Waves Calculator v = fλ
Calculate wave speed, frequency, wavelength, refraction, and standing waves with step-by-step solutions. Perfect for AP Physics 1 and AP Physics 2.
Wave Speed = Frequency × Wavelength
Solve for:
Quick Examples
Wave equation
GCSESpeed = Frequency × Wavelength
Period-frequency
GCSEPeriod = 1 / Frequency
Snell's law
GCSERefraction at boundary
Critical angle
GCSEFor total internal reflection
Refractive index
A-LevelSpeed of light / Speed in medium
Speed of sound
GCSESpeed of sound in air (T in °C)
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What is the Wave Equation?
The wave equation v = fλ is one of the most fundamental formulas in physics. It tells us that wave speed equals frequency times wavelength, and applies to ALL types of waves.
The Wave Equation
Speed (m/s) = Frequency (Hz) × Wavelength (m)
The SI unit of wave speed is meters per second (m/s), frequency is in Hertz (Hz), and wavelength is in meters (m).
Example: A wave with f = 500 Hz and λ = 0.68 m
v = 500 × 0.68 = 340 m/s
Transverse Waves
- • Oscillations perpendicular to direction of travel
- • Examples: light, water waves, EM waves
- • Can be polarised
- • Show crests and troughs
Longitudinal Waves
- • Oscillations parallel to direction of travel
- • Examples: sound, ultrasound, seismic P-waves
- • Cannot be polarised
- • Show compressions and rarefactions
Refraction & Snell's Law
When light passes from one medium to another, it changes speed and bends. This bending is called refraction. Snell's law describes exactly how much the light bends.
Snell's Law
n = refractive index, θ = angle from the normal
Denser to Less Dense
When light enters a less dense medium (lower n), it bends away from the normal and speeds up.
Example: Glass (n=1.5) to Air (n=1) - light bends away from normal.
Less Dense to Denser
When light enters a denser medium (higher n), it bends towards the normal and slows down.
Example: Air (n=1) to Water (n=1.33) - light bends towards normal.
| Material | Refractive Index (n) | Speed of Light |
|---|---|---|
| Vacuum | 1.00 | 3.00 × 10⁸ m/s |
| Air | 1.0003 | 2.998 × 10⁸ m/s |
| Water | 1.33 | 2.26 × 10⁸ m/s |
| Glass (crown) | 1.52 | 1.97 × 10⁸ m/s |
| 💎 Diamond | 2.42 | 1.24 × 10⁸ m/s |
Total Internal Reflection & Critical Angle
When light travels from a denser medium to a less dense one, something special happens at steep angles!
Critical Angle Formula
Only works when n₁ > n₂ (light going from denser to less dense)
Below Critical Angle
- • Light refracts at the boundary
- • Some light reflects, some transmits
- • Normal refraction occurs
At/Above Critical Angle
- • Light totally internally reflects
- • NO light transmits through boundary
- • 100% reflection back into medium
Applications of TIR
🔌 Optical Fibers
High-speed internet, endoscopes
💎 Diamonds
Sparkle from high n = 2.42
🔭 Binoculars
Prisms flip the image using TIR
🚗 Road Signs
Retroreflectors reflect headlights
Standing Waves & Resonance
Standing waves form when two waves of the same frequency travel in opposite directions and interfere. Unlike traveling waves, they appear to stay in place.
🎸 Strings
fₙ = nv/2L
Fixed at both ends
All harmonics (n = 1, 2, 3...)
🎺 Open Pipes
fₙ = nv/2L
Open at both ends
All harmonics (n = 1, 2, 3...)
🎷 Closed Pipes
fₙ = nv/4L
Closed at one end
Odd harmonics only (n = 1, 3, 5...)
Nodes
Points of zero displacement - the wave doesn't move here. Fixed ends of strings are always nodes.
Antinodes
Points of maximum displacement - maximum oscillation. Open ends of pipes are always antinodes.
Common Mistakes in Wave Problems
Avoid these frequent errors when solving waves questions in AP Physics 1, AP Physics 2, and high school physics exams:
Measuring angles from the surface
Angles in refraction must be measured from the normal (the perpendicular line to the surface), NOT from the surface itself.
Draw the normal first, then measure all angles from it.
Forgetting unit conversions
Using nm instead of m, or MHz instead of Hz gives answers that are orders of magnitude wrong.
Convert: nm ÷ 10⁹ → m, MHz × 10⁶ → Hz, kHz × 10³ → Hz
TIR in wrong direction
Total internal reflection only works when going FROM a denser medium TO a less dense medium (n₁ > n₂).
Check n values: TIR needs the first medium to have HIGHER n.
Using even harmonics for closed pipes
Closed pipes only have odd harmonics (1st, 3rd, 5th...). Using even harmonics gives wrong frequencies.
For closed pipes: only use n = 1, 3, 5, 7... in fₙ = nv/4L
Confusing wavelength and amplitude
Wavelength is the distance between peaks. Amplitude is the height/maximum displacement.
Wavelength = horizontal distance (peak to peak). Amplitude = vertical distance (rest to peak).
Using wrong v for waves
Sound in air ≈ 340 m/s. Light in vacuum = 3×10⁸ m/s. Using the wrong speed gives wrong wavelengths.
Check the wave type and medium. Sound ≠ Light speeds!
Worked Examples
Practice with these AP Physics 1 and AP Physics 2 style wave problems:
Example 1: Wave Speed
A wave has frequency 500 Hz and wavelength 0.68 m. Calculate its speed.
Solution:
Given: f = 500 Hz, λ = 0.68 m, v = ?
Formula: v = fλ
v = 500 × 0.68
v = 340 m/s ✓
Example 2: Angle of Refraction
Light travels from air (n=1) into glass (n=1.5) at 30° to the normal. Find the angle of refraction.
Solution:
n₁ sin θ₁ = n₂ sin θ₂
1 × sin(30°) = 1.5 × sin θ₂
0.5 = 1.5 × sin θ₂
sin θ₂ = 0.5 ÷ 1.5 = 0.333
θ₂ = sin⁻¹(0.333) = 19.5° ✓
Example 3: Critical Angle for Glass
Glass has refractive index 1.5 and air has 1.0. Calculate the critical angle.
Solution:
sin θc = n₂/n₁
sin θc = 1.0 / 1.5
sin θc = 0.667
θc = sin⁻¹(0.667) = 41.8° ✓
Any angle > 41.8° will cause total internal reflection
Example 4: Guitar String Frequency
A guitar string is 0.65 m long with wave speed 400 m/s. Calculate the fundamental frequency.
Solution:
For a string: f₁ = v/2L
f₁ = 400 / (2 × 0.65)
f₁ = 400 / 1.3
f₁ = 307.7 Hz ✓
2nd harmonic = 615.4 Hz, 3rd = 923.1 Hz, etc.
Frequently Asked Questions
What is the wave equation?
The wave equation is v = fλ where v is speed (m/s), f is frequency (Hz), and λ is wavelength (m). It applies to all waves.
How do I calculate wavelength?
Use λ = v/f. Divide the wave speed by the frequency. For sound in air at 340 m/s with f = 500 Hz: λ = 340/500 = 0.68 m.
What is Snell's law?
n₁ sin θ₁ = n₂ sin θ₂. It describes how light bends when crossing boundaries between media with different refractive indices.
What is the critical angle?
The angle of incidence that produces a 90° refracted ray. Beyond this angle, total internal reflection occurs. sin θc = n₂/n₁.
What is total internal reflection?
When light reflects completely at a boundary instead of refracting. Requires: n₁ > n₂ AND angle > critical angle. Used in optical fibers.
What is the speed of sound in air?
v = 331 + 0.6T where T is temperature in °C. At 20°C: v = 331 + 12 = 343 m/s.
What is the speed of light?
c = 3 × 10⁸ m/s in vacuum. In other media, v = c/n where n is the refractive index.
Is this calculator suitable for AP Physics 1 and 2?
Yes! It covers all wave topics with step-by-step solutions perfect for AP Physics 1 and AP Physics 2 exam preparation.
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