Difficulty: Intermediate | Prerequisites: basic wave concepts (speed, wavelength), understanding of transverse waves on springs/strings
These questions form the long-answer section of the PHYS 116 midterm and are worth the most points. They test your ability to reason about what happens when wave pulses travel along a medium, overlap with one another, and hit boundaries. The maths is minimal; the challenge is visualising the physics correctly. If you can draw accurate pulse shapes at specific times, you understand the material. If you cannot, no formula will save you.
Pulses travel at constant speed and pass through each other unchanged. When two pulses overlap, the displacement at each point is the sum of the individual displacements (superposition). At a fixed boundary a pulse reflects inverted; at a free boundary it reflects upright. These three rules are enough to solve every pulse-sketching problem on the exam.
Pulse
A single, short-duration disturbance that travels through a medium. Think of it as: one flick of a rope rather than a continuous shake.
Superposition principle
When two or more waves occupy the same region of space, the net displacement at each point is the algebraic sum of the individual displacements. In simple terms, this means the waves just add up, point by point.
Fixed end (rigid boundary)
A boundary where the medium cannot move. A pulse reflecting from a fixed end comes back inverted (flipped upside down).
Free end (loose boundary)
A boundary where the medium is free to move up and down. A pulse reflecting from a free end comes back upright (same orientation as the original).
Node
A point that remains at zero displacement. At a fixed boundary, the endpoint is always a node.
Transverse wave
A wave in which the displacement of the medium is perpendicular to the direction the wave travels. Pulses on a spring or string are transverse.
A pulse travels at constant speed v along the medium without changing shape (in an ideal, non-dispersive medium).
To find where a pulse is at time t, shift it by a distance v × t in its direction of travel.
The pulse shape does not change as it moves, only its position.
When two pulses travelling in opposite directions overlap, the displacement at every point is the sum of what each pulse would produce alone.
After passing through each other, both pulses continue unchanged, as though the other pulse were never there.
To find the net shape at a given time: draw each pulse at its position at that time, then add the displacements point by point.
Where one pulse is positive and the other is negative by the same amount, they cancel (destructive interference). Where both are positive, they add (constructive interference).
If you are given the total shape of the spring at some time and the shape of pulse 1 at that time, you can find pulse 2 by subtraction: pulse 2 = total shape minus pulse 1.
Step by step: (1) Work out where pulse 1 is at the given time (shift it by v × t). (2) At each position, subtract pulse 1's displacement from the total displacement. What remains is pulse 2.
A pulse hitting a fixed (rigid) boundary reflects back inverted. If the original pulse was above the equilibrium line, the reflected pulse is below it, and vice versa.
The reflected pulse travels back along the medium at the same speed.
During reflection, the boundary point stays at zero displacement (it is a node).
For the cups-on-a-spring problem: the incoming pulse pushes cups on one side outward. After reflecting from the fixed end, the pulse is inverted, so it pushes cups on the opposite side outward. Both rows of cups can be knocked over.
A pulse hitting a free (loose) boundary reflects back upright, keeping its original orientation.
The free end can move, so at the moment of reflection the displacement at the end is doubled (the incoming and reflected pulses overlap constructively at the boundary).
After the pulse fully reflects, it travels back with the same shape and same sign.
For a pulse of width 1 m travelling at 1 m/s towards a free end 3 m away at t = 0: by t = 3 s the leading edge has just reached the boundary. By t = 4 s the pulse has partially reflected. The reflected portion maintains its original orientation and overlaps with the still-incoming part at the boundary.
The formulas for this topic are minimal. The key relationships are:
\text{distance} = v \times ty_{\text{total}}(x, t) = y_1(x, t) + y_2(x, t)Fixed-end reflection: reflected pulse is inverted (multiply by −1).
Free-end reflection: reflected pulse keeps the same sign.
Reflection at fixed and free boundaries is how acoustic engineers design concert halls: sound waves reflecting off hard walls (fixed-like) versus soft absorbing surfaces behave differently. Superposition is the basis for noise-cancelling headphones, which generate an inverted copy of ambient sound so the two pulses cancel.
Students often think that when two pulses collide, they destroy each other or change shape permanently. They do not: after passing through one another, each pulse continues exactly as it was.
A common error is forgetting to invert the pulse at a fixed boundary. The inversion is the defining feature of fixed-end reflection.
Students sometimes confuse which boundary type produces inversion. Fixed end: inverted. Free end: upright. There is no middle ground.
When subtracting pulse 1 from a combined shape to find pulse 2, students occasionally subtract from the wrong baseline. The baseline is always the undisturbed equilibrium position (zero displacement), not the shape of pulse 1.
⚠️ The long-answer section (Questions 17 to 19) is worth 20 points total. These are the highest-value questions on the exam.
⚠️ You will be asked to sketch pulse shapes at specific times. Practice drawing these carefully with quantitatively correct positions (use d = vt to place them).
⚠️ The cups-on-a-spring problem tests whether you understand fixed-end reflection and inversion. Work through the logic: which side of the spring does the pulse displace before and after reflection?
⚠️ Free-end reflection problems require you to show the pulse coming back upright and overlapping with itself near the boundary. Draw the reflected portion clearly.
True or false: Two pulses travelling in opposite directions destroy each other when they meet. (False, they pass through each other and continue unchanged)
Fill in the blank: A pulse reflecting from a fixed end comes back ______. (inverted)
True or false: At a free end, the displacement is always zero. (False, a free end can move freely; that describes a fixed end)
Fill in the blank: The superposition principle says the total displacement equals the ______ of the individual displacements. (sum)
True or false: A pulse reflecting from a free end is inverted. (False, it reflects upright)
Q: Two pulses are created at opposite ends of a spring. Pulse 1 is shown at t = 0 and travels at 1 m/s. The combined shape at t = 3 s is given. How do you find pulse 2 at t = 3 s?
A: First, shift pulse 1 by 3 m (1 m/s × 3 s) in its direction of travel to find its position at t = 3 s. Then subtract pulse 1's displacement at each point from the total displacement shown. The remainder is pulse 2 at t = 3 s.
Q: A pulse on a spring approaches a fixed end. Cups are lined up on both sides. Which row(s) of cups get knocked over?
A: Both rows. The incoming pulse displaces the spring to one side, knocking over cups from that row. At the fixed end, the pulse reflects inverted, so it now displaces the spring to the opposite side on its way back, knocking over cups from the other row.
Q: A pulse travelling at 1 m/s approaches a free end. Sketch the spring at t = 4 s.
A: Determine when the pulse reaches the free end and how much has reflected by t = 4 s. The reflected portion keeps its original orientation (no inversion). Near the boundary, the incoming and reflected parts overlap, so the displacements add. Draw the reflected pulse moving in the opposite direction with the same shape and sign as the original.
Superposition is the same principle that explains standing waves (covered in the waves and sound notes). A standing wave is simply two continuous sinusoidal waves superposing, rather than individual pulses.
Reflection at boundaries connects to impedance matching in acoustics and electrical engineering. A fixed end is analogous to a short circuit; a free end is analogous to an open circuit.
These sketching problems also build the intuition you need for interference and diffraction, where many waves superpose at once.
wave pulse, superposition, principle of superposition, constructive interference, destructive interference, fixed end reflection, free end reflection, rigid boundary, loose boundary, inverted pulse, upright reflection, pulse sketching, transverse pulse, wave on a spring, cups knocked over, reflection at boundary, PHYS 116 midterm 1, University of Washington general physics