Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 64

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?

Short Answer

Expert verified
Answer: The next three lowest standing wave frequencies are 600 Hz, 900 Hz, and 1200 Hz. The lowest four standing wave frequencies with a node at the midpoint are 600 Hz, 1200 Hz, 1800 Hz, and 2400 Hz. The lowest four standing wave frequencies with only one side vibrating are 150 Hz, 300 Hz, 450 Hz, and 600 Hz.

Step by step solution

01

Identify the fundamental frequency

The fundamental frequency (also known as the first harmonic) is given as 300 Hz.
02

Calculate the frequencies of the next three harmonics

To calculate the frequencies of the next three harmonics (2nd, 3rd, and 4th harmonics), simply multiply the fundamental frequency by the harmonic number. 2nd harmonic frequency: \(300\,\text{Hz} \times 2 = 600\,\text{Hz}\) 3rd harmonic frequency: \(300\,\text{Hz} \times 3 = 900\,\text{Hz}\) 4th harmonic frequency: \(300\,\text{Hz} \times 4 = 1200\,\text{Hz}\) So, the next three lowest standing wave frequencies are 600 Hz, 900 Hz, and 1200 Hz. #b) Lowest four standing wave frequencies with a node at the midpoint#
03

Identify the harmonic pattern with a node at the midpoint

When a node is created at the midpoint of the string, only even harmonics can be produced because the string is essentially divided into two equal parts.
04

Calculate the lowest four even harmonics

We will use the same method as before, but only include even multiples of the fundamental frequency. 2nd harmonic frequency: \(300\,\text{Hz} \times 2 = 600\,\text{Hz}\) 4th harmonic frequency: \(300\,\text{Hz} \times 4 = 1200\,\text{Hz}\) 6th harmonic frequency: \(300\,\text{Hz} \times 6 = 1800\,\text{Hz}\) 8th harmonic frequency: \(300\,\text{Hz} \times 8 = 2400\,\text{Hz}\) The lowest four standing wave frequencies with a node at the midpoint are 600 Hz, 1200 Hz, 1800 Hz, and 2400 Hz. #c) Lowest four standing wave frequencies with only one side vibrating#
05

Identify the new fundamental frequency with only one side vibrating

When only one side of the string is allowed to vibrate, we effectively double the length of the vibrating section. Therefore, the new fundamental frequency will be half of the original fundamental frequency: \(150\,\text{Hz}\).
06

Calculate the lowest four harmonics with the new fundamental frequency

Now we will multiply the new fundamental frequency by the harmonic numbers to find the lowest four standing wave frequencies. 1st harmonic frequency: \(150\,\text{Hz} \times 1 = 150\,\text{Hz}\) 2nd harmonic frequency: \(150\,\text{Hz} \times 2 = 300\,\text{Hz}\) 3rd harmonic frequency: \(150\,\text{Hz} \times 3 = 450\,\text{Hz}\) 4th harmonic frequency: \(150\,\text{Hz} \times 4 = 600\,\text{Hz}\) The lowest four standing wave frequencies with only one side vibrating are 150 Hz, 300 Hz, 450 Hz, and 600 Hz.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) Write an equation for a surface seismic wave moving along the \(-x\) -axis with amplitude \(2.0 \mathrm{cm},\) period \(4.0 \mathrm{s}\) and wavelength $4.0 \mathrm{km} .\( Assume the wave is harmonic, \)x\( is measured in \)\mathrm{m},$ and \(t\) is measured in \(\mathrm{s}\). (b) What is the maximum speed of the ground as the wave moves by? (c) What is the wave speed?
A transverse wave on a string is described by $y(x, t)=(1.2 \mathrm{cm}) \sin [(0.50 \pi \mathrm{rad} / \mathrm{s}) t-(1.00 \pi \mathrm{rad} / \mathrm{m}) x]$ Find the maximum velocity and the maximum acceleration of a point on the string. Plot graphs for displacement \(y\) versus \(t\), velocity \(v_{y}\) versus \(t\), and acceleration \(a_{y}\) versus \(t\) at \(x=0.\)
A standing wave has wave number $2.0 \times 10^{2} \mathrm{rad} / \mathrm{m} .$ What is the distance between two adjacent nodes?
Two speakers spaced a distance \(1.5 \mathrm{m}\) apart emit coherent sound waves at a frequency of \(680 \mathrm{Hz}\) in all directions. The waves start out in phase with each other. A listener walks in a circle of radius greater than one meter centered on the midpoint of the two speakers. At how many points does the listener observe destructive interference? The listener and the speakers are all in the same horizontal plane and the speed of sound is \(340 \mathrm{m} / \mathrm{s} .\) [Hint: Start with a diagram; then determine the maximum path difference between the two waves at points on the circle. I Experiments like this must be done in a special room so that reflections are negligible.
A cord of length \(1.5 \mathrm{m}\) is fixed at both ends. Its mass per unit length is \(1.2 \mathrm{g} / \mathrm{m}\) and the tension is \(12 \mathrm{N} .\) (a) What is the frequency of the fundamental oscillation? (b) What tension is required if the \(n=3\) mode has a frequency of \(0.50 \mathrm{kHz} ?\)
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Particle Model of Matter

Read Explanation

Electricity and Magnetism

Read Explanation

Geometrical and Physical Optics

Read Explanation

Electricity

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free