GATE Mathematics Wave equation Syllabus

Wave equation syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Wave equation in GATE Mathematics, not a summary of it.

1. 1

   Cauchy problem and d'Alembert formula

   overview

   Examined as a single unit within Wave equation — no further topic split in the official outline.

2. 2

   Domains of dependence and influence

   overview

   Examined as a single unit within Wave equation — no further topic split in the official outline.

3. 3

   Non-homogeneous wave equation

   overview

   Examined as a single unit within Wave equation — no further topic split in the official outline.

Wave equation flashcards for GATE Mathematics

19 of 51 cards from the Wave equation deck — real questions with worked answers.

1. What is the standard form of the one-dimensional wave equation for displacement $u(x,t)$?

   $$\frac{\partial^{2} u}{\partial t^{2}} = c^{2}\,\frac{\partial^{2} u}{\partial x^{2}}$$ where $c>0$ is the wave speed.

2. In the 1D wave equation $u_{tt}=c^{2}u_{xx}$, what does the constant $c$ physically represent?

   The speed of propagation of the wave (e.g. $c=\sqrt{T/\rho}$ for a vibrating string with tension $T$ and linear density $\rho$).

3. How is the wave equation classified among second-order linear PDEs?

   It is a hyperbolic PDE.

4. For a general second-order PDE $Au_{xx}+Bu_{xy}+Cu_{yy}+\dots=0$, what discriminant condition makes it hyperbolic?

   $$B^{2}-4AC > 0$$ The wave equation satisfies this, so it is hyperbolic.

5. Write d'Alembert's general solution of $u_{tt}=c^{2}u_{xx}$ on the infinite line.

   $$u(x,t)=F(x-ct)+G(x+ct)$$ where $F$ and $G$ are arbitrary twice-differentiable functions.

6. In d'Alembert's solution $u=F(x-ct)+G(x+ct)$, what do the two terms represent?

   $F(x-ct)$ is a wave travelling to the right (positive $x$) and $G(x+ct)$ is a wave travelling to the left, both with speed $c$.

7. State d'Alembert's formula for the initial value problem $u_{tt}=c^{2}u_{xx}$, $u(x,0)=\phi(x)$, $u_t(x,0)=\psi(x)$.

   $$u(x,t)=\frac{1}{2}\big[\phi(x-ct)+\phi(x+ct)\big]+\frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds$$

8. What are the characteristic curves of the 1D wave equation $u_{tt}=c^{2}u_{xx}$?

   The straight lines $x-ct=\text{const}$ and $x+ct=\text{const}$.

9. Which change of variables reduces $u_{tt}=c^{2}u_{xx}$ to canonical form, and what is that form?

   With $\xi=x-ct$, $\eta=x+ct$ the equation becomes the canonical hyperbolic form $$u_{\xi\eta}=0.$$

10. What is the 'domain of dependence' of a point $(x_0,t_0)$ for the 1D wave equation?

    The interval $[x_0-ct_0,\; x_0+ct_0]$ on the initial line $t=0$; the solution at $(x_0,t_0)$ depends only on the data there.

11. What is the 'domain of influence' of a point $x_0$ on the initial line?

    The region $\{(x,t): x_0-ct\le x\le x_0+ct,\ t\ge 0\}$, i.e. the points reachable from $x_0$ at speed $c$.

12. Write the three-dimensional wave equation for $u(\vec{x},t)$.

    $$\frac{\partial^{2} u}{\partial t^{2}} = c^{2}\,\nabla^{2} u$$ where $\nabla^{2}$ is the Laplacian.

13. Express the 2D wave equation in Cartesian coordinates.

    $$u_{tt}=c^{2}\left(u_{xx}+u_{yy}\right)$$

14. What initial conditions are required to determine a unique solution of the wave equation?

    Two conditions: the initial displacement $u(x,0)=\phi(x)$ and the initial velocity $u_t(x,0)=\psi(x)$.

15. Why does the wave equation require two initial conditions rather than one?

    Because it is second order in time; the general solution involves two arbitrary functions, fixed by initial displacement and initial velocity.

16. State the wave equation for a vibrating string fixed at both ends $x=0$ and $x=L$.

    $u_{tt}=c^{2}u_{xx}$ for $0<x<L$, with boundary conditions $u(0,t)=0$ and $u(L,t)=0$.

17. Using separation of variables $u(x,t)=X(x)T(t)$, what ODEs arise from $u_{tt}=c^{2}u_{xx}$?

    $$X''+\lambda X=0,\qquad T''+c^{2}\lambda T=0$$ where $\lambda$ is the separation constant.

18. For a string fixed at $x=0,L$, what are the eigenvalues $\lambda_n$ from separation of variables?

    $$\lambda_n=\left(\frac{n\pi}{L}\right)^{2},\qquad n=1,2,3,\dots$$

19. For a string of length $L$ fixed at both ends, what are the spatial eigenfunctions (normal modes)?

    $$X_n(x)=\sin\!\left(\frac{n\pi x}{L}\right),\qquad n=1,2,3,\dots$$

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