How Does Energy Flux Balance in Connected Strings with Different Mass Densities?

[kevi555]

Hi there,

Two strings, tension T, mass densities mu1 and mu2, are connected. Consider a traveling incident wave on the boundary. Show that the energy flux of the reflected wave and the transmitted wave equals the energy flux of the incident wave. (The energy flux of a wave, the energy density times the wave speed, is proportional to (A^2)/v, where A is amplitude and v is wave speed.

I'm trying to solve it using the equations g_1= [(v_2-v_1)/(v_2+v_1)]f_1 for the reflected wave and f_2= [(2v_2)/(v_2+v_1)]f_1 for the transmitted wave.

Is that correct or should I use (1/2)(mu)((omega)^2)(A^2)(lambda) as the sum of both kinetic and potential energies in the string?

Any help or tips would be great! Thanks!

 

[jbsteele]

Using the equations you provided, you should be able to show that the energy flux of the reflected wave and the transmitted wave equals the energy flux of the incident wave. The equation for the energy flux of a wave is (A^2)/v, where A is amplitude and v is wave speed. To solve this problem, you need to calculate the amplitudes and wave speeds of the incident wave, reflected wave, and transmitted wave, and then substitute these values into the equation for energy flux. You can use the equations g_1= [(v_2-v_1)/(v_2+v_1)]f_1 for the reflected wave and f_2= [(2v_2)/(v_2+v_1)]f_1 for the transmitted wave to calculate the amplitudes and wave speeds. Once you have determined the amplitudes and wave speeds for each wave, you can calculate the energy flux for each wave and show that the sum of the energy fluxes of the reflected and transmitted waves is equal to the energy flux of the incident wave.

 

[thomate1]

Hi there,

Your approach to solving this problem seems to be on the right track. However, there are some key concepts that you may want to consider in order to accurately show that the energy flux of the reflected and transmitted waves equals the energy flux of the incident wave.

Firstly, it is important to understand that the energy flux of a wave is a measure of the rate at which energy is flowing through a given area. In the case of a string, this energy is carried by the motion of the particles in the string. Therefore, the energy flux can be calculated using the formula you mentioned: (A^2)/v, where A is amplitude and v is wave speed. This formula is derived from the fact that the energy of a wave is proportional to its amplitude squared and its speed.

Secondly, when considering the reflected and transmitted waves, it is important to remember that energy is conserved. This means that the total energy of the incident wave must be equal to the sum of the energies of the reflected and transmitted waves. In other words, the energy flux of the incident wave must be equal to the sum of the energy fluxes of the reflected and transmitted waves.

Finally, in order to accurately calculate the energy flux of the reflected and transmitted waves, you may need to use the equations that you mentioned, but also consider the concept of impedance matching. This concept states that when waves pass through a boundary between two different media, the waves will be reflected and transmitted in a way that minimizes the difference in impedance between the two media. This can affect the amplitude and speed of the reflected and transmitted waves, and therefore, their energy fluxes.

Overall, it is important to carefully consider the concepts of energy, energy flux, and impedance matching in order to accurately show that the energy flux of the reflected and transmitted waves equals the energy flux of the incident wave. I hope this helps and good luck with your calculations!