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Reading: Main 9.1.1 GEM 9.1.1 Taylor 16.1, 16.2, 16.3 (Thornton 13.4, 13.6, 13.7)

THE NON-DISPERSIVE WAVE EQUATION. Reading: Main 9.1.1 GEM 9.1.1 Taylor 16.1, 16.2, 16.3 (Thornton 13.4, 13.6, 13.7). Example: Non dispersive wave equation (a second order linear partial differential equation. Demonstrate that both standing and traveling waves satisfy this equation (HW).

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Reading: Main 9.1.1 GEM 9.1.1 Taylor 16.1, 16.2, 16.3 (Thornton 13.4, 13.6, 13.7)

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  1. THE NON-DISPERSIVE WAVE EQUATION Reading: Main 9.1.1 GEM 9.1.1 Taylor 16.1, 16.2, 16.3 (Thornton 13.4, 13.6, 13.7)

  2. Example: Non dispersive wave equation (a second order linear partial differential equation Demonstrate that both standing and traveling waves satisfy this equation (HW) PROVIDED w/k = v . Thus we recognize that v represents the wave velocity. (What does “velocity” mean for a standing wave?)

  3. SEPARATION OF VARIABLES: Assume solution Plug in and find Argue both sides must equal constant and set to a constant. For the moment, we’ll say the constant must be negative, and we’ll call it -k2 and this is the same k as in the wavevector.

  4. Separation of variables: Then And

  5. Example: Non dispersive wave equation (a second order linear partial differential equation) Separation of variables: Then With v = w/k, A, B, C, D arbitrary constants

  6. Example: Non dispersive wave equation Specifying initial conditions determines the (so far arbitrary) coefficients: In-class worksheet had different examples of initial conditions

  7. T=mg Generate standing waves in this system and measure values of w (or f) and k (or l). Plot wvs. k- this is the DISPERSION RELATION. Determine phase velocity for system.

  8. The dispersion relation is an important concept for a system in which waves propagate. It tells us the velocity at which waves of particular frequency propagate (phase velocity) and also the group velocity, or the velocity of a wave packet (superposition of waves). The group velocity is the one we associate with transfer of information (more in our QM discussion). Non dispersive: waves of different frequency have the same velocity (e.g. electromagnetic waves in vacuum) Dispersive: waves of different frequency have different velocity (e.g. electromagnetic waves in medium; water waves; QM)

  9. T q2 q1 y T x+Dx x x Waves in a rope: Application of Newton’s law results in the non dispersive wave equation! (for small displacements from equilibrium; cosq ≈ 1) Given system parameters: T = tension in rope; µ = mass per unit length of rope

  10. T (sometimes t) q2 q1 y T x+Dx x x Non disp wave eqn with v2=T/µ

  11. THE NON-DISPERSIVE WAVE EQUATION -REVIEW • (non-dispersive) wave equation • Separation of variables • initial conditions, • dispersion, dispersion relation • superposition, traveling wave <-> standing wave, • (non-dispersive) wave equation • Mathematical representations of the above

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