One dimensional lattice dispersion relation

Your Name]In summary, the conversation discusses an experiment involving a one-dimensional lattice held up by strings and a series of masses connected by springs. The goal is to find a relation between angular frequency and wave number for transverse oscillations. An equation has been found for longitudinal oscillations, but finding one for transverse oscillations is proving to be more difficult. Suggestions are given for checking the equation, considering forces acting on the masses, using approximations, and checking units and dimensions.
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I'm doing an experiment, in which I have a one dimensional lattice held up by strings. That is I have a series of n masses each of mass M each connected to each other by springs with spring constant C and unstretched length a. Each mass is suspended from the ceiling by a string of length L. I'm trying to find a relation that relates the angular frequency to the wave number for transverse oscillations. I found one for longitudinal oscillations, w^2=(g/L)+(4C/M)(sin(ka/2)^2, where w is the angular frequency and k is the wave number. For transverse oscillations things become more difficult. I know the relation has the form w^2=(g/L)+(4D/M)(sin(ka/2)^2, where D is a constant that involves C and is smaller than C. I'm guessing D has the form D=B(a/L)C, where B is a constant. You have to setup the equation of motion for transverse oscillations and assume a solution of the form y(x,t)=exp(i(k(na+x)-wt), where y is the transverse displacement. From there you make a bunch of approximations to get an equation of the above form. I've spent a lot of time working on this. I'll post more of the work I've done, when I have more time. I attached a pdf file with more info to the post.
 

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Thank you for sharing your experiment and progress with us. I am always interested in new findings and discoveries in the field of physics. I have read through your post and the attached pdf file, and I have a few suggestions that may help you in finding the relation between angular frequency and wave number for transverse oscillations.

Firstly, I would recommend checking your equation for longitudinal oscillations again. In the equation you provided, there seems to be a missing factor of 2 in the second term. It should be 2C instead of 4C. This may affect the relation you are trying to find for transverse oscillations.

Secondly, in order to find the relation for transverse oscillations, you will need to consider the forces acting on each mass in the lattice. This includes the tension in the string, the force due to gravity, and the force from the spring. From there, you can set up the equation of motion and use the given solution for y(x,t) to find the relation between angular frequency and wave number.

Additionally, you may want to consider using a small angle approximation for the sine term in your equation, as the oscillations in the lattice are likely to be small. This can simplify the equation and make it easier to work with.

Lastly, I would suggest checking your units and dimensions in your equation to ensure they are consistent. This can help in identifying any errors or missing factors.

I hope these suggestions are helpful to you in your experiment. Good luck with your research and I look forward to seeing your progress.
 

1. What is a one dimensional lattice dispersion relation?

A one dimensional lattice dispersion relation is a mathematical equation that describes the relationship between the energy and momentum of particles in a one dimensional lattice structure. It is used to understand the behavior of waves and particles in a lattice, such as in solid state physics and condensed matter physics.

2. How is the dispersion relation derived?

The dispersion relation is derived by applying the laws of conservation of energy and momentum to the lattice structure. This involves solving the equations of motion for the particles in the lattice and finding the relationship between their energy and momentum.

3. What does the dispersion relation tell us about the behavior of particles in a one dimensional lattice?

The dispersion relation tells us about the allowed energy and momentum states of particles in the lattice. It also provides information about the propagation of waves and the band structure of the lattice, which can affect the electrical and thermal conductivity of the material.

4. How does the dispersion relation change with different lattice structures?

The dispersion relation can vary depending on the type of lattice structure, such as a simple cubic lattice or a hexagonal lattice. Different lattice structures can lead to different band structures and energy-momentum relationships, which can affect the properties of the material.

5. What are some applications of the one dimensional lattice dispersion relation?

The dispersion relation is used in many fields, including solid state physics, materials science, and engineering. It can help researchers understand the behavior of materials and design new materials with specific properties, such as enhanced electrical conductivity or thermal insulation.

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