How to Solve: Wave Speed

How to Solve: Wave Speed

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you’ve ever wondered why a tsunami travels faster in deep ocean than near shore—or how your Wi-Fi signal slows down through walls—this is the formula that unlocks it. Master wave speed, and you’ll crush questions on sound, light, earthquakes, and even fiber-optic cables in your exam."

What You Need To Know First

Before diving into wave speed, make sure you understand: 1. What a wave is: A disturbance that transfers energy (not matter) through a medium (e.g., water, air, a rope). 2. Wavelength (λ): The distance between two identical points on a wave (e.g., crest to crest). 3. Frequency (f): How many waves pass a point per second (measured in Hertz, Hz).

If you’re shaky on these, pause and review them first—wave speed builds on them!

Key Vocabulary

Formulas To Know

1. Wave Speed Formula (Core Formula)

Formula: v = f × λ - v = wave speed (m/s) - f = frequency (Hz) - λ = wavelength (m)

Memorise This. It’s the foundation of all wave speed problems.

2. Wave Speed Using Period

Formula: v = λ / T - T = period (s) = time for one wave cycle

Given on exam sheet? Sometimes. If not, remember T = 1/f and substitute into v = f × λ.

3. Speed of Light in a Vacuum (Chemistry/Physics Crossover)

Formula: c = 3.0 × 10⁸ m/s - c = speed of light in a vacuum (constant)

Memorise This. Used in light/refraction problems.

4. Speed of Sound in Air (Approximate)

Formula: v ≈ 343 m/s (at 20°C, dry air)

Given on exam sheet? Usually. If not, memorise it for quick checks.

Step-by-Step Method

Follow these steps exactly for every wave speed problem.

1. Read the question carefully. Underline:
2. What’s given (e.g., frequency, wavelength, period)?

3. What’s asked (e.g., "Find the speed," "Find the wavelength")?

4. List the known values. Write them with units:

5. f = ____ Hz
6. λ = ____ m
7. T = ____ s

8. v = ____ m/s (if given)

9. Check units. Convert if needed:

10. Wavelength in meters (m).
11. Frequency in Hertz (Hz).

12. Speed in m/s.

13. Pick the right formula.

14. If you have f and λ → v = f × λ.
15. If you have T and λ → v = λ / T.

16. If you have v and need f or λ → Rearrange the formula.

17. Plug in the numbers. Write the formula, substitute, and solve.

18. Check your answer.

19. Does the speed make sense? (e.g., sound in air ≈ 343 m/s, light ≈ 3 × 10⁸ m/s).

20. Are the units correct?

21. Box your final answer. Include units!

Worked Example Using the Steps

Question: A sound wave has a frequency of 256 Hz and a wavelength of 1.34 m. Calculate its speed.

Step-by-Step Solution: 1. Read the question. Given: f = 256 Hz, λ = 1.34 m. Asked: Find v. 2. List known values:
- f = 256 Hz
- λ = 1.34 m 3. Check units: Already in Hz and m. Good. 4. Pick the formula: v = f × λ. 5. Plug in the numbers:
v = 256 Hz × 1.34 m
v = 343.04 m/s 6. Check answer: Close to 343 m/s (speed of sound in air). Makes sense! 7. Box the answer: v = 343 m/s (rounded to 3 significant figures).

Worked Examples

Example 1 – Basic (No Tricks)

Question: A wave on a string has a wavelength of 0.5 m and a frequency of 4 Hz. What is its speed?

Solution: 1. Given: λ = 0.5 m, f = 4 Hz. 2. Formula: v = f × λ. 3. Substitute: v = 4 Hz × 0.5 m = 2 m/s. 4. Answer: v = 2 m/s.

What we did and why: - Used the core formula v = f × λ because we had frequency and wavelength. - Units: Hz × m = m/s (correct).

Example 2 – Medium (Added Complication)

Question: A radio station broadcasts at 98.5 MHz. If the speed of light is 3.0 × 10⁸ m/s, what is the wavelength of the radio waves?

Solution: 1. Given: f = 98.5 MHz = 98.5 × 10⁶ Hz, v = 3.0 × 10⁸ m/s. 2. Rearrange v = f × λ to λ = v / f. 3. Substitute: λ = (3.0 × 10⁸ m/s) / (98.5 × 10⁶ Hz). 4. Calculate: λ ≈ 3.04 m. 5. Answer: λ = 3.04 m.

What we did and why: - Converted MHz to Hz (1 MHz = 1 × 10⁶ Hz). - Rearranged the formula to solve for wavelength. - Checked units: m/s ÷ Hz = m (correct).

Example 3 – Exam Style (Disguised, Time Pressure)

Question: A tsunami wave takes 15 minutes to travel 180 km. What is its speed in m/s?

Solution: 1. Trick: Time is in minutes, distance in km. Convert first!
- 15 minutes = 15 × 60 = 900 seconds.
- 180 km = 180,000 m. 2. Given: distance = 180,000 m, time = 900 s. 3. Formula: speed = distance / time (same as v = d/t). 4. Substitute: v = 180,000 m / 900 s = 200 m/s. 5. Answer: v = 200 m/s.

What we did and why: - Recognized that wave speed = distance/time for a single wave pulse. - Converted units to match (km → m, minutes → s). - Used the basic speed formula, not v = f × λ, because frequency wasn’t given.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Wave speed is just how fast a wave moves, and the core formula is v = f × λ. Remember: - Frequency (f) = waves per second (Hz). - Wavelength (λ) = distance between crests (m). - Period (T) = time for one wave (s), and T = 1/f.

If you’re given f and λ, multiply them. If you’re given T and λ, divide λ by T. If the question talks about distance and time (like a tsunami), use speed = distance/time.

Watch out for unit traps—convert km to m, minutes to seconds. And don’t assume all waves travel at light speed! Sound is way slower.

Tonight, write down the formula v = f × λ 5 times. Then try one problem from your textbook. You’ve got this!