How to Solve: Frequency and Wavelength

How to Solve: Frequency and Wavelength

Introduction

"If you’ve ever wondered why your favorite radio station cuts out in a tunnel or how doctors use X-rays to see broken bones, you’re already thinking about frequency and wavelength—two concepts that show up in every physics and chemistry exam. Master these, and you’ll solve wave problems in seconds."

What You Need To Know First

Before diving in, make sure you understand: 1. Waves – A disturbance that transfers energy (not matter) through space or a medium. 2. Speed – How fast a wave travels (distance per unit time). 3. Units – Meters (m) for wavelength, Hertz (Hz) for frequency, meters per second (m/s) for speed.

If any of these are unclear, pause and review them first.

Key Vocabulary

Formulas To Know

1. Wave Speed Equation (The Big One)

Formula: [ v = f \times \lambda ]

Variables: - ( v ) = wave speed (m/s) - ( f ) = frequency (Hz) - ( \lambda ) = wavelength (m)

Memorize this? ✅ YES – MEMORIZE THIS.

2. Relationship Between Frequency and Period

Formula: [ f = \frac{1}{T} ] or [ T = \frac{1}{f} ]

Variables: - ( f ) = frequency (Hz) - ( T ) = period (s)

Memorize this? ✅ YES – MEMORIZE THIS.

3. Speed of Light (For EM Waves)

Formula: [ c = f \times \lambda ] (Same as wave speed, but ( c ) is the speed of light in a vacuum: ( 3.00 \times 10^8 ) m/s.)

Memorize this? ❌ Given on exam sheet (but know ( c = 3.00 \times 10^8 ) m/s).

Step-by-Step Method

How to Solve Any Frequency/Wavelength Problem in 5 Steps

1. Identify what’s given and what’s asked.
2. Circle the values in the question.

3. Underline what you need to find (e.g., "Find the wavelength").

4. Choose the right formula.

5. If the question involves speed, frequency, or wavelength, use ( v = f \times \lambda ).

6. If it involves period, use ( f = \frac{1}{T} ) or ( T = \frac{1}{f} ).

7. Convert units if needed.

8. Frequency must be in Hz (1/s).
9. Wavelength must be in meters (m).
10. Speed must be in m/s.

11. Example: 500 kHz = 500,000 Hz.

12. Plug in the numbers and solve.

13. Rearrange the formula if needed (e.g., ( \lambda = \frac{v}{f} )).

14. Show every step—examiners love this!

15. Check your answer.

16. Does it make sense? (e.g., visible light wavelengths are ~400–700 nm).
17. Are the units correct?

Worked Example Using the Steps

Question: A sound wave travels at 340 m/s with a frequency of 170 Hz. What is its wavelength?

Solution: 1. Given: ( v = 340 ) m/s, ( f = 170 ) Hz
Asked: Find ( \lambda ).

1. Formula: ( v = f \times \lambda )
   Rearrange to ( \lambda = \frac{v}{f} ).

2. Units: Already in m/s and Hz—no conversion needed.

3. Plug in:
   ( \lambda = \frac{340}{170} = 2 ) m.

4. Check: Sound waves in air have wavelengths from ~17 mm to 17 m. 2 m is reasonable.

Answer: The wavelength is 2 meters.

Worked Examples

Example 1 – Basic (Direct Application)

Question: A radio station broadcasts at 98.5 MHz. If radio waves travel at the speed of light (( 3.00 \times 10^8 ) m/s), what is the wavelength of the signal?

Solution: 1. Given: ( f = 98.5 ) MHz, ( v = 3.00 \times 10^8 ) m/s
Asked: Find ( \lambda ).

1. Convert units:
   ( 98.5 ) MHz = ( 98.5 \times 10^6 ) Hz = ( 9.85 \times 10^7 ) Hz.

2. Formula: ( \lambda = \frac{v}{f} ).

3. Plug in:
   ( \lambda = \frac{3.00 \times 10^8}{9.85 \times 10^7} \approx 3.05 ) m.

4. Check: Radio wavelengths are typically 1–10 m. This fits.

Answer: The wavelength is 3.05 meters.

What we did and why: - Converted MHz to Hz because the formula requires Hz. - Used ( \lambda = \frac{v}{f} ) because we had speed and frequency. - Checked the answer against real-world values to ensure it made sense.

Example 2 – Medium (Missing Period)

Question: A wave has a period of 0.02 s. What is its frequency? If the wave speed is 150 m/s, what is its wavelength?

Solution: Part 1: Find frequency. 1. Given: ( T = 0.02 ) s
Asked: Find ( f ).

1. Formula: ( f = \frac{1}{T} ).

2. Plug in:
   ( f = \frac{1}{0.02} = 50 ) Hz.

Part 2: Find wavelength. 1. Given: ( v = 150 ) m/s, ( f = 50 ) Hz
Asked: Find ( \lambda ).

1. Formula: ( \lambda = \frac{v}{f} ).

2. Plug in:
   ( \lambda = \frac{150}{50} = 3 ) m.

Answer: - Frequency = 50 Hz - Wavelength = 3 meters

What we did and why: - First found frequency using the period because ( f ) and ( T ) are directly related. - Then used the wave speed equation to find wavelength. - Always solve for missing variables step by step.

Example 3 – Exam Style (Disguised Problem)

Question: A student observes that a wave in a ripple tank completes 10 cycles in 5 seconds. The distance between two consecutive crests is 4 cm. Calculate the speed of the wave.

Solution: 1. Given:
- 10 cycles in 5 s → Find frequency.
- Distance between crests = 4 cm → Wavelength.

1. Find frequency:
   ( f = \frac{\text{Number of cycles}}{\text{Time}} = \frac{10}{5} = 2 ) Hz.

2. Find wavelength:
   ( \lambda = 4 ) cm = 0.04 m (convert to meters!).

3. Formula: ( v = f \times \lambda ).

4. Plug in:
   ( v = 2 \times 0.04 = 0.08 ) m/s.

Answer: The wave speed is 0.08 m/s.

What we did and why: - The question didn’t directly give frequency or wavelength—we had to extract them from the description. - Always convert cm to m for consistency with SI units. - The formula ( v = f \times \lambda ) works for any wave, including water waves.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. Here’s the 60-second version:

1. The big formula: ( v = f \times \lambda ). Memorize it. If you know two, you can find the third.
2. Frequency and period are inverses: ( f = \frac{1}{T} ). If you see ‘cycles per second,’ that’s frequency. If you see ‘seconds per cycle,’ that’s period.
3. Units matter: Always convert to Hz, m, and m/s. No cm, no MHz—just base units.
4. Speed depends on the wave: Light = ( 3.00 \times 10^8 ) m/s. Sound in air = 340 m/s. Don’t mix them up.
5. Examiners love hiding frequency in ‘cycles per second’ or period. Extract it first, then solve.

Now go practice three problems tonight. Start with the basic ones, then try the disguised ones. You’ve got this!

Final Tip for Teachers: - Demo with a slinky or ripple tank to show wavelength and frequency visually. - Use a song’s frequency (e.g., A4 = 440 Hz) to make it relatable. - Time students on exam-style questions to build speed.

Final Tip for Students: - Write down the formula first before plugging in numbers. - Circle units in the question to avoid mistakes. - Check your answer against real-world values (e.g., visible light is ~400–700 nm).

Now go ace that exam! ?