How Can Bats Detect Ultrasonic Sound Waves at Extremely Low Intensities?

[roam]

Homework Statement

A bat can detect ultrasonic sound waves with an intensity as low as 1 × 10^–13 Wm^–2. An ultrasonic source emits sounds waves with a frequency of 54.0 kHz and a power of 1 × 10^–7 W. Take the speed of (ultra)sound in air as 340 ms^–1 and the density of air to be 1.2 kg/m³.

At a distance of 10.0 m from this source, calculate

(i) the displacement amplitude

(ii) the pressure amplitude of the ultrasonic wave

The Attempt at a Solution

For part (i) how can I use the equation $$\Delta P = \rho v \omega s_{max}$$ to find the displacement amplitude when I don't know what the pressure amplitude $$\Delta P$$ is? I can't find the pressure amplitude when I don't know the displacement amplitude.

Then I tried using the following formula to find the pressure amplitude

$$I=\frac{(\Delta P_{max})^2}{2\rho v}$$

But I ended up with $$8.67 \times 10 ^{-6}$$ which is wrong. The answer has to be 1.84 x 10^–12m and for part (ii) it is 0.255 mPa.

(P.S. I also know that the furthest distance at which a bat can detect this source is 282m. And the sound level in decibels is 19 dB.)

 

[anathema]

Hello!

To find the displacement amplitude, you can use the equation \Delta P = \rho v \omega s_{max} and rearrange it to solve for s_{max}. This will give you the displacement amplitude at a distance of 10.0 m from the source. The pressure amplitude, \Delta P, can then be found by substituting the calculated displacement amplitude into the formula I=\frac{(\Delta P_{max})^2}{2\rho v}. This will give you the pressure amplitude at a distance of 10.0 m.

As for the furthest distance at which a bat can detect this source, you can use the inverse square law, which states that the intensity of a sound wave decreases with the square of the distance from the source. So, if the bat can detect the sound at a distance of 10.0 m, the intensity at a distance of 282m would be 1/282^2 times the initial intensity. Using this, you can find the new intensity and then use the equation I=\frac{(\Delta P_{max})^2}{2\rho v} to calculate the pressure amplitude at the furthest distance.

For the sound level in decibels, you can use the formula L=10 log_{10}(\frac{I}{I_0}), where I is the intensity of the sound and I_0 is the reference intensity (usually taken as the threshold of human hearing, which is 1 x 10^-12 W/m^2). This will give you the sound level in decibels, which you can then compare to the given value of 19 dB to see if the bat can actually detect the sound at a distance of 282m.

Hope this helps!