Wave Problems (help!)

Question

1) a spring of length of 5.5 meters if held fixed at both ends and is vibrating at the third harmonic. determine the wavelength and frequency of the standing wave produced if the wave speed on the spring is 14.2 m/s

2) a pipe is resonating at the 1st haromonic at a frequency of 256 Hz. If the speed of sound in the air within the pipe is 342 m/s, what is the length of the pipe if (a) it is open at both ends, and (b) it is closed at one end and open at the other end

Answer 1

(1)

3rd harmonic means the length equals 3 half-wavelengths

L = 3(λ/2) ⇒

λ = (2/3)L = (2/3)(5.5 m) = 3.67 m

f = v/λ = (14.2 m/s) / (3.67 m) = 3.87 Hz

(2)

(a) 1st harmonic and open at both ends means the length equals 2 quarter-wavelengths, which is half a wavelength

L = λ/2 = (v/f)/2 = (1/2) [(342 m/s)/(256 Hz)] = 0.668 m

(b) 1st harmonic, open at one end and closed at the other end means the length equals 1 quarter-wavelength

L = λ/4 = (v/f)/4 = (1/4) [(342 m/s)/(256 Hz)] = 0.334 m