NDA Physics · Sound

How We Measure Sound — v = fλ, Speed, and the Frequency Bands

Three quantities describe any sound — frequency, wavelength, speed — tied by v = fλ. Speed is set by the medium (not by frequency, not by pressure at constant T). The named regions on the frequency axis (infrasonic, audible, ultrasonic) are just bands.

All Sound notes NDA Physics strategy

Why this matters

Now that you can identify sound, you need to MEASURE it. 13 PYQs cluster around three ideas: (1) the wave equation v = fλ and what hertz / period / decibel mean, (2) the standout result that speed depends on the MEDIUM only (it does not change with frequency, and at constant T it does not change with pressure either), (3) the bands and named scales NDA recycles — 20 Hz – 20 kHz audible, > 20 kHz ultrasonic, the Mach number, typical sound speeds in air / water / steel, plus the Richter scale GK item. Mostly EASY plus four MODERATE.

Concept 1 of 3

Frequency, period, wavelength — and v = fλ

Intuition

Frequency f is how many complete oscillations happen per second. Wavelength

\lambda

is how much distance one full oscillation occupies. Speed v is just "how much wave passes you each second" — so it must equal frequency multiplied by wavelength. This single relation

v = f\lambda

ties every sound problem together: give any two, the third is fixed.

Definition

Frequency $f$ : number of complete cycles per unit time. Unit: hertz (Hz) = 1 cycle / second = 1 s⁻¹. Equivalent units: s⁻¹, min⁻¹, kHz, MHz. Not a unit of frequency: decibel (dB) — dB measures intensity LEVEL, not frequency. Period $T = 1/f$ : time for one complete cycle. Wavelength $\lambda$ : spatial distance over which the wave repeats (compression-to-next-compression, in metres). Wave equation: $v = f\lambda$ — for ANY periodic wave, in any medium.

Wave equation

v = f\lambda

vwave speed (m/s)
ffrequency (Hz)
\lambdawavelength (m)

Worked example

A sound wave travels through air at 340 m/s with a wavelength of 0.85 m. Find its frequency and period.

Rearrange the wave equation for frequency: $f = v/\lambda$ .
$f = 340 / 0.85 = 400$ Hz.
Period is the reciprocal: $T = 1/f = 1/400 = 0.0025$ s $= 2.5$ ms.

Answer:

f = 400

Hz;

T = 2.5

ms.

Practice this conceptself-check · 4 quick reps

Try it yourself

An alternating current has frequency 3 Hz. What does that mean — and what is its period?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
If $f = 200$ Hz and $\lambda = 1.5$ m, find $v$ .
2.
If $v = 340$ m/s and $f = 170$ Hz, find $\lambda$ .
3.
If $f = 50$ Hz, what is the period?
4.
Which is NOT a unit of frequency: Hz, s⁻¹, min⁻¹, dB?

From the bank · past-year question

Example 1SoundMODERATE

A sound wave has a frequency of 1 kHz and wavelength 50 cm. How long will it take to travel 1 km?

[Q83 · Apr · 2022]

dB measures intensity LEVEL — not frequency, not amplitude

Decibel is a logarithmic ratio of two intensities (or two powers). It is a unit of intensity LEVEL only. Frequency uses Hz, s⁻¹, min⁻¹; amplitude uses Pa (pressure) or m (displacement). Any "which is NOT a unit of frequency" question puts dB as the answer.

3 Hz means 3 cycles per second — full cycles, not half or quarter

Hz is defined as one COMPLETE cycle per second. 3 Hz = 3 complete cycles per second, NOT 6 cycles and NOT 1.5 cycles.

Drill 3 more on frequency, period, wavelength — and v = fλ

Concept 2 of 3

Speed of sound depends on the MEDIUM — not on f, not on P (at constant T)

Intuition

Speed of sound is set by the MEDIUM the wave travels through — specifically by its elasticity and density. It does NOT depend on the wave's own frequency (a 100 Hz wave and a 10 kHz wave travel at the same speed in air), and at constant temperature it does NOT depend on pressure (doubling P doubles

\rho

too, and they cancel in the formula). It DOES grow with temperature — in a gas,

v \propto \sqrt{T}

Definition

For an ideal gas: $v = \sqrt{\gamma P / \rho} = \sqrt{\gamma RT/M}$ — depends on temperature $T$ and the gas's molar mass $M$ , but NOT on pressure $P$ at constant $T$ ( $P$ and $\rho$ move together). Across phases: $v_\text{solid} > v_\text{liquid} > v_\text{gas}$ (elasticity dominates). Frequency dependence: none — the wave equation $v = f\lambda$ holds because $\lambda$ adjusts; $v$ itself is fixed by the medium.

Speed of sound in an ideal gas

v = \sqrt{\dfrac{\gamma P}{\rho}} = \sqrt{\dfrac{\gamma R T}{M}}

\gammaadiabatic index ( $C_p/C_v$ )
Ppressure (Pa)
\rhodensity (kg/m^3)
Tabsolute temperature (K)
Mmolar mass (kg/mol)

Worked example

The speed of sound in a gas at 300 K is 330 m/s. If the temperature is raised to 1200 K (same gas), what is the new speed?

In a gas, speed depends on temperature as $v \propto \sqrt{T}$ (from $v = \sqrt{\gamma RT/M}$ ; the gas, hence $\gamma$ and $M$ , is unchanged).
Take the ratio: $v_2/v_1 = \sqrt{T_2/T_1} = \sqrt{1200/300} = \sqrt{4} = 2$ .
So $v_2 = 2 \times 330 = 660$ m/s.

Answer:

660

m/s — speed scales with

\sqrt{T}

, the one thing that does change it.

Practice this conceptself-check · 4 quick reps

Try it yourself

Two sound waves of frequency 100 Hz and 10 000 Hz travel through the same room at the same temperature. Which travels faster?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Does the speed of sound in air depend on the wave's frequency?
2.
Does the speed of sound in a gas depend on its pressure at constant temperature?
3.
Does the speed of sound in air increase or decrease with temperature?
4.
Rank speed of sound in steel, water, and air, fastest first.

From the bank · past-year question

Example 2SoundMODERATE

The speed of sound in a large ideal gas container is

x

. When pressure is doubled at constant temperature, speed becomes

y

. What is

x/y

[Q62 · Apr · 2026]

Pressure dependence trap — P only matters via T

"Pressure doubled, speed doubles" is the wrong intuition. The formula

v = \sqrt{\gamma P/\rho}

hides the fact that at constant temperature

P

and

\rho

are proportional, so the ratio is fixed. Speed only changes through TEMPERATURE:

v \propto \sqrt{T}

Frequency does NOT change the speed of sound

A 100 Hz sound and a 10 kHz sound travel at the same speed in the same air. Frequency only changes the WAVELENGTH (

\lambda = v/f

) — the same medium-set speed is shared.

Solid > Liquid > Gas (elasticity wins over density)

Naively you might expect dense materials to be slower, but for sound speed elasticity dominates: steel

\approx 5000

m/s, water

\approx 1500

m/s, air

\approx 340

m/s. The ordering is monotone.

Drill 2 more on speed of sound depends on the medium — not on f, not on p (at constant t)

Concept 3 of 3

Bands and scales — audible / infra / ultrasonic, Mach, sound speeds, decibel, Richter

Intuition

A handful of named bands and scales come up here every other paper — the audible frequency range (20 Hz – 20 kHz), the ultrasonic threshold, the Mach number labels (subsonic / sonic / supersonic / hypersonic), typical sound speeds in air / water / steel, plus what dB and Richter actually measure. Ultrasonic is just sound above the audible band — same speed in the same medium (per Subtopic 2's speed-medium-only result), shorter wavelength, higher frequency. Memorise the table once and these recall questions become a lookup.

Definition

Six clusters of named numbers, all tested at EASY level. Drill the boundary numbers (20 Hz, 20 kHz, Mach 1) cold; the typical-speed numbers (340 / 1500 / 5000 m/s) are the most-repeated quantitative recall in the chapter.

What	Value / range	Note
Audible frequency range (human ear)	20 Hz to 20 000 Hz	Drilled most years — memorise both endpoints
Infrasonic	< 20 Hz	Below the lower limit of human hearing — whales, earthquakes
Ultrasonic	> 20 000 Hz (> 20 kHz)	Bats, SONAR, medical imaging — applications in Subtopic 4
Ultrasonic vs audible (same medium)	Same speed, higher f, shorter $\lambda$	From $v = f\lambda$ : $v$ is medium-set; higher $f \Rightarrow$ shorter $\lambda$ Distractors pair higher frequency with higher SPEED — wrong; speed is set by the medium.
Speed of sound in air ( $20°$ C)	$\approx 340$ m/s	Standard round number — memorise
Speed of sound in water ( $20°$ C)	$\approx 1500$ m/s	Tested in 2019: distractors at 330 / 800 / 5000
Speed of sound in steel	$\approx 5000$ m/s	Solid > liquid > gas
Mach number	object speed / sound speed	Compares object's speed to local sound speed
Mach < 1	Subsonic	Most everyday motion (cars, propeller aircraft)
Mach = 1	Sonic / transonic	At the speed of sound — sonic boom region
Mach > 1	Supersonic	Faster than sound (fighter jets, Concorde) NDA 2017 tested exactly this — Mach > 1 means supersonic.
Mach > 5	Hypersonic	Re-entry vehicles, scramjets
Decibel (dB)	log scale of intensity ratio	Unit of intensity LEVEL — NOT a unit of frequency or amplitude
Richter scale	log scale of earthquake energy	Devised 1935 by C.F. Richter; no upper limit (though > 9.5 is rare)

The audible-range endpoints (20 Hz, 20 kHz) and the speed-in-water number (

\approx 1500

m/s) are the most-tested rows — they appear almost yearly.

Practice this conceptself-check · 6 quick reps

Try it yourself

Compared to audible sound waves at the same temperature in the same air, ultrasonic waves have ___ frequency, ___ wavelength, and ___ speed.

Practice — Level 1 (6 reps)

Quick reps to lock in the method. Try each, then check.

1.
The human audible frequency range is approximately ___ to ___ Hz.
2.
Sound waves above 20 kHz are called ___.
3.
Sound waves below 20 Hz are called ___.
4.
Approximate speed of sound in water at 20°C?
5.
A body with Mach number > 1 is called ___.
6.
Decibel measures ___ (frequency / amplitude / intensity level)?

From the bank · past-year question

Example 3SoundEASY

Which one of the following frequency ranges is sensitive to human ears?

[Q62 · Apr · 2018]

Audible range: 20 Hz to 20 kHz — NOT 0 Hz to 20 kHz

Distractors often use "0 – 200 Hz" or "200 – 20 000 Hz" or "2 000 – 20 000 Hz" to test whether you remember the LOWER endpoint (20 Hz). Below 20 Hz is infrasonic — you feel it as vibration but don't hear it as a tone.

Speed of sound in water $\approx 1500$ m/s, NOT 5000 m/s (that's steel)

Common distractor swaps water and steel speeds. Water sits in the middle: 1500 m/s. Steel

\approx 5000

m/s. Air

\approx 340

m/s.

Mach > 1 is SUPERsonic, not SUBsonic

Subsonic = slower than sound (Mach < 1). Supersonic = faster than sound (Mach > 1). Hypersonic kicks in around Mach 5. Easy to flip under exam pressure.

Ultrasonic does NOT travel faster than audible sound

Higher frequency does NOT imply higher speed — speed is set by the medium (this is the Subtopic 2 result). From

v = f\lambda

: higher

f

only shrinks

\lambda

;

v

is fixed.

Drill 5 more on bands and scales — audible / infra / ultrasonic, mach, sound speeds, decibel, richter

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Frequency, period, wavelength — and v = fλ
Wave equation
$v = f\lambda$
Speed of sound depends on the MEDIUM — not on f, not on P (at constant T)
Speed of sound in an ideal gas
$v = \sqrt{\dfrac{\gamma P}{\rho}} = \sqrt{\dfrac{\gamma R T}{M}}$

Reference tables (1)

Bands and scales — audible / infra / ultrasonic, Mach, sound speeds, decibel, Richter14 rows

What	Value / range	Note
Audible frequency range (human ear)	20 Hz to 20 000 Hz	Drilled most years — memorise both endpoints
Infrasonic	< 20 Hz	Below the lower limit of human hearing — whales, earthquakes
Ultrasonic	> 20 000 Hz (> 20 kHz)	Bats, SONAR, medical imaging — applications in Subtopic 4
Ultrasonic vs audible (same medium)	Same speed, higher f, shorter $\lambda$	From $v = f\lambda$ : $v$ is medium-set; higher $f \Rightarrow$ shorter $\lambda$ Distractors pair higher frequency with higher SPEED — wrong; speed is set by the medium.
Speed of sound in air ( $20°$ C)	$\approx 340$ m/s	Standard round number — memorise
Speed of sound in water ( $20°$ C)	$\approx 1500$ m/s	Tested in 2019: distractors at 330 / 800 / 5000
Speed of sound in steel	$\approx 5000$ m/s	Solid > liquid > gas
Mach number	object speed / sound speed	Compares object's speed to local sound speed
Mach < 1	Subsonic	Most everyday motion (cars, propeller aircraft)
Mach = 1	Sonic / transonic	At the speed of sound — sonic boom region
Mach > 1	Supersonic	Faster than sound (fighter jets, Concorde) NDA 2017 tested exactly this — Mach > 1 means supersonic.
Mach > 5	Hypersonic	Re-entry vehicles, scramjets
Decibel (dB)	log scale of intensity ratio	Unit of intensity LEVEL — NOT a unit of frequency or amplitude
Richter scale	log scale of earthquake energy	Devised 1935 by C.F. Richter; no upper limit (though > 9.5 is rare)

The audible-range endpoints (20 Hz, 20 kHz) and the speed-in-water number (

\approx 1500

m/s) are the most-tested rows — they appear almost yearly.

Watch out for (9)

dB measures intensity LEVEL — not frequency, not amplitude
→ Frequency, period, wavelength — and v = fλ
3 Hz means 3 cycles per second — full cycles, not half or quarter
→ Frequency, period, wavelength — and v = fλ
Pressure dependence trap — P only matters via T
→ Speed of sound depends on the MEDIUM — not on f, not on P (at constant T)
Frequency does NOT change the speed of sound
→ Speed of sound depends on the MEDIUM — not on f, not on P (at constant T)
Solid > Liquid > Gas (elasticity wins over density)
→ Speed of sound depends on the MEDIUM — not on f, not on P (at constant T)
Audible range: 20 Hz to 20 kHz — NOT 0 Hz to 20 kHz
→ Bands and scales — audible / infra / ultrasonic, Mach, sound speeds, decibel, Richter
Speed of sound in water $\approx 1500$ m/s, NOT 5000 m/s (that's steel)
→ Bands and scales — audible / infra / ultrasonic, Mach, sound speeds, decibel, Richter
Mach > 1 is SUPERsonic, not SUBsonic
→ Bands and scales — audible / infra / ultrasonic, Mach, sound speeds, decibel, Richter
Ultrasonic does NOT travel faster than audible sound
→ Bands and scales — audible / infra / ultrasonic, Mach, sound speeds, decibel, Richter

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1SoundEASY

The frequency (f), wavelength (

\lambda

) and speed (v) of a sound wave are related as

[Q133 · Apr · 2025]

Example 2SoundMODERATE

Which one of the following statements about the speed of sound waves is not correct?

[Q146 · Sep · 2022]

Example 3SoundEASY

Compared to audible sound waves, ultrasound waves have

[Q96 · Sep · 2019]

Example 4SoundEASY

The frequency of an alternating current is 3 Hz. It implies that

[Q64 · Apr · 2022]

Example 5SoundEASY

Which one among the following is true for the speed of sound in a given medium?

[Q149 · Apr · 2023]

Drill every past-year question on this subtopic

13 questions from the bank — paginated, with cart and Word-export support.

Drill the 13 questions