Please answer all part of the questions. Thanks. Two marks on a string are 4 met

Question

Please answer all part of the questions. Thanks.

Two marks on a string are 4 meters apart and are between two and three wavelengths apart. A harmonic wave is traveling to the right. At one moment in time, the two marks have the same speed but are oscillating in opposite directions The left mark has a positive displacement and moving down while the right mark has a negative displacement and moving up How is the displacement of the left mark "a _L" related to the displacement of the right mark "a _R"? After 0.1 second (which happens to be less than a period), the two marks have the same speed as before (t = 0) but now are oscillating in opposite directions as before. Further, the left mark has a negative displacement moving up while the right mark has a positive displacement moving down. How is the displacement of the left mark "a _L" related to the displacement of the right mark "a _R" now? What is the wavelength of the harmonic wave? What is the period of the harmonic wave? What is the speed of the harmonic wave?

Explanation / Answer

A travelling harmonic wave on a string can be expressed as undey
y = Aosin(wt - kx)

a. at given time
two marks have same speed u
velocity of left mark = -u
velocity of right mark = + u
also,
x coordinate of left mark can be assumed = 0
x coordinate of right mark can be assumed [2*lambda + dx] as it is more than 2*lambda and less than 3*lambda
so, at time t
y-left = Aosin(wt)
y-right = Aosin(wt - k(2*lambda + dx)) = Aosin(wt - 2*pi(2*lambda + dx)/lambda) = Aosin(wt - 2*pidx/lambda)
dx < lambda
so, 2*pi*dx/lambda < 2*pi

again
v = Vocos(wt - kx)
for this case
the phase difference between the right and the left particle is 180 degree
so that means
2*pi*dx/lambda = pi
dx = lambda/2

hence
y-left = Aosin(wt)
y-right = Aosin(wt -pi) = -Aosin(wt) = -y-left
b. After t = 0.1 s
since the phase difference between the pints will still be the same
aL = -ar [ is always true]
c. Wavelength = lambda
now diftance between marks = 4m = lambda/2
lambda = 8 m
d. Time period = T
consider left mark
at t = 0, velocity = -u
at t = 0.1, velocity = u
so, -u = Vosin(-kx)
u = Vosin(w*0.1 - kx)
comparing
-kx = w*0.1 - kx - pi
pi = 0.1*w
w = pi/0.1 = 2*pi/T
T = 2*pi*0.1/pi = 0.2s
e. speed = lambda/T = 8/0.2 = 40 m/s

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