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?