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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掳 Pa, determine the displacement amplitude that would be needed to produce a pressure amplitude of 3 脳 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:

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

$位=\frac{V}{f}$--(2)

v=1480 m/s, the speed of sound in the water.

f = 1000 Hz, the frequency of the given sound wave.

$\begin{array}{l}位=\frac{1480m/s}{1000Hz}\\ 位=1.48m\end{array}$

Therefore,
role="math" localid="1668152783392" $\begin{array}{l}k=\frac{2\mathrm{蟺}}{a}\\ =4.25m^{-1}\end{array}$

Now substitute the values of B, K, and P_max, into equation (1) to get A:

$\begin{array}{r}3脳{10}^{鈭�2}\mathrm{Pa}=\left(2.2脳{10}^9\mathrm{Pa}\right)脳\left(4.25{\mathrm{m}}^{鈭�1}\right)\\ A=\frac{3脳{10}^{鈭�2}\mathrm{Pa}}{\left(2.2脳{10}^9\mathrm{Pa}\right)脳\left(4.25{\mathrm{m}}^{鈭�1}\right)}\\ A=3.2脳{10}^{鈭�12}\mathrm{m}\end{array}$

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.