Problem 64 - College Physics

Question

A standing wave is formed in a stretched string that is \(3.0 \mathrm{~m}\) long. What are the wavelengths of (a) the first harmonic and (b) the second harmonic?

Step-by-Step Solution

Answer

The wavelengths of the harmonics are 6.0 m (1st) and 3.0 m (2nd).

Step 1: Understanding Harmonics in a Stretched String

A standing wave pattern results from the interference of two waves traveling in opposite directions in a string. Harmonics are specific patterns of these standing waves. The first harmonic, also known as the fundamental frequency, involves the entire string vibrating in one single loop. The second harmonic involves two loops.

Step 2: Calculating First Harmonic Wavelength

In the first harmonic, the string vibrates in one loop, meaning the length of the string (L = 3.0 m) is half of the wavelength (\lambda = 2L). Therefore, the wavelength of the first harmonic is given by \( \lambda_1 = 2 \times 3.0 \mathrm{~m} \).

Key Concepts

Harmonics

In the context of a stretched string, harmonics refer to the distinct patterns of vibration that occur when a standing wave is formed. These are the patterns you'll observe when multiple loops form along the length of the string. Each harmonic represents a different mode of vibration, and they are integral to understanding how standing waves behave.

• The first harmonic is the simplest form, where the string has one complete loop. It's also known as the fundamental frequency.
• The second harmonic, on the other hand, has two loops, which implies that the string vibrates in a pattern with more energy.

Discovering these harmonic patterns helps us understand the relationship between frequency, wavelength, and the physical properties of the string.

Wavelength

Wavelength is a key component in understanding standing waves. It defines the distance between two consecutive points that are in phase on a wave, such as from crest to crest or trough to trough. In the case of waves on a stretched string, knowing the wavelength helps determine how the waves will form harmonically.

• To calculate the wavelength for each harmonic, consider the relationship with the length of the string (L).
• For the first harmonic, the length of the string is half of the wavelength: \[\lambda_1 = 2L\]
• For the second harmonic, the string length equals the full wavelength: \[\lambda_2 = L\]

These relations are crucial for calculating the exact wavelengths of the harmonics.

Stretched String

The concept of a stretched string is fundamental to the study of standing waves and harmonics. When a string is under tension and fixed at both ends, it provides the potential for wave formation. This is because the tension and fixed boundaries allow waves to reflect back and forth, creating standing waves.

• A standing wave appears stationary while the actual waves are constantly moving.
• The points on the string that remain still are called nodes, while the points with maximum movement are called antinodes.

Understanding the dynamics of a stretched string helps in predicting how waves behave and the harmonics that are formed.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system like a stretched string can naturally vibrate. It is synonymous with the first harmonic and sets the stage for understanding other harmonics. When a string vibrates at the fundamental frequency:

• It completes one loop, which is the simplest form of vibration.
• This vibration occurs because the boundary conditions (nodes at both ends) support this wave pattern.

The fundamental frequency is critical because it determines the pitch of the sound produced by the vibrating string and serves as the basis for all subsequent harmonics.