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

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.