Problems

Problem 6.1 Consider the 1D wave equation, defined over the semi-infinite interval D=R⁺. Show that when the lossy boundary condition(6.22)is employed atx=0, withα=γ, left-going traveling waves are completely absorbed. (You may substitute the traveling wave solution (6.11) directly into the boundary condition.)

Problem 6.2 Consider the 1D wave equation, defined over the semi-infinite interval D=R⁺. Show that the nonlinear boundary condition

u_x(0, t )=α (u_t(0, t ))²sign(ut(0, t )) for α≥0 is dissipative, i.e.,

dH dt ≤0 If this condition is generalized to

u_x(0, t )=f (u_t(0, t ))

for some functionf, what property mustf satisfy in order that the system remain dissipative?

Problem 6.3 Consider the 1D wave equation, defined over the finite intervalD=U=[0,1], with fixed boundary conditionsu=0at both boundary points. Show that under these conditions, bounds (6.26)may be improved to

|u(x0, t )| ≤ 1 γ

H(0)

2 −→ uD≤ 1

γ H(0)

2

158 THE 1D WAVE EQUATION

Problem 6.4 Consider again the 1D wave equation, defined over the finite intervalU=[0,1].

Under the boundary conditions

u(0, t )=0 and ux(1, t )= −αu(1, t )

for some constantα >0, show that the modal frequenciesωmust satisfy the equation ω

γ α = −tan ω

γ

and, as such, must be determined numerically. Asω becomes large, show that the modal density approaches that of the wave equation under fixed/fixed, free/free, or fixed/free conditions.

Problem 6.5 Given the dispersion relation(6.43)for scheme(6.34)for the wave equation, show by direct calculation that the phase and group velocities, as functions of frequencyω, are equal to

vφ(ω)= hω

2 sin⁻¹(sin(ωk/2)/λ) vg(ω)=γ 1−sin²(ωk/2)/λ² cos(ωk/2)

Show that in the limit asωorkapproaches zero, the limiting value of both these expressions isγ, the wave speed for the continuous-time–space wave equation.

Problem 6.6 For the lossy numerical boundary condition (6.49)for scheme (6.34) for the 1D wave equation, applied at grid pointl=0, show that whenλ=1andα=γ, the condition reduces touⁿ₀⁺¹=uⁿ₁. See also Programming Exercise 6.3.

Problem 6.7 As has been seen in Section 6.2.7, from the bound on the energy for scheme(6.34) defined over the semi-infinite domainD=Z⁺, one may deduce that, under any conservative bound-ary condition atl=0, and under stability condition(6.40), one has the conditions

δ_t−uZ⁺ ≤ 2h δ_x+μ_t−uZ⁺ ≤

√2h

γ (6.68)

The first condition on its own serves to bound the rate of growth of the solution. If the condition at l=0is of Dirichlet type, i.e.,u0=0, then it is possible to determine a bound onuⁿ_l itself. This is a little trickier than in the continuous case, and requires a good number of the bounds and identities supplied in the previous chapter.

(a) First, show that ifu₀=0, one may writeu_l= 1, δx+u_[0,l−1], where “1” indicates here a sequence of ones and, thus, using the Cauchy–Schwartz inequality, that

|ul| ≤√

lhδx+u[0,l−1]≤√

lhδx+uZ⁺

(b) Using identity(2.7d), and the triangle inequality(5.22b), show furthermore that

|ul| ≤√ lh

δx+μt−uZ⁺+k

2δx+δt−uZ⁺

≤√ lh

δx+μt−uZ⁺+k

hδt−uZ⁺

(c) Use the separate bounds on δ_t−u_Z⁺ and μ_t−δ_x+u_Z⁺ given at the beginning of this problem to arrive at a final bound on|ul|in terms ofγ,λ,l, and the energyh.

Problem 6.8 Caution: This is a difficult, but instructive problem, in that some rather more advanced energy-based techniques for the analysis of numerical boundary conditions are introduced.

Consider scheme(6.34), defined overD=Z⁺, under the generalized Neumann boundary con-dition atl=0:

αδx−u0+(1−α)δx·u0=0

(Recall the example of numerical instability resulting from the use of this boundary condition, as described on page 140.) Using the results and definition in Problem 5.6 in the previous chapter, show that, by taking aninner product of the scheme withδ_t·u, the following energy balance may be obtained:

δ_t+h= −γ²(δ_t·u0) ((−1)δ_x−u0+(2−)δ_x·u0) where

h= 1 2

δt−u_Z+2

+γ²

2 δx+u, et−δx+u_Z+

Thus the boundary condition above may be considered conservative under aninner product with α=−1.

Go further, and show that

h≥ 1 2

δ_t−u_Z+2

−λ² 2

δ_t−u_Z+2

Show that, for non-negativity, beyond the CFL condition, the conditionλ≤√

must also be satis-fied, and, given the relationship betweenαand, determine the range ofαfor which this additional condition interferes with the CFL bound.

Problem 6.9 Consider, instead of the numerical radiation condition(6.50), the condition δ_x·u₀=α₁δ_t·u₀+α₂μ_t·u₀ (6.69) used in conjunction with scheme(6.34)over the domainD=Z⁺. Show that the conserved numerical energy will be of the formh+h_b, but with a different definition ofh_b from that which appears in(6.51). Show that, in this case, the energy remains non-negative (and the scheme stable) under the CFL condition(6.40), without any interfering condition for the boundary.

Problem 6.10 Considering the five-point stencil scheme (6.56)for the wave equation, show the stability conditions (6.58) and (6.59) through the determination of the maximum and minimum values of the functionF (p)over the unit intervalp∈[0,1], as in(6.57).

Problem 6.11 Show that the two implicit schemes below for the 1D wave equation, (θ+(1−θ )μx·) δt tu=γ²δxxu δt tu=γ²(φ+(1−φ)μt·) δxxu

in the free parametersθandφ, respectively, are equivalent whenθ=1+λ²(1−φ). The identities (5.14)and (2.7a)may be of use here.

Problem 6.12 Considering the implicit scheme (6.60)for the wave equation, show the stability conditions(6.62) through the determination of the maximum and minimum values of the function F (p)over the unit intervalp∈[0,1], as in(6.61).

Problem 6.13 Consider the wave equation with loss(6.64)defined over the domainD=U, under fixed boundary conditions. Extend the modal analysis of Section 6.1.11, and show that, for real wavenumbersβ_p=pπ, for integerp, a single solution is of the form

u_p(x, t )=e^σpt

a_pcos(ωpt )+b_psin(ωpt )

sin(pπ x) (6.70)

and determineσ_pandω_pin terms ofσ₀andγ.

160 THE 1D WAVE EQUATION

Problem 6.14 Consider the lossy wave equation(6.64), defined over the unit intervalD=U, with fixed boundary terminations.

(a) Assume allowed wavenumbersβpare of the formβp=pπ, and determine, from the disper-sion relation(6.65), an expression for the frequency, in Hertz, of thepth mode. (Here, assume the

“frequency” to refer to the imaginary part of the complex frequency variables.)

(b) Assuming thatγ =100andT₆₀=3, calculate the frequency of the first mode (these param-eters are typical of a musical string). By how many cents does it deviate from the frequency of the first mode in the lossless case?

Problem 6.15 The wave equation with a loss term exhibits a (very slight) degree of dispersion, so clearly a strict traveling wave solution does not exist, ruling out an efficient solution by digital waveguides. But dispersionless wave propagation can extend to the case of loss, if a slightly different model is employed. Consider the so-called distortionless transmission line equation

u_{t t} =γ²u_xx−2σ0u_t−σ₀²u (6.71) defined over the real lineD=R.

(a) Perform a dispersion analysis of this equation, by inserting a test solution of the form u(x, t )=e^st+jβx, and derive the characteristic equation relatings andβ.

(b) Solve for the rootss_±(β)of the characteristic equation, and express the real and imaginary parts of the roots separately, i.e., write s_±(β)=σ (β)±j ω(β). Show that, given this expression forω(β), the phase and group velocity are again constant.

(c) Show, by direct substitution, that damped traveling wave solutions of the form e^−σ0tu⁽⁺⁾ (x−γ t ) and e^−σ0tu⁽⁻⁾(x+γ t ) satisfy the distortionless transmission line equation (6.71), for arbitrary twice-differentiable distributionsu⁽⁺⁾andu⁽⁻⁾.

(d) By taking an inner product of (6.71)withu_toverD=R, find an expression for the energyH, and show that it is monotonically decreasing. Furthermore, find a bound onuRin terms of the initial energyH(0).

Problem 6.16 Consider the following approximation to the distortionless wave equation (6.71) presented in the previous problem:

δt tu=γ²δxxu−2σ0δt·u−σ₀²μt tu (6.72) defined over the infinite domainD=Z.

(a) Write the characteristic polynomial corresponding to scheme (6.72). Show that the stability condition onλ, the Courant number, is unchanged from the Courant–Friedrichs–Lewy condition (6.40).

(b) Write the explicit recursion in the grid functionuⁿ_l corresponding to the compact operator form given above. How does the computational footprint of the scheme simplify when λ²=1− σ₀²k²/4?

(c) Show that, under the assumption thatλ²=1−σ₀²k²/4, a lossy discrete traveling wave solu-tion exists. That is, given a traveling wave decomposisolu-tionuⁿ_l =u^(+),n_l +u^(−),n_l , and where the wave variables u^(+),n_l and u^(−),n_l are updated according to u^(+),n_l =ru^(+),n−1_l−1 , u^(−),n_l =ru^(+),n−1_l+1 , for some constantr, show that the scheme (6.72) is implied, and find the value ofr in terms ofσ₀ andk.

(d) The rate of loss in the scheme(6.72)is not quite equivalent to that of the partial differential equation (6.71). Show that scheme (6.72) solves equation (6.71) exactly, withσ0 replaced by a slightly different valueσ₀^∗, and find the value ofσ₀^∗in terms ofσ0andk.

(e) Sketch a digital waveguide representation of the discrete traveling wave solution, where, now, each delay must be followed by multiplication by a factor r. Is it possible to use a single

bidirectional delay line, and consolidate these multiplications into a single multiplicative factor, applied at a given point in the delay line (such as the end)? See also Programming Exercise 6.9.

Problem 6.17 Extend the energy analysis of the wave equation with loss (6.64) to the domain D=R⁺, and show that under the Dirichlet, Neumann, or lossy boundary condition(6.22)atx=0, the stored energyHwill be non-increasing.

Problem 6.18 Show that the roots of the characteristic equation(6.67)for the scheme(6.66)for the lossy wave equation are bounded by unity in magnitude under the conditionλ≤1.