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Chapter 16: Problem 111

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string producing a standing wave. The linear mass density of the string is \(\mu=0.075 \mathrm{kg} / \mathrm{m}\) and the tension in the string is \(F_{T}=5.00 \mathrm{N}\). The time interval between instances of total destructive interference is \(\Delta t=0.13 \mathrm{s}\). What is the wavelength of the waves?

Short Answer

Expert verified
The wavelength of the sinusoidal waves is approximately \(2.12 \, \mathrm{m}\).

Step by step solution

01

Find the wave speed v

First, let's find the wave speed, \(v\). The wave speed can be calculated using the formula: \[v = \sqrt{\frac{F_T}{\mu}}\] where \(F_T\) is the tension in the string and \(\mu\) is the linear mass density of the string.
02

Calculate the wave speed

Using the given values for tension and linear mass density, we can calculate the wave speed: \[v = \sqrt{\frac{5.00 \, \mathrm{N}}{0.075 \, \mathrm{kg/m}}} = \sqrt{\frac{5.00}{0.075}} \, \mathrm{m/s}\] Now, calculate the value inside the square root: \[\frac{5.00}{0.075} = 66.67\] So, \[v = \sqrt{66.67} \, \mathrm{m/s}\] which gives us: \[v \approx 8.16 \, \mathrm{m/s}\] So, the wave speed in the string is approximately \(8.16 \, \mathrm{m/s}\).
03

Find the frequency f

Next, let's calculate the frequency \(f\) of the standing wave. The relationship between the time interval \(\Delta t\) between instances of total destructive interference and the frequency is given by: \[f = \frac{1}{2\Delta t}\]
04

Calculate the frequency f

Using the given time interval of total destructive interference (\(\Delta t = 0.13 \, \mathrm{s}\)), we can find the frequency: \[f = \frac{1}{2(0.13 \, \mathrm{s})} = \frac{1}{0.26 \, \mathrm{s}} = 3.85 \, \mathrm{Hz}\] So, the frequency of the standing wave is \(3.85 \, \mathrm{Hz}\).
05

Find the wavelength 位

Finally, we can find the wavelength \(\lambda\). The relationship between wave speed \(v\), frequency \(f\), and wavelength \(\lambda\) is given by: \[v = f\lambda\]
06

Calculate the wavelength 位

Using the calculated values for wave speed \(v\) and frequency \(f\), we can find the wavelength \(\lambda\): \[8.16 \, \mathrm{m/s} = (3.85 \, \mathrm{Hz}) \lambda\] Now, divide both sides by 3.85 to solve for \(\lambda\): \[\lambda = \frac{8.16 \, \mathrm{m/s}}{3.85 \, \mathrm{Hz}} = 2.12 \, \mathrm{m}\] So, the wavelength of the sinusoidal waves is approximately \(2.12 \, \mathrm{m}\).

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Key Concepts

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

Wave Speed
The speed of a wave on a string is determined by the tension in the string and its linear mass density. The formula for wave speed is given by the square root of the ratio of the tension force (\(F_T\)) to the linear mass density (\(\mu\)). Mathematically, it is represented as:
  • \[v = \sqrt{\frac{F_T}{\mu}}\]
To find the wave speed, simply plug in the known values of the tension and linear mass density. For instance, with a string tension of 5.00 N and a linear mass density of 0.075 kg/m, the speed is calculated as \(\approx 8.16\) m/s. This value tells us how fast a wave can travel along the string, essentially describing its dynamic behavior.
Frequency Calculation
The frequency of a wave refers to how many cycles or waves pass a fixed point per second. In the context of standing waves, frequency is related to the occurrences of total destructive interference, which are points where waves cancel each other out completely.
For a standing wave, the frequency \(f\) can be derived knowing the time interval \(\Delta t\) between successive instances of total destructive interference:
  • \[f = \frac{1}{2\Delta t}\]
In this problem, the given \(\Delta t = 0.13\) s leads us to a frequency of \(3.85\) Hz. This indicates that 3.85 complete wave cycles occur each second.
Wavelength Determination
Wavelength (\(\lambda\)) is the spatial period of the wave 鈥� the distance over which the wave's shape repeats. It is a fundamental property of waves, interconnected with both speed and frequency.
Given the relationship:
  • \[v = f\lambda\]
You can rearrange the formula to solve for wavelength:
  • \[\lambda = \frac{v}{f}\]
Using the previously calculated wave speed (8.16 m/s) and frequency (3.85 Hz), you find the wavelength to be \(\approx 2.12\) m. This shows us the physical length of one complete wave cycle along the string.
Destructive Interference
Destructive interference happens when two waves meet in such a way that their crests and troughs are aligned oppositely, causing them to cancel each other out. In standing waves, this results in nodes 鈥� positions along the medium where no movement occurs.
The intervals of total destructive interference between waves traveling in opposite directions help define the frequency of the standing wave. In this problem, the frequency was calculated using the time interval between these moments, which is crucial for understanding the behavior and properties of the wave system.
Linear Mass Density
Linear mass density (\(\mu\)) is a property of a medium, measuring how much mass is distributed along a unit length. It plays a vital role in determining wave speed because it directly influences how waves propagate through the medium.
The wave speed formula 鈥� \(v = \sqrt{\frac{F_T}{\mu}}\) 鈥� shows that higher linear mass density results in slower wave speeds if tension is constant. For example, with a linear mass density of 0.075 kg/m used here, you can see how it contributes to calculating the speed. Knowing \(\mu\) is essential for optimizing materials and conditions when analyzing wave behavior.

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Most popular questions from this chapter

Does a sound wave move faster in seawater or fresh water, if both the sea water and fresh water are at the same temperature and the sound wave moves near the surface?$$\begin{array}{l}\left(\rho_{\mathrm{w}} \approx 1000 \frac{\mathrm{kg}}{\mathrm{m}^{3}}, \rho_{\mathrm{s}} \approx 1030 \frac{\mathrm{kg}}{\mathrm{m}^{3}}, B_{\mathrm{w}}=2.15 \times 10^{9} \mathrm{Pa}\right. \\\\\left.B_{\mathrm{s}}=2.34 \times 10^{9} \mathrm{Pa}\right) \end{array}$$

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