Energy and rate of energy transmitted by a given wave function

[nowtrams]

1.
The wave function for a wave on a taut string is given below, where x is in meters and t is in seconds.

y(x, t) = (0.300 m) sin(11πt - 3πx + π/4)​

(a) What is the average rate at which energy is transmitted along the string if the linear mass density is 75.0 g/m?

(b) What is the energy contained in each cycle of the wave?


2.
y(x,t) = Asin(kx-wt+phi)
E_^ = (1/2)uw²A²^
P = E_^/T = (1/2)uw²A²v
k = 2π/^
v = w/k
w = 2π/T = 2πf

A=amplitude, phi=phase contstant
E_^=total energy in one wavelength, u=linear mass density, ^=wavelength
P=power, T=period,
k=wave#
v=velocity
w=angular frequency, f=frequency


3.
given: y(x,t) = (0.300 m) sin(-3πx + 11πt + π/4), therefore A = 0.3m, k = -3π, w = -11π, phi = π/4; u = 75g/m

(a)
P = (1/2)uw²A²v and v = w/k
therefore:
P = (1/2)(75g/m)(-11π)²(0.3m)²(-11π/-3π)
P = 37.5*121π²*0.09(11/3) = 14660.2709

P = 14660.27W

(b)
E_^ = (1/2)uw²A²^ and k = 2π/^ so ^ = 2π/k
therefore:
E_^ = (1/2)(0.075kg/m)(-11π)²(0.3)²(2π/-3π)
E_^ = 0.0375*121π²*0.09(2/-3) = -2.6655038

E_^ = -2.67J

 

[member38644]

(negative sign indicates energy is being transmitted in the opposite direction)

Therefore, the average rate at which energy is transmitted along the string is 14660.27 watts and the energy contained in each cycle of the wave is -2.67 joules.