### Video Transcript

When a note on a piano is played, a
hammer inside the piano strikes one of the strings of the piano, producing a
standing wave on the string. Each of the following describes how
the fundamental frequency and fundamental wavelength of the string may change when
other properties of the string are changed.

Okay, before we get to these other
descriptions, let’s consider what’s happening in this situation. We have a string on a piano. Let’s say that this is it. And we know it’s fastened at both
ends. This string is set up to vibrate so
that when it does, it produces a note. In order to make this happen, a
device called a hammer strikes the string and produces a standing wave on it. In this context, a standing wave is
one that doesn’t exhibit any horizontal motion left or right, but only vertical
motion up and down.

As this string vibrates, we’re
specifically interested in the fundamental frequency and fundamental wavelength of
the string. The fundamental wavelength of a
string is the longest wavelength that can fit on it. The two factors determining this
wavelength are the endpoints of the string, which in this case are both fixed or
fastened. And then, given that, the only
other factor affecting the fundamental wavelength is the string length, what we can
call capital 𝐿.

If this string were oscillating at
its fundamental frequency, then the corresponding wavelength of the string would
make it look like this at one moment in time. As the string continue to
oscillate, it would then move down to this dashed position and then back to this
original one and so forth, on and on. Looking at this string, we can tell
that it’s not able to fit an entire wavelength along its length. To do that, to fit an entire
wavelength, we would have to go out to a length twice 𝐿. It’s this distance which is equal
to what we can call 𝜆, the fundamental wavelength of this string.

Seeing how 𝜆 and 𝐿 relate in the
sketch, we can write an equation describing this. The fundamental wavelength 𝜆 of a
string that’s fastened at both ends is equal to two times the length of the string
𝐿. This relationship shows us what
physical parameters the wavelength 𝜆 depends on. It depends only on the length of
the string.

Now, let’s think about the
fundamental frequency of this string. We know that when it comes to waves
in general, the frequency of a wave — we can call it 𝑓 — is equal to the speed of
the wave 𝑣 divided by its wavelength. And we’ve just seen that in this
case, 𝜆 is equal to two times 𝐿. So, we can substitute that in for
𝜆 in our equation for 𝑓. It’s worth mentioning that these
two relationships for fundamental frequency and fundamental wavelength apply
specifically in the case of a string that’s fixed at both ends.

All right, now that we know what
parameters the wavelength and the frequency depend on, let’s move on to considering
the following descriptions of how frequency and wavelength may change when other
string properties change.

Description (a) Frequency: Lower,
Wavelength: Longer. (b) Frequency: Lower, Wavelength:
Stays the same. (c) Frequency: Stays the same,
Wavelength: Shorter. (d) Frequency: Higher, Wavelength:
Shorter. (e) Frequency: Higher, Wavelength:
Stays the same. When a piano string is replaced
with another string of greater density but is placed under the same tension and has
the same length, which of the above describes how the fundamental frequency and
fundamental wavelength of the string change?

Okay, so, as we saw, in this
example, we’re working with a piano string. And we know that when this string
vibrates, it has certain properties known as its fundamental frequency and
fundamental wavelength. And we saw that if the length of
this string is 𝐿, then the fundamental wavelength 𝜆 is equal to two times 𝐿 and
the fundamental frequency 𝑓 is equal to the wave speed 𝑣 divided by two 𝐿.

Now, in this description, we’re
told that this piano string is replaced with another string of greater density. So, let’s say that this is our new,
more dense string. But we’re told that it’s under the
same tension and has the same length as the original one. The question is, how does this
change to a more dense string affect the fundamental frequency and fundamental
wavelength of the string? In other words, how did the
fundamental frequency and wavelength of this thicker string compared with this
thinner one?

To answer this question, we can
rely on these equations which show us what physical parameters 𝑓 and 𝜆 depend
on. We see that 𝜆, the fundamental
wavelength of a string, only depends on one physical parameter, the string
length. Since we’re explicitly told that
this more dense string has the same length as the original one, that means the
fundamental wavelength of our new string will be the same as that of the old
one. And this is because the lengths of
these two strings are the same.

Knowing this, we can look over our
five answer options and eliminate any which do not claim that the wavelength stays
the same. This includes options (a), (c), and
(d). These all say that the fundamental
wavelength either gets longer or shorter, which we know is not true.

Next, we can consider how this
change in string density might affect the fundamental frequency 𝑓. We see that this value depends on
two physical parameters, wave speed and string length. We’ve already seen that string
length is constant, but what about the characteristic speed of this standing
wave? When a wave is moving along a
string, its speed — we can call it 𝑣 — is equal to the square root of the force of
tension along the string — we’ve called it 𝐹 sub 𝑇 — divided by the string density
𝜇.

Now, in our description, we’re told
that this new string does have a greater density but it has the same tension as the
original one. So, looking at our equation for
wave speed, we see that 𝐹 sub 𝑇 isn’t changing as we moved to the new string but
𝜇 is. It’s going up. Now, if the string density
increases, and it’s in the denominator of this equation, then that tells us that
increasing 𝜇 will decrease the wave speed 𝑣. And then, looking over at our
equation for fundamental frequency, we see that if wave speed 𝑣 goes down, then the
fundamental frequency 𝑓 must go down as well. We know this because the string
length 𝐿 is constant. And so, if 𝑣 gets smaller, 𝑓 must
too.

Therefore, while the fundamental
wavelength of this string doesn’t change, its fundamental frequency does. It decreases. This agrees with our answer option
(b), which says that the fundamental frequency will get lower while the fundamental
wavelength will stay the same. This is what will happen if our
string is replaced by another string of greater density but under the same tension
and with the same length.