GATE Mathematics Wave equation Flashcards

24 sample cards from the Wave equation deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the speed of propagation of information in the wave equation, and what principle does it embody?

   Finite speed $c$: disturbances travel at speed exactly $c$, reflecting finite propagation speed (no instantaneous spreading).

2. Contrast the wave equation with the heat equation regarding propagation speed.

   The wave equation (hyperbolic) has finite propagation speed $c$, while the heat equation (parabolic) has infinite propagation speed — disturbances are felt instantly everywhere.

3. What is Huygens' principle in odd space dimensions $\geq 3$ for the wave equation?

   A sharp signal propagates exactly on the wavefront with no trailing tail (sharp Huygens principle); in $3$D the solution at a point depends only on data on a sphere, not its interior.

4. In which spatial dimensions does the strong Huygens principle hold for the wave equation?

   In odd dimensions $\geq 3$ (notably $3$D). It fails in even dimensions (e.g. $2$D), where waves leave a trailing tail.

5. State Kirchhoff's formula (solution of the 3D wave equation) in words.

   The solution at $(\vec{x},t)$ is given by spherical means of the initial data over the sphere of radius $ct$ centered at $\vec{x}$.

6. What is Poisson's spherical-mean (Kirchhoff) solution for $u_{tt}=c^2\nabla^2 u$ in 3D with $u(\vec x,0)=\phi$, $u_t(\vec x,0)=\psi$?

   $$u(\vec x,t)=\frac{\partial}{\partial t}\!\left(t\,\overline{\phi}\right)+t\,\overline{\psi},$$ where $\overline{\phi},\overline{\psi}$ are averages over the sphere of radius $ct$ about $\vec x$.

7. Write the spherically symmetric 3D wave equation for $u(r,t)$.

   $$u_{tt}=c^{2}\left(u_{rr}+\frac{2}{r}u_r\right)$$ Equivalently $(ru)_{tt}=c^{2}(ru)_{rr}$.

8. For spherically symmetric 3D waves, what substitution reduces the equation to a 1D wave equation?

   Let $v=ru$. Then $v_{tt}=c^{2}v_{rr}$, the standard 1D wave equation.

9. What boundary condition corresponds to a 'fixed' (clamped) end of a string?

   A Dirichlet condition: $u=0$ at that end, e.g. $u(0,t)=0$.

10. What boundary condition corresponds to a 'free' end of a string?

    A Neumann condition: zero slope, $u_x=0$ at that end, e.g. $u_x(L,t)=0$.

11. For a semi-infinite string $x\ge 0$ with fixed end $u(0,t)=0$, how is d'Alembert's solution obtained?

    By odd extension of the initial data to $x<0$; the incident wave reflects with inverted sign at $x=0$.

12. How does a wave reflect at a fixed (Dirichlet) boundary?

    It reflects with a phase inversion (sign change / $180^\circ$ phase shift).

13. How does a wave reflect at a free (Neumann) boundary?

    It reflects without phase inversion (same sign).

14. What does the principle of superposition state for the linear wave equation?

    Any linear combination $\alpha u_1+\beta u_2$ of solutions is again a solution, since the equation is linear and homogeneous.

15. Write the inhomogeneous (forced) 1D wave equation.

    $$u_{tt}-c^{2}u_{xx}=f(x,t)$$ where $f$ is an external forcing/source term.

16. State Duhamel's principle solution for $u_{tt}-c^2u_{xx}=f(x,t)$ with zero initial data.

    $$u(x,t)=\frac{1}{2c}\int_{0}^{t}\int_{x-c(t-s)}^{x+c(t-s)} f(y,s)\,dy\,ds$$

17. What is a standing wave, expressed as a product solution?

    $u(x,t)=\sin(kx)\cos(\omega t)$ — spatial and temporal parts separate; nodes stay fixed in space while amplitude oscillates.

18. What is the dispersion relation for the classical wave equation $u_{tt}=c^2u_{xx}$?

    $$\omega=\pm c\,k$$ Phase velocity $\omega/k=c$ is independent of $k$, so the equation is non-dispersive.

19. Why is the standard wave equation called 'non-dispersive'?

    Because all frequencies travel at the same phase speed $c$ (since $\omega=ck$), so a wave packet keeps its shape without spreading.

20. Write a plane-wave (complex exponential) solution of the wave equation and the condition it must satisfy.

    $u=e^{i(kx-\omega t)}$ is a solution provided $\omega^{2}=c^{2}k^{2}$, i.e. $\omega=\pm ck$.

21. What is the telegrapher's / Klein-Gordon-type modification of the wave equation, and how does it differ?

    $$u_{tt}-c^{2}u_{xx}+m^{2}u=0$$ It is dispersive — its dispersion relation $\omega^{2}=c^{2}k^{2}+m^{2}$ makes phase speed depend on $k$.

22. Define phase velocity and group velocity for a wave $\omega(k)$.

    Phase velocity $v_p=\dfrac{\omega}{k}$; group velocity $v_g=\dfrac{d\omega}{dk}$. For $u_{tt}=c^2u_{xx}$ both equal $c$.

23. State the uniqueness result for the wave equation initial-boundary value problem.

    With given initial displacement, initial velocity, and (Dirichlet/Neumann) boundary data, the solution is unique — typically proved via the energy method (energy conservation forces a zero-data solution to vanish).

24. What is the relationship between the 1D wave equation and a pair of first-order transport equations?

    $u_{tt}-c^2u_{xx}=(\partial_t-c\partial_x)(\partial_t+c\partial_x)u=0$, factoring into left- and right-moving transport operators with speeds $\pm c$.

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