The equation for the vibration of a string, fixed at both ends vibrating in its third harmonic, is given
y = (0.4 cm) sin [(0.314 cm−1)x]cos[(600 π s−1)t]
i. What is the frequency of vibration?
ii. What are the positions of the nodes?
iii. What is the length of the string?
iv. What is the wavelength and the speed of two travelling waves that can interfere to give this vibration?
300 Hz, 0, 10 cm, 20 cm, 30 cm, L = 30 cm, λ = 20 cm, v = 60 ms-1
Given y = (0.4 cm) sin [(0.314 cm−1)x]cos[(600 π s−1)t]
From the equation ω = 600 π
f = ω2π = 300 Hz
k = 0.314
λ= 2πk = 2 × 3.140.314 = 20cm;v = fλ = 6000 cm s−1 as it is the third harmonic
f = f3 = 3v2L
300 = 3 × 60002 × L
L = 30 cm
If we draw this we can see the position of nodes
Nodes occur at the ends and 10 cm and 20 cm. two identical waves travelling in opposite direction would create such standing wave their speed and wavelengths would be 6000
cm s−1 = 60 ms−1 and 20
cm.as,y1 = 0.2 sin(0.314 × −600 π t)
y2 = 0.2 sin(0.314 × +600 π t + π)
would add up to given equation.