The problem describes a scenario where sound waves are used for ocean exploration. The frequency of the sound wave used by a ship is 40,000 Hz. The questions are: (a) Can this sound wave be heard by the human ear? (b) Explain the reason for your answer to (a). (c) If the speed of sound in seawater is $1500 \, \text{m/s}$, what is the wavelength of the wave used in the exploration?

1. Problem Description

The problem describes a scenario where sound waves are used for ocean exploration. The frequency of the sound wave used by a ship is 40,000 Hz. The questions are:

(a) Can this sound wave be heard by the human ear?

(b) Explain the reason for your answer to (a).

(c) If the speed of sound in seawater is $1500 \, \text{m/s}$, what is the wavelength of the wave used in the exploration?

2. Solution Steps

(a) The range of human hearing is typically between 20 Hz and 20,000 Hz. Since 40,000 Hz is outside this range, the sound wave cannot be heard by the human ear.

(b) The human ear can only detect sound waves with frequencies between 20 Hz and 20,000 Hz. A frequency of 40,000 Hz is higher than the upper limit of human hearing. Such sounds are called ultrasounds and are inaudible.

(c) The relationship between the speed of a wave ($v$), its frequency ($f$), and its wavelength ($\lambda$) is given by:

$v = f \lambda$

We are given that $v = 1500 \, \text{m/s}$ and $f = 40,000 \, \text{Hz}$. We need to find $\lambda$. Rearranging the formula, we get:

$\lambda = \frac{v}{f}$

Substituting the given values:

$\lambda = \frac{1500 \, \text{m/s}}{40000 \, \text{Hz}} = \frac{1500}{40000} \, \text{m} = \frac{15}{400} \, \text{m} = \frac{3}{80} \, \text{m}$

$\lambda = 0.0375 \, \text{m}$

3. Final Answer

(a) No, this sound wave cannot be heard by the human ear.

(b) The frequency of the sound wave (40,000 Hz) is higher than the upper limit of human hearing (20,000 Hz).

(c) The wavelength of the wave is $0.0375 \, \text{m}$.