A plane sound wave in air at 20°C, with wavelength 0.645 m, is incident on a smooth surface of water at 25°C, at an angle of incidence of 3.65°. How do you determine the angle of refraction for the sound wave and the wavelength of the sound in water?

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A plane sound wave in air at 20°C, with wavelength 0.645 m, is incident on a smooth surface of water at 25°C, at an angle of incidence of 3.65°. How do you determine the angle of refraction for the sound wave and the wavelength of the sound in water?

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Answer 1 · 2016-01-19T03:29:55

Angle of refraction $θ_{r} \approx {0.8}^{o}$ $theta_rapprox 0.8^o$
Wavelength in water $\approx 0.1478 m$ $approx0.1478m$

Explanation:

In air at $20^{o} C$ $20^o C$ the speed of sound $v_{s}^{a i r} = 343 m / s$ $v_s^(air) = 343m//s$ .
In fresh water at $25^{o} C$ $25^oC$ the speed of sound $v_{s}^{w a t e r} = 1497 m / s$ $v_s^(water)=1497 m//s$

Using the following relation between angle of incidence $θ_{i}$ $theta _i$ , angle of refraction $θ_{r}$ $theta_r$ , wavelengths and velocity of the sound wave in different media

$(\frac{{sin θ}_{i}}{{sin θ}_{r}}) = \frac{λ_{a i r}}{λ_{w a t e r}} = \frac{v_{w a t e r}}{v_{a i r}}$ $(sin theta _i/sin theta_r)=lambda_(air)/lambda_(water)=v_(water)/v_(air)$

For calculating angle of refraction the above equation between angle and velocity becomes

$(\frac{{sin θ}_{i}}{{sin θ}_{r}}) = \frac{v_{w a t e r}}{v_{a i r}}$ $(sin theta _i/sin theta_r)=v_(water)/v_(air)$

$(\frac{sin 3.65}{{sin θ}_{r}}) = \frac{1497}{343}$ $(sin 3.65/sin theta_r)=1497/343$
$θ_{r} = {sin}^{- 1} (0.06366 \times \frac{343}{1497})$ $theta_r=sin^(-1)(0.06366times 343/1497)$
$θ_{r} = {sin}^{- 1} (0.01459)$ $theta_r=sin^(-1)(0.01459)$
2. To calculate the wavelength in water

$\frac{λ_{a i r}}{λ_{w a t e r}} = \frac{v_{w a t e r}}{v_{a i r}}$ $lambda_(air)/lambda_(water)=v_(water)/v_(air)$
Inserting the values

$\frac{0.645}{λ_{w a t e r}} = \frac{1497}{343}$ $0.645/lambda_(water)=1497/343$
$λ_{w a t e r} = 0.645 \times \frac{343}{1497}$ $lambda_(water)=0.645 times343/1497$

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A plane sound wave in air at 20°C, with wavelength 0.645 m, is incident on a smooth surface of water at 25°C, at an angle of incidence of 3.65°. How do you determine the angle of refraction for the sound wave and the wavelength of the sound in water?

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Explanation:

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