# Energy and rate of energy transmitted by a given wave function

1. Oct 28, 2009

### 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