Question

The following equation with respect to \(\varphi(x,t)\), where \(a\) is a non-zero constant, represents ________.
\[ \frac{\partial \varphi}{\partial t} + a\frac{\partial \varphi}{\partial x} = 0 \]

• A. linear wave propagation
• B. transient heat conduction
• C. Newton's law of cooling
• D. radiative transfer

Hint:

Solve this first order PDE by the method of characteristics. The solution turns out to be any function of (x minus at), which is the mathematical signature of a wave moving at constant speed without changing shape.

Answer 1

The Correct Option is A

Step 1: Identify the type and order of the equation.
The equation \(\frac{\partial \varphi}{\partial t} + a\frac{\partial \varphi}{\partial x} = 0\) is a first order, linear, homogeneous partial differential equation in the two independent variables \(x\) and \(t\), with constant coefficient \(a\). This particular form is called the linear advection equation, or the one way wave equation.

Step 2: Solve it using the method of characteristics.
Along a curve in the \(x\)-\(t\) plane where \(x\) and \(t\) change together such that \(\frac{dx}{dt} = a\), the chain rule gives
\[ \frac{d\varphi}{dt} = \frac{\partial \varphi}{\partial t} + \frac{dx}{dt}\frac{\partial \varphi}{\partial x} = \frac{\partial \varphi}{\partial t} + a\frac{\partial \varphi}{\partial x} \]
which is exactly the left side of the given PDE. Since the PDE says this equals zero, \(\varphi\) does not change at all along any curve satisfying \(\frac{dx}{dt}=a\), meaning \(\varphi\) is constant along the straight lines \(x - at = \text{constant}\). These lines are called the characteristics of the equation.

Step 3: Write the general solution.
Since \(\varphi\) only depends on which characteristic line it sits on, the general solution has the form
\[ \varphi(x,t) = f(x - at) \]
for an arbitrary function \(f\), fixed by whatever initial shape \(\varphi(x,0) = f(x)\) is given.

Step 4: Interpret this solution physically.
At time \(t=0\), the profile is \(f(x)\). At a later time \(t\), the profile is \(f(x-at)\), the same shape \(f\) simply shifted to the right by a distance \(at\) if \(a>0\), or to the left if \(a<0\). The waveform keeps its exact shape and just translates rigidly at constant speed \(a\), with no spreading, no decay, and no change in profile. This rigid, undistorted translation of a fixed shape at constant speed is exactly what wave propagation means, and because the governing equation is linear, it is specifically linear wave propagation.

Why the other options are wrong:
Transient heat conduction is governed by the heat, or diffusion, equation \(\varphi_t = \kappa\varphi_{xx}\), which has a second order spatial derivative and describes smoothing and spreading of a temperature profile over time, not translation, so it does not match this first order equation. Newton's law of cooling is an ordinary differential equation in time alone, \(dT/dt = -k(T-T_\infty)\), with no spatial derivative or dependence on \(x\) at all, so it cannot match a PDE involving \(\partial\varphi/\partial x\). Radiative transfer equations involve integral terms over direction and frequency and usually a fourth power temperature term from the Stefan-Boltzmann law, nothing like this simple linear first order balance.

Final Answer:
The equation describes a fixed profile translating at constant speed \(a\), which is linear wave propagation.
\[ \boxed{\text{linear wave propagation}} \]