Question

Consider the motion of a stretched string, whose displacement $y(x,t)$ obeys the differential equation $$\frac{1}{c^2}\frac{{\partial }^{2}y}{\partial t^2}=\frac{{\partial }^{2}y}{\partial x^2}$$ The boundary conditions are $(a)$ the ends of the string are fixed; $y(0,t)=y(L,t)=0$ (b) the shape of the string is given at $t=0;y(x,0)=f(x)$ (c) the shape of the string is given at another time $T:y(x,T)=g(x)$. This is an example of the application of a Dirichlet boundary condition on a closed boundary (in the $x-t$ plane) to a hyperbolic differential equation. Does the problem have a solution? Discuss separately the cases: (a) $\frac{cT}{L}=$ integer (b) $\frac{cT}{L}=$ rational number (c) $\frac{cT}{L}=$ irrational number

Short answer

The problem has a solution when $\frac{cT}{L}$ is an integer or a rational number, but not when it is an irrational number.

Step-by-step solution

- Analyze the Differential Equation

The given partial differential equation is a classic wave equation: $$\frac{1}{c^2}\frac{{\partial }^{2}y}{\partial t^2}=\frac{{\partial }^{2}y}{\partial x^2}$$This describes the propagation of waves in a medium with wave speed $c$.

- Examine Boundary Conditions

The boundary conditions specify that the ends of the string are fixed: $$y(0,t)=y(L,t)=0$$The initial shape of the string is given by:$$y(x,0)=f(x)$$And the shape at another time $T$ is given by:$$y(x,T)=g(x)$$.

- Case (a): $\frac{cT}{L}=\text{integer}$

When $\frac{cT}{L}=n$, where $n$ is an integer, the system has a periodic solution. The wave can be expressed as a sum of standing wave solutions. Therefore, the original problem has a solution as the standing wave modes, characterized by integer multiples of $\frac{c}{L}$, can satisfy the given initial conditions.

- Case (b): $\frac{cT}{L}=\text{rational number}$

When $\frac{cT}{L}$ is a rational number, we can write it as $\frac{cT}{L}=\frac{p}{q}$ where $p$ and $q$ are integers. This solution still exhibits periodicity although over a longer period. Rational ratios of $\frac{cT}{L}$ indicate the existence of a quasi-periodic solution, implying that the original problem has a solution.

- Case (c): $\frac{cT}{L}=\text{irrational number}$

When $\frac{cT}{L}=\alpha$ is an irrational number, the wave does not repeat periodically. Irregular intervals produced by irrational numbers imply that no standing wave solutions fully satisfy the time period $T$. Therefore, no exact solution exists for the problem under these conditions because the exact initial and boundary conditions cannot be consistently met at all required points.

Key concepts

Boundary Conditions

Boundary conditions are crucial when solving partial differential equations (PDEs) because they specify the behavior of the solution at the boundaries of the domain. In this problem, we have a string fixed at both ends, meaning the displacement is zero at these points. Mathematically, this is expressed as:

$$y(0,t)=y(L,t)=0$$
These conditions ensure that the string doesn't move at its endpoints, simulating a physical string tightly secured at both ends.

Additionally, the problem gives us initial shape conditions of the string:

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• At time t = 0: $y(x,0)=f(x)$
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• At another time t = T: $y(x,T)=g(x)$

The first initial condition gives the initial displacement shape, and the second one gives the shape at a later time.

Partial Differential Equations

Partial differential equations involve functions of several variables and their partial derivatives. In our exercise, we deal with the wave equation, which is a second-order linear PDE given by:

$$\frac{1}{c^2}\frac{{\text{∂}}^{2}y}{\text{∂}{t}^2}=\frac{{\text{∂}}^{2}y}{\text{∂}{x}^2}$$

This specific PDE describes wave propagation in a medium where $c$ is the wave speed.

Key aspects of solving PDEs involve understanding the nature of the equation and utilizing initial and boundary conditions to determine the specific solution. In our case, the boundary conditions ensure the displacement at the ends of the string is zero, while initial conditions provide information about the shape of the string at specific times.

Wave Propagation

Wave propagation refers to the movement of waves through a medium. For our string, we analyze how waves travel along its length, governed by the wave equation. The solutions to the wave equation can reveal different behaviors based on the ratio $\frac{cT}{L}$:

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• **Integer $\frac{cT}{L}$**: The system has a periodic solution resembling standing waves. These waves can be expressed as sums of sine and cosine functions (modes), satisfying both boundary and initial conditions.
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• **Rational $\frac{cT}{L}$**: The system behaves quasi-periodically. We can still find solutions, but the periods are longer and more complex than in the integer case.
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• **Irrational $\frac{cT}{L}$**: The solutions do not repeat periodically. This irregularity means we cannot find standing wave solutions that match all boundary and initial conditions consistently.

Understanding wave propagation helps us predict the behavior of the string over time, which is essential for solving physical problems like this one.