Physics of Sound: Formula Summary and Key Concepts

Sound: Formula Summary

1. Speed of Sound

The speed of sound depends on the medium through which it travels. In air, it varies with temperature, while in solids and liquids, it depends on material properties.

• In air (approximation): $v = 331 + 0.60 T_c$   (m/s), where $T_c$ is temperature in Celsius.

• General formula for ideal gas: $v = \sqrt{\frac{3RT}{M}}$ R: universal gas constant, T: absolute temperature, M: molar mass.

• In solids and liquids: $v = \sqrt{\frac{B}{\rho}}$ B: bulk modulus, \rho: density.

2. Wave Relationships

Sound waves obey the general wave relationship between speed, frequency, and wavelength.

• $v = f \lambda$

• Where:

  • v: speed

  • f: frequency

  • \lambda: wavelength

3. Sound Intensity & Power

Sound intensity measures the power per unit area carried by a sound wave.

• Intensity: $I = \frac{P}{A}$ P: power, A: area.

• Reference intensity: $I_0 = 1.0 \times 10^{-12} \text{ W/m}^2$

4. Sound Level (Decibels)

Sound level in decibels (dB) quantifies intensity on a logarithmic scale.

• Decibel level: $\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$   dB

• Difference in decibels: $\Delta \beta = 10 \log_{10} \left( \frac{I_2}{I_1} \right)$

5. Spherical Spreading of Sound

Sound intensity decreases with distance from a point source due to spreading over a sphere.

• $I = \frac{P}{4 \pi r^2}$ r: distance from source.

6. Interference of Sound Waves

When two sound waves overlap, they can interfere constructively or destructively depending on their path difference.

• Constructive interference: $\Delta L = n \lambda$

• Destructive interference: $\Delta L = (n + \frac{1}{2}) \lambda$ n: integer

7. Beats

Beats occur when two sound waves of slightly different frequencies interfere, producing a fluctuating sound intensity.

• $f_{\text{beat}} = |f_1 - f_2|$

• Example: Tuning musical instruments by listening for beats between two notes.

8. Doppler Effect

The Doppler effect describes the change in frequency observed when the source or observer is moving.

• General formula: $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$ f': observed frequency, f: source frequency, v: speed of sound, v_o: observer speed, v_s: source speed. Top sign: observer moving toward/source moving away Bottom sign: observer moving away/source moving toward

• Application: Ambulance siren pitch changes as it passes by.

9. Shock Waves & Mach Number

Shock waves occur when an object moves faster than the speed of sound, forming a cone-shaped wave front.

• Mach number: $\text{Mach} = \frac{v_{\text{object}}}{v_{\text{sound}}}$

• Shock cone angle: $\sin \theta = \frac{v_{\text{sound}}}{v_{\text{object}}}$

• Example: Sonic boom produced by supersonic aircraft.

10. Standing Waves in Air Columns

Standing waves form in pipes, with patterns depending on whether ends are open or closed.

• Open pipe (both ends open): $f_n = \frac{nv}{2L}$,   $n = 1, 2, 3, \ldots$

• Closed pipe (one end closed): Only odd harmonics: $f_n = \frac{nv}{4L}$,   $n = 1, 3, 5, \ldots$

11. Water Waves, Boundary Effects (AP Focus)

When waves encounter boundaries, they may reflect or transmit, sometimes with a phase change.

• Reflection (phase reversal at fixed end): $A \rightarrow -A$

• Transmission into new medium (string comparison): $v = \sqrt{\frac{T}{\mu}}$ T: tension, \mu: mass per unit length

Summary Table: Key Sound Concepts

Topic	Main Formula	Key Variables
Speed of Sound (air)	$v = 331 + 0.60 T_c$	$T_c$ = temperature (°C)
Wave Relationship	$v = f \lambda$	$f$ = frequency, $\lambda$ = wavelength
Intensity	$I = \frac{P}{A}$	$P$ = power, $A$ = area
Decibel Level	$\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$	$I$ = intensity, $I_0$ = reference intensity
Doppler Effect	$f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$	$v_o$ = observer speed, $v_s$ = source speed
Standing Waves (open pipe)	$f_n = \frac{nv}{2L}$	$n$ = harmonic number, $L$ = length

Additional info: These notes summarize essential formulas and concepts for the study of sound in college-level physics, including wave behavior, intensity, interference, and applications such as the Doppler effect and standing waves in pipes.