For the transverse wave with displacement y = 8 m sin (2pi/1m x + 4 pi/1 s t) wi

Question

For the transverse wave with displacement y = 8 m sin (2pi/1m x + 4 pi/1 s t) with tension 2N, what is the linear mass density, mu, in kilograms/meter? Two waves of amplitude lm that are equal in every way have a phase difference (relative phase) of pi/3 radians, what is the amplitude of the superposition of the two waves in meters? A string has two ends 2 meters apart that are fixed, it's tension is 5 Newtons, it's mass density is 3grams/meter what is the frequency (in Hertz) that corresponds to the largest wavelength?

Explanation / Answer

4. y = 8m sin(2pi*x/1m + 4pi*t/1s)
y = Asin(wt - kx)
comparing, w = 4pi per s
x = -2pi per m
now, c = w/k = -4pi/2pi = -2m/s
|C| = sqroot(T/mu)
2 = sqroot(2/mu)
mu = 0.5kg/m

5. y1 = Asin(wt - kx )
y2 = Asin(wt - kx + pi/3)
y1 + y2 = A[sin(wt - kx) + sin(wt - kx + pi/3)] = A[2sin(wt - kx + wt - kx + pi/3)/2 cos(wt - kx - wt + kx - pi/3)/2] = 2Asin(wt - kx +pi/6)cos(pi/6)
y1 + y2 = 1.732sin(wt-kx+pi/6) [ A = 1m]
6. l = 2m
T = 5N
mu = 0.003 kg/m

v = sqroot(T/mu) = 40.824 m/s
for largest wavelength, lambda = l/2 = 1m
and v = lambda*f
40.824 = 1*f
f = 40.824 Hz

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