Superposition And Reflection Of Pulses Homework Answers

Superposition and Reflection of Pulses Homework Answers

Understanding how pulses interact when they meet and how they behave at boundaries is a cornerstone of wave physics. The principle of superposition tells us that when two or more pulses occupy the same medium at the same time, the resulting displacement at any point is simply the algebraic sum of the individual displacements. Reflection, on the other hand, describes what happens when a pulse reaches the end of a medium and is sent back, often with a change in polarity depending on whether the end is fixed or free. Below you will find a detailed walk‑through of the key concepts, step‑by‑step solutions to typical homework problems, and practical tips for checking your work.

1. Core Concepts ### 1.1 Principle of Superposition

The superposition principle states that for linear media (e.g., a taut string, a spring, or an air column) the net wave function (y_{\text{net}}(x,t)) is the sum of the individual wave functions:

[ y_{\text{net}}(x,t) = \sum_{i} y_i(x,t) ]

Because the governing wave equation is linear, this sum is also a valid solution. When two pulses overlap, you can picture the medium’s displacement as the “height” you would get if you added the heights of each pulse point‑by‑point.

1.2 Reflection at Boundaries

When a pulse reaches the end of a medium, part of its energy is reflected back into the medium. The nature of the reflection depends on the boundary condition:

The reflected pulse travels back with the same speed and shape (in an ideal, lossless medium) but may be flipped upside‑down if the end is fixed.

1.3 Constructive vs. Destructive Interference

When two pulses overlap, the superposition can lead to:

• Constructive interference – displacements add, producing a larger amplitude.
• Destructive interference – displacements subtract, possibly canceling each other out completely if the pulses are equal in magnitude and opposite in sign.

These phenomena are the basis for standing waves, beats, and many practical applications such as noise‑cancelling headphones.

2. Typical Homework Problems and Solutions

Below are three representative problems that frequently appear in introductory physics assignments. Each problem is broken into (a) conceptual reasoning, (b) mathematical setup, and (c) final answer with a brief check.

Problem 1 – Two Identical Pulses Approaching Each Other

Statement:
Two identical triangular pulses, each of amplitude (A = 2.0\ \text{cm}) and base width (w = 4.0\ \text{cm}), travel toward each other on a string with speed (v = 5.0\ \text{m/s}). At (t = 0) the leading edges of the pulses are separated by a distance (d = 6.0\ \text{cm}). Sketch the string’s shape at (t = 2.0\ \text{ms}) and indicate the maximum displacement during overlap.

Solution:

(a) Conceptual reasoning Each pulse travels a distance (x = vt). After (t = 2.0\ \text{ms} = 2.0\times10^{-3}\ \text{s}),

[ x = (5.0\ \text{m/s})(2.0\times10^{-3}\ \text{s}) = 1.0\times10^{-2}\ \text{m}=1.0\ \text{cm}. ]

Thus each pulse has moved 1.0 cm toward the center. Initially the gap between the leading edges was 6.0 cm; after each pulse advances 1.0 cm, the gap shrinks to (6.0 - 2(1.0) = 4.0\ \text{cm}). The pulses are still separated, but their trailing edges are now closer.

(b) Mathematical setup
Define the left‑moving pulse (y_L(x,t)) and right‑moving pulse (y_R(x,t)) as triangular functions:

[ y_{L,R}(x,t) = \begin{cases} A\left(1 - \dfrac{|x - x_{0,R/L}(t)|}{w/2}\right), & |x - x_{0,R/L}(t)| \le w/2\[6pt] 0, & \text{otherwise} \end{cases} ]

where (x_{0,L}(t) = -d/2 + vt) (starting left of centre) and (x_{0,R}(t) = +d/2 - vt).

The net displacement is (y_{\text{net}} = y_L + y_R).

(c) Final answer
At (t = 2.0\ \text{ms}) the pulses have not yet overlapped; the string shows two separate triangles, each still of height (2.0\ \text{cm}). The maximum displacement during the entire interaction will occur when the pulses fully overlap (complete constructive interference). When the centers coincide, the amplitudes add:

[ y_{\max}= A + A = 2A = 4.0\ \text{cm}. ]

Check: The pulses each travel 1.0 cm in 2 ms, leaving a 4 cm gap—consistent with the sketch. The maximum possible displacement (4 cm) is twice the individual amplitude, as expected for perfect constructive superposition.

Problem 2 – Reflection from a Fixed End

Statement:
A rectangular pulse of height (H = 3.0\ \text{cm}) and length (L = 5.0\ \text{cm}) travels toward a fixed end of a string. Draw the pulse shape just before it reaches the end, immediately after reflection, and after it has traveled back a distance equal to its original length.

Solution:

(a) Conceptual reasoning
At a fixed end the transverse displacement must be zero. To satisfy this condition, the incoming pulse is reflected with an inverted polarity (flipped upside‑down). The shape and size are preserved in an ideal lossless string.

(b) Mathematical setup
Let the incoming pulse be described by

[ y_{\text{in}}(x,t) = \begin{cases} H, & 0 \le x - (vt - x_0)