Waves, Sound & Optics — Problem set, Grade 12

Connect wave patterns, sound, electromagnetic radiation, interference, and refraction.

Review the key ideas

Wave Properties

Waves propagate disturbances and energy; material can oscillate locally. Define spatial wavelength, time period and amplitude separately and state the model for v=fλ.

Sound

Sound needs a medium. Frequency strongly influences pitch, while intensity and loudness differ. Decibel ratios, Doppler shifts and standing waves require stated conditions.

Light & the Electromagnetic Spectrum

Light is an electromagnetic wave obeying c = f λ; the spectrum runs from low-energy radio waves to high-energy gamma rays; and light shows wave-particle duality, with each photon carrying energy E = h f.

Reflection, Refraction & Optics

Light reflects at equal angles from the normal and refracts (bends) when its speed changes between media (n = c/v, Snell's law); lenses and mirrors form real or virtual images predicted by 1/f = 1/dₒ + 1/dᵢ.

Superposition, Standing Waves & Resonance

Interference adds local displacement; boundary conditions select standing-wave modes.

Doppler Effect & Relative Motion

Relative motion changes observed frequency; use a formula appropriate to the wave and medium.

Problem 1

A wave has a frequency of 12 Hz and a wavelength of 2.5 m. Find its speed.
Answer and explanation

30 m/s

v = f × λ = 12 × 2.5 = 30 m/s.

Problem 2

A wave has a period of 0.01 s. Find its frequency.
Answer and explanation

100 Hz

f = 1 ÷ T = 1 ÷ 0.01 = 100 Hz.

Problem 3

A wave travels at 6 m/s with a wavelength of 1.5 m. Find its frequency.
Answer and explanation

4 Hz

f = v ÷ λ = 6 ÷ 1.5 = 4 Hz.

Problem 4

You shout toward a cliff and hear the echo exactly 2 s later. Using the speed of sound as 340 m/s, how far away is the cliff?
Answer and explanation

340 m

In 2 s the sound makes a round trip: total distance = 340 × 2 = 680 m. The cliff is half of that, since the sound goes there and back: 680 ÷ 2 = 340 m.

Problem 5

A sound wave in air has a wavelength of 1.7 m. Using the speed of sound as 340 m/s, find its frequency.
Answer and explanation

200 Hz

f = v ÷ λ = 340 ÷ 1.7 = 200 Hz.

Problem 6

Order these electromagnetic waves from LOWEST to HIGHEST energy: microwaves, ultraviolet, gamma rays, infrared.
Answer and explanation

Microwaves, infrared, ultraviolet, gamma rays

Energy increases as wavelength decreases along the spectrum. Of these four, the order from lowest to highest energy is microwaves → infrared → ultraviolet → gamma rays.

Problem 7

Ultraviolet light has a wavelength of 3 × 10⁻⁷ m. Using c = 3 × 10⁸ m/s, find its frequency.
Answer and explanation

1 × 10¹⁵ Hz

f = c ÷ λ = (3 × 10⁸) ÷ (3 × 10⁻⁷) = 1 × 10¹⁵ Hz.

Problem 8

A type of glass has an index of refraction of 1.5. Using c = 3 × 10⁸ m/s, how fast does light travel inside it?
Answer and explanation

2 × 10⁸ m/s

Rearrange n = c ÷ v to v = c ÷ n = (3 × 10⁸) ÷ 1.5 = 2 × 10⁸ m/s.

Problem 9

An object is placed 4 cm in front of a converging lens with a focal length of 12 cm (closer than the focal point). Find the image distance and describe the image.
Answer and explanation

−6 cm: an upright, magnified virtual image (as in a magnifying glass).

1 ÷ dᵢ = 1 ÷ f − 1 ÷ dₒ = 1 ÷ 12 − 1 ÷ 4 = 1 ÷ 12 − 3 ÷ 12 = −2 ÷ 12 = −1 ÷ 6, so dᵢ = −6 cm. The negative image distance means the image is virtual, upright, and enlarged — exactly how a magnifying glass works when the object is inside the focal length.

Problem 10

Two loudspeakers play the same tone in phase. At a point where sound from both arrives crest-on-crest, is the sound louder or quieter, and what is this called?
Answer and explanation

Louder — constructive interference.

When the two waves arrive in phase (crest-on-crest), superposition adds their amplitudes, producing a larger-amplitude — and therefore louder — sound. This reinforcement is called constructive interference.

Problem 11

A car horn emits 600 Hz while the car approaches you at 40 m/s. Using the speed of sound as 340 m/s, what frequency do you hear? (f′ = f × v ÷ (v − vₛ))
Answer and explanation

680 Hz

The source approaches, so use the minus sign: f′ = f × v ÷ (v − vₛ) = 600 × 340 ÷ (340 − 40) = 600 × 340 ÷ 300 = 600 × 1.13333 = 680 Hz.

Problem 12

f=5 Hz,λ=2 m. Find speed and period.
Answer and explanation

v=fλ=10 m/s andT=1/f=.20 s.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 13

Chapter review: A wave has period0.020 s and wavelength0.80 m. Find frequency and speed. If frequency doubles in the same nondispersive medium, find the new wavelength.
Answer and explanation

f=50 Hz and v=40 m/s. At100 Hz with the same40 m/s speed, λ=0.40 m.

f=50 Hz and v=40 m/s.

At100 Hz with the same40 m/s speed, λ=0.40 m.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 14

An echo returns after.50 s at 340 m/s. Find wall distance.
Answer and explanation

Round-trip 170 m; one-way distance 85 m.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 15

Chapter review: Two sound intensity levels differ by15 dB. Find their intensity ratio. A separate echo returns in0.30 s at340 m/s; find reflector distance.
Answer and explanation

The intensity ratio is10^1.5≈31.6. Echo distance=340×0.30/2=51 m; the two calculations use different models and quantities.

The intensity ratio is10^1.5≈31.6.

Echo distance=340×0.30/2=51 m; the two calculations use different models and quantities.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 16

λ=3 m in vacuum. Find frequency withc=3×10⁸ m/s.
Answer and explanation

f=c/λ=1×10⁸ Hz. Use meters with SI c.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 17

Chapter review: A vacuum photon has wavelength400 nm. Find frequency and energy using c=3.00×10⁸ m/s and h=6.626×10⁻³⁴ J·s.
Answer and explanation

λ=4.00×10⁻⁷ m; f=7.50×10¹⁴ Hz and E=4.97×10⁻¹⁹ J per photon. This is not total beam energy without photon number.

λ=4.00×10⁻⁷ m; f=7.50×10¹⁴ Hz and E=4.97×10⁻¹⁹ J per photon.

This is not total beam energy without photon number.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 18

Light enters a higher-index medium obliquely. Explain direction.
Answer and explanation

It slows in the simple model and bends toward the normal. Frequency stays continuous across a stationary boundary; wavelength changes.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 19

Chapter review: A thin converging lens has f=12 cm and a real object at36 cm. Find image distance and magnification; describe the image.
Answer and explanation

1/di=1/12−1/36=1/18, so di=18 cm and m=−18/36=−0.50. It is real, inverted and half the object height in this paraxial model.

1/di=1/12−1/36=1/18, so di=18 cm and m=−18/36=−0.50.

It is real, inverted and half the object height in this paraxial model.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 20

Fixed string fundamental 20 Hz. Find third harmonic.
Hints

Write the relationship first; substitute with consistent units.

Answer and explanation

60 Hz

fn=nf 1=3×20.

Problem 21

Why state boundary conditions?
Answer and explanation

Fixed, free, open, and closed boundaries allow different mode patterns.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 22

Chapter review: A1.5 m string fixed at both ends has wave speed30 m/s. Find its fundamental and third-harmonic frequencies and the third-harmonic wavelength.
Answer and explanation

f1=v/(2L)=10 Hz; f3=30 Hz. λ3=2L/3=1.0 m, with fixed-end nodes in this ideal string model.

f1=v/(2L)=10 Hz; f3=30 Hz. λ3=2L/3=1.0 m, with fixed-end nodes in this ideal string model.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 23

400 Hz source recedes at 20 m/s; v=340 m/s. Find fobs.
Hints

Write the relationship first; substitute with consistent units.

Answer and explanation

377.78 Hz

400×340/360≈377.78.

Problem 24

Can the sound-source equation be used unchanged for starlight?
Answer and explanation

No. Light requires electromagnetic relativistic Doppler relationships.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 25

Chapter review: A600 Hz source recedes at30 m/s from a stationary observer in still air with sound speed330 m/s. Find received frequency and compare with the approach case at the same speed.
Answer and explanation

Receding:600×330/360=550 Hz. Approaching:600×330/300=660 Hz. The shifts are not symmetric about600 in this finite-speed source model.

Receding:600×330/360=550 Hz.

Approaching:600×330/300=660 Hz.

The shifts are not symmetric about600 in this finite-speed source model.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.
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