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Chapter 13: Problem 68

Will a standing wave be formed in a 4.0 -m length of stretched string that transmits waves at a speed of \(12 \mathrm{~m} / \mathrm{s}\) if it is driven at a frequency of (a) \(15 \mathrm{~Hz}\) or (b) \(20 \mathrm{~Hz} ?\)

Short Answer

Expert verified
Standing waves form at both 15 Hz and 20 Hz frequencies.

Step by step solution

01

Understanding the Need for a Standing Wave

A standing wave is formed when waves reflect back on themselves and interfere. For a standing wave to form on a string of length \(L\), the condition is \(L = n \times \frac{\lambda}{2}\), where \(n\) is a positive integer, and \(\lambda\) is the wavelength.
02

Calculate the Wavelength

The wavelength \(\lambda\) can be found from the wave speed \(v\) and frequency \(f\) using the formula \(\lambda = \frac{v}{f}\). With \(v = 12 \mathrm{~m/s}\), calculate the wavelength for both frequencies given.
03

Check Frequency (a) 15 Hz

Calculate \(\lambda\) for \(f = 15 \mathrm{~Hz}\): \[\lambda = \frac{12 \mathrm{~m/s}}{15 \mathrm{~Hz}} = 0.8 \mathrm{~m}\] Evaluate \(L = 4.0 \mathrm{~m} = n \times \frac{0.8 \mathrm{~m}}{2}\). Possible values of \(n\) are \(n = 1, 2, 3, 4, 5\), which are all integers, indicating a standing wave can form.
04

Check Frequency (b) 20 Hz

Calculate \(\lambda\) for \(f = 20 \mathrm{~Hz}\): \[\lambda = \frac{12 \mathrm{~m/s}}{20 \mathrm{~Hz}} = 0.6 \mathrm{~m}\]Evaluate \(L = 4.0 \mathrm{~m} = n \times \frac{0.6 \mathrm{~m}}{2}\). Determine if \(n\) yields integers. For \(n = 1, 2, 3, \,\dots\), this also forms integer values, indicating standing waves.

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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 a key concept when discussing waves on a string. It tells us how fast the wave travels through the medium. In this context, the medium is a stretched string.
The formula to calculate wave speed is:
  • Wave Speed (v) = Distance traveled by a wave per unit time.
For the exercise in question, the speed of the wave is given as 12 m/s.
Wave speed can be impacted by several factors:
  • The tension in the string
  • The mass per unit length of the string
Understanding wave speed helps determine how quickly waves move along the string, which is crucial for identifying standing wave formations.
Wavelength
Wavelength, denoted by the Greek letter \(\lambda\), represents the distance between consecutive crests (or troughs) of a wave. This measure is critical for understanding wave interactions.
  • Formula: \(\lambda = \frac{v}{f}\) where \(v\) is wave speed and \(f\) is frequency.
  • For a wave to fit a string of length \(L\) and form standing waves, multiples of half-wavelengths must match \(L\).
In the exercise at hand, different frequencies resulted in different wavelengths:
  • A 15 Hz frequency yielded a wavelength of 0.8 m.
  • A 20 Hz frequency produced a wavelength of 0.6 m.
These wavelengths help assess whether complete cycles can fit perfectly in a 4.0 m string, thus forming standing waves.
Frequency
Frequency is the number of complete wave cycles passing a point per second. Measured in hertz (Hz), it directly impacts the wavelength and the behavior of waves on a string.
In the example challenge, two frequencies are analyzed: 15 Hz and 20 Hz.
  • A higher frequency (20 Hz) means waves with shorter wavelengths (0.6 m) when speed is constant.
  • A lower frequency (15 Hz) results in longer wavelengths (0.8 m).
The frequency determines how many half-wavelengths fit within the string's length, essential for standing wave formations. Frequency is vital for sound propagation and wave interference analysis, especially in musical instruments.
Interference
Interference is a phenomenon where two or more waves superpose to form a resultant wave. This can occur constructively or destructively, depending on the phase relationship.
  • Constructive Interference: Occurs when peaks coincide, intensifying the wave.
  • Destructive Interference: Occurs when peaks meet troughs, reducing or canceling the wave.
In standing waves, constructive interference leads to nodes and antinodes that stabilize certain standing positions on the string.
Understanding interference helps in predicting patterns and point stability within a wave structure, crucial for applications like acoustics and optical technologies.
Wave Reflection
When waves hit a boundary or end of a medium, they can reflect back. Reflection is a change in direction of the wave, remaining in the same medium.
  • On a fixed end, the wave inverts upon reflection.
  • On a free end, the wave maintains its original orientation.
Wave reflection is a fundamental component in the formation of standing waves, as it causes waves to continuously reflect and overlap on a string. This overlap can lead to the creation of nodes (points of no movement). Understanding reflection aids in grasping wave behaviors and interactions, such as how waves can maintain stability within certain physical constraints, a principle seen in instruments and architectural acoustics.

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Most popular questions from this chapter

A piece of steel string is under tension. (a) If the tension doubles, the transverse wave speed (1) doubles, (2) halves, (3) increases by \(\sqrt{2},\) (4) decreases by \(\sqrt{2}\). Why? (b) If the linear mass density of a 10.0 -m length of string is \(0.125 \mathrm{~kg} / \mathrm{m}\) and it is under a tension of \(9.00 \mathrm{~N}\), what is the transverse wave speed in the string? (c) What are its waves' natural frequencies?

\(\bullet\bullet\) The range of sound frequencies audible to the human ear extends from about \(20 \mathrm{~Hz}\) to \(20 \mathrm{kHz}\). If the speed of sound in air is \(345 \mathrm{~m} / \mathrm{s},\) what are the wavelength limits of this audible range?

\(\bullet\bullet\bullet\) The forces acting on a simple pendulum are shown in \(\nabla\) Fig. \(13.26 .\) (a) Show that, for the small angle approximation \((\sin \theta \approx \theta),\) the force producing the motion has the same form as Hooke's law. (b) Show by analogy with a mass on a spring that the period of a simple pendulum is given by \(T=2 \pi \sqrt{L / g}\). [Hint: Think of the effective spring constant.]

To study the effects of acceleration on the period of oscillation, a student puts a grandfather clock with a \(0.9929-\mathrm{m}-\) long pendulum inside an elevator. Find the period of the grandfather clock (a) when the elevator is stationary, (b) when the elevator is accelerating upward at \(1.50 \mathrm{~m} / \mathrm{s}^{2}\) (c) when the elevator is accelerating downward at \(1.50 \mathrm{~m} / \mathrm{s}^{2},\) (d) when the cable on the elevator breaks and the elevator simply falls, and (e) when the elevator is moving upward at a constant speed of \(5.00 \mathrm{~m} / \mathrm{s}\).

A university physics professor buys \(100 \mathrm{~m}\) of string and determines its total mass to be \(0.150 \mathrm{~kg} .\) This string is used to set up a standing wave laboratory demonstration between two posts \(3.0 \mathrm{~m}\) apart. If the desired second harmonic frequency is \(35 \mathrm{~Hz},\) what should be the required string tension?
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