A sound wave of frequency of 2 kHz has a wavelength of 35 cm in a given medium. How long will it take to travel a distance of 2.1 km through the medium?

Correct — C, 3.0 s. One relation carries the whole item: wave speed equals frequency times wavelength. Convert both given quantities into base units before multiplying — 2 kHz is 2000 Hz, and 35 cm is 0.35 m — and the speed in that medium is 2000 × 0.35 = 700 metres per second. The journey is 2.1 km, which is 2100 m, so the time is 2100 ÷ 700 = 3.0 seconds. Note that the frequency never has to be used again after it has produced the speed; once the speed is known the question is ordinary distance-over-speed arithmetic. The whole difficulty sits in the units, since kilohertz, centimetres and kilometres are three different scales in one short stem.

• (a)30 s — Ten times too large, and it comes from a decimal slip in the wavelength — reading 35 cm as 0.035 m gives a speed of 70 m/s, and 2100 ÷ 70 = 30. Checking the speed for plausibility catches this at once: 70 m/s is a fifth of the speed of sound in air.
• (b)2.1 s — This lifts the distance figure straight out of the stem. It would be right only if the medium carried sound at exactly 1000 m/s, which is not what 2000 Hz and 35 cm give.
• (d)4.1 s — It answers to a speed of about 512 m/s, which no combination of the two given quantities produces. The digits are a rearrangement of the numbers in the stem rather than the result of any workable slip.

For any wave the speed equals the frequency multiplied by the wavelength, because a full wavelength passes a fixed point once every period, and the frequency is the number of periods per second. The frequency of a sound is fixed by whatever is vibrating and does not change when the wave crosses into another medium; the speed does change, and the wavelength adjusts to keep the product equal to the new speed. That is why a wavelength is only meaningful once the medium has been named, exactly as this stem names it.

The printed booklet sets the distance as 2·1 km with a raised dot for the decimal point, an older typesetting convention that UPSC still uses; it means the same as 2.1 km. Sound travels at about 343 m/s in air at room temperature, near 1,500 m/s in water and around 5,000 m/s in steel, so the 700 m/s implied here belongs to a hypothetical medium denser than air and thinner than water. Carrying those three benchmarks into the hall is worth more than it looks: a computed speed that falls wildly outside them is a signal to recheck the unit conversion before spending time on the division.

• Wave speed equals frequency multiplied by wavelength, for sound and for every other wave.
• One kilohertz is 1,000 hertz, and 35 cm is 0.35 m — both conversions are needed before multiplying.
• The speed here works out to 700 m/s, and the time to cover 2,100 m at that speed is 3.0 s.
• Frequency is set by the source and does not change with the medium; speed and wavelength both do.
• Sound moves at roughly 343 m/s in air at room temperature, about 1,500 m/s in water and around 5,000 m/s in steel.

• Multiplying 2000 by 35 and leaving the answer in centimetres per second while the distance is in kilometres.
• Assuming the frequency changes when the sound enters a different medium.
• Reaching for the speed of sound in air when the stem has already supplied the data to compute the speed in this medium.

Asked as a one-formula numerical item wrapped in three unit conversions — the physics is a single line and the marks are lost on the arithmetic.

• practice — not a real PYQ

  A sound wave of frequency 500 Hz has a wavelength of 0.6 m in a medium. The speed of sound in that medium is

  • (a)150 m/s
  • (b)300 m/s
  • (c)600 m/s
  • (d)1200 m/s

  Answer(b) 300 m/s — speed is frequency times wavelength, so 500 × 0.6 = 300 metres per second.

• practice — not a real PYQ

  A sound wave passes from air into water. Which one of the following stays unchanged?

  • (a)Speed
  • (b)Wavelength
  • (c)Frequency
  • (d)All three

  Answer(c) Frequency — it is fixed by the vibrating source, while the speed rises in water and the wavelength stretches to match.