- #1

TRB8985

- 74

- 15

- Homework Statement
- See message body for full problem statement.

- Relevant Equations
- ω_n = 2πf_n ; k_n = 2π/λ_n ; f_n = v/2L ; L = n*λ/2

Good evening,

I'm working on the following problem and running into a little trouble:

Part (a) and (b) were super easy, but I have a question on part (c).

I'm trying to graph the total wave at ##t=0##, and it says I should get something that looks like this:

My graph doesn't even remotely look like that, though. It's more like this:

For context, here's what I did.

At ##t=0##, I switched out ##sin(\omega t)## for ##\cos(\omega t)##, and the first wave function reduced to this: $$y_1(x,0) = sin(k_1x)$$ Since I'm told that ##\omega_1 = vk_1##, I rewrote ##y_1## in the following form: $$=sin(\frac {\omega_1}{v}x)$$ $$=sin(\frac {2\pi f_1}{v}x)$$ $$=sin(\frac {2\pi \cdot \frac {v}{2L}}{v}x)$$ $$=sin(\pi \cdot \frac {x}{L})$$ The process was similar with the 2nd wave function, where I used the expression of the

2nd fundamental frequency (##f_2 = 2 \cdot \frac {v}{2L}##) to get a result of: $$y_2(x,0)=sin(2\pi \cdot \frac {x}{L})$$ My thought was to plot ##x## in terms of ##L## in order to eliminate ##L## in the trig argument.

The Excel spreadsheet I used had a pattern like this: $$

\begin{array}{|c|c|c|c|}

\hline x(L) & y_1(x,0) & y_2(x,0) & y_1 + y_2 \\

\hline 0L & 0 & 0 & 0 \\

\hline L/4 & \frac {1}{\sqrt{2}} & 1 & \frac {1}{\sqrt{2}} + 1 \\

\hline L/2 & 1 & 0 & 1 \\

\hline 3L/4 & \frac {1}{\sqrt{2}} & -1 & \frac {1}{\sqrt{2}} - 1 \\

\hline ... & ... & ... & ... \\

\hline 13L/4 & -\frac {1}{\sqrt{2}} & 1 & -\frac {1}{\sqrt{2}} + 1 \\

\hline

\end{array} $$

Then with that ready, the last step was to graph ##y_1 + y_2## vs. ##x##, leading to the funky-looking plot above.

I'm not entirely sure where I went wrong... was my approach totally out in left-field?

Would appreciate any feedback on why my plot looks so wild compared to the expected one.

Thank you!

I'm working on the following problem and running into a little trouble:

**
**Part (a) and (b) were super easy, but I have a question on part (c).

I'm trying to graph the total wave at ##t=0##, and it says I should get something that looks like this:

My graph doesn't even remotely look like that, though. It's more like this:

For context, here's what I did.

At ##t=0##, I switched out ##sin(\omega t)## for ##\cos(\omega t)##, and the first wave function reduced to this: $$y_1(x,0) = sin(k_1x)$$ Since I'm told that ##\omega_1 = vk_1##, I rewrote ##y_1## in the following form: $$=sin(\frac {\omega_1}{v}x)$$ $$=sin(\frac {2\pi f_1}{v}x)$$ $$=sin(\frac {2\pi \cdot \frac {v}{2L}}{v}x)$$ $$=sin(\pi \cdot \frac {x}{L})$$ The process was similar with the 2nd wave function, where I used the expression of the

2nd fundamental frequency (##f_2 = 2 \cdot \frac {v}{2L}##) to get a result of: $$y_2(x,0)=sin(2\pi \cdot \frac {x}{L})$$ My thought was to plot ##x## in terms of ##L## in order to eliminate ##L## in the trig argument.

The Excel spreadsheet I used had a pattern like this: $$

\begin{array}{|c|c|c|c|}

\hline x(L) & y_1(x,0) & y_2(x,0) & y_1 + y_2 \\

\hline 0L & 0 & 0 & 0 \\

\hline L/4 & \frac {1}{\sqrt{2}} & 1 & \frac {1}{\sqrt{2}} + 1 \\

\hline L/2 & 1 & 0 & 1 \\

\hline 3L/4 & \frac {1}{\sqrt{2}} & -1 & \frac {1}{\sqrt{2}} - 1 \\

\hline ... & ... & ... & ... \\

\hline 13L/4 & -\frac {1}{\sqrt{2}} & 1 & -\frac {1}{\sqrt{2}} + 1 \\

\hline

\end{array} $$

Then with that ready, the last step was to graph ##y_1 + y_2## vs. ##x##, leading to the funky-looking plot above.

I'm not entirely sure where I went wrong... was my approach totally out in left-field?

Would appreciate any feedback on why my plot looks so wild compared to the expected one.

Thank you!