12.1 Waves & Sound

12.1 Waves & Sound

Quick Answer: Wave MCQs on the PAF Aeronautical Engineering initial test reward clean use of v = fλ, Doppler sign sense, beat frequency, and intensity/decibel ideas. Physics is commonly near ~30% of the academic battery at FSc Pre-Engineering depth—heavier than GD Pilot—so treat oscillations and sound as core, not side topics.

Waves connect mechanics to acoustics and, later, to optics. Selection-centre e-testing formats are not fully standardised in one public PDF for every induction cycle; coaching reports often describe roughly ~50 physics MCQs in about ~25 minutes. Always verify current instructions on joinpaf.gov.pk and your registration slip.

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.

Exam Habits That Save Marks

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 MCQs on an FSc-depth aeronautical paper become routine calculations rather than guesswork.