Question

Humans can hear sounds with frequencies up to about \(20.0 \mathrm{kHz},\) but dogs can hear frequencies up to about \(40.0 \mathrm{kHz} .\) Dog whistles are made to emit sounds that dogs can hear but humans cannot. If the part of a dog whistle that actually produces the high frequency is made of a tube open at both ends, what is the longest possible length for the tube?

Short answer

Answer: The longest possible length of the tube is approximately 4.3 mm.

Step-by-step solution

Write down the formula for the fundamental frequency of a tube open at both ends

The formula for the fundamental frequency (\(f_1\)) of a tube open at both ends is: \[f_1 = \frac{v}{2L}\] where: - \(f_1\) is the fundamental frequency, - \(v\) is the speed of sound, - \(L\) is the length of the tube.

Substitute the maximum frequency a dog can hear for the fundamental frequency

The maximum frequency a dog can hear is \(40.0 \mathrm{kHz}\). Substitute this value for \(f_1\) in the formula: \[\frac{v}{2L} = 40,000 \,\text{Hz}\]

Find the speed of sound in the air

To solve the equation for the length of the tube, we need to know the speed of sound in the air. Consider the speed of sound in air at room temperature (\(20^\circ C\)) to be \(v = 343 \,\text{m/s}\).

Solve the equation for the length of the tube

Now substitute the speed of sound into the equation and solve for the length of the tube: \[\frac{343}{2L} = 40,000\] First, multiply both sides of the equation by 2: \[343 = 80,000L\] Now, divide both sides by 80,000: \[L = \frac{343}{80,000} \,\text{m}\]

Calculate the longest possible length of the tube

Finally, calculate the value of \(L\): \[L = \frac{343}{80,000} = 0.0042875 \,\text{m}\] The longest possible length for the tube that can produce a frequency only dogs can hear is approximately \(0.0043 \,\text{m}\) or \(4.3 \,\text{mm}\).