b) Draw a snapshot of the wave at t = 0 and a ”history graph” (i.e. graph of D(x = 0, t) vs.
t) at x = 0. Show 2 cycles and label them appropriately.
Since the question asks for a snapshot at t=0, we need to find the wave function at that specific time. From the given equation of the wave, D(x, t) = sin(2π(x - t)), we can substitute t=0 to find the snapshot at t=0.
D(x, 0) = sin(2π(x - 0))
D(x, 0) = sin(2πx)
To draw the snapshot of the wave at t=0, we need to plot the function D(x, 0) = sin(2πx).
Additionally, we need to draw a "history graph," which is a graph of D(x=0, t) vs. t. From the wave equation given, we can substitute x=0 to find the history graph at x=0.
D(0, t) = sin(2π(0 - t))
D(0, t) = sin(-2πt)
To draw the history graph, we need to plot the function D(0, t) = sin(-2πt) against the time variable t.
However, since we are asked to show 2 cycles on both graphs, we can choose a wavelength for our wave. Let's say one wavelength corresponds to 2 units in the x-direction.
Now we can plot the snapshot and history graphs:
Snapshot of the wave at t=0:
The graph will have two cycles from x=0 to x=4 (since one wavelength is 2 units). The amplitude is 1, so the wave oscillates between -1 and 1 on the y-axis. The graph will look like this:
/\ /\
/ \ / \
/ \ / \
____/______\_______________/______\_____________________
0 1 2 3 4 5 6 7 8 9 10
History graph at x=0:
The graph will also have two cycles from t=0 to t=1 (since one wavelength is 0.5 units). The amplitude is 1, so the wave oscillates between -1 and 1 on the y-axis. The graph will look like this:
2
| /\ /\
| / \ / \
| / \ / \
|/ \ / \
____/________\_________/________\____________________________
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0