Wave Equation and Energy Methods

Consider the wave equation in a two-dimensional quadrant

with boundary conditions \(\partial _x u(0,y,t)=0\) and \(u(x,0,t)=0\) and initial condition \(u_0\), supported in the ball

Describe, precisely and with careful justification, the support of \(u(x, y, t)\) at all positive times \(t > 0\). Note: The support of a function \(f(x,y)\) is the closure of the set \(\{(x,y): f(x,y)\ne 0\}\).

1. Show that the wave equation \(u_{tt}=c^2u_{xx}\) \((-\infty <x<\infty ,\ t>0)\) conserves energy \[ E=\int _{-\infty }^{\infty }\left [(u_t)^2+c^2(u_x)^2\right ]d x. \]
2. Show the uniqueness: If two solutions of the wave equation satisfy the same initial conditions, then the solutions are identical.

1. Why does the heat/diffusion equation imply infinite propagation speed?
2. Why does the wave equation imply finite propagation speed?

Suppose someone treats one of the variables in the Laplace equation as time and tries to solve the evolution problem \[ \begin {aligned} u_{tt}+u_{xx}&=0\quad (0<x<l,\ t>0),\\ u(0,t)&=u(l,t)=0,\qquad u(x,0)=\phi (x),\qquad u_t(x,0)=\psi (x), \end {aligned} \] similar to the wave equation. Show that this problem has no continuous dependence on data, even if the time \(t\) belongs to a finite interval \(0<t<T\) (\(T\) is a positive constant).

Explain the fundamental difference of wave equations in 1D, 2D, and 3D: In 3D — unlike 1D and 2D — any signal (with finite support) propagates leaving no trace behind it.

Suppose that the initial value problem for the wave equation \[ \begin {aligned} \frac {\partial ^2u}{\partial t^2}-c^2\nabla ^2u&=0, &&x\in \mathbb {R}^n,\quad t>0,\\ u(0,x)&=g(x),\\ \frac {\partial u}{\partial t}(0,x)&=0, &&x\in \mathbb {R}^n \end {aligned} \] has a solution \(u_g\). Find (and verify) the solution \(u_h\) of the same equation subject to the initial conditions \[ \begin {aligned} u(0,x)&=g(x),\\ \frac {\partial u}{\partial t}(0,x)&=h(x), &&x\in \mathbb {R}^n \end {aligned} \] in terms of \(u_g\), \(g\) and \(h\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac {1}{2}[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\abs {x}<a\), \(u(x,0)=0\) if \(x>\abs {a}\); \(u_t(x,0)=0\)
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\abs {x}<a\), \(u_t(x,0)=0\) if \(x>\abs {a}\)

Consider a string clamped at the end points \(a\) and \(b\), say, with \(u(a,t)=0\) and \(u(b,t)=0\), where \(u(x,t)\) is the string’s deviation from the horizontal rest position. The kinetic energy of the string is \[ E_k(t)=\frac {1}{2}\int _a^b \rho u_t^2d s, \] where \(\rho (x,t)\) is the mass density and \(ds=\sqrt {1+u_x^2}d x\) is an infinitesimal arc length. The potential energy consists of the sum of the energy due to the stretching of the string, and the work done against a load: \[ E_p(t)=\int _a^b \left (d(\sqrt {1+u_x^2}-1)-lu\sqrt {1+u_x^2}\right )d x. \] Here \(d(x,t)\) is the string’s elastic coefficient and \(l(x,t)\) is the load on the string. The classical action for the continuum model is \[ J=\int _{t_1}^{t_2}[E_k(t)+E_p(t)]d t. \]

1. Substitute the integral expressions for \(E_k\) and \(E_p\) into the action \(J\), and consider the variations \(u+\epsilon \psi \) such that \(\psi \) vanishes at the end points \(a,b\) and at the initial and final times \(t_1,t_2\). Calculate the first variation \(\delta J\).
2. After integrating by parts terms of the form \(u_t\psi _t\) and \(u_x\psi _x\), and using the boundary conditions for the variation \(\psi \), show that Hamilton’s principle \(\delta J=0\) yields the following PDE for \(u\): \[ (\rho u_t)_t-(1+u_x^2)^{-1/2} \left [d(1+u_x^2)^{-1/2}u_x\right ]_x-l=0. \]
3. Show that if \(\rho ,d\) are constants, there is no load (\(l=0\)), and the small slope approximation \(\abs {u_x}\ll 1\) holds, then the PDE of part (b) reduces to the classical one-dimensional wave equation. Write down an expression for the wave speed.

Suppose the wave equation \[ \frac {\partial ^2u}{\partial t^2}-c^2\nabla ^2u=0,\qquad x\in \mathbb {R}^n,\qquad t>0, \] subject to initial conditions \(u(x,0)=0\), \(\frac {\partial u}{\partial t}(x,0)=h(x)\), for \(x\in \mathbb {R}^n\) has a solution denoted by \(u_h\). Find (and verify) the solution of the same equation subject to the initial conditions \(u(x,0)=g(x)\), \(\frac {\partial u(x,0)}{\partial t}=h(x)\), for \(x\in \mathbb {R}^n\) in terms of \(u_h\) and \(u_g\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac {1}{2}[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\abs {x}<a\), \(u(x,0)=0\) if \(x>\abs {a}\); \(u_t(x,0)=0\)
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\abs {x}<a\), \(u_t(x,0)=0\) if \(x>\abs {a}\)

1. Consider a string clamped at the end points \(a,b\) with \(u(a,t)=u_a\), \(u(b,t)=u_b\), where \(u(x,t)\) is the string’s deviation from the horizontal rest position. The kinetic energy of the string is \[ E_k=\frac {1}{2}\int _a^b \rho u_t^2\dd s,\qquad \dd s=\sqrt {1+u_x^2}\dd x, \] where \(\rho (x,t)\) is the mass density. The potential energy due to stretching of the string is \[ E_p=\int _a^b k\left (\sqrt {1+u_x^2}-1\right )\dd x, \] where \(k(x,t)\) is the elastic coefficient.

   1. Find the first variation of the action \[ A=\int _0^T(E_k-E_p)\dd t. \]
   2. Integrating by parts, imposing the boundary conditions, and equating the first variation to zero, derive the PDE \[ \left (\rho u_t\sqrt {1+u_x^2}\right )_t = \left (k(1+u_x^2)^{-1/2}u_x\right )_x. \] Assuming \(\rho ,k\) are constants and using the approximation \(\abs {u_x}\ll 1\) derive the classical one-dimensional wave equation.
2. Consider the functional \[ Y[u]=\int _D\left (\frac {1}{2}\abs {\nabla u}^2+\frac {1}{2}u^2+fu\right )\dd x, \] where \(D\) is a bounded domain in \(\mathbb {R}^n\) and \(f\) is a bounded, continuous function in \(D\). Suppose that there exists a minimizer \(u^*\) of \(Y[u]\) in the Sobolev space \(H_1(D)\). Use this to prove existence and uniqueness of the weak formulation of the PDE (for which \(u\in C^1(D)\)) \[ -\nabla ^2u+u=-f,\qquad x\in D\subset \mathbb {R}^n,\qquad \partial _nu=0,\qquad x\in \partial D. \] [Hint: show that the weak formulation can be expressed as \((u,\psi )_{H_1(D)}=0\) for all \(\psi \in H_1(D)\).]

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\norm {x}<a\), \(u(x,0)=0\) if \(x>\norm {a}\); \(u_t(x,0)=0\).
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\norm {x}<a\), \(u_t(x,0)=0\) if \(x>\norm {a}\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\norm {x}<a\), \(u(x,0)=0\) if \(x>\norm {a}\); \(u_t(x,0)=0\).
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\norm {x}<a\), \(u_t(x,0)=0\) if \(x>\norm {a}\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Briefly explain the concepts of domain of dependence, and finite propogation speed, and prove that these properties follow d’Alembert’s Formula.
3. Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)=0\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s Formula as a starting point, find a closed form solution for \(u(x,t)\) with \(x,t>0\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\quad x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)]+\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Briefly explain the concepts of domain of dependence, and finite propogation speed, and prove that these properties follow d’Alembert’s Formula.
3. Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)=0\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s Formula as a starting point, find a closed form solution for \(u(x,t)\) with \(x,t>0\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\quad x\in \mathbb {R}. \] along with the intial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s formula \[ u(x,t)=\frac 12\bigl (g(x+ct)+g(x-ct)\bigr ) +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Briefly explain the concepts of domain of dependence, and finite propagation speed, and prove that these properties follow from d’Alembert’s formula.
3. Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s formula as a starting point, find a closed-form solution for \(u(x,t)\), with \(x,t>0\).

(a) Consider the three–dimensional wave equation \[ \frac {\partial ^2u}{\partial t^2}=c^2\nabla ^2u \] with Cauchy data \[ u(\mathbf {x},0)=0,\qquad \frac {\partial u}{\partial t}(\mathbf {x},0)=g(\mathbf {x}) \]

1. Show that radially symmetric solutions are in the form of outgoing and incoming waves \[ u(r,t)=\frac {1}{r}F(r\pm ct). \] [Hint: Perform the substitution \(v=ur\).]
2. Writing the general solution to the Cauchy problem as a superposition of outgoing waves \[ u(\mathbf {x},t)=\int _{\mathbb {R}^3}\frac {\delta (r-ct)}{r}f(x',y',z')\dd x'\dd y'\dd z', \] where \(r^2=(x-x')^2+(y-y')^2+(z-z')^2\), derive the retarded potential solution \[ \begin {aligned} u(\mathbf {x},t)=ct\int _0^{2\pi }\int _0^\pi &f(x+ct\sin \theta \cos \phi ,\ y+ct\sin \theta \sin \phi ,\\ &z+ct\cos \theta )\sin \theta \dd \theta \dd \phi \end {aligned} \] and show that \(f=g/(4\pi c)\).
3. Assuming that the initial data \(g\) is only nonzero in a bounded domain \(D\), use a graphical construction to derive Huygen’s principle.

Consider the wave equation in three spatial dimensions \[ u_{tt}-\Delta u=0 \quad \text {in } \mathbb {R}^3\times (0,\infty ). \]

1. Explain the concept of domain of dependence for solutions of the wave equation, and describe what this has to do with finite propagation speed.
2. Sketch a proof, either using a representation formula or energy methods, that solutions \(u\in C^2\) of the wave equation have an explicitly defined domain of dependence.

Let \(\mathbb {R}_+=(0,\infty )\). Consider the problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in } \mathbb {R}_+\times \{t>0\},\\ u(x,0) &= g(x),\quad u_t(x,0)=h(x), && x\in \mathbb {R}_+ && \text {(initial conditions)},\\ u_x(0,t) &= 0, && t>0 && \text {(boundary condition)}. \end {aligned} \] Assume that \(g,h\in C^1(\overline {\mathbb {R}_+})\), and that \(g_x(0)=h_x(0)=0\).

1. Find an explicit formula for the solution \(u(x,t)\) (d’Alembert’s formula is a good place to start).
2. Suppose that for \(0\le t<\frac {1}{2}\) and \(x>0\), we know \(u(x,t)=g(x+t)\), and assume \(\operatorname {supp}g\subset (1,2)\). Thus, initially a wave is travelling toward the origin \(x=0\). What happens to the wave after it hits the origin, say for \(t>2\)? Is it absorbed, reflected, damped, inverted? Calculate \(u(x,3)\).

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2}, \qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=f(t). \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\widehat {F}(s)\sinh sx}{\sinh sl} \] where \(\widehat {F}(s)\) is the Laplace transform of \(f(t)\).
2. Suppose that \(\phi _0(x,t)\) is the solution when \(f(t)=1\). Show that the solution for general \(f\) is given by \[ \phi (x,t)=\int _0^t \frac {\partial \phi _0}{\partial t'}(x,t')f(t-t')\,dt'. \]

Distributions. Suppose \(f(\xi ),g(\xi )\in C^2_0(\mathbb {R})\) are compactly supported twice continuously differentiable functions. Then \[ u(x,t)=f(x+t)+g(x-t) \tag {2} \] is a classical solution of the wave equation \[ u_{tt}-u_{xx}=0\qquad \text {for }-\infty <x,t<\infty . \tag {3} \] Show that if \(f(\xi ),g(\xi )\in C_0(\mathbb {R})\) are merely compactly supported continuous functions, then (2) still defines a solution to (3) in the sense of distributions.

Wave Propagation. Consider the equation \[ u_{tt}-u_{xx}+\lambda u=0,\quad x\in \mathbb {R},\quad t>0, \] where \(\lambda >0\) is fixed.

1. Explain what is meant by a dispersive solution, and find a dispersion relation for the equation.
2. Define an “energy” functional \(E(t)\) for this equation and show that each solution \(u(x,t)\) with finite initial energy conserves energy. State conditions under which uniqueness of solutions holds.

Wave Equation. Let \(f(x,t)\in C^\infty (\mathbb {R}^3\times \mathbb {R})\) and \(R>0\) be fixed. Suppose that the support of \(f\) is contained in the cylinder \[ \operatorname {spt} f\subset \{(x,t)\in \mathbb {R}^3\times \mathbb {R}:\lvert x\rvert \le R\text { and }t\ge 0\}. \] Show that there exists a solution \(u(x,t)\) to the inhomogeneous wave equation \((\spadesuit )\). Show that the solution satisfies \(u(x,t)=0\) whenever \(\lvert x\rvert \ge ct+R\). \[ \begin {cases} \dfrac {\partial ^2u}{\partial t^2}=c^2\Delta u+f(x,t), & \text {if }x\in \mathbb {R}^3\text { and }t\ge 0,\\ u(x,0)=0,\\ \dfrac {\partial u}{\partial t}(x,0)=0, & \text {for }x\in \mathbb {R}^3. \end {cases} \qquad (\spadesuit ) \]

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2},\qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=1. \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\sinh sx}{s\sinh sl}. \]
2. Obtain the solution \(\phi (x,t)\) by inverting the Laplace transform using a Bromich integral and show that it can be expressed as the infinite series \[ \phi (x,t)=\frac {x}{l}+\sum _{n=1}^{\infty }\frac {2(-1)^n}{n\pi } \sin \left (\frac {n\pi x}{l}\right )\cos \left (\frac {n\pi t}{l}\right ). \]

1. State d’Alembert’s formula for the solutions \(u\) of the wave equation \(u_{tt}=u_{xx}\).
2. Consider the initial boundary value problem \[ \begin {aligned} u_{tt}&=u_{xx} &&\text {for }0<x<\pi ,\ t>0,\\ u(x,0)&=f(x),\qquad u_t(x,0)=0 &&\text {for }0<x<\pi ,\\ u(0,t)&=0,\qquad u(\pi ,t)=0 &&\text {for }t\ge 0. \end {aligned} \] Use d’Alembert’s formula and the parallelogram rule to define a weak solution solution to this problem. HINT: divide the region \(R=\{(x,t):0<x<\pi ,\ t>0\}\) into subregions. What compatibility conditions are needed upon \(f\) to ensure that the above solution is \(C\), is \(C^1\) and is \(C^2\)?
3. Show that the energy \[ E(t)=\frac {1}{2}\int _0^\pi (u_t^2+u_x^2)\,dx \] is conserved, and use this to prove that a \(C^2\) solution to the above initial boundary value problem is unique.

Suppose \(u(x,t): x\in \mathbb {R}\), \(t\in [0,\infty )\) is of class \(C^\infty \) and that \(u\) is a solution of \[ \begin {cases} u_{tt}=u_{xx}\\ u(x,0)=g(x)\\ u_t(x,0)=h(x). \end {cases} \] Suppose \(g\) and \(h\) are in \(C_0^\infty \). (\(C^\infty \) functions with compact support.) Define \[ k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\,dx\quad \text {(kinetic energy)} \] and \[ p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\,dx\quad \text {(potential energy)} \] Prove that \(k(t)+p(t)\) is constant and that for sufficiently large times \(t\), \(k(t)=p(t)\). Hint: d’Alembert’s solution.

Solve the wave equation \[ u_{tt}=c^2u_{xx}+e^{2x},\qquad u(x,0)=0,\qquad u_t(x,0)=0 \] on \(-\infty <x<\infty \), \(0\le t<\infty \).

Consider the wave equation \(u_{tt}=u_{xx}\).

1. State and derive the D’Alembert formula for the solutions \(u\) of this equation.
2. Use the method of separation of variables to solve the initial-boundary value problem \[ u_{tt}=u_{xx}\quad \text {for }0<x<1,\ t>0, \] \[ u(x,0)=f(x),\quad u_t(x,0)=0\quad \text {for }0\le x\le 1, \] \[ u(0,t)=u(1,t)=0\quad \text {for }t\ge 0. \] Define what you mean by a solution.
3. Show how the solution of b) can be written in the D’Alembert form.