Question

A transverse harmonic wave travels on a rope according to the following expression:
y(x, t) = A cos(kx − ωt − φ)
The mass density of the rope is μ = 0.113 kg/m. x and y are measured in meters and t in seconds. Using the graphs at of y vs t at x=0 and y vs x at t=0 shown below, answer the following questions:

y vs t @x 0 0.15 0.1 0.05 0 time[sec] 0.25 0.05 0.1 0.15 0.2 0.05 0.15 yvs x @ t=0 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 x(m) 4

(a) What is the value for the wave number, k, for the expression y(x,t)?

(b) What is the value for the angular frequency, ω, for the expression y(x,t)?

(c) What is the value in units of radians for the phase shift, ϕ, for the expression y(x,t)?

(d) What is the speed of the wave?

(e) What is the tension in the rope?

(f) Which direction does it travel? (up, left, right, down)?

Answer 1

We have, y(x, t) = A cos (kx - wt - phi). mass density, mu = 0.113 kg/m. From the given graphs, Following are the answers, Since, lambda = 4m. So, wave number, k = (2 pi/lambda) = (2 pi/4) or, K = 1.570796 m^-1/sec. Period, T = 0.2 sec, So, angular frequency, w = (2 pi/T) = (2 pi/0.2) or, w = 31.4159 rad/s Phase sift, phi = (pi/2) = 1.570796 rad speed of wave, v = (w/k) = 31.4159/1.570796 or, v = 19.99998 m/s. Tension, T = mu v^2 = 0.113 times (19.99998)^2. or, T = 45.19991 N It travels along positive x-axis i.e. right.