“Towards means higher”
Every Doppler question
Any approach raises the pitch, any recession lowers it. Work out the direction first, then choose the signs to match that answer. If your formula disagrees with the direction, the signs are wrong.
Class 11 Physics · Gujarat Board, CBSE, GUJCET and JEE
Waves formulas
Waves carries the most formulas of any Class 11 physics chapter, and the most marks. Sound, strings, pipes, beats and Doppler all sit here, and the Doppler sign convention is the single most common place students lose an easy three marks.
17 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Wave basics
| What | Formula | Symbols and notes |
|---|---|---|
| Wave equation | v = fλ | speed, frequency, wavelength, for every wave there is |
| Progressive wave | y = A sin(ωt − kx) | moving in the positive x direction |
| Wave number | k = 2π/λ | also called the propagation constant |
| Speed on a string | v = √(T/μ) | T tension, μ mass per unit length |
| Speed of sound in a gas | v = √(γp/ρ) | Laplace's correction; Newton's version used p/ρ and was wrong |
| Speed of sound and temperature | v = 331 + 0.61t m s−1 | t in °C, for air |
Stationary waves
| What | Formula | Symbols and notes |
|---|---|---|
| Stationary wave | y = 2A sin(kx) cos(ωt) | nodes stay still, antinodes move most |
| Distance between nodes | λ/2 | and node to next antinode is λ/4 |
| String fixed at both ends | fn = n·v/2L | all harmonics present, n = 1, 2, 3, … |
| Open organ pipe | fn = n·v/2L | all harmonics present |
| Closed organ pipe | fn = (2n − 1)v/4L | odd harmonics only, and the fundamental is half that of an open pipe |
| End correction | closed pipe: L + 0.6r | r is the radius of the pipe |
Beats and Doppler
| What | Formula | Symbols and notes |
|---|---|---|
| Beat frequency | fbeat = |f₁ − f₂| | audible when the difference is under about 10 Hz |
| Doppler, general | f' = f(v ± vo)/(v ∓ vs) | upper signs for approach, lower for recession |
| Source approaching, observer still | f' = f·v/(v − vs) | pitch rises |
| Observer approaching, source still | f' = f(v + vo)/v | pitch rises, but by a different amount |
| Intensity | I ∝ A² | and I ∝ 1/r² for a point source |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Closed pipes are odd”
Harmonics in pipes
A closed pipe gives odd harmonics only, and its fundamental is half that of an open pipe of the same length.
“Node to node is half”
Spacing in a stationary wave
Node to node is λ/2, node to antinode is λ/4. Almost every stationary wave numerical comes down to this.
“Tighter and lighter is faster”
Speed on a string
v = √(T/μ), so more tension speeds a wave up and more mass per unit length slows it down. That is what tuning a guitar does.
Where marks go
The mistakes that cost the most.
- Mixing up the source and observer speeds in the Doppler formula. The observer's speed changes the top, the source's changes the bottom.
- Treating beats as a sum of frequencies. Beat frequency is the difference, and only the difference.
- Using all harmonics for a closed pipe. Only the odd ones exist there.
- Forgetting that sound cannot travel through vacuum while light can, because sound needs a medium.
- Thinking the speed of sound depends on its frequency or loudness. It depends on the medium and its temperature.
Test yourself
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