1.
Predict what you think the shape of the string will look like as the wave pulses continue to move on the string. Sketch a few diagrams showing the string at a few different instants in time. Explain your reasoning.
Section 17.1 Wave Superposition
Exercises Warm-up Activity
Two pulses approach each other on the string below.
When two or more waves meet, they do not typically impact each other, but they do work together to influence the resulting displacement of the medium.
Principle 17.1.3. Wave Superposition.
The total displacement caused by multiple waves can be determined by adding the displacements caused by each wave. The displacements have to be added separately at every point in space and time:
\begin{equation*}
y(x,t) = y_1(x,t) + y_2(x,t) + \dots
\end{equation*}
Exercises Activities
The diagram below shows two wave pulses at \(t = 0\) moving in opposite directions on a string with wave speed 2 mm/s. Each grid box represents 1 mm.
1.
Sketch the wave pulses at \(t = 4 \text{ s}\text{.}\) Use your sketches and wave superposition to sketch the shape of the string at this instant in time.
Answer.
Two waves adding up like this, in such a way that their displacements reinforce each other, is known as constructive interference.
The two wave pulses overlap and add by superposition. Note that the displacements add separately at every position, so in the outer regions, where one pulse has a displacement of zero, the spring matches the displacement of the other pulse—only in the region where both pulses have nonzero displacement does the spring have a greater displacement!
The diagram below shows two wave different pulses at \(t = 0\) moving in opposite directions on a string with wave speed 2 mm/s. Each grid box represents 1 mm.
2.
Sketch the wave pulses at \(t = 4 \text{ s}\text{.}\) Use your sketches and wave superposition to sketch the shape of the string at this instant in time.
Answer.
The peaks of the pulses overlap at this location. Since the pulses are antisymmetric, every upward point on the left wave cancels the corresponding downward point on the right wave, which means the string will be entirely flat!
Two waves adding up like this, in such a way that their displacements cancel each other, is known as destructive interference.
References References
[1]
Superposition animation Pulse Superposition Figure courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State. licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
