The problem shown in the image is asking for the initial conditions of the one-dimensional wave equation using D'Alembert's solution. The equation is:

$u_{tt} = c^2 u_{xx}$

Where:

• $u(x, t)$ represents the deflection of the vibrating string at position $x$ and time $t$.
• The length of the string is $L = 1$, and the wave speed is $c = 1$.
• The initial velocity is zero, and the initial deflection is given by a function $f(x)$.

The question provides several conditions to choose from, asking for the correct initial conditions for D'Alembert's solution.

Let's analyze each option:

1. $u(x, 0) = f(x), \quad \frac{\partial u(x, 0)}{\partial t} = 0$

   • This implies that the initial displacement is $f(x)$ and the initial velocity is zero.

2. $u(x, 0) = f(x), \quad \frac{\partial u(x, t)}{\partial t} = 0$

   • This seems like a misinterpretation because it's asserting that the time derivative is zero for all $t$, which would mean no motion occurs at all times.

3. $u(x, 0) = f(x), \quad \frac{\partial u(x, 0)}{\partial t} = 0$

   • This matches the conditions for a wave with initial displacement $f(x)$ and zero initial velocity.

4. $u(x, t) = f(x), \quad \frac{\partial u(x, t)}{\partial t} = 0$

   • This suggests that $u(x, t)$ is always equal to $f(x)$ and there's no variation in time, which does not align with the behavior of a vibrating string.

Correct Answer:

The correct initial conditions for the wave equation are:

$u(x, 0) = f(x), \quad \frac{\partial u(x, 0)}{\partial t} = 0$

This corresponds to option (a) and (c), as they both describe initial deflection given by $f(x)$ and zero initial velocity.

Would you like further details on how D'Alembert's solution works, or explanations on the other options?

Five Related Questions:

1. How does D'Alembert's solution handle boundary conditions for a vibrating string?
2. What physical scenarios correspond to different initial velocity conditions in the wave equation?
3. How can we solve the wave equation with arbitrary initial conditions using D'Alembert's formula?
4. What role does the wave speed $c$ play in the solution of the wave equation?
5. How does the initial deflection $f(x)$ affect the solution to the wave equation over time?

Tip:

For solving wave equations, always carefully check the initial conditions, as they determine how the wave will evolve over time.