Problem 69 - College Physics

Question

Two waves of equal amplitude and frequency of \(250 \mathrm{~Hz}\) travel in opposite directions at a speed of \(150 \mathrm{~m} / \mathrm{s}\) in a string. If the string is \(0.90 \mathrm{~m}\) long, for which harmonic mode is the standing wave set up in the string?

Step-by-Step Solution

Answer

The standing wave is set up in the 3rd harmonic mode.

Step 1: Formula for wave speed and wavelength

The speed of a wave (v) is related to its frequency (f) and wavelength (λ) by the formula: \[ v = f \times \lambda \]Here, the wave speed \(v = 150 \mathrm{~m/s}\) and the frequency \(f = 250 \mathrm{~Hz}\).

Step 2: Calculate the wavelength

Using the formula from Step 1:\[\lambda = \frac{v}{f} = \frac{150}{250} = 0.6 \mathrm{~m}\]Thus, the wavelength of each wave is \(0.6 \mathrm{~m}\).

Step 3: Determine the condition for standing waves in a string

Standing waves are set up in a string when the length of the string (L) is an integer multiple of half the wavelength:\[ L = n \left(\frac{\lambda}{2}\right) \] where \(n\) represents the mode or harmonic number.

Step 4: Solve for the harmonic mode

Given that the string length is \(0.9 \mathrm{~m}\), substitute the values into the standing wave condition:\[0.9 = n \left(\frac{0.6}{2}\right)\]This simplifies to:\[0.9 = n \times 0.3\]Solving for \(n\), we get:\[n = \frac{0.9}{0.3} = 3\]Thus, the harmonic mode for the standing wave is \(n = 3\).

Key Concepts

Wave Speed

Understanding wave speed is essential when studying waves. The wave speed is represented by the symbol \(v\) and tells us how fast a wave travels through a medium. It is determined by the properties of the medium through which the wave is moving and is typically measured in meters per second (m/s). It is crucial to distinguish that wave speed is not the speed of individual particles within the wave, but rather the speed at which the overall wave pattern progresses.

Wave speed can be expressed mathematically by the relationship between the frequency \(f\) of the wave and its wavelength \(λ\):

• Formula: \(v = f \times λ\)
• Where \(v\) is the wave speed, \(f\) is the frequency in hertz (Hz), and \(λ\) is the wavelength in meters (m).

By using this formula, we can calculate either the wave speed, frequency, or wavelength if two of these quantities are known. In our specific problem, the wave speed of 150 m/s and a frequency of 250 Hz are given. These values allow us to deduce the wave's wavelength, which is an important aspect of determining other wave properties.

Harmonic Mode

The concept of a harmonic mode is key to understanding how standing waves form within a medium such as a string. A standing wave is produced when two waves of the same frequency travel in opposite directions and interfere with each other. This leads to points of constructive and destructive interference, forming a pattern of nodes (points of no displacement) and antinodes (points of maximum displacement).

The harmonic mode or harmonic number \(n\) corresponds to the number of half-wavelengths that fit into the length of the string. The nth harmonic for a string is given by the equation:

• \(L = n \left(\frac{λ}{2}\right)\)

Here, \(L\) denotes the length of the string, and \(λ\) is the wavelength. The integer \(n\) indicates the harmonic mode or number. The string's fundamental frequency (first harmonic) corresponds to \(n = 1\), the second harmonic at \(n = 2\), and so on.

In this problem, the wave fits three half-wavelengths along the string, indicating it is in the third harmonic mode. Understanding this relationship is crucial as it allows us to predict how the wave patterns will evolve in various conditions.

Wavelength

The wavelength of a wave, represented by the symbol \(λ\), is the distance over which the wave's shape repeats. It is a crucial characteristic that influences various phenomena tied to wave mechanics. Wavelength is usually measured in meters and can be thought of as the distance between consecutive points that are in phase – such as crest to crest or trough to trough.

Wavelength plays an integral role in the formula that describes wave speed:

• \(v = f \times λ\)

This means that if you know the frequency and the speed of a wave, you can calculate its wavelength. In our task, after determining the wave speed (150 m/s) and the frequency (250 Hz), we calculated the wavelength to be 0.6 m using the formula \(λ = \frac{v}{f}=\frac{150}{250}\). Understanding how wavelength ties into the nature of standing waves further helps us determine the harmonics.

The wavelength affects the pattern of standing waves and the harmonic modes that can exist in any bounded medium. Recognizing these factors is vital for comprehending how waves behave under different constraints and setups.