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Chapter 11: Problem 19

What is the speed of the wave represented by \(y(x, t)=\) $A \sin (k x-\omega t),\( where \)k=6.0 \mathrm{rad} / \mathrm{cm}\( and \)\omega=5.0 \mathrm{rad} / \mathrm{s} ?$

Short Answer

Expert verified
Answer: The speed of the wave is \(\frac{1}{120}~\frac{\mathrm{m}}{\mathrm{s}}\).

Step by step solution

01

Identify the given values in the problem

The given values in the problem are: - Angular frequency, \(\omega=5.0 \mathrm{rad} / \mathrm{s}\) - Wave number, \(k=6.0 \mathrm{rad} / \mathrm{cm}\)
02

Convert the wave number to SI units

To make our calculations more consistent, we will convert the wave number from rad/cm to rad/m. To do this, we need to multiply the wave number by the conversion factor of 100 cm/m. \(k = 6.0 \mathrm{rad} / \mathrm{cm} \times 100 \mathrm{cm} / \mathrm{m} = 600 \mathrm{rad} / \mathrm{m}\)
03

Calculate the speed of the wave using the formula

Now that we have the wave number in SI units and the angular frequency, we will use the wave speed formula to determine the speed of the wave. \(v = \frac{\omega}{k} = \frac{5.0 \mathrm{rad} / \mathrm{s}}{600 \mathrm{rad} / \mathrm{m}}\)
04

Simplify the expression and solve for the speed

Simplify the expression and solve for \(v\): \(v = \frac{5.0}{600} \frac{\mathrm{rad} / \mathrm{s}}{\mathrm{rad} / \mathrm{m}} = \frac{1}{120} \frac{\mathrm{m}}{\mathrm{s}}\)
05

Write down the final answer

The speed of the wave represented by the given equation is: \(v = \frac{1}{120} \frac{\mathrm{m}}{\mathrm{s}}\)

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Most popular questions from this chapter

A guitar string has a fundamental frequency of \(300.0 \mathrm{Hz}\) (a) What are the next three lowest standing wave frequencies? (b) If you press a finger lightly against the string at its midpoint so that both sides of the string can still vibrate, you create a node at the midpoint. What are the lowest four standing wave frequencies now? (c) If you press hard at the same point, only one side of the string can vibrate. What are the lowest four standing wave frequencies?
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A transverse wave on a string is described by the equation $y(x, t)=(2.20 \mathrm{cm}) \sin [(130 \mathrm{rad} / \mathrm{s}) t+(15 \mathrm{rad} / \mathrm{m}) x].$ (a) What is the maximum transverse speed of a point on the string? (b) What is the maximum transverse acceleration of a point on the string? (c) How fast does the wave move along the string? (d) Why is your answer to (c) different from the answer to (a)?
For a transverse wave on a string described by $y(x, t)=(0.0050 \mathrm{m}) \cos [(4.0 \pi \mathrm{rad} / \mathrm{s}) t-(1.0 \pi \mathrm{rad} / \mathrm{m}) x]$ find the maximum speed and the maximum acceleration of a point on the string. Plot graphs for one cycle of displacement \(y\) versus \(t\), velocity \(v_{y}\) versus \(t\), and acceleration \(a_{y}\) versus \(t\) at the point \(x=0.\)
A seismic wave is described by the equation $y(x, t)=(7.00 \mathrm{cm}) \cos [(6.00 \pi \mathrm{rad} / \mathrm{cm}) x+(20.0 \pi \mathrm{rad} / \mathrm{s}) t]\( The wave travels through a uniform medium in the \)x$ -direction. (a) Is this wave moving right ( \(+x\) -direction) or left (-x-direction)? (b) How far from their equilibrium positions do the particles in the medium move? (c) What is the frequency of this wave? (d) What is the wavelength of this wave? (e) What is the wave speed? (f) Describe the motion of a particle that is at \(y=7.00 \mathrm{cm}\) and \(x=0\) when \(t=0 .\) (g) Is this wave transverse or longitudinal?
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