1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

228 Chapter 3 The Wave Equation It is actually possible to find the general solution of this last equation. Put in another form ...

3.3 d’Alembert’s Solution 229 If we divide through the second equation bycand integrate, it becomes ψ(x)−φ(x)=G(x)+A, 0 <x< ...

230 Chapter 3 The Wave Equation At the second endpoint, a similar calculation shows that f ̃(a+ct)+f ̃(a−ct)+G ̃(a+ct)−G ̃(a−ct) ...

3.3 d’Alembert’s Solution 231 Similarly, iff(x)≡0, sketchG(x)and its even periodic extensionG ̄e(x). Then sketch the graphs ofG ...

232 Chapter 3 The Wave Equation 9.Sketch the solution of the vibrating string problem, Eqs. (2)–(5), at times ct=0, 0. 1 a,0. 3 ...

3.4 One-Dimensional Wave Equation: Generalities 233 3.4 One-Dimensional Wave Equation: Generalities As for the one-dimensional h ...

234 Chapter 3 The Wave Equation The functionw(x,t), being the difference betweenu(x,t)andv(x),satisfies the initial value–bounda ...

3.4 One-Dimensional Wave Equation: Generalities 235 and its two initial conditions, yet to be satisfied, are w(x, 0 )= ∑∞ n= 1 a ...

236 Chapter 3 The Wave Equation 4.Althoughu(x,t)has no limit ast→∞, show that the following general- ized limit is valid: v(x)=T ...

3.5 Estimation of Eigenvalues 237 see that the frequencies of vibration areλnc/ 2 π,n= 1 , 2 , 3 ,....Thuswemust find the eigenv ...

238 Chapter 3 The Wave Equation Example. Estimate the first eigenvalue of φ′′+λ^2 φ= 0 , 0 <x< 1 , φ( 0 )=φ( 1 )= 0. Let u ...

3.6 Wave Equation in Unbounded Regions 239 This method of estimating the first eigenvalue is calledRayleigh’s method, and the ra ...

240 Chapter 3 The Wave Equation Consider the problem ∂^2 u ∂x^2 = 1 c^2 ∂^2 u ∂t^2 ,^0 <t,^0 <x, (1) u(x, 0 )=f(x), 0 < ...

3.6 Wave Equation in Unbounded Regions 241 solution of the wave equation can come to our aid again here. We know that the soluti ...

242 Chapter 3 The Wave Equation Figure 5 Solution of Eqs. (1)–(4) withg(x)≡0. On the left are graphs of fo(x+ct)(solid) and offo ...

3.6 Wave Equation in Unbounded Regions 243 u( 0 ,t)=h(t), 0 <t. (10) Asuis to be a solution of the wave equation, it must hav ...

244 Chapter 3 The Wave Equation Figure 6 Graphs ofh(t)andφ(q)for semi-infinite string with time-varying boundary condition. Figu ...

3.6 Wave Equation in Unbounded Regions 245 EXERCISES Derive a formula similar to Eq. (6) for the case in which the boundary con ...

246 Chapter 3 The Wave Equation Show that this is correct. You will need Leibniz’s rule (see the Appendix) to differentiate the ...

Miscellaneous Exercises 247 More information about the Rayleigh quotient and estimation of eigenval- ues is inBoundary and Eigen ...