The transverse displacement of a string (clamped at its both ends) is given by where and are in m and in s. The length of the string is 1.5 m and its mass is kg.
Does the function represent a travelling wave or a stationary wave?
The function represents a stationary wave because the equation has the form y(x,t) = A sin(kx) cos(ωt), where the spatial part sin(kx) and temporal part cos(ωt) are separated. This is characteristic of stationary waves formed by superposition of two travelling waves.
Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?
Wavelength = 3 m, Frequency = 60 Hz, Speed = 180 m/s for each travelling wave.
Determine the tension in the string.
Tension in the string = 648 N
Explanation
This question tests understanding of stationary waves, wave superposition, and the relationship between wave speed and string tension. The given equation y(x,t) = 0.06 sin(2πx/3) cos(120πt) is a product of separate spatial and temporal functions, which is the defining characteristic of stationary waves. Using the trigonometric identity sin A cos B = (1/2)[sin(A+B) + sin(A-B)], the wave can be decomposed into two travelling waves moving in opposite directions. The wave parameters are extracted by comparing with standard forms, and tension is calculated using the wave speed formula for stretched strings.