# 2.E: The Classical Wave Equation (Exercises)

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Solutions to select questions can be found online.

## 2.1A

Find the general solutions to the following differential equations:

1. $$\dfrac{d^{2}y}{dx^{2}} - 4y = 0$$
2. $$\dfrac{d^{2}y}{dx^{2}} - 3\dfrac{dy}{dx} - 54y = 0$$
3. $$\dfrac{d^{2}y}{dx^{2}} + 9y = 0$$

## 2.1B

Find the general solutions to the following differential equations:

1. $$\dfrac{d^{2}y}{dx^{2}} - 16y = 0$$
2. $$\dfrac{d^{2}y}{dx^{2}} - 6\dfrac{dy}{dx} + 27y = 0$$
3. $$\dfrac{d^{2}y}{dx^{2}} + 100y = 0$$

## 2.1C

Find the general solutions to the following differential equations:

1. $$\dfrac{dy}{dx} - 4\sin(x)y = 0$$
2. $$\dfrac{d^{2}y}{dx^{2}} - 5\dfrac{dy}{dx}+6y = 0$$
3. $$\dfrac{d^{2}y}{dx^{2}} = 0$$

## 2.2A

Practice solving these first and second order homogeneous differential equations with given boundary conditions:

1. $$\dfrac{dy}{dx} = ay$$ with $$y(0) = 11$$
2. $$\dfrac{d^2y}{dt^2} = ay$$ with $$y(0) = 6$$ and $$y'(0) = 4$$
3. $$\dfrac{d^2y}{dt^2} + \dfrac{dy}{dt} - 42y = 0$$ with $$y(0) = 2$$ and $$y'(0) = 0$$

## 2.3A

Prove that $$x(t)$$ = $$\cos(\theta$$) oscillates with a frequency

$\nu = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}} \nonumber$

Prove that $$x(t)$$ = $$\cos(\theta$$) also has a period

$T = {2\pi}\sqrt{\dfrac{m}{k}} \nonumber$

where $$k$$ is the force constant and $$m$$ is mass of the body.

## 2.3B

Try to show that

$x(t)=\sin(\omega t)\nonumber$

oscillates with a frequency

$\nu = \omega/2\pi\nonumber$

Explain your reasoning. Can you give another function of x(t) that have the same frequency.

## 2.3C

Which two functions oscillate with the same frequency?

1. $$x(t)=\cos( \omega t)$$
2. $$x(t)=\sin (2 \omega t)$$
3. $$x(t)=A\cos( \omega t)+B\sin( \omega t)$$

## 2.3D

Prove that $$x(t) = \cos(\omega(t))$$ oscillates with a frequency

$\nu = \dfrac{\omega}{2\pi} \nonumber.$

Prove that $$x(t) = A \cos(\omega(t) + B \sin(\omega(t))$$ oscillates with the same frequency:

$\nu = \dfrac{\omega}{2\pi}. \nonumber$

## 2.4

Show that the differential equation:

$\dfrac{d^2y}{dx^2} + y(x) = 0\nonumber$

has a solution

$y(x)= 2\sin x + \cos x \nonumber$

## 2.7

For a classical harmonic oscillator, the displacement is given by

$\xi (t)=v_0 \sqrt{\dfrac{m}{k}} \sin \sqrt{\dfrac{k}{m}} t\ \nonumber$

where $$\xi=x-x_0$$. Derive an expression for the velocity as a function of time, and determine the times at which the velocity of the oscillator is zero.

## 2.11

Verify that

$Y(x,t) = A \sin \left(\dfrac{2\pi }{\lambda}(x-vt) \right)\nonumber$

has a frequency $$\nu$$ = $$v$$/$$\lambda$$ and wavelength $$\lambda$$ traveling right with a velocity $$v$$.

## 2.13A

Explain (in words) how to expand the Hamiltonian into two dimensions and use it solve for the energy

## 2.13B

Given that the Schrödinger equation for a two-dimensional box, with sides $$a$$ and $$b$$, is

$\dfrac{∂^2 Ψ}{∂x^2} + \dfrac{∂^2 Ψ}{∂y^2} +\dfrac{(8π^2mE) }{h^2}Ψ(x,y) = 0 \nonumber$

and it has the boundary conditions of

$$Ψ(0,y)= Ψ (a,y)=0$$ and $$Ψ(o,x)= Ψ(x,b)=0$$

for all $$x$$ and $$y$$ values, show that

$E_{2,2}=\left(\dfrac{h^2}{2ma^2}\right)+\left(\dfrac{h^2}{2mb^2}\right). \nonumber$

## 2.14

Explain, in words, how to expand the Schrödinger Equations into a three-dimensional box

## 2.18

Solving for the differential equation for a pendulum gives us the following equation,

$\phi(x)= c_1\cos {\sqrt{\dfrac{g}{L}}} +c_2\sin {\sqrt{\dfrac{g}{L}}} \nonumber$

Assuming $$c_1=2$$, $$c_3= 5$$, $$g=7$$ and $$L=3$$, what is the position of the pendulum initially? Does this make sense in the real world. Why or why not? (We can ignore units for this problem).

## 2.23

Consider a Particle of mass $$m$$ in a one-dimensional box of length $$a$$. Its average energy is given by

$\langle{E}\rangle = \dfrac{1}{2m}\langle p^2\rangle\nonumber$

Because

$\langle{p}\rangle\ = 0\nonumber$

$\langle p^2\rangle = \sigma^{2}_{p}\nonumber$

where $$\sigma_p$$ can be called the uncertainty in $$p$$. Using the Uncertainty Principle, show that the energy must be at least as large as $$\hbar/8ma^2$$ because $$\sigma_x$$, the uncertainty in $$x$$, cannot be larger than $$a$$.

## 2.33

Prove $$y(x, t) = A\cos[2π/λ(x - vt)]$$ is a wave traveling to the right with velocity $$v$$, wavelength $$λ$$, and period $$λ/v$$.

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