Chapter 26, Problem 50A

Suppose your friend is foolish enough to play his keyboard instrument underwater, where the speed of sound is 1,500 m/s.

a. Show that the wavelength of the middle-C tone in water would be 5.86 m.

b. Explain why middle C (or any other tone) has a longer wavelength in water than in air.

To determine

To show that the wavelength of middle-C tone in water is 5.86 m where the speed of sound in water is 1,500 ${\mathrm{ms}}^{-1}$ .

Answer to Problem 50A

The wavelength of middle-C tone in water is 5.86 m where the speed of sound in water is 1,500 ${\mathrm{ms}}^{-1}$ .

Explanation of Solution

Given:

Frequency ( $\mathrm{f}$ ) of middle- C keyboard tone = 256 Hz

Speed of sound in water ( $\mathrm{s}$ ) = 1500 ${\mathrm{ms}}^{-1}$

Formula used:

  $\mathrm{Wavelength}(\lambda )ofsoundwaveinwater=\frac{speed(s)ofsoundwaveinwater}{frequency(f)}$

Calculation:

Put the given values and we get;

  $\mathrm{Wavelength}(\lambda )ofsoundwaveinwater=\frac{1500m{s}^{-1}}{256Hz}=5.86m$

  $\mathrm{Wavelength}(\lambda )ofsoundwaveinwater=5.86m$

Conclusion:

Thus, the wavelength of middle-C tone in water is 5.86 m where the speed of sound in water is 1,500 ${\mathrm{ms}}^{-1}$ .

To determine

The reason that middle C tone has longer wavelength in water than in air.

Answer to Problem 50A

The speed of a sound wave in water (1,500 ${\mathrm{ms}}^{-1}$ ) is greater than speed of a sound wave in air ( $340m{s}^{-1}$ ) that’s why the middle C tone has longer wavelength in water than in air.

Explanation of Solution

Introduction:

For any wave, the relation between speed, frequency and wavelength is given as follows;

  $\begin{array}{l}speed=frequency\times wavelength\\ wavelength=\frac{speed}{frequency}\end{array}$

Since the frequency of middle-C tone (256 Hz) remain same in both air and water so wavelength becomes directly proportional to speed of the wave. And it is known that the speed of a sound wave in water (1,500 ${\mathrm{ms}}^{-1}$ ) is greater than speed of a sound wave in air ( $340m{s}^{-1}$ ). Hence the middle C tone has longer wavelength in water than in air.

Conclusion:

Thus, the wavelength of sound wave in air is 1.33 m for the frequency 256 Hz.