Q2E - University Physics with Modern Physics

Question

For sound waves in air with frequency 1000 Hz, a displacement amplitude of 1.2 x 108 m produces a pressure amplitude of 3.0 × 10-² Pa. Water at 20°C has a bulk modulus of 2.2 x 10⁹ Pa, and the speed of sound in water at this temperature is 1480 m/s. For 1000-Hz sound waves in 20°C water, what displacement amplitude is produced if the pressure amplitude is 3.0 x 102 Pa? Explain why your answer is much less than 1.2 x 10-8 m.

Step-by-Step Solution

Answer

A=3.2 x 10-12 m is the displacement amplitude is produced if the pressure amplitude is 3.0 x 102 Pa. The reason for A being smaller than 1.2 x 10-8 Is that Bk for water is much larger than Bk for air

Step 1: Determination of the Displacement Amplitude

A sound wave with a frequency of 1000 Hz travels through the water with a speed of 1480 m/s. If the bulk modulus of the water is 2.2 x 10° Pa, determine the displacement amplitude that would be needed to produce a pressure amplitude of 3 × 10^-2 Pa. 
 
 The relation that describes the pressure amplitude for a sound wave is as follows:
 
 $P_{max}=B\mathrm{kA}$--(1)

 In order to make use of equation (1) first calculate k:

$k=\frac{2\mathrm{\pi }}{\lambda }$
 So, the relation between the wavelength and the frequency of a sound wave is given by the following equation:

$\lambda =\frac{V}{f}$--(2) 
 
 v=1480 m/s, the speed of sound in the water. 
 
 f = 1000 Hz, the frequency of the given sound wave. 
 
$\begin{array}{l}\lambda =\frac{1480m/s}{1000Hz}\\ \lambda =1.48m\end{array}$ 
 

Therefore,
$\begin{array}{l}k=\frac{2\mathrm{\pi }}{a}\\ =4.25m^{-1}\end{array}$ 

Now substitute the values of B, K, and P_max, into equation (1) to get A: 
 
 $\begin{array}{r}3\times {10}^{-2}\mathrm{Pa}=\left(2.2\times {10}^9\mathrm{Pa}\right)\times \left(4.25{\mathrm{m}}^{-1}\right) \\ A=\frac{3\times {10}^{-2}\mathrm{Pa}}{\left(2.2\times {10}^9\mathrm{Pa}\right)\times \left(4.25{\mathrm{m}}^{-1}\right)}\\ A=3.2\times {10}^{-12}\mathrm{m} \end{array}$

Step 2: Explanation for Amplitude being less than 1.2 x 10-8 m

A Is Inversely proportional to Bk, so, the reason for A being smaller than (1.2 x 10-8m) is that Bk for water is much larger than Bk for air. 
 
 Therefore, A=3.2 x 10-12 m is the displacement amplitude is produced if the pressure amplitude is 3.0 x 102 Pa. The reason for A being smaller than 1.2 x 10-8 Is that Bk for water is much larger than Bk for air.