Waves and Sound: Quick Revision

A brief note on Waves and Sound. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• $v = f\lambda$. The medium sets the speed; the source sets the frequency. On a string, $v = \sqrt{\dfrac{T}{\mu}}$.
• Two waves in step: whole wavelengths of path difference are loud, odd half wavelengths are quiet. Beats: $|f_1 - f_2|$ each second.
• Standing waves: a string fixed at both ends or an open pipe gives $f_n = \dfrac{nv}{2L}$; a pipe closed at one end gives only odd multiples of $\dfrac{v}{4L}$.
• Doppler: motion towards each other raises the pitch: $f' = f\,\dfrac{v \pm v_o}{v \mp v_s}$.

The facts at a glance

• Wave speed: $v = f\lambda$
• Beats: $\lvert f_1 - f_2 \rvert$ each second
• String, open pipe: $f_n = \dfrac{nv}{2L}$
• Closed pipe: $\dfrac{v}{4L}, \dfrac{3v}{4L}, \dfrac{5v}{4L}, \ldots$
• Resonance tube: $\lambda = 2(l_2 - l_1)$
• Moving source: $f' = f\,\dfrac{v}{v \mp v_s}$

A worked example

A string $0.75\ \text{m}$ long has a tension of $81\ \text{N}$ and a mass of $0.01\ \text{kg}$ for each metre. Find the wave speed and its first three harmonics.

1. The speed. $v = \sqrt{\dfrac{81\ \text{N}}{0.01\ \text{kg/m}}} = 90\ \text{m/s}$.
2. The fundamental. $f_1 = \dfrac{90\ \text{m/s}}{2 \times 0.75\ \text{m}} = 60\ \text{Hz}$.
3. The harmonics. $120\ \text{Hz}$ and $180\ \text{Hz}$.

Practice with new numbers

A program to try

Mistakes to avoid

• Thinking a higher frequency travels faster. It has a shorter wavelength instead.
• Taking the average frequency as the beats. The beats are the difference.
• Using $\dfrac{v}{2L}$ for a closed pipe. It is $\dfrac{v}{4L}$, with odd harmonics only.
• Putting the Doppler signs the wrong way. Towards each other always raises the pitch.

Check yourself

1. A pipe $0.17\ \text{m}$ long is closed at one end. Sound travels at $340\ \text{m/s}$. What is its lowest note?

2. Forks of $384\ \text{Hz}$ and $381\ \text{Hz}$ sound together. How many beats do you hear each second?

3. Why does an approaching car's horn sound higher than it really is?

4. What is the wavelength of a $1360\ \text{Hz}$ note in air, where sound travels at $340\ \text{m/s}$?