Problem 68 - College Physics

Question

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) \(20 \mathrm{~Hz} ?\)

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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.