. A rope has a length of 86cm. If the fundamental frequency is 230 Hz, determine and draw the first 3 harmonics.

Here's how to determine and draw the first three harmonics of a vibrating rope:**Understanding Harmonics**Harmonics are integer multiples of the fundamental frequency (f1). They represent the different standing wave patterns that can be established on the rope.* **First Harmonic (Fundamental Frequency, f1):** The simplest standing wave with one antinode (point of maximum displacement) in the middle and two nodes (points of zero displacement) at the ends.* **Second Harmonic (f2):** Two antinodes and three nodes. The frequency is twice the fundamental frequency (f2 = 2 * f1).* **Third Harmonic (f3):** Three antinodes and four nodes. The frequency is three times the fundamental frequency (f3 = 3 * f1).**Calculations**Given:* Fundamental frequency (f1) = 230 HzWe can calculate the frequencies of the other harmonics:* Second Harmonic (f2) = 2 * f1 = 2 * 230 Hz = 460 Hz* Third Harmonic (f3) = 3 * f1 = 3 * 230 Hz = 690 Hz**Drawings**Here's how to draw the first three harmonics:1. **First Harmonic (f1 = 230 Hz):**``` A / \ / \ N-----N ``` * A = Antinode * N = Node2. **Second Harmonic (f2 = 460 Hz):**``` A A / \ / \ / \ / \ N-----N-----N```3. **Third Harmonic (f3 = 690 Hz):**``` A A A / \ / \ / \ / \ / \ / \ N-----N-----N-----N```**Relationship to Rope Length**The wavelength (λ) of the fundamental frequency is related to the length (L) of the rope by: λ1 = 2L. Since the rope is 86 cm long, the wavelength of the fundamental frequency is 2 * 86 cm = 172 cm. The wavelengths of the higher harmonics are related to the fundamental wavelength: λn = λ1 / n, where n is the harmonic number. While the rope length is important for determining the fundamental frequency, it's not directly used in drawing the *shape* of the harmonic waves. The drawings focus on the relative positions of nodes and antinodes.