Question

A guitar string has a fundamental frequency of \(300.0 \mathrm{Hz}\) (a) What are the next three lowest standing wave frequencies? (b) If you press a finger lightly against the string at its midpoint so that both sides of the string can still vibrate, you create a node at the midpoint. What are the lowest four standing wave frequencies now? (c) If you press hard at the same point, only one side of the string can vibrate. What are the lowest four standing wave frequencies?

Short answer

Answer: The next three lowest standing wave frequencies are 600 Hz, 900 Hz, and 1200 Hz. The lowest four standing wave frequencies with a node at the midpoint are 600 Hz, 1200 Hz, 1800 Hz, and 2400 Hz. The lowest four standing wave frequencies with only one side vibrating are 150 Hz, 300 Hz, 450 Hz, and 600 Hz.

Step-by-step solution

Identify the fundamental frequency

The fundamental frequency (also known as the first harmonic) is given as 300 Hz.

Calculate the frequencies of the next three harmonics

To calculate the frequencies of the next three harmonics (2nd, 3rd, and 4th harmonics), simply multiply the fundamental frequency by the harmonic number. 2nd harmonic frequency: \(300\,\text{Hz} \times 2 = 600\,\text{Hz}\) 3rd harmonic frequency: \(300\,\text{Hz} \times 3 = 900\,\text{Hz}\) 4th harmonic frequency: \(300\,\text{Hz} \times 4 = 1200\,\text{Hz}\) So, the next three lowest standing wave frequencies are 600 Hz, 900 Hz, and 1200 Hz. #b) Lowest four standing wave frequencies with a node at the midpoint#

Identify the harmonic pattern with a node at the midpoint

When a node is created at the midpoint of the string, only even harmonics can be produced because the string is essentially divided into two equal parts.

Calculate the lowest four even harmonics

We will use the same method as before, but only include even multiples of the fundamental frequency. 2nd harmonic frequency: \(300\,\text{Hz} \times 2 = 600\,\text{Hz}\) 4th harmonic frequency: \(300\,\text{Hz} \times 4 = 1200\,\text{Hz}\) 6th harmonic frequency: \(300\,\text{Hz} \times 6 = 1800\,\text{Hz}\) 8th harmonic frequency: \(300\,\text{Hz} \times 8 = 2400\,\text{Hz}\) The lowest four standing wave frequencies with a node at the midpoint are 600 Hz, 1200 Hz, 1800 Hz, and 2400 Hz. #c) Lowest four standing wave frequencies with only one side vibrating#

Identify the new fundamental frequency with only one side vibrating

When only one side of the string is allowed to vibrate, we effectively double the length of the vibrating section. Therefore, the new fundamental frequency will be half of the original fundamental frequency: \(150\,\text{Hz}\).

Calculate the lowest four harmonics with the new fundamental frequency

Now we will multiply the new fundamental frequency by the harmonic numbers to find the lowest four standing wave frequencies. 1st harmonic frequency: \(150\,\text{Hz} \times 1 = 150\,\text{Hz}\) 2nd harmonic frequency: \(150\,\text{Hz} \times 2 = 300\,\text{Hz}\) 3rd harmonic frequency: \(150\,\text{Hz} \times 3 = 450\,\text{Hz}\) 4th harmonic frequency: \(150\,\text{Hz} \times 4 = 600\,\text{Hz}\) The lowest four standing wave frequencies with only one side vibrating are 150 Hz, 300 Hz, 450 Hz, and 600 Hz.