(a) The wave speed (v) is given as 12m/s and the wavelength (λ) is 0.4m, so the frequency (f) can be calculated as:
v = fλ
f = v/λ = 12/0.4 = 30 Hz
The period (T) is the inverse of the frequency, so:
T = 1/f = 1/30 = 0.0333 s
The wave number (k) can be calculated as:
k = 2π/λ = 2π/0.4 = 15.708 rad/m
(b) The wave function can be written as:
y(x, t) = A sin(kx - ωt + φ)
where A is the amplitude, ω is the angular frequency, and φ is the phase constant. Since the x = 0 end of the string has zero displacement and is moving upward at t = 0, the phase constant is 0. At t = 0, the wave function can be written as:
y(x, 0) = A sin(kx)
The amplitude is given as 0.05m, so:
0.05 = A sin(k*0)
sin(0) = 0, so A = 0.05. Therefore, the wave function is:
y(x, t) = 0.05 sin(15.708x - 188.5t)
(c) To find the transverse displacement of the wave at x = 0.25m and t = -0.15s, we can substitute these values into the wave function:
y(0.25, -0.15) = 0.05 sin(15.7080.25 - 188.5(-0.15))
y(0.25, -0.15) = 0.05 sin(0.5906)
y(0.25, -0.15) ≈ 0.029 m
Therefore, the transverse displacement of the wave at x = 0.25m and t = -0.15s is approximately 0.029m.
(d) The point at x = 0.25m has zero displacement when the argument of the sine function is an integer multiple of π, or:
15.708(0.25) - 188.5t + φ = nπ
where n is an integer. We want to find the time at which this equation is satisfied. Rearranging, we get:
t = (15.708*0.25 + φ - nπ)/188.5
The phase constant is 0, so we can simplify the equation to:
t = (15.708*0.25 - nπ)/188.5
For n = 0, the time is:
t = (15.708*0.25 - 0)/188.5 ≈ 0.0208 s
Therefore, the time required for the point at x = 0.25m to have zero displacement is approximately 0.0208s.