12.1 Waves & Sound

12.1 Waves & Sound

Quick Answer: This wave review uses v = fλ, Doppler sign sense, beat frequency, and intensity/decibel ideas. Oscillations and sound are useful technical foundations, without an asserted official PAF weight or route comparison.

Waves connect mechanics to acoustics and, later, to optics. Current public PAF sources do not publish a stable physics item count, timing, weight, or confirmed subject allocation for this route, so this section is presented only as independent technical preparation.

What a Wave Transfers

A wave is a disturbance that transfers energy through space without permanently transferring the medium’s mass. Particles (or fields) oscillate about equilibrium; the pattern of disturbance travels.

Sound in air is a longitudinal mechanical wave. Light is a transverse electromagnetic wave. Confusing those two is a classic trap on FSc-style papers.

Progressive vs Stationary Waves

A progressive (travelling) wave carries energy from one place to another. A stationary (standing) wave forms when two identical waves travel in opposite directions and interfere; nodes (zero amplitude) and antinodes (maximum amplitude) appear at fixed positions. Standing-wave wavelength on a string fixed at both ends satisfies L = n(λ/2) for harmonic number n = 1, 2, 3, …

The Master Relation: v = fλ

For any periodic wave:

$$v = f\lambda$$

where v is wave speed (m/s), f is frequency (Hz), and λ is wavelength (m). Also:

$$T = \frac{1}{f}, \quad \omega = 2\pi f$$

Wave speed in a medium is set by the medium (and tension/linear density for strings), not by how hard you “push” once the wave has left the source. Frequency is usually fixed by the source; wavelength then adjusts so that v = fλ holds.

Worked example — wavelength from speed and frequency

A sound wave travels at 340 m/s with frequency 680 Hz. Find λ.

$\lambda = \frac{v}{f} = \frac{340}{680} = 0.50\ \text{m}$

Worked example — string wave speed

On a taut string, $v = \sqrt{T/\mu}$. If tension T doubles and linear density μ is unchanged, speed increases by $\sqrt{2}$ (about 1.41×). Frequency of a given harmonic rises by the same factor if length is fixed, because fₙ = n v/(2L).

Sound Speed and Pitch

In dry air near 0 °C, sound speed is about 331 m/s; near 20 °C it is about 343–344 m/s. Approximate temperature dependence often taught at FSc level:

$$v \approx 331 + 0.6\,\theta_C \quad (\text{m/s})$$

Pitch tracks frequency; loudness relates to intensity (and the ear’s response); quality/timbre depends on overtone mix. Speed is essentially independent of frequency and amplitude for ordinary audible sound in air—so a louder shout does not reach you sooner than a soft one from the same distance.

Doppler Effect

When source and observer move relative to the medium, the observed frequency changes:

$$f' = f\,\frac{v \pm v_o}{v \mp v_s}$$

Use the sign convention carefully (FSc textbooks state it in words):

• Numerator: + v_o if observer moves toward the source; − v_o if away.
• Denominator: − v_s if source moves toward the observer; + v_s if away.

Worked example — approaching source

A siren of frequency 1000 Hz approaches a stationary observer at 34 m/s. Take v = 340 m/s.

$f' = 1000\,\frac{340}{340 - 34} = 1000\,\frac{340}{306} \approx 1111\ \text{Hz}$

Frequency rises when the source approaches. For aircraft and radar intuition: closing range raises returned frequency; opening range lowers it—same physical idea as acoustic Doppler, different wave type.

Beats

Two close frequencies sounding together produce beats. Beat frequency:

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

Worked example: Tuning forks 256 Hz and 260 Hz sounded together give 4 beats per second. If the 260 Hz fork is loaded with wax (frequency falls) and beats become 6 s⁻¹, the loaded frequency is 250 Hz (farther from 256), not 266 Hz—because wax lowers frequency.

Intensity and Decibels

Intensity I is average power per unit area (W/m²). For a spherical point source in free space:

$$I = \frac{P}{4\pi r^2}$$

so I ∝ 1/r². Also, for a given wave, I ∝ A² (amplitude squared).

Human hearing spans an enormous intensity range, so we use intensity level:

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

Worked example — inverse-square

If intensity at 2 m from a small source is I, at 6 m it is I/9 because distance triples and intensity falls as 1/r².

Worked example — decibels

If I = 10⁻⁶ W/m², then β = 10 log₁₀(10⁻⁶/10⁻¹²) = 10 log₁₀(10⁶) = 60 dB.

Reliable Practice Habits

1. Identify wave type (longitudinal vs transverse; mechanical vs EM) before picking a formula.
2. Keep v = fλ dimensionally consistent—λ in metres, f in hertz.
3. For Doppler, sketch who moves toward whom; do not memorise a random sign pattern without the picture.
4. Beats use the absolute difference of frequencies.
5. Intensity ratios become logs in decibels; distance scaling uses 1/r², not 1/r.

Master these five habits and most waves-and-sound exercises become routine calculations rather than guesswork.