Lecture 24: 5 November 2010 As in the notes from the previous lecture, considered the general case of the scalar wave equation t=a∂/v∂, splitting this up ast2∂u/2∂u)=(a·with non-constant coefficients a,b>0: bu and ∂u/∂t=bunder a modified inner product 〈w,w'〉 = ∫(uu'/b+v·v'/a). Example: a stretched string with a non-constant mass density ρ(x) and tension T(x), which corresponds to b=1/ρ, a=T, u=∂h/∂t and v=T∂h/∂x. In this case, we immediately get conservation of kinetic+potential energy: ∫[ρ(∂h/∂t)2 + T(∂h/∂x)2] (= 2×energy, not that constant factors matter in conservation laws). Example: pressure waves in a fluid or gas. In this case, u=P (pressure), v is a velocity, a=1/ρ (ρ is density), and b=K (bulk modulus: dP=-KdV/V, relating change in pressure dP to fractional change in volume dV/V). Again this gives a wave equation, with a conserved kinetic+potential energy ∫(ρ|v|2+P2/K). Traveling waves: D'Alembert's solution. Considered the 1d scalar wave equation c2∂2u/∂x2=∂2u/∂t2 on an infinite domain with a constant coefficient c. Showed that any f(x) gives possible solutions u(x,t)=f(x±ct). This is called D'Alembert's solution, and describes the function f(x) "moving" to the left or right with speed c. That is, wave equations have travelling solutions, and the constant c can be interpreted as the speed of these solutions. Conversely, if at a fixed point we have a time-dependence g(t), then a solution is u(x,t)=g(t±x/c). Hence, if we look for oscillating solutions g(t)=e-iωt for a frequency ω, then the solutions are planewaves ei(kx-ωt) where ω=±ck. 2π/k is a spatial wavelength λ, and ω/2π is a frequency f, and from this we find that λf=c, a relation you may have seen before. (This whole situation is greatly complicated by the fact that, in realistic wave equations, c depends on ω. We will return to this soon.) ·v. As in the notes, showed that the resulting D operator is still anti-Hermitian butMIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:
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