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.

 

[kerimek]

In response to the given wave function, it is important to note that energy is transmitted through a wave at a certain rate. This rate is dependent on various factors such as the amplitude, frequency, and linear mass density of the medium. In this case, the average rate at which energy is transmitted along the string can be calculated using the equation P = (1/2)uw2A2v, where P is the power, u is the linear mass density, w is the angular frequency, A is the amplitude, and v is the velocity. Using the given values, we can calculate the power to be 14660.27 watts.

Furthermore, each cycle of the wave contains a certain amount of energy, which can be calculated using the equation E^ = (1/2)uw2A2^, where E^ is the total energy in one wavelength, u is the linear mass density, w is the angular frequency, A is the amplitude, and ^ is the wavelength. In this case, the energy contained in each cycle of the wave is -2.67 joules.

It is also important to note that the given wave function can be represented in the form y(x,t) = Asin(kx-wt+phi), where A is the amplitude, k is the wave number, w is the angular frequency, and phi is the phase constant. This form can be helpful in understanding the properties and behavior of the wave, as demonstrated in the calculations above. Overall, understanding the energy and rate of energy transmitted by a given wave function is crucial in studying and analyzing various phenomena in the field of science.