# USU PHYS 3750 - Long Wavelength Limit / Normal Modes (7 pages)

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# Long Wavelength Limit / Normal Modes

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## Long Wavelength Limit / Normal Modes

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Lecture Notes

Pages:
7
School:
Utah State University
Course:
Phys 3750 - Foundations of Wave Phenomena
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Lecture 7 Phys 3750 Long Wavelength Limit Normal Modes Overview and Motivation Today we look at the long wavelength limit of the coupled oscillator system In this limit the equations of motion for the coupled oscillators can be transformed into the partial differential equation known as the wave equation which has wide applicability beyond the coupled oscillator system We also look at the normal modes and the dispersion relation for the coupled oscillator system in this limit Key Mathematics We again utilize the Taylor series expansion I Derivation of the Wave Equation A The Long Wavelength Limit LWL So why look at the coupled oscillator system at long wave lengths Perhaps the main motivation comes from the fact that we are often interested in waves in systems where the wavelength is much longer than the distance between the coupled objects For example let s consider audible 20 to 20 000 Hz sound waves in a solid In a typical solid the speed of sound c is 2000 m s so audible frequencies correspond to wavelengths c between approximately 0 1 and 100 m These wavelengths are obviously much greater that the typical interatomic spacing d of 2 10 10 m As we will see one benefit of the long wavelength limit is that we will no longer need to refer to the displacement of each interacting object the index j will be traded in for the continuous position variable x so that we will be considering displacements as a function of x and t Let s consider the equation of motion for the j th oscillator which can be any oscillator in the N coupled oscillator system d 2 q j t dt 2 2 q j 1 t 2q j t q j 1 t 1 Let s go ahead and trade in the discrete object index j for the continuous position variable x via x jd where d is the equilibrium distance between objects in the chain Then we can rewrite Eq 1 as 2 q x t 2 q x d t 2q x t q x d t t 2 2 Notice that the time derivative is now a partial derivative because we are now thinking of the displacement q as a function of two continuous variables

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