Short Answer

Expert verified
Amplitude is 3. Period is 2. Frequency is 0.5 Hz. Wavelength is 2. Wave velocity is approximately 0.159.

Step by step solution

01

Identify the amplitude

The amplitude of a sine wave is the coefficient in front of the sine function. Here, the amplitude is 3. The equation of the wave is given by: \( y = 3 \, \sin \pi \left( x - \frac{1}{2} t \right) \).
02

Determine the period

The period of a sine function \( \sin(kx) \) can be found using the formula \( T = \frac{2\pi}{k} \). In this equation, \( k = \pi \), so the period is \[ T = \frac{2\pi}{\pi} = 2 \].
03

Calculate the frequency

The frequency is the reciprocal of the period. \( f = \frac{1}{T} \). Since the period \( T \) is 2, the frequency \[ f = \frac{1}{2} = 0.5 \, \text{Hz} \].
04

Find the wave velocity

The wave velocity \( v \) can be found using the formula \( v = \frac{\text{wavelength}}{\text{period}} \). Here, let's first find the wavelength. We have \( k = \pi \) and \( \omega = \frac{1}{2} \), so \( v = \frac{\omega}{k} \) which results in \[ v = \frac{0.5}{\pi} \approx 0.159 \].
05

Find the wavelength

Using \( k = \pi \), the wavelength \( \lambda \) is given by \( \lambda = \frac{2\pi}{k} \). Therefore, \[ \lambda = \frac{2\pi}{\pi} = 2 \].
06

Plotting \( y \) as a function of \( x \)

For each \( t = 0, 1, 2, 3 \), substitute the value into the wave equation \( y = 3 \sin(\pi (x - \frac{1}{2}t)) \). Plot \( y \) against \( x \) for \( x = 0, \frac{1}{2}, 1, \frac{3}{2}, 2 \).
07

Plotting \( y \) as a function of \( t \)

For each \( x = 0, \frac{1}{2}, 1, \frac{3}{2}, 2 \), substitute the value into the wave equation \( y = 3 \sin(\pi (x - \frac{1}{2}t)) \). Plot \( y \) against \( t \) for \( t = 0, 1, 2, 3 \).

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Key Concepts

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

Amplitude
Amplitude is one of the core properties of a wave. It represents the maximum displacement of the wave from its equilibrium position. In simpler terms, it's the 'height' of the wave. The larger the amplitude, the taller the wave.

In the given equation, the amplitude is represented by the coefficient in front of the sine function. For the wave equation,
\( y = 3 \sin \pi \left( x - \frac{1}{2} t \right) \),
the amplitude is clearly 3. This means the wave will oscillate between +3 and -3 units.
This amplitude tells us about the energy carried by the wave: higher amplitude means higher energy.
Period
The period of a wave is the time it takes to complete one full cycle of oscillation. It tells us how long it takes for the wave to start repeating itself.

For sine waves, the period can be found using the formula:
\[ T = \frac{2\pi}{k} \]
where \( k \) is the coefficient inside the sine function.

For the given wave equation,
\( y = 3 \sin \pi \left( x - \frac{1}{2} t \right) \),
the coefficient is \( \pi \). Substituting, we get the period as
\( T = \frac{2\pi}{\pi} = 2 \).
This means that every 2 units of time, the wave pattern repeats itself.
Frequency
Frequency is the number of cycles a wave completes in one unit of time. It is the reciprocal of the period and is measured in Hertz (Hz).

For the given wave, once we have the period \( T \), we can find the frequency using:
\[ f = \frac{1}{T} \]
From the previous section, we know the period \( T \) is 2 units.
So, the frequency \( f = \frac{1}{2} = 0.5 \) Hz.
This means the wave completes half a cycle per unit of time.
Wave Velocity
Wave velocity is the speed at which the wave propagates through space. It can be computed if we know both the wavelength and the period.

The formula for wave velocity is:
\[ v = \frac{\text{wavelength}}{\text{period}} \]
In our given problem, the period \( T \) is 2 units and the wavelength needs to be found first.
Using the provided details,
we calculate the velocity using another relation:
\( v = \frac{\omega}{k} = \frac{0.5}{\pi} \approx 0.159 \).
This is the speed at which the wave travels.
Wavelength
The wavelength is the spatial period of the wave, which is the distance over which the wave's shape repeats.

In mathematical terms, if \( k \) is the spatial frequency (the coefficient of \( x \) in the sin function), then the wavelength \( \lambda \) is given by:
\[ \lambda = \frac{2\pi}{k} \]
For our given wave,
\( y = 3 \sin \pi \left( x - \frac{1}{2} t \right) \),
we have \( k = \pi \). Therefore,
\[ \lambda = \frac{2\pi}{\pi} = 2 \]
This means every 2 units of distance, the wave pattern repeats itself.
Sine Wave
A sine wave is a mathematical curve that describes a smooth periodic oscillation. It is one of the simplest forms of waves, described by the equation:
\( y = A \sin(Bx + C) \),
where:
- \( A \) is the amplitude,
- \( B \) is the wave number (related to wavelength),
- and \( C \) is the phase shift.

Our given wave equation,
\( y = 3 \sin \pi \left( x - \frac{1}{2} t \right) \),
fits this form, making it a sine wave.
Sine waves are important in many fields including physics, engineering, and signal processing because they describe a lot of natural phenomena and can simplify complex problems.

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Most popular questions from this chapter

Find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot \(f(x)\) and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section). $$f(x)=\left\\{\begin{array}{lr} 3 x^{2}+2 x^{3}, & -1< x< 0 \\ 3 x^{2}-2 x^{3}, & 0< x< 1 \end{array}\right.$$

You are given a complex function \(z=f(t) .\) In each case, show that a particle whose coordinate is (a) \(x=\operatorname{Re} z,\) (b) \(y=\operatorname{Im} z\) is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion. $$z=2 e^{i \pi t}$$

(a) Let \(f(x)\) on \((0,2 l)\) satisfy \(f(2 l-x)=f(x),\) that is, \(f(x)\) is symmetric about \(x=l .\) If you expand \(f(x)\) on \((0,2 l)\) in a sine series \(\sum b_{n} \sin \frac{n \pi x}{2 l},\) show that for even \(n, b_{n}=0 .\) Hint: Note that the period of the sines is \(4 l .\) Sketch an \(f(x)\) which is symmetric about \(x=l,\) and on the same axes sketch a few sines to see that the even ones are antisymmetric about \(x=l\). Alternatively, write the integral for \(b_{n}\) as an integral from 0 to \(l\) plus an integral from \(l\) to \(2 l,\) and replace \(x\) by \(2 l-x\) in the second integral. (b) Similarly, show that if we define \(f(2 l-x)=-f(x),\) the cosine series has \(a_{n}=0\) for even \(n\).

Find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot \(f(x)\) and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section). $$f(x)=\left(x^{2}-\pi^{2}\right)^{2}, \quad-\pi

You are given \(f(x)\) on an interval, say \(0< x< b\). Sketch several periods of the even function \(f_{c}\) of period \(2 b,\) the odd function \(f_{s}\) of period \(2 b,\) and the function \(f_{p}\) of period \(b\), each of which equals \(f(x)\) on \(0< x< b\). Expand each of the three functions in an appropriate Fourier series. $$f(x)=\left\\{\begin{array}{ll} 1, & 0< x< 1 \\ 0, & 1< x< 3 \end{array}\right.$$
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