Question

In a medium sound travel 2 km in 3 sec and in air, it travels 3 km in 10 sec. The ratio of the wavelengths of sound in the two media is:
A) 1:8
B) 1:18
C) 8:1
D) 20:9

Answer 1

Hint: We need to relate wavelength with speed of sound and for that we know that speed of sound indifferent medium is directly proportional to the wavelength of sound ie; \[v \propto \lambda \].

Formula used:
\[\
  \dfrac{{{v_m}}}{{{v_a}}} = \dfrac{{{\lambda _m}}}{{{\lambda _a}}} \\
  v \propto \lambda \\
\ \]
where v = speed of sound f= frequency \[\lambda \]= wavelength of sound wave

Complete step by step solution:
We have given sound travel in medium with 2km in 3s
ie; ${v_m}$ = speed of sound in medium=distance/time
\[\
   \Rightarrow \dfrac{{2km}}{{3sec}} \\
   \Rightarrow \dfrac{{2000}}{3}m/s\;\;\;\; ………………………...(1)
 \ \]
And sound travel in air with 3km in 10s
ie; ${v_a}$ = speed of sound in air
=\[\dfrac{3}{{10}}km/\sec \;\;\;\;\;\] ………………….(2)
We know that speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. The speed of sound varies greatly depending upon the medium it is traveling through.

Wavelength of sound- wavelength is defined as the distance between adjacent identical parts of a wave.

Relationship between the speed of sound,
 \[v = f\lambda \]
So from here when frequency is constant we can write above equation as
 \[v \propto \lambda \]
So relation between speed of sound and wavelength of sound in a medium
\[{v_m} \propto {\lambda _m}\] ……………………………….(3)
relation between speed of sound and wavelength of sound in air
\[{v_a} \propto {\lambda _a}\] ……………………………..(4)

From (3) and (4) ratio of the wavelengths of sound in the two media is
 \[\dfrac{{{v_m}}}{{{v_a}}} = \dfrac{{{\lambda _m}}}{{{\lambda _a}}}\]
Putting the value of speed of sound in both the medium from equation (1) and (2)

\[\dfrac{{{\lambda _m}}}{{{\lambda _a}}} = \dfrac{{2000/3}}{{3000/10}}\]
= \[\dfrac{{2/3}}{{3/10}}\]
After further calculation
\[\dfrac{{{\lambda _m}}}{{{\lambda _a}}} = \dfrac{{20}}{9}\]

Therefore option (D) is correct.

Note:Speed of sound also depends on density and is inversely proportional to it while it depends on temperature and is directly proportional to it ie; \[v \propto \sqrt T \] speed of sound changes 0.66m/s with the change in 1°C of temperature.