A "right moving" solution to the wave equation is:
$$ f_R(z,t) = A \cos(kz – \omega t + \delta)$$
Which of these do you prefer for a "left moving" soln?
1. $f_L(z,t) = A \cos(kz + \omega t + \delta)$
2. $f_L(z,t) = A \cos(kz + \omega t - \delta)$
3. $f_L(z,t) = A \cos(-kz – \omega t + \delta)$
4. $f_L(z,t) = A \cos(-kz – \omega t - \delta)$
5. more than one of these!
(Assume $k, \omega, \delta$ are positive quantities)
Note:
* All of them could be because cos(x) = cos(-x)
Two different functions $f_1(x,t)$ and $f_2(x,t)$ are solutions of the wave equation.
$$\dfrac{\partial^2 f}{\partial x^2} = \dfrac{1}{c^2}\dfrac{\partial^2 f}{\partial t^2}$$
Is $(A f_1 + B f_2 )$ also a solution of the wave equation?
1. Yes, always
2. No, never
3. Yes, sometimes depending on $f_1$ and $f_2$
Note:
* Correct answer: A
Two traveling waves 1 and 2 are described by the equations:
$$y_1(x,t) = 2 \sin(2x – t)$$
$$y_2(x,t) = 4 \sin(x – 0.8 t)$$
All the numbers are in the appropriate SI (mks) units.
Which wave has the higher speed?
1. 1
2. 2
3. Both have the same speed
Note:
* Correct Answer: B
Two impulse waves are approaching each other, as shown. Which picture correctly shows the total wave when the two waves are passing through each other?
<img src="./images/two_waves.png" align="center" style="width: 400px";/>
Note:
* Correct Answer: D
A solution to the wave equation is:
$$f(z,t) = A \cos(kz – \omega t + \delta)$$
* What is the speed of this wave?
* Which way is it moving?
* If $\delta$ is small (and >0), is this wave "delayed" or "advanced"?
* What is the frequency?
* The angular frequency?
* The wavelength?
* The wave number?
A solution to the wave equation is:
$$f(z,t) = Re\left[A e^{i(kz – \omega t + \delta)}\right]$$
* What is the speed of this wave?
* Which way is it moving?
* If $\delta$ is small (and >0), is this wave "delayed" or "advanced"?
* What is the frequency?
* The angular frequency?
* The wavelength?
* The wave number?
A complex solution to the wave equation in 3D is:
$$\widetilde{f}(\mathbf{r},t) = \widetilde{A}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$$
* What is the speed of this wave?
* Which way is it moving?
* Why is there no $\delta$?
* What is the frequency?
* The angular frequency?
* The wavelength?
* The wave number?