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Chapter 15: Problem 45

(1) In an earthquake, it is noted that a footbridge oscillated up and down in a one-loop (fundamental standing wave) pattern once every 1.5 \(\mathrm{s}\) . What other possible resonant periods of motion are there for this bridge? What frequencies do they correspond to?

Short Answer

Expert verified
The possible resonant periods are 0.75 s, 0.5 s, 0.375 s, etc., with corresponding frequencies 1.333 Hz, 2 Hz, 2.667 Hz, etc.

Step by step solution

01

Understand the Problem

The problem states that a footbridge oscillated in a fundamental standing wave pattern with a period of 1.5 seconds. Our task is to find other possible resonant periods and their corresponding frequencies.
02

Identify the Fundamental Period

The fundamental period of a standing wave is given as 1.5 seconds. This period represents the first harmonic of the system, also known as the fundamental frequency.
03

Calculate the Fundamental Frequency

The frequency is the reciprocal of the period. Therefore, the fundamental frequency \(f_1\) can be calculated as:\[\text{Fundamental Frequency} = \frac{1}{1.5} = 0.6667 \text{ Hz}\]
04

Identify Harmonics and Resonant Periods

Resonant frequencies occur at harmonics of the fundamental frequency. If the fundamental frequency is \(f_1\), the harmonics can be expressed as \(f_n = n \times f_1\), where \(n\) is a positive integer.
05

Calculate Frequencies of Higher Harmonics

Using the formula \(f_n = n \times f_1\), calculate the frequencies:1. Second harmonic: \(f_2 = 2 \times 0.6667 = 1.3334 \text{ Hz}\).2. Third harmonic: \(f_3 = 3 \times 0.6667 = 2.0001 \text{ Hz}\).3. Fourth harmonic: \(f_4 = 4 \times 0.6667 = 2.6668 \text{ Hz}\), and so on.Continue this calculation for higher harmonics as needed.
06

Calculate Corresponding Periods

Find the periods for these frequencies by taking their reciprocal. Specifically:1. Second harmonic period: \(\frac{1}{f_2} = \frac{1}{1.3334} \approx 0.75 \text{ s}\).2. Third harmonic period: \(\frac{1}{f_3} = \frac{1}{2.0001} \approx 0.5 \text{ s}\).3. Fourth harmonic period: \(\frac{1}{f_4} = \frac{1}{2.6668} \approx 0.375 \text{ s}\), and so on.

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Key Concepts

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

Harmonics
Harmonics are an essential concept in understanding standing waves, such as those observed in bridges during an earthquake. The fundamental frequency of a system characterizes the simplest wave pattern, often called the first harmonic. When a bridge oscillates in this pattern, it moves in one complete loop. This fundamental oscillation period is a starting point in assessing harmonics.
  • The first harmonic is the basis of the wave pattern and exhibits the longest wavelength.
  • Higher harmonics build on this, representing more complex patterns with shorter wavelengths.
To calculate harmonics, we use the relationship that for any harmonic number \(n\), the frequency \(f_n\) is \(n\) times the fundamental frequency \(f_1\). For example, the second harmonic is twice the fundamental frequency, leading to increased complexity and more nodes in the wave pattern. Understanding harmonics allows us to predict how different parts of a structure might respond to various frequencies and helps in designing structures to withstand such stresses.
Resonant Frequency
Resonant frequencies are the natural frequencies at which a system tends to oscillate. For the footbridge, when shaken at its resonant frequency, it exhibits large amplitude oscillations. The fundamental frequency of 0.6667 Hz represents the simplest form of this resonance. However, there are other resonant frequencies based on harmonics.
  • These occur at multiples of the fundamental frequency, so in general, \(f_n = n \times f_1\).
  • An example is the second harmonic, which resonates at 1.3334 Hz, followed by the third at 2.0001 Hz.
Predicting a structure's resonant frequencies helps in avoiding potential destructive oscillation during an earthquake. Engineers can use this information to create designs that minimize resonant amplification, thereby protecting structures from excessive vibrations.
Oscillation Period
The oscillation period of a wave is the time it takes to complete one full cycle. In the context of a footbridge during an earthquake, this period corresponds to the standing wave pattern observed. The fundamental period of 1.5 seconds represents the time for the bridge to complete one oscillation in its simplest form.
  • Inverse of frequency: The period \(T\) is the reciprocal of the frequency \(f\), such that \(T = \frac{1}{f}\).
  • For example, for a frequency of 1.3334 Hz (second harmonic), the period is approximately 0.75 seconds.
Understanding the oscillation period is vital for analyzing how quickly a system responds to an external force. By knowing the periods of various harmonics, engineers can better prepare for potential resonance disasters by predicting how long these oscillations will affect the structure under stress.

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Most popular questions from this chapter

A wave with a frequency of \(220 \mathrm{~Hz}\) and a wavelength of \(10.0 \mathrm{~cm}\) is traveling along a cord. The maximum speed of particles on the cord is the same as the wave speed. What is the amplitude of the wave?

(II) A ski gondola is connected to the top of a hill by a steel cable of length \(660 \mathrm{~m}\) and diameter \(1.5 \mathrm{~cm} .\) As the gondola comes to the end of its run, it bumps into the terminal and sends a wave pulse along the cable. It is observed that it took \(17 \mathrm{~s}\) for the pulse to return, \((a)\) What is the speed of the pulse? (b) What is the tension in the cable?

(I) A fisherman notices that wave crests pass the bow of his anchored boat every \(3.0 \mathrm{~s}\). He measures the distance between two crests to be \(8.0 \mathrm{~m}\). How fast are the waves traveling?

A sinusoidal traveling wave has frequency \(880 \mathrm{~Hz}\) and phase velocity \(440 \mathrm{~m} / \mathrm{s}\). ( \(a\) ) At a given time, find the distance between any two locations that correspond to a difference in phase of \(\pi / 6\) rad. \((b)\) At a fixed location, by how much does the phase change during a time interval of \(1.0 \times 10^{-4} \mathrm{~s} ?\)

A highway overpass was observed to resonate as one full loop \(\left(\frac{1}{2} \lambda\right)\) when a small earthquake shook the ground vertically at \(3.0 \mathrm{~Hz}\). The highway department put a support at the center of the overpass, anchoring it to the ground as shown in Fig. \(15-41 .\) What resonant frequency would you now expect for the overpass? It is noted that earthquakes rarely do significant shaking above 5 or \(6 \mathrm{~Hz}\). Did the modifications do any good? Explain.
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