A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 m/s. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

Solution 40E Problem (a) Step 1: Length of the rope L = 1.50 m Speed of the travelling wave v = 48.0 m/s To find the wavelength and frequency of the fundamental Step 2: The expressions for Wavelength and frequency = 2L n n nv f n 2L Step 3: For fundamental n = 1 Wavelength = 2*1.5 1 1 = 3.00 m 1 Step 4: 1 48.0 Frequency f =1 2*1.5 f = 16.00 Hz 1 Problem (b) Step 1: To find the wavelength and frequency of the second overtone, For second overtone the harmonic is third, therefore n = 3 Step 2: Wavelength = 2L n n 2*1.5 3 3 3 1.00 m Step 3: Frequency f n nv 2L 3*48.0 f 3 2*1.5 f = 48.00 Hz 3 Problem (c) Step 1: To find the wavelength and frequency of the fourth harmonic, For fourth harmonic n = 4 (overtone is three) Step 2: Wavelength 2L n n = 2*1.5 4 4 40.75 m