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Chapter 16: Problem 9

A wave is described by \(y=(2.00 \mathrm{cm}) \sin (k x-\omega t),\) where \(k=2.11 \mathrm{rad} / \mathrm{m}, \omega=3.62 \mathrm{rad} / \mathrm{s}, x\) is in meters, and \(t\) is in seconds. Determine the amplitude, wavelength, frequency, and speed of the wave.

Short Answer

Expert verified
The amplitude is \(0.02 \mathrm{m}\), the wavelength is \(2.98 \mathrm{m}\), the frequency is \(0.58 \mathrm{Hz}\), and the speed is \(1.73 \mathrm{m/s}\).

Step by step solution

01

Determine the Amplitude

The amplitude of the wave is the coefficient of the sinusoidal function in the given equation. Therefore, the amplitude is \(2.00 \mathrm{cm}\), or converted to meters \(0.02 \mathrm{m}\).
02

Determine the Wavelength

The wavelength can be found using the wave number \(k\) by rearranging the formula \(k = \frac{2\pi}{\lambda}\) to give \(\lambda = \frac{2\pi}{k}\). Substituting \(k = 2.11 \mathrm{rad/m}\) gives \(\lambda \approx \frac{2\pi}{2.11} \approx 2.98 \mathrm{m}\).
03

Determine the Frequency

The frequency can be found using the angular frequency \(\omega\) by rearranging the formula \(\omega = 2\pi f\) to give \(f = \frac{\omega}{2\pi}\). Substituting \(\omega = 3.62 \mathrm{rad/s}\) gives \(f \approx \frac{3.62}{2\pi} \approx 0.58 \mathrm{Hz}\).
04

Determine the Speed

The speed of the wave is the ratio of the wavelength to the period (or the inverse of the frequency). Using the formula \(v = \lambda f\) with \(\lambda = 2.98 \mathrm{m}\) and \(f = 0.58 \mathrm{Hz}\), you get \(v \approx 2.98 * 0.58 \approx 1.73 \mathrm{m/s}\).

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Key Concepts

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

Amplitude
Amplitude refers to the maximum extent of a wave as it oscillates from its equilibrium position. It is an indicator of the wave's strength or intensity. You can imagine it as the wave's height when it reaches its peak.
In the equation provided, the amplitude is the constant in front of the sine function. For our wave equation, the amplitude is clearly stated as 2.00 cm. Recognizing this is crucial because amplitude helps determine the energy being carried by the wave. But why convert it to meters? Since most scientific equations use the International System of Units (SI), we convert centimeters to meters, giving us an amplitude of 0.02 m.
Wavelength
Wavelength is the distance between two consecutive similar points of a wave, such as crest to crest or trough to trough. It is a measure of the repeating pattern of the wave over space.
The formula for the relationship (known as the wave number) is\(k = \frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength. Rearranging this formula helps us find the wavelength from the wave number. So, when \(k = 2.11 \text{ rad/m}\), use \(\lambda = \frac{2\pi}{k}\). Calculating this, the resulting wavelength is approximately 2.98 meters, which illustrates how far one cycle of the wave stretches out along its path.
Frequency
Frequency defines how often the wave completes a cycle per unit of time and is measured in Hertz (Hz). It tells us how quick the oscillations are occurring.
From the given equation, the angular frequency \(\omega = 3.62 \text{ rad/s}\). The connection between angular frequency and regular frequency is modeled by the equation \(\omega = 2\pi f\). By manipulating it to \(f = \frac{\omega}{2\pi}\), we can find the frequency by plugging in the value of \(\omega\). Using these clues, we find that the frequency is approximately 0.58 Hz. This frequency is essential for understanding how rapid the wave vibrates over time.
Wave Speed
Wave speed is the rate at which the wave travels through the medium. Knowing the wave's speed can be crucial for many practical applications like measuring oceanographic movements or data transmission.
Using the relationship \(v = \lambda f\), where \(v\) represents the wave speed, you can calculate how fast the wave traverses its medium. For the given data, with \(\lambda = 2.98 \text{ m}\) and \(f = 0.58 \text{ Hz}\), the wave speed would be calculated as \(v \approx 2.98 \times 0.58\), yielding approximately 1.73 meters per second. This speed demonstrates how swiftly a wave propagates, affecting both the energy it can transmit and the distance it can cover within a time frame.

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Most popular questions from this chapter

A student taking a quiz finds on a reference sheet the two equations $$f=1 / T \text { and } v=\sqrt{T / \mu}$$ She has forgotten what \(T\) represents in each equation. (a) Use dimensional analysis to determine the units required for \(T\) in each equation. (b) Identify the physical quantity each \(T\) represents.

(a) Evaluate \(A\) in the scalar equality \((7+3) 4=A\). (b) Evaluate \(A, B,\) and \(C\) in the vector equality \(7.00 \hat{\mathbf{i}}+\) \(3.00 \hat{\mathbf{k}}=A \hat{\mathbf{i}}+B \hat{\mathbf{j}}+C \hat{\mathbf{k}} .\) Explain how you arrive at the answers to convince a student who thinks that you cannot solve a single equation for three different unknowns. (c) What If? The functional equality or identity \(A+B \cos (C x+D t+E)=(7.00 \mathrm{mm}) \cos (3 x+4 t+2)\) is true for all values of the variables \(x\) and \(t,\) which are measured in meters and in seconds, respectively. Evaluate the constants \(A, B, C, D,\) and \(E .\) Explain how you arrive at the answers.

In a region far from the epicenter of an earthquake, a seismic wave can be modeled as transporting energy in a single direction without absorption, just as a string wave does. Suppose the seismic wave moves from granite into mudfill with similar density but with a much lower bulk modulus. Assume the speed of the wave gradually drops by a factor of \(25.0,\) with negligible reflection of the wave. Will the amplitude of the ground shaking increase or decrease? By what factor? This phenomenon led to the collapse of part of the Nimitz Freeway in Oakland, California, during the Loma Prieta earthquake of 1989.

A two-dimensional water wave spreads in circular ripples. Show that the amplitude \(A\) at a distance \(r\) from the initial disturbance is proportional to \(1 / \sqrt{r}\). (Suggestion: Consider the energy carried by one outward- moving ripple.)

When a particular wire is vibrating with a frequency of \(4.00 \mathrm{Hz},\) a transverse wave of wavelength \(60.0 \mathrm{cm}\) is produced. Determine the speed of waves along the wire.
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