GATE BT 2019 • Q22
MCQ • 1 Mark
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Fundamentals of Differential Equations
Question Statement:
Which one of the following equations represents a one-dimensional wave equation?
(B) atz
Options:
-
(B) atz
032 = 02224
-
(C) B2U = C202 D)
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2019 Q.22)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Partial Differential Equations (PDEs)
- Classical linear partial differential equations in one spatial dimension:
- **1D Wave Equation**: Hyperbolic PDE governing acoustic and electromagnetic propagation:
$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
where $c$ is the wave propagation speed.
- **1D Heat / Diffusion Equation**: Parabolic PDE:
$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
- **1D Laplace Equation**: Elliptic PDE:
$\frac{\partial^2 u}{\partial x^2} = 0$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Diffusion equation form.
- **Option (B) [∂²u/∂t² = c² ∂²u/∂x²] is CORRECT**: Standard second-order hyperbolic one-dimensional wave equation.
- **Option (C) is INCORRECT**: Incorrect derivative orders.
- **Option (D) is INCORRECT**: Erroneous formulation.
**3. Final Answer**:
- **Correct Choice**: **B**
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