Subtopic: Conservation Equations of Mass, Momentum & Energy
Question Statement:
The following equation with respect to $\phi$(x, t), where a is a non-zero constant,
represents ________.
$\partial \phi \partial$t + a $\partial \phi \partial$x = 0
Options:
(A) linear wave propagation
(B) transient heat conduction
(C) Newton's law of cooling
(D) radiative transfer
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.13)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the hyperbolic partial differential equation $\frac{\partial^2 \phi}{\partial t^2} = a^2 \frac{\partial^2 \phi}{\partial x^2}$, where $a \ne 0$ is a physical constant.
**2. Step-by-Step Mathematical Derivation**:
1. **Identification of PDE Canonical Form**:
The equation is the classic 1-D second-order linear wave equation:
\frac{\partial^2 \phi}{\partial t^2} - a^2 \frac{\partial^2 \phi}{\partial x^2} = 0
2. **d'Alembert's General Solution**:
Factoring the operator into $\left(\frac{\partial}{\partial t} - a\frac{\partial}{\partial x}\right)\left(\frac{\partial}{\partial t} + a\frac{\partial}{\partial x}\right)\phi = 0$ yields d'Alembert's solution:
\phi(x, t) = f(x - at) + g(x + at)
which represents two non-dispersive waves propagating in opposite directions at constant phase speed $c = a$.
3. **Comparison with Alternative Physical Equations**:
- Transient heat conduction is parabolic: $\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$ (first derivative in time).
- Newton's law of cooling is an ODE: $\frac{dT}{dt} = -k(T - T_\infty)$.
- Radiative transfer follows the Stefan-Boltzmann or photon transport radiative transfer equation.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) linear wave propagation**: ✅ **CORRECT**. The second derivative in both time and space with positive $a^2$ defines hyperbolic wave propagation at speed $a$.
- **(B) transient heat conduction**: ❌ **INCORRECT**. Heat diffusion contains a first-order time derivative $\partial T/\partial t$, causing dissipation rather than non-dispersive propagation.
- **(C) Newton's law of cooling**: ❌ **INCORRECT**. Governing an ordinary differential equation with respect to lumped body temperature.
- **(D) radiative transfer**: ❌ **INCORRECT**. Involves integro-differential radiation intensity transport.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Second derivative in time ($u_{tt}$) paired with second derivative in space ($u_{xx}$) with same sign indicates hyperbolic wave propagation. First derivative in time ($u_t = lpha u_{xx}$) indicates parabolic diffusion.