Standing waves on a string
- 1. Standing waves on a string Key concepts: -‐ a string that is plucked with both ends fixed results in standing waves on the string (aka normal modes) o amplitudes are 0 at x=0 and x=L (L = length) -‐ fundamental frequency aka first harmonic: the lowest frequency corresponding to the longest wavelength (λ = 2L) -‐ wavelength: 𝜆! = 2𝐿/𝑚 where m is a positive, nonzero integer -‐ fundamental frequency: 𝑓! = ! ! = ! !! 𝑣 = ! !! ! ! , where v = wavespeed, T = tension, 𝜇 = linear mass density -‐ frequency: fm = mf1 Example: A guitar string is 0.61m long, has a fundamental frequency of 500Hz, and a tension kept at 80.0N a) Find the wave speed of the string (hint: find linear mass density) b) Find the wavelengths and frequencies for the 2nd, 3rd, and 4th normal modes of vibration ANSWER a) 𝑣 = ! ! 𝑓! = ! !! ! ! → 500 = ! !(!.!"!) ∗ !".! ! 𝜇 = 0.0013kg/m 𝑣 = 𝑇 𝜇 = 80.0 0.013 = 245m/s b) for the 2nd harmonic: wavelength: 𝜆! = 2𝐿/𝑚 à 𝜆! = !! ! = !(!.!") ! = 0.61𝑚 frequency: fm = mf1 à 𝑓! = 2 500 = 1000Hz For the 3rd harmonic: wavelength: 𝜆! = 2𝐿/𝑚 à 𝜆! = !! ! = !(!.!") ! = 0.41𝑚 frequency: fm = mf1 à 𝑓! = 3 500 = 1500Hz For the 4th harmonic: wavelength: 𝜆! = 2𝐿/𝑚 à 𝜆! = !! ! = !(!.!") ! = 0.31𝑚 frequency: fm = mf1 à 𝑓! = 4 500 = 2000Hz