Question

A long taut string is plucked at its centre. The pulse travelling on it can be described as $y(x, t) = e^{-(x + 2t)^2} + e^{-(x - 2t)^2}$. Draw the shape of the string at time t = 0, a short time after t = 0 and a long time after t = 0.

Answer 1

At $t=0$: A single, symmetric Gaussian hump centered at $x=0$ with peak height 2. A short time after $t=0$: The single hump starts to split into two overlapping humps, with a reduced peak at $x=0$. A long time after $t=0$: Two distinct, well-separated Gaussian humps, each with peak height approximately 1, located at $x \approx -2t$ and $x \approx 2t$.

Answer 2

The wave pulse is described by $y(x, t) = e^{-(x + 2t)^2} + e^{-(x - 2t)^2}$. This function is a superposition of two Gaussian pulses: $y_1(x, t) = e^{-(x + 2t)^2}$ and $y_2(x, t) = e^{-(x - 2t)^2}$. The first pulse travels in the negative x-direction with speed $v=2$, and the second pulse travels in the positive x-direction with speed $v=2$.

At $t = 0$, $y(x, 0) = e^{-x^2} + e^{-x^2} = 2e^{-x^2}$. This is a single, symmetric Gaussian hump centered at $x=0$ with a peak height of 2.

A short time after $t = 0$ (e.g., $t = \Delta t > 0$), $y(x, \Delta t) = e^{-(x + 2\Delta t)^2} + e^{-(x - 2\Delta t)^2}$. The two pulses are close to each other and overlap significantly. The overall shape is still roughly a single hump, but the peak height at $x=0$ is reduced, and the shape starts to flatten or develop a dip in the middle.

A long time after $t = 0$ (e.g., $t = T > 0$), $y(x, T) = e^{-(x + 2T)^2} + e^{-(x - 2T)^2}$. The two pulses are well-separated and do not overlap. The string's shape consists of two distinct Gaussian humps, each with a peak height of approximately 1, located around $x = -2T$ and $x = 2T$.