. 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:<br /><br />**Understanding Harmonics**<br /><br />Harmonics are integer multiples of the fundamental frequency (f₁). They represent the different standing wave patterns that can be established on the rope.<br /><br />* **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.<br />* **Second Harmonic (f₂):** Two antinodes and three nodes. The frequency is twice the fundamental frequency (f₂ = 2 * f₁).<br />* **Third Harmonic (f₃):** Three antinodes and four nodes. The frequency is three times the fundamental frequency (f₃ = 3 * f₁).<br /><br />**Calculations**<br /><br />Given:<br /><br />* Fundamental frequency (f₁) = 230 Hz<br /><br />We can calculate the frequencies of the other harmonics:<br /><br />* Second Harmonic (f₂) = 2 * f₁ = 2 * 230 Hz = 460 Hz<br />* Third Harmonic (f₃) = 3 * f₁ = 3 * 230 Hz = 690 Hz<br /><br />**Drawings**<br /><br />Here's how to draw the first three harmonics:<br /><br />1. **First Harmonic (f₁ = 230 Hz):**<br /><br />```<br /> A<br /> / \<br /> / \<br /> N-----N <br />```<br /> * A = Antinode<br /> * N = Node<br /><br />2. **Second Harmonic (f₂ = 460 Hz):**<br /><br />```<br /> A A<br /> / \ / \<br /> / \ / \<br /> N-----N-----N<br />```<br /><br />3. **Third Harmonic (f₃ = 690 Hz):**<br /><br />```<br /> A A A<br /> / \ / \ / \<br /> / \ / \ / \<br /> N-----N-----N-----N<br />```<br /><br />**Relationship to Rope Length**<br /><br />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: λ_n = 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.<br />