Understanding the Sound Wave Problem

The question asks us to find the wavelength of a sound wave. We are given the wave's speed and information about how many crests and troughs it produces over a specific period.

Here's the information provided:

• The speed of the sound wave, denoted by $v$, is $330$ m/s.
• The number of crests produced is 20.
• The number of troughs produced is 20.
• The time interval during which this occurs, denoted by $t$, is $0.1$ second.

Calculating the Wave Frequency

To find the wavelength, we first need to determine the frequency ($f$) of the sound wave. A complete wave cycle includes one crest and one trough. Therefore, observing 20 crests and 20 troughs means that 20 full wave cycles have occurred within the given time.

The formula for frequency is the number of waves (or cycles) divided by the time taken:
$f = \frac{\text{Number of waves}}{\text{Time taken}}$

Using the given values:
Number of waves = 20
Time taken, $t = 0.1$ s

Now, we calculate the frequency:
$f = \frac{20}{0.1 \text{ s}}$
$f = 200$ Hz

So, the frequency of the sound wave is 200 Hertz.

Determining the Wavelength

We know the relationship between the speed ($v$), frequency ($f$), and wavelength ($\lambda$) of a wave is given by the formula:
$v = f \lambda$

To find the wavelength ($\lambda$), we can rearrange this formula:
$\lambda = \frac{v}{f}$

Now, we substitute the known values into the equation:
Speed, $v = 330$ m/s
Frequency, $f = 200$ Hz

$\lambda = \frac{330 \text{ m/s}}{200 \text{ Hz}}$

Calculating the result:
$\lambda = \frac{33}{20}$ m
$\lambda = 1.65$ m

Thus, the wavelength of the sound wave is 1.65 m.