GATE Mathematics Wave equation Syllabus

What are the natural (angular) frequencies of vibration of a string fixed at both ends?

$$\omega_n=\frac{n\pi c}{L},\qquad n=1,2,3,\dots$$

For the vibrating string, how are the coefficients $a_n$ determined from the initial displacement $\phi(x)$?

$$a_n=\frac{2}{L}\int_{0}^{L}\phi(x)\sin\!\frac{n\pi x}{L}\,dx$$

For the vibrating string, how are the coefficients $b_n$ determined from the initial velocity $\psi(x)$?

$$b_n=\frac{2}{n\pi c}\int_{0}^{L}\psi(x)\sin\!\frac{n\pi x}{L}\,dx$$

What is the fundamental frequency (first harmonic) of a string of length $L$ with wave speed $c$?

$$f_1=\frac{c}{2L}\qquad\text{(angular: }\omega_1=\tfrac{\pi c}{L}\text{)}$$

Define the term 'normal mode' for the vibrating string.

A standing-wave solution $u_n(x,t)=\sin\!\frac{n\pi x}{L}\big(a_n\cos\omega_n t+b_n\sin\omega_n t\big)$ in which every point oscillates with the same frequency $\omega_n$.

What is the energy (conserved quantity) associated with the 1D wave equation on a string?

$$E(t)=\frac{1}{2}\int\left(u_t^{2}+c^{2}u_x^{2}\right)dx,$$ which is constant in time for fixed/free ends.

State the conservation of energy property for the wave equation.

For solutions with suitable boundary conditions, the total energy $\frac{1}{2}\int (u_t^{2}+c^{2}u_x^{2})\,dx$ is conserved (independent of $t$).

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).

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.

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.

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.

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 $\vec{x}$.

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 $\vec x$.

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 $(ru)_{tt}=c^{2}(ru)_{rr}$.

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.

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

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

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

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

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 $x=0$.

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

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

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

It reflects without phase inversion (same sign).

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.

Write the inhomogeneous (forced) 1D wave equation.

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

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

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

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.

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.

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.

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. $\omega=\pm ck$.

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 $k$.

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 $c$.

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).

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 $\pm c$.