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Chapter 15: Problem 38

A 1.50 -m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 \(\mathrm{m} / \mathrm{s}\) . What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

Short Answer

Expert verified
Fundamental: \( \lambda = 3.00 \text{ m}, f = 16.0 \text{ Hz} \); 2nd overtone: \( \lambda = 1.00 \text{ m}, f = 48.0 \text{ Hz} \); 4th harmonic: \( \lambda = 0.75 \text{ m}, f = 64.0 \text{ Hz} \).

Step by step solution

01

Understanding the Problem

We are tasked with finding the wavelength and frequency of transverse waves on a rope stretched between two supports. Given the length of the rope is 1.50 m, with a wave speed of 48.0 m/s, and we need to determine the fundamental, second overtone, and fourth harmonic.
02

Find the Wavelength of the Fundamental

The fundamental frequency (first harmonic) corresponds to half a wavelength fitting the length of the rope. Therefore, the wavelength \( \lambda_1 \) is given by:\[ \lambda_1 = 2 \cdot \text{Length of the rope} = 2 \cdot 1.50 \text{ m} = 3.00 \text{ m} \]
03

Calculate the Frequency of the Fundamental

The frequency \( f_1 \) of the fundamental is calculated using the wave equation \( v = f \lambda \), where \( v \) is the wave speed. Plug in the values:\[ \begin{align*}f_1 &= \frac{v}{\lambda_1} \f_1 &= \frac{48.0 \text{ m/s}}{3.00 \text{ m}} \f_1 &= 16.0 \text{ Hz}\end{align*} \]
04

Wavelength and Frequency of the Second Overtone

The second overtone corresponds to the third harmonic, where 1.5 wavelengths fit along the rope length. The wavelength \( \lambda_3 \) is:\[ \lambda_3 = \frac{2}{3} \cdot \text{Length of the rope} = \frac{2}{3} \cdot 1.50 \text{ m} = 1.00 \text{ m} \] The frequency \( f_3 \) is:\[ \begin{align*}f_3 &= \frac{v}{\lambda_3} \f_3 &= \frac{48.0 \text{ m/s}}{1.00 \text{ m}} \f_3 &= 48.0 \text{ Hz}\end{align*} \]
05

Wavelength and Frequency of the Fourth Harmonic

The fourth harmonic corresponds to two full wavelengths fitting into the rope. The wavelength \( \lambda_4 \) is:\[ \lambda_4 = \frac{1}{2} \cdot \text{Length of the rope} = \frac{1}{2} \cdot 1.50 \text{ m} = 0.75 \text{ m} \] The frequency \( f_4 \) is:\[ \begin{align*}f_4 &= \frac{v}{\lambda_4} \f_4 &= \frac{48.0 \text{ m/s}}{0.75 \text{ m}} \f_4 &= 64.0 \text{ Hz}\end{align*} \]

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Key Concepts

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

Transverse Waves
Transverse waves are a key concept in wave mechanics and are crucial for understanding how energy travels through mediums like ropes or strings. In transverse waves, the motion of the particles of the medium is perpendicular to the direction of the wave's travel.
Picture the ripples on a water surface: as the wave passes, the water moves up and down, while the wave itself travels horizontally across the surface. This is similar to how a transverse wave moves through a rope.
  • The amplitude is the height of the wave above or below its rest position.
  • The crest is the highest point, and the trough is the lowest.
  • Wavelength is the distance between two consecutive crests or troughs.
  • The speed of the wave depends on the tension and mass per unit length of the rope.
Understanding these fundamental characteristics helps in analyzing the behavior of the wave when it is produced on a real physical medium, such as a rope or string.
Wavelength Calculation
Calculating the wavelength of transverse waves involves understanding the relationship between the wave speed, wavelength, and frequency. The wave equation, expressed as \( v = f \lambda \), plays a fundamental role here. **\( v \)** represents the wave speed, **\( f \)** the frequency, and **\( \lambda \)** the wavelength.
Let's break down the process:
  • The wave speed, \( v \), is usually given. In this scenario, it was calculated as 48.0 m/s, determined by the tension in the rope.
  • The fundamental frequency involves half a wavelength fitting the length of the rope. Using the relationship, \( \lambda_1 = 2 \cdot L \), we calculate the wavelength for the fundamental frequency.
  • For the second overtone, or third harmonic, one and a half wavelengths fit into the rope, given by \( \lambda_3 = \frac{2}{3} \cdot L \).
  • The fourth harmonic involves two full wavelengths within the rope, expressed as \( \lambda_4 = \frac{1}{2} \cdot L \).
These steps reveal how different harmonics require distinct fractions of the rope length for calculating the appropriate wavelengths.
Harmonic Frequencies
Harmonic frequencies are integral to understanding how waves split themselves into different modes, or patterns, when confined within a medium like a rope or a string. A harmonic is essentially a resonance condition where the wave fits seamlessly within the boundaries of the rope.
Let's delve into the specifics:
  • The fundamental frequency, often referred to as the first harmonic, is the simplest form, with the wave fitting in such that one half of the wavelength equals the length of the rope.
  • The second overtone is equivalent to the third harmonic. Here, the condition modifies to allow one and a half wavelengths to fit along the rope, resulting in a higher frequency.
  • The fourth harmonic condition requires two full wavelengths along the rope, producing yet a higher frequency than the previous conditions.
The determination of these harmonic frequencies involves using the wave speed and wavelength to compute different frequencies via the formula \( f = \frac{v}{\lambda} \).This understanding allows us to predict the behavior of the wave as it interacts with the physical constraints imposed by the rope's length and tension.

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Most popular questions from this chapter

A water wave traveling in a straight line on a lake is described by the equation $$ y(x, t)=(3.75 \mathrm{cm}) \cos \left(0.450 \mathrm{cm}^{-1} x+5.40 \mathrm{s}^{-1} t\right) $$ where \(y\) is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

Tuning an Instrument. A musician tunes the C.string of her instrument to a fundamental frequency of 65.4 \(\mathrm{Hz}\) . The vibrating portion of the string is 0.600 \(\mathrm{m}\) long and has a mass of 14.4 \(\mathrm{g}\) . (a) With what tension must the musician stretch it? (b) What percent increase in tension is needed to increase the frequency from 65.4 \(\mathrm{Hz}\) to 73.4 \(\mathrm{Hz}\) , corresponding to a rise in pitch from \(\mathrm{C}\) to \(\mathrm{D} ?\)

Two traveling waves moving on a string are identical except for opposite velocitles. They obey the equation \(y(x, t)=\) \(A \sin (k x \pm \omega t),\) where the plus-or-minus sign in the argument depends on the direction the wave is traveling. (a) Show that the vibrating string is described by the equation \(y_{\text { net }}(x, t)=\) \(2 A \sin k x \cos \omega t,\) (Hint: Use the trigonometric formulas for \(\sin (a \pm b) . )\) (b) Show that the string never moves at the places along it for which \(x=n \lambda / 2,\) where \(n\) is a nonnegative integer.

Waves of Arbitrary Shape. (a) Explain why any wave described by a function of the form \(y(x, t)=f(x-v t)\) moves in the \(+x\) -direction with speed \(v\) . (b) Show that \(y(x, t)=f(x-v t)\) satisfies the wave equation, no matter what the functional form of \(f\) . To do this, write \(y(x, t)=f(u),\) where \(u=x-v t\) . Then, to take partial derivatives of \(y(x, t),\) use the chain rule: $$ \begin{array}{l}{\frac{\partial y(x, t)}{\partial t}=\frac{d f(u)}{d u} \frac{\partial u}{\partial t}=\frac{d f(u)}{d u}(-v)} \\ {\frac{\partial y(x, t)}{\partial x}=\frac{d f(u)}{d u} \frac{\partial u}{\partial x}=\frac{d f(u)}{d u}}\end{array} $$ (c) A wave pulse is described by the function \(y(x, t)=D e^{-(B x-C)^{2}}\) where \(B, C,\) and \(D\) are all positive constants. What is the speed of this wave?

Guitar String. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(\mathbf{B}_{3}\) (frequency 245 \(\mathrm{Hz} )\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 \(\mathrm{m} / \mathrm{s}\) , find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?
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