Reflected and Transmitted Waves Phase changes

In summary, when considering an infinitely long continuous string with tension \tau and a mass M attached at x=0, reflection and transmission occur at x=0 when a wave train with velocity \frac{\omega}{k} is incident from the left. The coefficients R and T are given by Sin[theta]^2 and Cos[theta]^2 respectively, where Tan[theta] = m*(omega)/(2*tau*k). To find the phase changes for the reflected and transmitted waves, use the boundary condition on the derivatives of the wave functions at x=0 and the complex solution found in the first part of the problem. The phase changes will be dependent on the phase of the incident wave and can be determined by finding the ratio of
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Homework Statement



Consider an infinitely long continuous string with tension [tex]\tau[/tex]. A mass M is attached to the string at x=0. If a wave train with velocity [tex]\frac{\omega}{k}[/tex] is incident from the left, show that reflection and transmission occur at x=0 and that the coefficients R and T are given.
Consider carefully the boundary condition on the derivatives of the wave functions at x=0. What are the phase changes for the reflected and transmitted waves?

The Attempt at a Solution



I got the first part of the problem using a boundary condition derived from Newton's second law. And showed that Sin[theta]^2 = R and Cos[theta]^2 = T, where Tan[theta] = m*(omega)/(2*tau*k). I don't know where to start to find the phase changes. I think it has something to do with the ratio of the complex to the real part of the solution that I found in the first part, but again not sure how to go about this.
 
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Hello there,

Thank you for your post. It seems like you have made some good progress on the problem so far. To find the phase changes for the reflected and transmitted waves, you will need to use the boundary condition on the derivatives of the wave functions at x=0. This boundary condition is that the derivatives of the wave functions must be equal at the point of attachment of the mass. This means that the phase changes for the reflected and transmitted waves will be dependent on the phase of the incident wave.

To find the phase changes, you will need to use the complex solution that you found in the first part of the problem. You can use the complex solution to find the ratio of the complex to the real part of the solution, as you mentioned. This will give you the phase changes for the reflected and transmitted waves.

I hope this helps. If you need further assistance, please let me know. Good luck with your problem!
 

1. What causes phase changes in reflected and transmitted waves?

The phase changes in reflected and transmitted waves are caused by the difference in the medium through which the waves are traveling. When a wave reaches the boundary between two mediums, some of the wave is reflected back and some is transmitted through. The change in medium results in a change in the speed of the wave, which causes a phase shift.

2. How do the amplitudes of reflected and transmitted waves compare to the incident wave?

The amplitudes of the reflected and transmitted waves are dependent on the properties of the medium they are traveling through. In general, the amplitude of the reflected wave will be equal to the incident wave if the boundary is rigid, and it will be equal to the negative of the incident wave if the boundary is free. The amplitude of the transmitted wave will be determined by the impedances of the two mediums.

3. What is the relationship between the angle of incidence and the angle of reflection?

The angle of incidence and the angle of reflection are equal but opposite in direction. This is known as the law of reflection, which states that the angle of incidence, the angle of reflection, and the normal (a line perpendicular to the boundary) all lie in the same plane. This relationship holds true for both reflected and transmitted waves.

4. Can phase changes occur in all types of waves?

Yes, phase changes can occur in all types of waves, including electromagnetic waves, sound waves, and water waves. Whenever a wave reaches a boundary between two mediums, there is a potential for a phase change to occur. The amount of phase shift will depend on the properties of the two mediums.

5. How do phase changes affect the interference of waves?

Phase changes can have a significant impact on the interference of waves. When two waves with different phases meet, they can either reinforce or cancel each other out, depending on the phase difference. This is known as interference. In the case of reflected and transmitted waves, the phase changes at the boundary can result in constructive or destructive interference, altering the overall amplitude of the waves.

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