Question

An infinitely long string on which waves travel at speed \(c\) has an initial displacement $$ y(x)= \begin{cases}\sin (\pi x / a), & -a \leq x \leq a \\ 0, & |x|>a\end{cases} $$ It is released from rest at time \(t=0\), and its subsequent displacement is described by \(y(x, t)\). By expressing the initial displacement as one explicit function incorporating Heaviside step functions, find an expression for \(y(x, t)\) at a general time \(t>0\). In particular, determine the displacement as a function of time (a) at \(x=0\), (b) at \(x=a\), and (c) at \(x=a / 2\).

Short answer

For \(x=0\): \(y(0, t) = \sin (\pi ct / a)\). For \(x=a\): \(y(a, t) = 0\). For \(x=a/2\): \(y(a/2, t) = \sin( \pi /2 (1 - 2ct / a) )\).

Step-by-step solution

Write Initial Displacement Using Heaviside Functions

The initial displacement can be described using the Heaviside step function, denoted by the symbol \(H(x)\). This step function is defined as \(H(x) = 1\) for \(x \geq 0\) and \(H(x) = 0\) for \(x < 0\). Therefore, the initial displacement can be rewritten as: \[ y(x) = \sin \left( \pi x / a \right) \left[ H(x + a) - H(x - a) \right] \]

Use D'Alembert's Solution for Wave Equation

In the case of an infinitely long string, wave solutions can be described using D'Alembert's formula. For the initial displacement \(y(x) = f(x)\) and initial velocity \(y_t(x) = 0\), the general solution is: \[ y(x, t) = \frac{1}{2} [f(x - ct) + f(x + ct)] \]

Apply Initial Condition to General Solution

Since the initial displacement \(y(x)\) involves Heaviside functions, substitute the initial displacement function into D'Alembert's solution: \[ y(x, t) = \frac{1}{2} \left[ \sin \left( \pi (x - ct) / a \right) \left( H(x - ct + a) - H(x - ct - a) \right) + \sin \left( \pi (x + ct) / a \right) \left( H(x + ct + a) - H(x + ct - a) \right) \right] \]

Displacement at Specific Points (a) x=0

At \(x=0\), simplify the expression: \[ y(0, t) = \frac{1}{2} \left[ \sin(-\pi ct / a) \left( H(-ct + a) - H(-ct - a) \right) + \sin(\pi ct / a) \left( H(ct + a) - H(ct - a) \right) \right] \] Simplifying further, noting that \(\sin(-\theta) = -\sin(\theta)\): \[ y(0, t) = \sin (\pi ct / a) \]

Displacement at Specific Points (b) x=a

At \(x=a\), simplify the expression: \[ y(a, t) = \frac{1}{2} \left[ \sin(\pi (a - ct) / a) \left( H(a - ct + a) - H(a - ct - a) \right) + \sin(\pi (a + ct) / a) \left( H(a + ct + a) - H(a + ct - a) \right) \right] \] Assessing Heaviside functions, it simplifies to: \[ y(a, t) = \frac{1}{2} \left[ \sin(\pi (1 - ct / a)) + \sin(\pi (1 + ct / a)) \right] = 0 \]

Displacement at Specific Points (c) x=a/2

At \(x=a/2\), simplify the expression: \[ y(a/2, t) = \frac{1}{2} \left[ \sin(\pi (a/2 - ct) / a) \left( H(a/2 - ct + a) - H(a/2 - ct - a) \right) + \sin(\pi (a/2 + ct) / a) \left( H(a/2 + ct + a) - H(a/2 + ct - a) \right) \right] \] Assessing Heaviside functions and simplifying: \[ y(a/2, t) = \sin(\pi (1/2) \left(1 - ct / a \right)) = \sin(\pi / 2 \left(1 - 2ct / a \right)) \]

Key concepts

Heaviside Step Function

The Heaviside step function, denoted by the symbol \(H(x)\), is a useful mathematical tool in wave equations. It's defined as \(H(x) = 1\) for \(x \geq 0\) and \(H(x) = 0\) for \(x < 0\). This makes it ideal for handling functions that change values abruptly. In our problem, the initial displacement can be described as:
\[ y(x) = \sin \left( \frac{\pi x}{a} \right) \left[ H(x + a) - H(x - a) \right]\]
This form allows us to clearly represent the displacement for different regions of \(x\). Without the Heaviside function, it would be challenging to piece together a single coherent function to describe the wave displacement over its entire range.

D'Alembert's Solution

D'Alembert's solution comes into play when we need to solve wave equations, particularly for infinitely long strings. The general solution for an initial displacement \(y(x) = f(x)\) and initial velocity \(y_t(x) = 0\) is given by:
\[ y(x, t) = \frac{1}{2} \left[ f(x - ct) + f(x + ct) \right] \]
This solution indicates how the initial displacement evolves over time as it propagates both forwards and backwards along the string. Essentially, D'Alembert's formula splits the initial wave into two components moving in opposite directions, each maintaining its shape.
For our case, where the initial displacement includes the Heaviside function, substituting \(f(x)\) into this formula allows us to determine how the wave's displacement changes over time.

Initial Displacement

Initial displacement represents the initial shape or configuration of the wave at \(t=0\). In our problem, the initial displacement is given by:
\[ y(x) = \begin{cases}\sin \left(\frac{\pi x}{a}\right), & -a \leq x \leq a \ 0, & |x| > a\end{cases} \]
By incorporating the Heaviside step function, we can rewrite this as:
\[ y(x) = \sin \left(\frac{\pi x}{a} \right) \left[ H(x + a) - H(x - a) \right]\]
This refined expression aids in applying D'Alembert's solution to find the wave's displacement at any future time \(t > 0\) by providing a clear, explicit function for initial conditions.

Wave Propagation

Wave propagation deals with how a wave travels through a medium over time. In the context of our exercise, it concerns how the initial displacement on an infinitely long string evolves. Using D'Alembert's solution, we plug in initial conditions to observe how the wave's profile changes with time.
At specific points, such as \(x = 0\), \(x = a\), and \(x = a/2\), D'Alembert's formula helps us compute the wave's displacement. For instance, at \(x = 0\):
\[ y(0, t) = \sin \left(\frac{\pi ct}{a}\right)\]
This implies that at the center of the string, the wave oscillates sinusoidally with time. Similarly, for other points like \(x = a\) and \(x = a/2\), specific calculations show how the initial wave configuration influences future displacements and helps us understand wave dynamics better.