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Topic 18 of 20

Sound & Acoustics

Sound is a longitudinal mechanical wave — a traveling pressure variation in a medium. From the Doppler effect of a passing ambulance to the resonance of a concert hall, the physics of sound underlies all of acoustics, musical instruments, and medical ultrasound.

Key Concepts

Sound as a Pressure Wave

Sound propagates as alternating compressions (high pressure) and rarefactions (low pressure). It requires a material medium — sound cannot travel through a vacuum.

Speed of Sound

In air at 20°C:

v \approx 343

m/s. General formula:

v = \sqrt{B/\rho}

(B = bulk modulus, ρ = density). Sound travels faster in denser (stiffer) media: faster in water (~1480 m/s) and steel (~5000 m/s) than in air.

Intensity and Decibels

Intensity

I = P/A

(W/m²). Sound level in decibels:

\beta = 10\log_{10}(I/I_0)

dB, where

I_0 = 10^{-12}

W/m² is the threshold of hearing. Each 10 dB increase represents a 10× increase in intensity.

Doppler Effect

Apparent shift in frequency when source and observer have relative motion. Source moving toward observer → higher frequency; moving away → lower frequency. Same principle applies for moving observer.

Beats

Two waves with slightly different frequencies

f_1

and

f_2

produce a periodic amplitude variation at the beat frequency

f_\text{beat} = |f_1 - f_2|

. Used to tune musical instruments.

Resonance in Pipes

Open pipe (both ends open): resonant frequencies

f_n = nv/2L

, all harmonics. Closed pipe (one end closed):

f_n = nv/4L

, odd harmonics only (

n = 1, 3, 5, ...

Key Equations

Sound intensity level

\beta = 10\log_{10}\!\left(\frac{I}{I_0}\right) \text{ dB}, \quad I_0 = 10^{-12} \text{ W/m}^2

Logarithmic scale; 0 dB = threshold of hearing, 120 dB = threshold of pain.

Intensity vs. distance (point source)

I = \frac{P_\text{source}}{4\pi r^2}

Intensity falls as 1/r² from a point source in an open 3D space.

Doppler effect

f_O = f_S\,\frac{v \pm v_O}{v \mp v_S}

Upper signs when source/observer approach each other; lower when receding. v is sound speed.

Beat frequency

f_\text{beat} = |f_1 - f_2|

The perceived "wah-wah" frequency when two close frequencies interfere.

Pipe resonances

f_n = \frac{nv}{2L}\,(\text{open}), \quad f_n = \frac{(2n-1)v}{4L}\,(\text{closed})\quad n=1,2,3,...

Open pipes support all harmonics; closed pipes support only odd harmonics.

Worked Example

Doppler Effect: Approaching Train

Problem

A train horn emits 400 Hz. The train moves at 30 m/s toward a stationary observer. Speed of sound = 340 m/s. What frequency does the observer hear?

Solution

Observer is stationary ( $v_O = 0$ ), source approaches ( $v_S = 30$ m/s toward). Use lower sign in denominator:

f_O = f_S\,\frac{v}{v - v_S} = 400\times\frac{340}{340 - 30} = 400\times\frac{340}{310}

f_O \approx 400\times1.097 \approx 439 \text{ Hz}

Answer The observer hears approximately 439 Hz — higher than the emitted 400 Hz.

Practice

Exercises

7 problems

1 of 7

A sound wave in air has frequency $f = 440$ Hz and the speed of sound is $v = 343$ m/s. What is the wavelength (in m)?

Your answer

2 of 7

A sound intensity is $I = 0.001$ W/m². What is the sound level (in dB)? Use $I_0 = 10^{-12}$ W/m².

Your answer

3 of 7

An ambulance with a $600$ Hz siren moves toward a stationary observer at $25$ m/s. What frequency (in Hz) does the observer hear? Use $v_\text{sound} = 340$ m/s.

Your answer

4 of 7

After the ambulance passes and is moving **away** at $25$ m/s, what frequency (in Hz) does the stationary observer hear?

Your answer

5 of 7

An open organ pipe has length $L = 0.85$ m. What is its fundamental frequency (in Hz)? Use $v = 343$ m/s.

Your answer

6 of 7

Two tuning forks produce frequencies of $440$ Hz and $444$ Hz. What is the beat frequency (in Hz)?

Your answer

7 of 7

A closed (one end open) pipe resonates at its third harmonic at $f = 510$ Hz. What is the length (in m) of the pipe? Use $v = 340$ m/s.

Your answer

Key Takeaways

Sound is a longitudinal pressure wave; it travels faster in stiffer, less dense media.
The decibel scale is logarithmic: +10 dB = 10× intensity; +20 dB = 100× intensity.
Doppler: moving source compresses or stretches the wavelengths; moving observer encounters waves at a different rate.
Beat frequency $= |f_1 - f_2|$ : the slower the beat, the closer the two frequencies are.
Open pipe: all harmonics. Closed pipe (one end): odd harmonics only. This shapes the timbre of musical instruments.