Question

Two pulses of equal width and equal amplitude approach each other on opposite sides of a spring (i.e., the pulses displace the spring in opposite directions). The snapshots below were taken at equal time intervals. Is the behavior of the spring consistent with the principle of superposition? If so, which quantity is "added" in this case? If not, explain why not

Answer 1

y_total = 2A cos (-wt) sin (kx)

the amount that is added is the amplitude of the pulses.

Step-by-step explanation:

The superposition principle states that the resulting pulse is the sum of the incident pulses

y_total = y₁ + y₂

let's apply this expression to our case

the two pulses are identical except that they are inverted, so their expression is

y₁ = A sin (kx-wt)

y₂ = A sin (kx + wt)

we look for the resulting

y_total = A (sin (kx-wt) + sin (kx + wt))

let's use the trigonometric relationship

sin a + sin b = 2 cos (a-b) / 2 sin (a + b) / 2

y_total = 2A cos (-wt) sin (kx)

therefore we see that the amplitude of the resulting pulse is twice the amplitude of each wave, so the amount that is added is the amplitude of the pulses.

In the images to be consistent the following characteristics should be seen

- before reaching the pulses at the junction point, you must see two pulses of the same amplitude and width

- a pulse with twice the amplitude and the same width must be seen at the junction point

- a time after passing the junction two pulses that move away of equal width and width