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Magazine Chapter 17: Problem 30
Two wires, each of length 1.2 m, are stretched between two fixed supports. On wire A there is a second-harmonic standing wave whose frequency is 660 Hz. However, the same frequency of 660 Hz is the third harmonic on wire B. Find the speed at which the individual waves travel on each wire.
Short Answer
Expert verified
Wire A speed: 792 m/s; Wire B speed: 528 m/s.
Step by step solution
01
Understanding Harmonics
Standing waves are formed at specific frequencies known as harmonics. The second harmonic has two loops, and the third harmonic has three loops. These harmonics are related to the wave speed, frequency, and length of the wire.
02
Identifying the Wave Formula
The general formula for the frequency of the n-th harmonic is given by \( f_n = \frac{n \cdot v}{2L} \) for wire harmonics, where \( n \) is the harmonic number, \( v \) is the wave speed, and \( L \) is the length of the wire.
03
Solving for the Wave Speed on Wire A
For wire A at the second harmonic, where \( n = 2 \), the frequency \( f = 660 \, \text{Hz} \), and length \( L = 1.2 \, \text{m} \), solve for the wave speed \( v \):\[v = \frac{2Lf}{n} = \frac{2 \times 1.2 \, \text{m} \times 660 \, \text{Hz}}{2} = 792 \, \text{m/s}.\]
04
Solving for the Wave Speed on Wire B
For wire B at the third harmonic, where \( n = 3 \), the frequency \( f = 660 \, \text{Hz} \), and length \( L = 1.2 \, \text{m} \), solve for the wave speed \( v \):\[v = \frac{2Lf}{n} = \frac{2 \times 1.2 \, \text{m} \times 660 \, \text{Hz}}{3} = 528 \, \text{m/s}.\]
05
Conclusion on Wave Speeds
From the calculations, the wave speed on wire A is 792 m/s and on wire B is 528 m/s.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Speed
Wave speed is an essential concept in physics, particularly when discussing waves on strings or wires. It indicates how quickly the wave travels along the medium. In the context of the problem with the two wires, knowing the wave speed helps determine how harmonics behave and relate to each other in different setups.
Given the formula for harmonic frequency, we use it to derive the wave speed. For any wire with a fixed length, if you know the harmonic frequency and number, you can easily calculate the speed using the formula:
Given the formula for harmonic frequency, we use it to derive the wave speed. For any wire with a fixed length, if you know the harmonic frequency and number, you can easily calculate the speed using the formula:
- \[v = \frac{2Lf}{n}\]
- \(v\) is the wave speed
- \(L\) is the length of the wire
- \(f\) is the frequency
- \(n\) is the harmonic number
Standing Waves
Standing waves are a fascinating phenomenon involving waves traveling in opposite directions. They appear as waves that are stationary, with certain points called nodes remaining still while others, known as antinodes, vibrate with maximum amplitude.
These waves occur when two identical waves interfere with each other under specific conditions. When a wave reflects back upon reaching a fixed endpoint like a clamped wire, it can interfere with incoming waves to form standing waves.
Standing waves support certain frequencies called harmonics, and understanding these can help us predict how different systems behave based on their dimensions and tension.
These waves occur when two identical waves interfere with each other under specific conditions. When a wave reflects back upon reaching a fixed endpoint like a clamped wire, it can interfere with incoming waves to form standing waves.
Standing waves support certain frequencies called harmonics, and understanding these can help us predict how different systems behave based on their dimensions and tension.
Harmonics
Harmonics are the specific frequencies at which standing waves form. Each harmonic corresponds to a particular pattern of nodes and antinodes:
- The first harmonic (or fundamental frequency) has one half-wavelength fitting within the wire's length.
- The second harmonic doubles the number of antinodes, producing two loops within the same length.
- The third harmonic has three loops, and so on.
Frequency
Frequency is the number of oscillations or cycles a wave completes in one second, measured in Hertz (Hz). It's a critical property that relates closely to wave speed and harmonics. Every harmonic has a specific frequency it resonates at, and in many practical situations, like musical instruments, this determines the pitch we hear.
The frequency formula, as used in this exercise, is:
The frequency formula, as used in this exercise, is:
- \[f_n = \frac{n \cdot v}{2L}\]
- As the number of harmonics increases, frequency does too, assuming constant wave speed and length.
- A higher frequency indicates a higher pitch sound if the waves are sound waves on instruments.
Wave Formula
The wave formula is a mathematical expression used to relate wave properties like speed, frequency, and wavelength. When discussing harmonics, the specific formula employed is tailored to standing waves on strings:
By rearranging this expression, we can solve for different unknowns, like calculating wave speed when given frequency and harmonic information. Thus, mastering this formula allows one to unravel complex wave systems simply by knowing a few variables.
- General formula for harmonic frequency: \[f_n = \frac{n \cdot v}{2L}\]
By rearranging this expression, we can solve for different unknowns, like calculating wave speed when given frequency and harmonic information. Thus, mastering this formula allows one to unravel complex wave systems simply by knowing a few variables.
