Video Transcript
A sound source generates 90
oscillations each three seconds. Given that sound waves travel at
340 meters per second in air, calculate the distance between the centers of a
rarefaction and a successive compression.
To get started, let’s refresh our
memories about the properties of sound waves. A sound wave is a longitudinal
wave, which means that its medium — in this case, air — oscillates parallel to the
direction of overall wave propagation. We can draw a diagram to help
illustrate this. Sound waves travel through a medium
like air by means of pressure differences that cause air molecules to oscillate back
and forth along the same direction that the wave is traveling.
Rarefaction refers to the areas of
the wave with low particle density, and compression refers to the areas of high
particle density. We should recall that the
wavelength of such a wave, represented by 𝜆, refers to the distance between either
successive centers of compression or successive centers of rarefaction. But in this question, we’ve been
asked to find the distance between the centers of a rarefaction and a successive
compression, so that would be half of a wavelength, or 𝜆 divided by two.
Now, let’s look at the values we
were given in the question. We’re told the speed of a sound
wave as well as the number of oscillations this wave completes in three seconds. With this information, we can
calculate the wave’s frequency. Frequency, 𝑓, gives the number of
complete wave cycles per second, so we have that 𝑓 equals 90 cycles divided by
three seconds, which is equal to 30 cycles per second, or 30 hertz. At this point, it’ll be helpful to
recall the wave speed formula: 𝑣 equals 𝑓 times 𝜆, where 𝑣 is the speed of the
wave, 𝑓 is its frequency, and 𝜆 is its wavelength.
To answer this question, we’ll need
to solve for the wavelength, so let’s rearrange this equation to make 𝜆 the
subject. To do this, we simply divide both
sides by 𝑓 so that frequency cancels out of the numerator and denominator, leaving
𝜆 by itself. Thus, the expression can be written
as 𝜆 equals 𝑣 divided by 𝑓. We also need to remember that this
question is asking for half of a wavelength. We really want to find 𝜆 divided
by two, so let’s divide both sides of this equation by two.
We can now substitute in the values
for 𝑣 and 𝑓. But before we calculate, let’s take
a moment to check out the units here. We have meters per second divided
by hertz. Recall that hertz is equivalent to
inverse seconds. So, units of per second will cancel
out of the numerator and denominator, leaving only meters. This is a good sign, since
wavelength is a distance measurement.
Finally, grabbing a calculator, we
get a result of 5.666 and so on meters. Choosing to round this to two
decimal places, we have our final answer. The distance between the centers of
a rarefaction and a successive compression in this sound wave is 5.67 meters.
