Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

Short answer

The fundamental has \( \lambda = 3.00 \) m and \( f \approx 20.67 \) Hz; the second overtone has \( \lambda = 1.00 \) m and \( f = 62.0 \) Hz; the fourth harmonic has \( \lambda = 0.75 \) m and \( f \approx 82.67 \) Hz.

Step-by-step solution

Understanding Wave Properties on a String

For a string fixed at both ends, the wavelengths of the standing waves are given by \( \lambda_n = \frac{2L}{n} \), where \( L = 1.50 \) m is the length of the string, and \( n \) is the harmonic number. The speed \( v \) of waves on the string is 62.0 m/s.

Calculating Wavelength for the Fundamental Frequency

The fundamental frequency, also known as the first harmonic, corresponds to \( n = 1 \). Use the formula \( \lambda_1 = \frac{2 \times 1.50}{1} = 3.00 \text{ m} \).

Finding Fundamental Frequency from Wave Speed and Wavelength

The frequency \( f_n \) of a wave is given by \( f_n = \frac{v}{\lambda_n} \). For the fundamental frequency: \( f_1 = \frac{62.0}{3.00} \approx 20.67 \text{ Hz} \).

Calculating Wavelength for the Second Overtone

The second overtone is the third harmonic, which corresponds to \( n = 3 \). Use the formula \( \lambda_3 = \frac{2 \times 1.50}{3} = 1.00 \text{ m} \).

Finding Frequency for the Second Overtone

For the second overtone: \( f_3 = \frac{62.0}{1.00} = 62.0 \text{ Hz} \).

Calculating Wavelength for the Fourth Harmonic

The fourth harmonic corresponds to \( n = 4 \). Use the formula \( \lambda_4 = \frac{2 \times 1.50}{4} = 0.75 \text{ m} \).

Finding Frequency for the Fourth Harmonic

For the fourth harmonic: \( f_4 = \frac{62.0}{0.75} \approx 82.67 \text{ Hz} \).

Key concepts

Fundamental Frequency

In the context of waves on a string, the fundamental frequency is the simplest form of vibration, where the string has the least amount of nodes and antinodes required to sustain a wave. This is also referred to as the first harmonic. The fundamental frequency is crucial because it sets the baseline for understanding other harmonics or overtones in a wave system. It is represented when the string completes exactly one half of a wavelength within the span of the string length.
To calculate the fundamental frequency (\(f_1\)), you use the equation \(f_1 = \frac{v}{\lambda_1}\). Here, \(v\) is the wave speed, and \(\lambda_1\) is the wavelength of the fundamental frequency. This calculation helps us understand how fast the wave oscillates at its simplest harmonic state.

Harmonics

Harmonics refer to the multiple frequencies at which a system, such as a string, naturally oscillates. Beyond the fundamental frequency, there exist other harmonics that make the string vibrate in more complex patterns. These are the 2nd, 3rd, etc., harmonics, also called overtones.
The general expression for the wavelength of harmonics in a string fixed at both ends is given by \(\lambda_n = \frac{2L}{n}\), where \(L\) is the length of the string, and \(n\) is the harmonic number. Harmonics are integral because they embody the different modes of vibration a string can possess. Each mode comes with its own unique wavelength and frequency. For example, the third harmonic has three half-wavelength segments in the length of the string, leading to a different wavelength and frequency than the fundamental.

Wave Speed

Wave speed is an essential property that defines how quickly a wave travels through a medium. In this problem, we consider transverse waves on a string. The wave speed (\(v\)) depends on both the tension of the string and its linear density, although these specifics are not covered in our direct calculation.
To find the wave speed, the equation \(v = f \times \lambda\) is utilized, where \(f\) is the frequency and \(\lambda\) is the wavelength of the wave. This relation highlights the interdependence of wave properties and allows us to determine any one property if the others are known. In our problem, the speed is given by 62.0 m/s, helping us calculate frequencies and wavelengths of different harmonics.

Wave Properties

Wave properties encompass various aspects such as wavelength, frequency, amplitude, and speed. Each of these contributes to the behavior and characteristics of the wave.

• Wavelength (\(\lambda\)): This is the distance over which the wave's shape repeats. Shorter wavelengths mean more waves are packed into a given space.
  
• Frequency (\(f\)): This defines how many wave cycles pass a point in a given time period, usually a second. It is measured in hertz (Hz).
  
• Amplitude: Though not directly covered in the problem, amplitude represents the wave's height and is related to the wave's energy.
  
• Speed (\(v\)): This is the rate at which a wave propagates through a medium. Relating to all other properties through the formula \(v = f \times \lambda\), it underscores the interconnectedness of the elements.
  

Understanding these properties allows us to dissect and predict wave behavior in different mediums and situations.