Wave Equation: Traveling Solutions, Sources, Modes, and Boundaries

All flashcards

What is the scalar wave equation in three dimensions?

It is $u_{tt}=c^2\Delta u$, where $u$ is a scalar field, $c$ is the propagation speed, and $\Delta$ is the spatial Laplacian.

What does the Laplacian measure in the scalar wave equation?

It combines spatial second derivatives. In Cartesian three-dimensional coordinates, $\Delta u=u_{xx}+u_{yy}+u_{zz}$, describing spatial curvature of the field.

What is the superposition principle?

For a linear wave equation, sums and constant multiples of solutions are also solutions. This lets complex waves be constructed from simpler wave components.

How does a scalar wave field differ from a vector wave field?

A scalar field has one value at each point, such as pressure. A vector field assigns a vector at each point, such as an electric field or displacement.

What roles do initial and boundary conditions play?

Initial conditions specify the field and its time rate of change at a starting time. Boundary conditions specify behavior at the edge of a domain. Together they help select a particular solution.

Quiz questions

1. For the one-dimensional wave equation on an unrestricted line, which expression gives the general solution?Answer: $u(x,t)=F(x-ct)+G(x+ct)$, with profiles traveling in opposite directions.
2. In the ideal three-dimensional model of a spherically symmetric outgoing wave, how does its amplitude vary with radius?Answer: It includes a factor of $1/r$.
3. What does a Green's function represent for a wave equation?Answer: The response to a concentrated impulse source.
4. What condition models a fixed endpoint of a string?Answer: The displacement vanishes at the endpoint.
5. In the simple scalar interface model, what does a negative reflection coefficient indicate when $c_2<c_1$?Answer: The reflected component undergoes a phase reversal of $180^\circ$.