Beats, Resonance and the Doppler Effect: Quick Revision

A brief note on Beats, Resonance and the Doppler Effect. Read it before a test. Try each question at the end before you open its answer.

Beats

• Two close frequencies drift in and out of step.
• Beat frequency: $f_b = |f_1 - f_2|$. Beat period: $T_b = 1/f_b$.
• Pitch heard: $\dfrac{f_1 + f_2}{2}$.
• Wax lowers a fork's frequency. Filing raises it. Tightening raises a string's.
• Beats faster after a change: the two frequencies moved apart.

The resonance tube

• Node at the water, antinode a distance $e$ above the open end.
• $l_1 + e = \lambda/4$, $l_2 + e = 3\lambda/4$: only odd quarter waves.
• Speed of sound: $v = 2f(l_2 - l_1)$. The end correction cancels.
• End correction: $e = \dfrac{l_2 - 3l_1}{2} \approx 0.3d$.

The Doppler effect

• $f' = f\,\dfrac{v \pm v_o}{v \mp v_s}$. Upper signs when that body moves towards the other.
• Source moving: the wavelength changes. Listener moving: the rate of meeting crests changes.
• Speeds are measured relative to the air.
• Echo from a reflector coming at $u$: $f\,\dfrac{v + u}{v - u}$.

A worked example

A fire engine with a $600\ \text{Hz}$ siren passes you at $15\ \text{m/s}$.

1. Coming. $f' = 600\ \text{Hz} \times \dfrac{340\ \text{m/s}}{325\ \text{m/s}} = 628\ \text{Hz}$.
2. Going. $f' = 600\ \text{Hz} \times \dfrac{340\ \text{m/s}}{355\ \text{m/s}} = 575\ \text{Hz}$.
3. The drop. About $53\ \text{Hz}$, heard the moment it passes.

Practice with new numbers

A program to try

The program finds the two notes of the fire engine.

Mistakes to avoid

• Adding frequencies for beats. Take the difference.
• Taking $l_2 - l_1$ as $\lambda$. It is $\lambda/2$.
• Swapping the Doppler signs. Towards always raises the note.
• A one-step echo. The reflector is a listener, then a source.

Check yourself

1. Forks of $315\ \text{Hz}$ and $318\ \text{Hz}$ sound together. What is the beat period?

2. A tube resounds at $l_1$ and $l_2$. How much is $l_2 - l_1$, in wavelengths?

3. A siren comes towards you, then goes away. When does its note drop?

4. Why does an echo from a car coming towards you sound higher than the sound that was sent?