3. If the 3rd harmonic has a frequency of 415 Hz and the wire has a length of 120 cm, determine the number of nodes , antinodes , standing waves , wavelength, and the speed of the wave.

Here's the solution:<br /><br />**1. Understanding Harmonics:**<br /><br />The 3rd harmonic means the string is vibrating in a pattern with three loops. A harmonic is a multiple of the fundamental frequency (the lowest frequency at which the string can vibrate).<br /><br />**2. Nodes and Antinodes:**<br /><br />* **Nodes:** Points on the string that do not move (zero displacement).<br />* **Antinodes:** Points on the string with maximum displacement.<br /><br />For the 3rd harmonic:<br /><br />* **Nodes:** There are 4 nodes (including the two fixed ends).<br />* **Antinodes:** There are 3 antinodes.<br /><br />**3. Standing Waves:**<br /><br />The 3rd harmonic itself represents a standing wave. So, there's one standing wave present, specifically the 3rd harmonic standing wave.<br /><br />**4. Wavelength (λ):**<br /><br />In the 3rd harmonic, three half-wavelengths fit within the length (L) of the string:<br /><br />3(λ/2) = L<br />λ = 2L/3<br />λ = (2 * 120 cm) / 3<br />λ = 80 cm = 0.8 m<br /><br />**5. Speed of the Wave (v):**<br /><br />We can use the wave equation:<br /><br />v = fλ<br /><br />where:<br /><br />* v is the speed of the wave<br />* f is the frequency<br />* λ is the wavelength<br /><br />v = 415 Hz * 0.8 m<br />v = 332 m/s<br /><br />**Summary of Answers:**<br /><br />* **Nodes:** 4<br />* **Antinodes:** 3<br />* **Standing Waves:** 1 (the 3rd harmonic)<br />* **Wavelength:** 80 cm or 0.8 m<br />* **Speed of the wave:** 332 m/s<br />