Question

24\. Suppose that the solution to the wave equation \(\frac{2^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}\) is given by \(y=\Psi(t+x)-\Psi(t-x)\). Show that the initial conditions \(y(0, x)=f(x), y^{\prime}(0, x)=g(x)\) and the condition \(y(t, 0)=y(t, l)=0\) for all \(t\) lead to the requirements that \(f(x)\) and \(g(x)\) are odd functions of period \(2 l\) (d'Alembert).

Short answer

Question: Show that under the given initial and boundary conditions, the functions \(f(x)\) and \(g(x)\) are odd functions with a period of \(2l\). Solution: Based on our analysis, we have shown that under the given initial and boundary conditions, both functions \(f(x)\) and \(g(x)\) are odd functions and have a period of \(2l\).

Step-by-step solution

Apply Initial Conditions

We are given two initial conditions: 1. \(y(0, x)=f(x)\) 2. \(y'(0, x)=g(x)\) First, for the initial condition \(y(0, x)=f(x)\), substitute \(t=0\) in the given solution: $$f(x)=\Psi(x)-\Psi(-x)$$ For the second initial condition, we need the derivative of \(y\) with respect to \(t\). To achieve that, we differentiate the given solution with respect to \(t\): $$\frac{\partial y}{\partial t}=\frac{\partial\Psi(t+x)-\partial\Psi(t-x)}{\partial t}=\Psi'(t+x)-(-\Psi'(t-x))$$ Now, using the initial condition \(y'(0, x)=g(x)\) by substituting \(t=0\): $$g(x)=\Psi'(x)+\Psi'(-x)$$

Apply Boundary Conditions

We are given two boundary conditions: 1. \(y(t,0)=0\) 2. \(y(t,l)=0\) Let's apply these boundary conditions one by one to find additional conditions on \(\Psi\). First, for \(y(t,0)=0\), we substitute \(x=0\) in the given solution: $$0=\Psi(t)-\Psi(t)$$ This condition holds true because \(\Psi(t)-\Psi(t)=0\). For the second boundary condition, \(y(t,l)=0\), we substitute \(x=l\) in the given solution: $$0=\Psi(t+l)-\Psi(t-l)$$ In addition to the given boundary conditions, we know that \(y(t,0)=y(t,2l)=0\) according to our requirement that \(f(x)\) and \(g(x)\) have a period of \(2l\). By plugging these values into the given solution, we get another requirement: $$0=y(t,2l)=\Psi(t+2l)-\Psi(t-2l)$$

Find requirements on \(\Psi\) for odd periodic functions

We derived the following equations for \(f(x)\) and \(g(x)\) in terms of \(\Psi\): 1. \(f(x)=\Psi(x)-\Psi(-x)\) 2. \(g(x)=\Psi'(x)+\Psi'(-x)\) Also, we derived another requirement on \(\Psi\): \(\Psi(t+2l)-\Psi(t-2l)=0\). Now we need to show that \(f(x)\) and \(g(x)\) are odd functions of period \(2l\). An odd function satisfies this condition: \(f(-x)=-f(x)\). Let's check if this is true for our expressions of \(f(x)\) and \(g(x)\). Checking for \(f(x)\): $$f(-x)=\Psi(-x)-\Psi(x)=-\left(\Psi(x)-\Psi(-x)\right)=-f(x)$$ Checking for \(g(x)\): $$g(-x)=\Psi'(-x)+\Psi'(x)=-\left(\Psi'(x)+\Psi'(-x)\right)=-g(x)$$ These results show that both \(f(x)\) and \(g(x)\) are odd functions. To show that both functions have a period of \(2l\), we need to check if \(f(x+2l)=f(x)\) and \(g(x+2l)=g(x)\). We know that \(\Psi(t+2l)-\Psi(t-2l)=0\). Let's check the periodicity of \(f(x)\) and \(g(x)\). For \(f(x)\): $$f(x+2l)=\Psi(x+2l)-\Psi(-x+2l)=\Psi(x)-\Psi(-x+2l-2l)$$ Since \(\Psi(t+2l)-\Psi(t-2l)=0\), we have: $$f(x+2l)=\Psi(x)-\Psi(-x)=f(x)$$ This shows that \(f(x)\) is a periodic function with a period of \(2l\). For \(g(x)\): $$g(x+2l)=\Psi'(x+2l)+\Psi'(-x+2l)=\Psi'(x)+\Psi'(-x)+\Psi'(2l)-\Psi'(-2l)=\Psi'(x)+\Psi'(-x)$$ Again, this shows that \(g(x)\) is a periodic function with a period of \(2l\). #Conclusion#Our analysis shows that under the given initial and boundary conditions, \(f(x)\) and \(g(x)\) indeed are odd functions with a period of \(2l\).

Key concepts

d'Alembert's Solution

The wave equation is a fundamental partial differential equation used to describe waves. D'Alembert's solution is a powerful way to solve this equation. It expresses the solution as a combination of two arbitrary functions. Specifically, if a wave equation is written as \( \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \), the d'Alembert solution can be written as:

• \( y(x, t) = f(x - ct) + g(x + ct) \)

This approach describes waves traveling in opposite directions: one function represents a wave moving to the right and the other to the left. The functions \( f \) and \( g \) are determined by initial and boundary conditions.

Initial Conditions

Initial conditions provide specific information about the state of the system at the start of observation, time \( t = 0 \). For waves, these might be the initial shape, \( y(0, x) = f(x) \), and the initial velocity, \( y'(0, x) = g(x) \).

By substituting \( t = 0 \) into the d'Alembert solution, you can solve for \( f(x) \) and \( g(x) \). This helps define the arbitrary functions in the solution so they represent the actual wave's starting shape and speed. Without these conditions, the original problem remains unspecific as many waves could fit a single equation.

Boundary Conditions

Boundary conditions are essential in establishing how wave functions behave at specific points in space. They determine how solutions conform at boundaries, like ends of a string.

For boundary conditions \( y(t, 0) = 0 \) and \( y(t, l) = 0 \), the wave must be zero at \( x = 0 \) and \( x = l \) at all times. These conditions guide the waveform and restrict the functions \( \Psi \) to satisfy:

• \( \Psi(t+l) = \Psi(t-l) \)

These constraining requirements ensure that the wave matches physical constraints, like fixed ends on a string.

Odd Functions

An odd function is symmetric relative to the origin. For a function \( f(x) \) to be odd, it must satisfy \( f(-x) = -f(x) \).

In our wave equation, if \( f(x) = \Psi(x) - \Psi(-x) \), it is easy to see:

• \( f(-x) = \Psi(-x) - \Psi(x) = -f(x) \)

Similarly, for \( g(x) = \Psi'(x) + \Psi'(-x) \), the odd symmetry will also show that \( g(-x) = -g(x) \). Oddness here reflects underlying physical properties, like equal and opposite actions, that facilitate periodic extensions of solution behaviors.

Periodic Functions

Periodic functions repeat their values in regular intervals or periods. For a function to be periodic with a period of \( 2l \), it should hold that \( f(x + 2l) = f(x) \) and similarly for \( g(x) \).

In our wave solutions, as shown, both functions \( f(x) \) and \( g(x) \) equate back to themselves over any shift equal to twice the boundary distance \( l \). This periodic nature stems from the boundary condition extensions, effectively modeling repetitive or cycle-bound behaviors seen in many wave phenomena, like strings fixed on ends or electromagnetic waves reflecting between surfaces.