Question

A guitar emits a 440 Hz tone. How far apart are the crests in room temperature air (20 °C)? How long does it take the string to vibrate from crest position to trough position (across twice the amplitude)?

Answer 1

We are asked to determine the distance between crest of a sound wave that has a frequency of 440 Hz. To do that we will determine the wavelength using the following formula:

`\lambda=(c)/(f)`

Where:

`\begin{gathered} \lambda=\text{ wavelength} \\ c=\text{ speed of sound} \\ f=\text{ frequency} \end{gathered}`

The speed of sound at 20°C is equal to:

`c=343\text{ m/s}`

Now, we substitute the values:

`\lambda=\frac{343\text{ m/s}}{440Hz}`

Solving the operations:

`\lambda=0.78m`

Therefore, the distance from crest to crest is 0.78 meters.

Now, we are asked to determine the time it takes the string to go from the crest to the trough. To do that we will determine the period first using the following formula:

`T=(1)/(f)`

Now, we substitute the frequency:

`T=(1)/(440Hz)=0.0023s`

Now, since the time it takes to go to crest to trough is half the time it takes the wave to go from crest to crest we divide the result by two:

`T_{(1)/(2)}=0.0012s`

Therefore, the time is 0.0012 seconds.