1. Strings

2. Masses on a String  Couple n equal masses on a massless string. Displacements  i Separation a Constraints  0 =  n+1 = 0 Potential from string tension  The longitudinal problem is similar Displacements in x Replace tension with elastic springs x 11 Transverse vibration, n segments 00 22 33 nn  n+1 a

3. Small Displacements  There is potential energy from each segment. Dependence only on nearest coordinatesDependence only on nearest coordinates Tension  times extensionTension  times extension Elements 2  /a on diagonalElements 2  /a on diagonal Elements –  /a off diagonalElements –  /a off diagonal  The kinetic energy is from motion of masses . Matrix is diagonalMatrix is diagonal

4. Large Matrix  The direct solution is not generally possible.  If there is a solution it is an harmonic oscillator. Each row related to the previous oneEach row related to the previous one  The eigenvalue equation reduces to three terms.

5. Fixed Boundaries  The eigenvalue equation gives a result based on .  The phase difference  depends on initial conditions. Pick sin for 0Pick sin for 0 Find the other end pointFind the other end point Requires periodicityRequires periodicity  Substitute to get eigenfrequencies. Integer m gives values for Integer m gives values for 

6. Standing Wave  The  are the eigenfrequencies.  Components of the eigenvectors are similar. All fall on a sine curveAll fall on a sine curve Wavelength depends on m.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  repeats after n intervals. Require whole number of wavelengthsRequire whole number of wavelengths Integer m for solutions with that periodInteger m for 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.  The phase and group velocity follow from the form of the eigenfrequencies. Phase velocityPhase velocity Approximate for m << n.Approximate for m << n. Group velocityGroup velocity Wave Velocity