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Waves

Waves. Masses on a String. Equal masses on a massless string Displacements y i Separation a Constraints y 0 = y n+1 = 0 Potential from string tension Longitudinal problem is similar Displacements in x Replace tension with elastic springs. y 1. y n. y 0. y n+1. y 2. y 3. a.

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Waves

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  1. Waves

  2. Masses on a String • Equal masses on a massless string • Displacements y i • Separation a • Constraints y0 = y n+1 = 0 • Potential from string tension • Longitudinal problem is similar • Displacements in x • Replace tension with elastic springs y1 yn y0 yn+1 y2 y3 a x Transverse vibration, n segments

  3. Potential energy per segment V. Assume dependence only on nearest coordinates Tension k times extension Elements 2k /a on diagonal Elements –k/a off diagonal Kinetic energy from motion of masses m. Matrix is diagonal Small Displacements

  4. The direct solution is not generally possible. Solution is harmonic oscillator. Each row related to the previous one. Eigenvector equation reduces to three terms. Large Matrix

  5. Fixed Boundaries • The eigenvalue equation gives a result based on f. • Phase difference f depends on initial conditions. • Pick sin for 0 • Find the other end point • Requires periodicity • Substitute to get eigenfrequencies • Integer m gives values for w

  6. Standing Wave • The w are the eigenfrequencies. • Components of the eigenvectors are similar. • All fall on a sine curve • Wavelength depends on m. • The eigenvectors define a series of standing waves.

  7. Periodic Boundaries • To simulate an infinite string, use boundaries that repeat. • Phase f repeats after n. • Require whole number of wavelengths • Integer m gives possible solutions with that period. • Substitute to get eigenfrequencies as before

  8. Traveling Wave • In a traveling wave the initial point is not fixed. • Other points derive from the initial point as before. • The position can be expressed in terms of the unit length and wavenumber.

  9. Wave Velocity • The phase and group velocity follow from the form of the eigenfrequencies. • Phase velocity • Approximate for m << n. • Group velocity next

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