E. Waves and Sound · Waves and SoundWaves at Boundaries: Reflection and Transmission
E2.2 E2.1 A1.5 A1.12About 65 minutes
Learning goals- I can predict how a pulse reflects from a fixed end and from a free end.
- I can describe the reflected and transmitted pulses when a wave moves into a slower or faster medium.
- I can explain that energy is shared between the reflected and transmitted waves.
- I can use reflection (echoes) to calculate distances.
Words to know
boundary
The place where one medium meets another, or where a medium ends.
fixed end
An end of a medium that cannot move (e.g. a rope tied to a wall). Pulses reflect inverted.
free end
An end that can move freely (e.g. a rope on a ring that slides on a pole). Pulses reflect upright.
reflection
The bouncing back of a wave from a boundary.
transmission
The passing of part of a wave into the next medium.
inverted
Upside down: a crest becomes a trough.
echo
A reflected sound wave heard separately from the original sound.
What happens at the end of the rope?
Shout across a frozen lake toward a rocky shore and you may hear your voice come back a second later. Send a pulse down a rope tied to a fence and it comes racing back to your hand. Waves do interesting things when they hit a boundary: the end of a medium or the place where one medium meets another. Understanding boundaries is the key to echoes, ultrasound images and, two lessons from now, standing waves in musical instruments.
Reflection at a fixed end
Tie a rope to a wall and send an upward pulse. When the pulse reaches the wall, the rope pulls up on the wall. By Newton's third law (from the Forces unit), the wall pulls down on the rope. That downward pull starts a pulse travelling back. So at a fixed end the reflected pulse is inverted: a crest comes back as a trough. Its speed is unchanged (same rope), and if little energy is lost, its size is about the same.
Reflection at a free end
Now put the end of the rope on a light ring that slides up and down a smooth pole. When the upward pulse arrives, the end is free to swing up even farther, then it pulls the rope back down behind it. This sends a pulse back that is upright. At a free end the reflected pulse is not inverted.
When a wave meets a new medium
Most boundaries are partly fixed and partly free. Tie a light rope to a heavier rope. When a pulse reaches the knot, part of the energy is transmitted into the second rope and part is reflected back.
Light → heavy (into a slower medium): the heavy rope is harder to move, so it acts a bit like a fixed end.
- Reflected pulse: inverted, smaller.
- Transmitted pulse: upright, slower, and shorter (narrower), because λ = v/f and the speed is lower.
Heavy → light (into a faster medium): the light rope is easy to move, so it acts a bit like a free end.
- Reflected pulse: upright, smaller.
- Transmitted pulse: upright, faster and longer (wider).
The transmitted pulse is always upright. Only the reflected pulse can flip. The bigger the difference between the two media, the more energy is reflected and the less is transmitted.
| Boundary | Reflected pulse | Transmitted pulse |
|---|
| Fixed end | inverted, same speed | none |
| Free end | upright, same speed | none |
| Into slower medium (light → heavy) | inverted, smaller | upright, slower, shorter λ |
| Into faster medium (heavy → light) | upright, smaller | upright, faster, longer λ |
Echoes and ultrasound
Sound reflects too. An echo is a reflected sound you hear separately from the original. Because the sound goes to the wall and back, the distance to the wall is
d = v t ÷ 2
where t is the time between the sound and the echo. Medical ultrasound works the same way: a probe sends very high-frequency sound into the body, and part of it reflects at every boundary between different tissues. The timing of the echoes builds an image. You will see more about this in the lesson on echolocation.
Worked exampleExample 1: Predicting reflections
A crest is sent along a light spring that is attached to a much heavier spring. (a) Describe the reflected pulse. (b) Describe the transmitted pulse. (c) What would change if the far end of the light spring were held tightly in a clamp instead?
- Light → heavy means the wave is entering a slower medium, which acts partly like a fixed end.
- (a) Reflected: inverted (a trough), smaller than the original, same speed as the original.
- (b) Transmitted: upright (a crest), slower, narrower, smaller.
- (c) A clamp is a fixed end: all the energy reflects (ideally), so the reflected pulse is an inverted trough about the same size, and there is no transmitted pulse.
(a) inverted and smaller (b) upright, slower and narrower (c) a full-size inverted reflection and no transmitted pulse.
Worked exampleExample 2: When does the pulse come back?
A 4.5 m rope is tied to a post. A pulse travels along it at 6.0 m/s. You send a crest at t = 0. When does the reflection reach your hand, and is it a crest or a trough?
- The pulse travels to the post and back: total distance 2 × 4.5 m = 9.0 m.
- t = d/v = 9.0 m ÷ 6.0 m/s = 1.5 s.
- The post is a fixed end, so the reflection is inverted: a trough.
After 1.5 s, as a trough.
Worked exampleExample 3: Echo from a cliff
A hiker on Georgian Bay claps and hears the echo from a cliff 1.2 s later. The speed of sound is 340 m/s. How far away is the cliff?
- The sound travels to the cliff and back, so the one-way distance is half the total.
- Total distance = v t = 340 m/s × 1.2 s = 408 m.
- d = 408 ÷ 2 = 204 m ≈ 2.0 × 10² m.
Just for funFunny-but-true: if you shout at a cliff and hear the echo exactly 1 second later, the cliff is only about 170 m away. Sound has to make the round trip, so the cliff is always closer than it sounds!
Hands-on activityFixed, free and two-rope boundaries
You need: a skipping rope (or 3 m of rope), about 2 m of light string or thread, a heavier rope or a garden hose, a heavy chair, and a partner if possible.
- Fixed end: tie the rope tightly to the chair leg. Lay it on a smooth floor and pull it just tight. Send one sharp sideways pulse to your left. Watch the reflection. Which side of the rope is it on when it comes back?
- Nearly free end: tie the light string between the end of the rope and the chair. The string is so light that the rope's end is almost free. Repeat. Which side is the reflection on now?
- Light → heavy: tie your rope to the hose (or heavier rope). Send a pulse from the light end. Look for a small inverted reflection and a smaller, slower pulse carrying on into the hose.
- Heavy → light: send the pulse from the hose end instead.
- Make a table of what you saw and compare it with the summary table above.
What to look for: with the fixed end, the pulse comes back on the opposite side (inverted). With the string, it comes back on the same side (upright). With two ropes, the reflection flips only when going from light to heavy. Reflections on a floor are fast; a slow-motion phone video helps a lot.
Your turn
1A crest reaches a fixed end. Describe the reflected pulse.
A trough (inverted), moving back at the same speed, about the same size.
2A crest reaches a free end. Describe the reflected pulse.
A crest (upright), moving back at the same speed.
3A pulse passes from a heavy rope into a light rope. Is the transmitted pulse upright or inverted? Faster or slower?
4Why is the reflected pulse smaller than the original when a wave meets a new medium?
Some of the energy is transmitted into the new medium, so less energy is reflected.
5A pulse moves from a light rope into a heavy rope. Is the transmitted pulse wider or narrower than the incoming one? Explain.
Narrower: it moves more slowly, so in the same amount of time it covers less distance (λ = v/f, smaller v gives smaller λ).
6You clap and hear an echo 0.80 s later. The speed of sound is 343 m/s. How far away is the wall?
d = vt/2 = 343 × 0.80 ÷ 2 ≈ 137 m (about 1.4 × 10² m).
More practice: warm-up
1What word describes a pulse that has been flipped upside down?
2Name the two things that can happen to a wave when it reaches a new medium.
It can be reflected and/or transmitted (usually both).
3Is the transmitted pulse ever inverted?
No, the transmitted pulse is always upright.
A reflected sound heard separately from the original sound.
5Does a reflected pulse travel faster, slower or at the same speed as the incoming pulse?
The same speed, because it is in the same medium.
More practice: on your own
1A trough is sent toward a fixed end. Describe the reflection.
It comes back as a crest (a fixed end inverts any pulse).
2A 6.0 m spring is clamped at the far end. A pulse travels at 4.0 m/s. How long after sending it does the reflection return?
t = 2 × 6.0 ÷ 4.0 = 3.0 s.
3A ship's sonar sends a sound pulse straight down. The echo returns after 0.40 s. Sound travels at 1530 m/s in sea water. How deep is the water?
d = vt/2 = 1530 × 0.40 ÷ 2 = 306 m ≈ 3.1 × 10² m.
4Explain, using forces, why a fixed end inverts a pulse.
The rope pulls up on the fixed support as the crest arrives; by Newton's third law the support pulls down on the rope with an equal force. That downward pull sends a downward pulse (a trough) back along the rope.
5A pulse moves from rope A into rope B. The reflected pulse is upright. Which rope is the faster medium?
An upright reflection means the boundary acts like a free end, so rope B is lighter and is the faster medium.
Challenge
1You stand between two cliffs and clap. You hear one echo after 1.0 s and another after 1.6 s. Sound travels at 340 m/s. How far apart are the cliffs?
Distance to cliff 1 = 340 × 1.0 ÷ 2 = 170 m. Distance to cliff 2 = 340 × 1.6 ÷ 2 = 272 m. The cliffs are on opposite sides, so they are 170 + 272 = 442 m ≈ 4.4 × 10² m apart.
2A pulse with a width of 40 cm travels at 5.0 m/s on a light rope and passes into a heavy rope where the speed is 2.0 m/s. Estimate the width of the transmitted pulse.
The time to pass a point stays the same (set by the source): 0.40 m ÷ 5.0 m/s = 0.080 s. In the heavy rope: width = 2.0 m/s × 0.080 s = 0.16 m (16 cm).
3An ultrasound pulse travels at 1540 m/s in soft tissue. An echo from a boundary inside the body returns 65 μs after the pulse is sent. How deep is the boundary?
d = vt/2 = 1540 × (65 × 10⁻⁶) ÷ 2 = 0.050 m ≈ 5.0 cm.
Quiz
1. A crest reflects from a fixed end. It returns as…
2. A pulse moves from a light rope into a heavy rope. The transmitted pulse is…
3. A pulse moves from a heavy rope into a light rope. The reflected pulse is…
4. Why is a partly reflected pulse smaller than the incoming pulse?
5. An echo returns 2.0 s after a shout. With v = 340 m/s, the wall is…
6. Which quantity is the same for the incoming and transmitted parts of a periodic wave crossing a boundary?
Big idea: expectation E2.2 asks students to conduct inquiries with springs or strings to see how waves behave; this lesson covers reflection and transmission, which are needed for standing waves and resonance later in this unit.
How to help: The rope activity is the best way to learn this; a slow-motion phone video lets your teen see which side the reflection is on. Ask them to predict before each trial.
Common difficulties:
- Forgetting that the transmitted pulse is always upright.
- Mixing up which boundary inverts. Memory hook: fixed flips, and "slow is like fixed".
- Forgetting to halve the round-trip distance in echo problems.