How Do Mass Attachments Affect Wave Reflection and Transmission on a String?

In summary, the conversation discusses reflection and transmission occurring at x=0 when a wave train is incident on an infinitely long continuous string with tension \tau. The coefficients R and T are given by equations i and ii, and the phase changes for the reflected and transmitted waves can be found using equations iii and iv. The boundary condition on the derivatives of the wave functions at x=0 is carefully considered, and it is noted that the densities on either side of the mass are the same. The second boundary condition is given by an expression involving Newton's Second Law and the tension of the string.
  • #1
piano.lisa
34
0

Homework Statement


Consider an infinitely long continuous string with tension [tex]\tau[/tex]. A mass [tex]M[/tex] 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?

Homework Equations


i. [tex]R = sin^2\theta[/tex]
ii. [tex]T = cos^2\theta[/tex]
iii. [tex]tan\theta = \frac{M\omega^2}{2k\tau}[/tex]
iv. [tex]\psi_1(x,t) = \psi_i + \psi_r = Ae^{i(\omega t - kx)} + Be^{i(\omega t + kx)}[/tex]
v. [tex]\psi_2(x,t) = \psi_t[/tex], however, I do not know what this is.

** note ** [tex]\psi_i[/tex] is the incident wave, [tex]\psi_r[/tex] is the reflected wave, and [tex]\psi_t[/tex] is the transmitted wave

The Attempt at a Solution


I am used to dealing with situations where the string is of 2 different densities, therefore, [tex]\psi_t[/tex] will have a different value for k than [tex]\psi_i[/tex]. However, in this case, the densities are the same on either side of the mass, and the only obstruction is the mass. If I knew how to find an equation for [tex]\psi_2(x,t)[/tex], then I could potentially solve the rest of the problem.
Thank you.
 
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  • #2
I still haven't reached any solution to my problem.

Any help is appreciated.
 
  • #3
EDIT: Ignore this post. The result leads nowhere.

According to my calculations, this is true: [tex]\frac{d^2\psi_1}{dt^2}(0,t) = \frac{d^2\psi_2}{dt^2}(0,t)[/tex]

Do you see why?

Hint: Apply Newton's (2nd?) Law to the central mass and find an expression for the net force on [tex]M[/tex]
 
Last edited:
  • #4
I realize this is due in about ~1/2 hour, but the second boundary condition is given by:
[tex]M\frac{d^2\psi_1}{dt^2} = M\frac{d^2\psi_2}{dt^2} = \tau \left( \frac{d\psi_1}{dx} - \frac{d\psi_2}{dx}\right) (0,t)[/tex]
 

Related to How Do Mass Attachments Affect Wave Reflection and Transmission on a String?

What is reflection and transmission?

Reflection and transmission are two processes that occur when a wave encounters a boundary between two different mediums. Reflection is the bouncing back of the wave when it hits the boundary, while transmission is the passing through of the wave into the second medium.

How is the amount of reflection and transmission determined?

The amount of reflection and transmission depends on the properties of the two mediums, such as their densities and elasticity, as well as the angle at which the wave hits the boundary. These factors determine how much of the wave is reflected and how much is transmitted.

What is the law of reflection?

The law of reflection states that the incident angle of a wave is equal to the reflected angle. This means that if a wave hits a boundary at a 30 degree angle, it will be reflected at a 30 degree angle.

How does the wavelength of a wave affect reflection and transmission?

The wavelength of a wave does not affect the amount of reflection and transmission, but it can affect the angle at which the wave is reflected. Longer wavelengths tend to be reflected at smaller angles, while shorter wavelengths are reflected at larger angles.

What are some real-life examples of reflection and transmission?

Reflection and transmission can be observed in many everyday situations, such as light reflecting off a mirror, sound waves being transmitted through a wall, or radio waves being reflected off the ionosphere. They are also important concepts in fields such as optics, acoustics, and electromagnetism.

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